dynamic analyses of idealised solar panels with a bilinear snubber

Dynamic analyses of idealised solar

panels with a bilinear snubber

M.W.J.E. Wijckmans

Report nr. TUE: W F W 95.071

Professor: Coaches:

Prof. dr. ir. D.H. van Campen, TUE Dr. ir. A. de Kraker, TUE Ir. J.J. Wijker, TUD/Fokker Space tk Systems Dr. Ir. R.H.B. Fey, TNO Ir. E.L.B. van de Vorst, TUE

Eindhoven University of Technology Department of Mechanical Engineering Division of Fundamental Mechanics Thesis for the engineering degree, June 1995

1

Preface

I would like t o thank, all the people who reviewed this report, my parents who were a tremendous support during my study and Chris, Frank, Marcel and Patrick. They all had their own contribution t o this report. And last but not least, I would like to thank They en Ontij. Here it was underlined once more tha t there is more in life than working.

ii

Summary

The subject of this report is the long term behaviour of idealised solar panels with a bilinear snubber, which are excited by a base acceleration. Bilinear snubbers are used to reduce the response of the solar panels and to meet the criterion that it must be possible to unfold the wings without any special adjustments. These locally app!ied one-sided snubbers cause !ma! ncnlinectrities. I I I ~ luedmeu buidi pmeis are mocieiieci using finite eiements and further reduced by a Compo- nefit Mode Synthesis method, based on rigid body modes, free-interface eigenmodes and flexibility modes. To perform nonlinear dynamic analyses of such systems with local nonlinearities, special calculation techniques are necessary. These techniques have been developed at the TNO Centre for Mechanical Engineering. The steady-state behaviour is investigated by calculating periodic solutions at various excitation frequencies. Nonlinear phenomena like subharmonic and superharmonic resonances are found. In this report is tried to answer the question whether it is possible to rely on a linear calculation instead of a nonlinear calculation. This is done by comparing the system response of linear and nonlinear models. Further will be discussed whether the introduction of local nonlinearities can be seen as some sort of damping. The influence of the snubber damping and stiffness on the system response has been examined. Also the responses for various types of backlash have been calculated. A model with a free backlash gives a completely different response than a model with a snubber under preload. It is tried to predict the minimum required preload of the snubber, to be allowed to use linear dynamic analyses. But the consequence of introducing a preload of the snubber, is the fact that the system response is excitation dependant. The responses of the nonlinear and the linear model are comparable if the preload of the snubber is large enough. But it is not advisable to rely on a cheaper linear calculation solely. Nevertheless, the bilinear eigenfrequencies of the model with a bilinear snubber can be estimated by linear calculations only. Measurements of an experimental setup have been done t o show tha t the numerically found nonlinear phenomena do exist in experiments.

ml-.. 2 3 - - 1 : - - 3 - - l c - ~.

3 %(a) w 1.1 - A-l a a a' a''

f e

f b i

fn A t T.? dsnub

ksnub

Fsnub 2

W

Y1

Y2 U

- K - M b E 9 h I k L

P m

a!

complex unity constant real part of a imaginary part of a absolute value of a inverse of matrix A first time derivative of a second time derivative of a first dimensionless time derivative of a second dimensionless time derivative of a excitation frequency i-th bilinear eigenfrequency n-th eigenfrequency excitation amplitude time excitation period snubber damping snubber stiffness snubber force snubber compression angular frequency displacement of node 51 displacement of node 151 prescribed base displacement stiffness matrix of linear system mass matrix of linear system width Young-modulus gravity height second moment of inertia stiffness length mass density dimensionless stiffness

... 111

iv

,B mass ratio y dimensionless mass S dimensionless backlash < modal damping factor C q û dimensioniess time

dimensionless damping factor of free-interface eigenmode dimensidess d,of ,of the system

1 Introduction 1

2 Model assumptions 2

3 Finite element model 4

4 Linear. dynamic analyses 6 4.1 Damped. linear model without snubber . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.2 Damped. linear model with snubber . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Nonlinear dynamic analyses 13 5.1 Review of some nonlinear phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 Estimation of the bilinear eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . 22 5.3 Damped. bilinear model of snubber . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3.1 Harmonic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.3.2 Superharmonic resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3.3 Subharmonic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3.4 Other points of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3.5 The six and eight dof model compared . . . . . . . . . . . . . . . . . . . . . . 33

6 Experimental verification 35 6.1 Determination of model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6.2 Linear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.3 Nonlinear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7 Parameter variations 45 7.1 Snubber damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.2 Modal damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.3 Snubber stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.4 Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7.4.1 Free backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 7.4.2 Preload . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7.5 Linear and nonlinear model compared . . . . . . . . . . . . . . . . . . . . . . . . . . 59

8 Conclusions and recommendations 62

A Dimensioning the linear model E4

V

CON TENTS vi

B Eigenfrequencies depending on 01

C The dimensionless bilinear two dof model

D Design of the experimental model

E Response of bilinear eight dof mode!

F Measuring-ins t rument s

G Response of bilinear model with a preload

H DIANA-input files and user-supplied routines

References

70

75

78

81

88

92

9 4

95

Introduction

During the launching of satellites, the solar panels of the spacecraft body are in a folded position to save space on behalf of the fairing and to make sure that the panels survive the vibration loads. Because these solar panels are excited during the flight, vibrations will occur in the panels. When the excitation is too severe the panels may struck each other or the stresses in the panels may become too high. This may lead to damage of the layer of solar cells on the surface of the panels. The application of one-sided snubbers can increase the performance of the solar panels or even avoid these problems. The advantage of one-sided snubbers is that the wings can still be unfolded without any special adjustments. These locally applied one-sided snubbers cause local nonlinearities. Although the nonlinearity is local, the overall dynamic response of the system may change drastically. In this thesis for the engineering degree the dynamic behaviour of solar panels, which are excited by a prescribed harmonic base acceleration with an amplitude of 1 g, will be examined. Especially in spacecraft industries, constructions with a minimum mass are designed. This can lead to slightly damped constructions, like solar panels. Mostly the dynamic analyses are limited t o linear analyses, because they are fast and thus cheap. It will be shown that , certainly for slightly damped structures, linear dynamic analyses do not fulfil anymore. Nonlinear analyses are done with tools developed at the TNO Centre for Mechanical Engineering and implemented in the FE-software package DIANA. The solar panels have been modelled as a 2-dimensional structure consisting of finite element beams, discrete springs, dampers and masses. The assumptions that have been made are described in chapter 2, while chapter 3 deals with the modelling. In chapter 4 linear dynamic analyses are performed t o get a better idea of the dynamic behaviour of the structure. In chapter 5 the nonlinear model is examined. Therefore the degrees of freedom of the model are reduced by a Component Mode Synthesis method, based on rigid body modes, free-interface eigenmodes and flexibility modes. First, some nonlinear phenomena are introduced and a n estimation of the bilinear eigenfrequencies is made. After that a complete analysis of the nonlinear model is given. The responses of a six and eight dof model have been compared to show how accurate the reduction is. Chapter 6 deals with an experimental setup. Measurements have been done to verify the numerical results and t o show that the numerically found nonlinear phenomena do exist in experiments. Next some parameter variations are performed. The influence of the snubber damping and stiffness on the system response has been examined. Further the influence of backlash is examined. Due to the backlash the system response is excitation and frequency dependant. A model with free backlash gives a completely different system response than a system with a snubber under preload. Finally the responses of the linear and the nonlinear model are compared to be able to conclude whether it is admissible to rely on linear calculations solely.

1

Model assumptions

In order to get more knowledge about solar arrays of satellites, the working of some parts of these arrays will be discussed in more detail. Each satellite contains two identical wings which consist of one or more solar panels, covered with a layer of solar cells. The application of a lightweight sandwich structure for the solar panels has led to a stiff structure with a low mass. A retracted wing, i.e. a wing in folded position, is attached to the body of the spacecraft by several hold-down and release systems. At the same time these hold-down and release systems clench the solar panels t o each other. In figure A.2 (Appendix A) can be seen how all solar panels of a retracted wing are attached to a satellite. Besides this all panels are connected to each other by deployment mechanisms, around which the panels can rotate while spreading the wings. Figure A.3 shows the deployed wing. To reduce the vibrations of the wing in folded position, a snubber, i.e. a piece of rubber, can be placed between a deployment mechanism. The figures A.4 and A.5 in Appendix A show some pictures of snubbers mounted on the yoke and on a deployment mechanism. The foldable Y-configuration of the yoke consists of two arms and can be seen in both figure A.2 and figure A.3. In order to model the wing, the following assumptions have been made:

o In this report only an idealised configuration of a solar wing will be examined. This type contains two panels per wing, instead of three panels its shown in figure A.2 and A.3.

o The dynamic problems of the retracted wing, like too high acceleration levels, occur at that side of the wing where the yoke is mounted to the first solar panel. This is the framed part in figure A.2, which is called the hang over. The other parts are not modelled here because the lowest eigenfrequency of these parts are considerably higher than the lowest eigenfrequencies of the framed part.

o The lowest eigenmode will occur in the length and not in the width of the panel. Therefore only a small part of the wing is considered, which means tha t also only one leg of the yoke is modelled.

o The panels are excited by an acceleration of the spacecraft perpendicular to the surface of a panel. Because there are only forces acting perpendicular to the panel and the thickness/length- ratio is small (3%), pure bending is considered.

o The hold-down and release system is regarded as an infinite stiff restraint. The base-points of both panels are considered to be in-phase. This means that the restraint of every panel can be modelled as one and the same body. The excitation is applied to this restraint.

2

CHAPTER 2. MODEL ASSUMPTIONS 3

outer panel

node 151

node 5 1

Figure 2.1: Model of the idealised wing.

Deployment mechanism and one leg of the yoke have been modelled as a dummy mass of 0.5 kg. This is the effective mass of one leg of the yoke, which is attached to the inner panel.

The snubber will be modelled as a discrete spring, both linear and bilinear. In the bilinear case also discrete bilinear damping is taken into account. Only one snubber will be modelled, i.e. the snubber between the two panels. Thus there is no snubber between the inner panel and the spacecraft body. The snubber is assumed to be massless, so no impact phenomena will occur.

To compare the results of the numerical model with the experimental results of the idealised wing, i t is important tha t the lowest eigenmode of the numerical model and the idealised wing are the same. This means close io 50 Hz. Iïì appeïìdix A al! relevant dimensiûns of the idealised wing are mentioned. Also the dimensions of the numerical model have been calculated.

All these assumptions have led to the model shown in figure 2.1. The motive to perform dynamic analyses of this model is the following: when no snubber is applied, the model consists of two separate panels. As a dummy is attached to only one panel, both panels have a different dynamic behaviour. A base excitation in an eigenfrequency of one of the panels leads to an amplification of the displacement of that particular panel and the panels might struck each other. At the same time, too high stresses might occur in such solar panel. This can be avoided by the application of a linear snubber, attached to both panels. This leads to a different dynamic behaviour of the model. The lowest eigenmode is an in-phase motion. T h u s no contact between the panels will appear. The second eigenmode is an out-of-phase motion. The displacement response of both panel ends is reduced by the damping property of the snubber. But to meet the criterion that it must be possible t o unfold the wings without any special adjustments, bilinear (one-sided) snubbers are applied. In the following chapters both the linear and the bilinear model are analysed.

Finite element model

This chapter describes in detail how the finite element model (FEM) is made and checked. This is done in the FE-software package DIANA1. As pure bending in only one direction is considered, both panels are build up by 2-dimensional finite Euler beam elements. This means tha t no shear and transverse shear deformation are taken into account. Every element consists of two nodes with 3 degrees of freedom (dof) per node (2 translations, 1 rotation). In DIANA this finite element is called 'LGBEN'. For further information is referred to the DIANA User's Manual [ll. The DIANA-input file is given in Appendix H. Mesh refinement has been used to make sure that the results of a calculation of the eigenmodes are accurate enough. One panel (length is 500 mm.) has been modelled by both 25 equal and 50 equal elements. As the first 6 eigenmodes are exactly the same for both models, it can be concluded that 25 elements per panel lead to a convergent solution. The outer panel is modelled by 50 elements of equal dimensions. In order to verify the numerical results by comparing them with analytic results, some linear static and linear dynamic analyses have been made:

o In the static situation a constant force acting on one node is considered. The analytical and numerical results were exactly the same. Further the residual forces were zero in every node. Except for the reaction force which was equal to the applied load.

o In the dynamic case, eigenvalues of the outer panel have been calculated both numerically and analytically. According to Den Hurtog [2] the eigenfrequencies of a one-sided clamped beam are:

[Hz], for transverse vibrations EI

f n =

In table 3.1 the analytic and numerical calculated eigenfrequencies are compared. The difference between these values are defined as: relative error = (analytic value - numerical value)/analytic value*lOO%

'developed by TNO Building and Construction Research in Delft, the Netherlands

4

C H A P T E R 3. FINITE ELEMENT MODEL 5

I 11 I 2546 I 2546 I 0.0 I - fL

Table 3.1: Analytic and numerical calculated eigenfrequencies.

From table 3.1 it can be concluded that the first eigenfrequency of the finite element approach is nearly exact. The higher frequencies of the outer panel however show some differences, although small. As explained in the previous chapter, the inner panel will be modelled the same as the outer panel, only extended with a dummy mass, which represents a part of the yoke. The dummy mass has been modelled as a short beam with corresponding stiffness and mass. Out of mesh refinement it appeared tha t 25 elements of equal dimensions led to a convergent solution of the eigenfrequencies of the inner panel. Like the outer panel, the inner panel has been modelled with 50 elements of equal dimensions. The finite element approach resulted in an eigenfrequency of 30.95 Hz. According to Den Hurtog [a] the lowest eigenfrequency of a one-sided clamped beam with tip-mass m is:

= 29.06 HZ 3.03EI (m+0.227pAL)L3 (3.3)

The higher approximation (relative error = -6.5 %) of the finite element approach is due to the fact that formula 3.3 does not include the relative high stiffness of the dummy mass.

Another possibility is t o model the dummy mass as a tip-mass in a single node. Formula 3.3 is only valid for relative large point-masses compared to the beam mass. In this case the ratio between the point-mass and the beam mass is in the order of 0.48 and the finite element approach gives an eigenfrequency of 28.69 Hz (relative error = 1.3 %). For iarger point-masses the analytic and numerical values are more equal. For a point-mass of 2 kg formula 3.3 gives a lowest eigenfrequency of 16.46 Hz, while a finite element approach resulted i n a eigenfrequency of 16.35 Hz (relative error = 0.67 %). In fact formula 3.3 can only be used to determine an approximate value of the eigenfrequency. Although it was not possible to check the modelling of the dummy mass more accurate, the model of the dummy mass as a short beam is used to perform dynamic analyses, as the yoke of the idealised wing introduces some additional stiffness too.

Linear,

4

dynamic analyses

To get a better understanding of the dynamic behaviour of the model and before performing nonlinear dynamic analyses, the linear model is examined. This chapter is divided into two parts. First the model without snubber is discussed. Next the model with snubber will be examined. Both models are still linear.

4.1 Damped, linear model without snubber

In figure 4.1 the first six eigenmodes are sketched.

1st. eigenmode: 30.950 Hz

node 151 151 B~~ -- -. node 5 1

I *. -. . I

2nd. eigenmode: , ~

49.334 HZ ,*'

3th. eigenmode:

4th. eigenmode:

5th. eigenmode: 782.54 Hz


Figure 4.1: Scaled eigenmodes of linear model without snubber.

Figure 4.2 shows the bode diagram of the displacements of node 51 and node 151 for a 1 g. base excitation. Node 51 is at the end of the inner panel, while node 151 is at the end of the outer panel. The peaks of the amplitude and the jump i n the phase are relatively sharp, because there is a low modal damping of 0.5% (this is a good approximation for aluminium, see chapter 6) of the first 12 eigenmodes. For low frequencies, both models have a phase difference of -180 degrees, compared to

6

C H A P T E R 4. LINEAR, DYNAMIC ANALYSES 7

the prescribed base acceleration. As also the prescribed base displacement has a phase difference of -180 degrees, compared to the prescribed base acceleration, both the displacements of node 51 and 151 and the prescribed base displacement are in-phase. When the second eigenfrequency is passed both nodes are in-phase with the prescribed base acceleration and thus they move out-of-phase with the prescribed displacement.

I 1 ............................ /.I.. \ ....__.....__, . .... ................................................. 4

, I I I I I I 10 20 30 40 50 60 70 80 90 100

1 O-;

frequency [Hz]

-200; 1 10 20 30 40 50 60 70 80 90 100

frequency [Hz]

Figure 4.2: Bode diagram of linear model without snubber for 1g. base excitation.

4.2 Damped, linear model with snubber

In this subsection first a discrete two dof model will be derived to examine the influence of the snubber stiffness on the two eigenfrequencies. Next these eigenfrequencies will be compared with numerical obtained results from DIANA.

To determine a discrete two dof model, each beam has been modelled by a discrete spring and a discrete mass in a single node at the end of the beam (node 51 or node 151), as can be seen in figure 4.3.

k

Figure 4.3: Discrete single dof model.

As an estimation, the beam stiffness IC is calculated from formula 4.1, which is based on a static reduction, of the stiffness matrix of the beam, to the single degree of freedom at the end of the beam:


- 2.36 t lo4 N/m (4.1) 3EI L3

k = kl = k2 = - -

Note that the subscript 1 refers to the inner panel and subscript 2 refers to the outer panel. The discrete masses ml and m2 of the beams wiii De chosen so that they fuifii equation 4.2 and 4.3. This means tha t ml and m2 are fitted on the eigenfrequencies using a beam stiffness obtained by a static reduction.

- 49.34 Hz (4.3)

In which f 1 and f 2 are the lowest eigenfrequencies of the two beam-configurations mentioned in chapter 3: f 1 is equal to the finite element approach of the dummy mass modelled as one single point mass and f 2 is equal to the lowest eigenfrequency listed in table 3.1. Combining equation 4.1, 4.2 and 4.3 results in:

ml = 0.727 k g , m2 = 0.246 k g .

In the undamped case the linear model with snubber can be represented by a two dof model with two springs (one for every beam) and one spring representing the snubber stiffness ksnub. See figure 4.4. When the base is excited with a displacement u, the equations of motion for the masses ml and m2 are:

/ / / / / / / / / / / / / / / / /

Figure 4.4: Discrete two dof model.

In which y1 the displacement of node 51 and y2 the displacement of node 151, with respect to the initial position of each mass, represents. Both equations can be conveniently arranged in matrix form:

CHAPTER 4. LINEAR, DYNAMIC ANALYSES 9

w i t h M = [ ml o ] : mass-matrix, 0 m2

1

1 -ksnub ksnub f k 2 1 - K = i kï T ksnub -ksnuó I : stiffness-matrix,

y = [ i: ] : matrix with dof and N

(4.7)

(4.8)

(4.9)

u is the prescribed base displacement. (4.10)

In Appendix B these equations have been worked out further and the eigenfrequencies have been calculated as a function of the ratio a of the linear snubber stiffness ksnub and the beam stiffness I C , obtained from a static reduction according t o equation 4.1, thus a=*. Figure 4.5 shows the dependency of the two lowest eigenfrequencies on cy. Due to the snubber, the eigenfrequencies of the model have changed to higher values (especially the second one). Figure 4.6 shows the scaled lowest four eigenmodes of the model with a linear snubber for cy = 4. These eigenmodes remain the same for values of a close to 4. From figure 4.5 it can be concluded that:

lim fi = 35.08Hz and,

lim f2 = co. a+m

a-+m

alpha [-]

Figure 4.5: Two lowest eigenfrequencies depending on a.

In table 4.1 the analytic and numerical calculated eigenfrequencies are listed for several snubber stiffness values.

C H A P T E R 4 . LINEAR, D Y N A M I C ANALYSES 10

Table 4.1: Anaiytic and numerical calculated eigenfrequencies depending on a.

The difference between the numerical and analytic obtained value for an infinite stiff snubber, is due to the fact that only two dof have been taken into account. For a model with more degrees of freedom, the third eigenfrequency will become lower than the analytic calculated second eigenfrequency, which is nothing more than the eigenfrequency of the infinite stiff snubber. Thus the second eigenfrequency will be finite. The other differences in table 4.1 are due to the estimation of the beam stiffness k . In fact the beam stiffness kl and k2 in equation 4.2 and 4.3 must be derived from a dynamic reduction of each beam to one dof. This will result in different values for the stiffness k1 and k2, the masses ml and m2 and thus different analytic calculated eigenfrequencies. In Appendix B the difference between a static and a dynamic reduction is explained. In table 4.1 can be seen tha t (according to DIANA) the lowest eigenfrequericy for a n infinite snubber stiffness is 37.08 Hz. This means tha t a higher first eigenfrequency, obtained by changing snubber parameters, is only possible by moving the snubber towards the hold-down system.

1 st. eigenmode: 36.650Hz _.

2nd. eigenmode: 115.3 1 Hz



Figure 4.6: Scaled eigenmodes of linear model with snubber ( (Y =4).

Figure 4.7 shows the bode diagram for a linear snubber stiffness of 1.6*104 N/m, i.e. a = 0.678. Again the peaks are relatively sharp because of the low modal damping. For node 151 an anti- resonance occurs at 53 Hz, for node 51 this happens around 69 Hz. An anti-resonance is due to


the snubber. When the excitation frequency is close to the anti-resonance, the internal force in the snubber is almost equal to the mass inertia force of one of the panels. Thus the resulting force on a panel is close t o zero, because the snubber force balances the mass inertia force of the panel. As a result the beam displacement is small i n an anti-resonance. Co the response is close to zero, while both nodes are vibrating out-of-phase. Figure 4.8 shows the response for a linear stiffness of 9.45"104 N/m. According to tabìe 4.i it can be seen that the second eigenÎrequency has become higher than 100 Hz.

node 51 (-) and node 151 (--)

O 10 20 30 40 50 60 70 80 90 100 excitation frequency [Hz] node 51 (-) and node 151 (--)

200[ I I I I I I I I I

.- s 0 ' a v) m 'n-100 i'.i .... ....

... , ............

...............

. . . . . . . . . . . . . .

.........

-I : \ j I ;

. _ < . . . . . . . . . . . . . . . . . . . . . . . . .... . . . . . . : I : I : 1 ' I \ : I : I : I : : I : I : I I ._-d

-200; I I I I , 10 20 30 40 50 60 70 80 90 100

excitation frequency [Hz]

Figure 4.7: Bode diagram of linear model with snubber (ksnab=1.6* lo4 N/m) for lg. base excitation.

CHAPTER 4. LINEAR, DYNAMIC ANALYSES

v> 0 !? o)

............................ . . i < . " .... " < . i . . . .....{.. . . . . . . j . . . . . . . . j . . . . . . ; . . . . . . . -

node 51 (-) and node 151 (--) 10.'

......... :.. . . . . . . : . . . . .

. . . . . . . . . . . . . . . t A ...:......... ; .............................

:._ . . . ._:. . . . . . :. . . . . : . . . . : . .

.................. ............................................. $03 U ZI - ............ - I=

-*o$ !O ;o ;o i o ;e i' ;e e' ;o Id' excitation frequency [Hz]

12

Figure 4.8: Bode diagram of linear model with snubber (ksnub=9.45* lo4 N/m) for lg. excitation.

base

Nonlinear dynamic analyses

The advantage of a bilinear snubber, like a one-sided linear spring, is the fact t ha t the wings can still be unfolded without any special adjustments. But one-sided springs introduce physical local nonlinearities, which may cause a drastic change of the overall dynamic response of the system. The term local nonlinearities means that the system degrees of freedom is large and there are only a few dof for the nonlinear elements. This requires special calculation techniques like those developed at the TNO Centre for Mechanical Engineering and implemented in DIANA as described by Fey [5] :

e Periodic solutions will be calculated by solving two-point boundary value problems, by means of a time discretization technique. The differential equations can be turned into algebraic equations using difference schemes. For instance, a period can be divided in 400 discretization points. So n differential equations are replaced by ni; 400 algebraic equations. Algebraic equations are solved using a damped Newton method. To keep the calculation time accept- able, it is necessary to have a limit number of dof. When analysing finite element models with many elements and thus many dof, it is necessary to reduce the number of dof.

o A path-following technique is used to investigate how a periodic solution is influenced by a change of the excitation frequency f e .

o Sometimes it is difficult t o formulate a starting solution which leads to a stable subharmonic solution instead of an unstabie harmonic solution. To find a usefui start vaiue, numericai integration can be used, which will only converge to the stable (sub)harmonic solution. Next, a branch of solutions can be found by the path-following method. The CPU-time consuming numerical integration is also applied to identify quasi-periodic behaviour.

As it is impossible to uncouple the nonlinear differential equations, these techiques are very time consuming. Therefore it is necessary to reduce the system degrees of freedom. Only the linear part is reduced. A straightforward simple reduction of the nonlinear set of equations of motion is impossible, because the eigenvalues and eigenvectors are state dependant.

The complete model has been divided into two substructures; each panel is a substructure. Further each substructure has been reduced by a Component Mode Synthesis method based on rigid body modes, free-interface eigenmodes and flexibility modes for each boundary degree of freedom. A flexibility mode represents the static response, resulting from an unit load acting on one of the boundary dof. Because we are only interested in the response up to 100 Hz, only the two lowest free-interface eigenmodes of each substructure have been taken into account. According to de

13

CHAPTER 5. NONLINEAR DYNAMIC ANALYSES 14

eigenfreq. [Hz] 11 unreduced model 1 red. six dof model 1 red. eight dof model fi 30.950 I 30.950 I 30.950

Kraker et al. [4] flexibility modes appear to be important for accurate eigenmodes of the reduced coupled model and for the frequency response near anti-resonance. In figure 5.1, it can be seen how the reduction looks like.

red. ten dof model 30.950

panel Reduction 4

L restraint

2 free-interface eigenmodes 1 rigid body mode I !

I flexibility modes

Figure 5.1: Model reduction.

This results in five dof instead of 103 dof per substructure. The complete model has due to the restraint eight dof, which is an enormous reduction. Table 5.1 compares the eigenfrequencies of the unreduced and reduced model.

Table 5.1: Eigenfrequencies of unreduced and reduced model without snubber.

For the eight dof model, the first four eigenfrequencies are practically the same. This is sufficient, as we are only interested up to 100 Hz. But as can be seen in the next paragraphs, local nonlinearities introduce higher frequency components like superharmonic resonances, which origin from higher eigenfrequencies. Therefore i t is necessary to examine the influence of taking into account higher eigenmodes. Introducing more dof by taking into account more eigenmodes means that eigenmodes of the reduced model become more accurate and the reduced system contains more modes. Con- sequently superharmonics may change to other frequencies and new superharmonic resonances are introduced. Perhaps a six dof model is accurate enough, b u t to examine this, the response has been compared with the eight dof model. The six dof model has been reduced the same as the eight dof model,


except for the eigenmodes. Now for every substructure one eigenmode has been taken into account. The eigenfrequencies of this reduced six dof model are also listed in table 5.1. I t can be seen that now only the lowest two eigenfrequencies are equal to the unreduced model.

C H A P T E R 5. NONLINEAR DYNAMIC ANALYSES 16

5.1 Review of some nonlinear phenomena

Before discussing the nonlinear dynamic behaviour, it is necessary to define and explain some nonlinear phenomena properly. Examples will be given by means of the steady-state behaviour of an eight dof model. Steady-state behaviour is the long term behaviour of a periodically excited nonconservative dynamic system after the transient response has damped out. First the terms subharmonic and superharmonic resonance are cieñneci, next the difference between iocai and giobai stability are explained.

For linear systems, the steady-state response and the excitation are both periodic with the same frequency. In the nonlinear case however, the period of the steady-state response can differ from the excitation period. Cubharmonic, superharmonic, quasi-periodic and chaotic behaviour can be distinguished. The first two phenomena will be explained by means of the figures 5.2 and 5.3, which show the response of an eight dof model with a bilinear snubber. In Appendix E both figures are displayed on A3-format as well. As the response of nonlinear systems is not always harmonic, the response will be shown as the maximum of the absolute value of the response, depending on the excitation frequency. This is in contrast with linear systems, where it is sufficient to show the amplitude of the response.

o Subharmonic resonance means that a system is oscillating with a lower frequency than the excitation frequency. A system excited with a frequency f e in a l/n-subharmonic resonance will oscillate with a frequency of f e l n . For some values of the excitation frequency a periodic solution can become marginally stable. When this bifurcation value is passed, the local stability of a steady-state, the number of steady-states and the type of steady-state behaviour may change. Three types of bifurcations can be distinguished: cyclic fold, flip and Neimark bifurcations. The flip bifurcation, also called a period-doubling bifurcation, and the cyclic fold bifurcation are often responsible for the appearance of subharmonic solutions. Bifurcations can be determined from the so-called Floquet multipliers. See Fey [5]. Another way to identify subharmonic solutions is by making a Poincaré section of the phase-plane. A Poincaré section, which is a state-space stroboscopically lighted at every excitation period, of a subharmonic solution of order l / n will contain n points. The frequency spectrum of a l/n-subharmonic solution will show at least two dominant peaks. One at the excitation frequency f e and one at the frequency feln. In practice a subharmonic resonance may occur at multiples of the bilinear eigenfrequencies. In figure 5.2 a resonance peak occurs around 70 Hz. In figure 5.4 the frequency spectrum of the response of node 51 is shown for an excitation at 71.94 Hz. The highest peak occurs at 35.97 Hz, which is the half of the excitation frequency. So this is a 1/2-subharmonic solution.

o S u p e r h a r m o n i c resonance [5, 6, 7, 81 is the phenomenon in which one or more higher harmonics cause resonance in a (sub) harmonic response. Superharmonic resonance, also called i n t e r n a l resonance, arise in nonlinear systems when the natural frequencies become commensurable. Then energy from some eigenmodes is transmitted t o other eigenmodes through various nonlinear i n t e rna l coupling mechanisms. Energy is continuously exchanged between the coupled modes during the ensuing motion. If damping is present in the system, then the energy is continuously decreasing. So 'internal' means that modes excite each other. This in contrast with a forced (external) excitation. This phenomenon does not occur in linear systems, because the modes of linear systems are uncoupled. The total response of a system is equal t o the summation of the solution of the free-oscillation and the particular


solution. In the linear case the free-oscillating term will always damp out. So the steady-state is equal to the particular solution. For nonlinear systems [û] at a superharmonic resonance the free-oscillating term does, in contrast with linear systems, not decay t o zero in spite of the presence of damping. This is due to the fact that , for every period again, the system is internally excited. The Poincaré section of a superharmonic resonance only shows one single point. But a Îrequency spectrum of a superharmonic resonance not only shows a peak at the excitation frequency, but also shows a peak at the internal excited eigenmode. When considering figure 5.2 again, it can be seen that there is a resonance peak at 32.02 Hz. Figure 5.5 gives the frequency spectrum of a second superharmonic resonance at 32.02 Hz. In the harmonic case the higher components are not dominant, but for the superharmonic resonance, there is also a dominant peak at 64.04 Hz. This means tha t for an excitation frequency of 32.02 Hz also the second eigenfrequency is excited.

Nonlinear dynamic systems have as important feature that the frequency spectrum of the response of the system not only has a component for the excitation frequency, but also higher frequencies do occur. Subharmonics contain not only the excitation frequency f e but also the frequency fe ln . And the spectrum of superharmonics shows not only peaks for f e but also for a dominant frequency rife, if rife N f b t . For an excitation G e , the response is for example:

Ue = g cos (27rfet) yr = A COS ( 2 r f . t 4- 4 a ) 4- B COS ( 2 T f n t 4- 46)

( 5 4 (5.2) (5.3) =+ yr = - A ( 2 ~ f e ) ~ COS ( 2 T f e t + 4 a ) - B ( 2 T f n ) 2 COS (2Tfet -I- db)

When we look at the displacement response yr it is possible that the term A overrules the higher frequency component B, but if f n is greater than f e , it is possible when we look a t the acceleration response yr that the higher frequency component B ( 2 ~ f , ) ~ overrules A ( 2 ~ f , ) ~ . Thus the displacement response does not necessarily show superharmonic resonance, while the acceleration spectrum does show superharmonic peaks. Still these resonances can be detected in the displacement response, because it is difficult to follow these parts of the branch with path-following. This leads to a frequency step reduction, so the calculated solutions will be closer together. A subharmonic branch however will not be seen better in the acceleration response, because f n N f e / n is always lower than f e . In that case the opposite might happen. Aii these things mentioned here can be seen when the figures 5.2 and 5.3 are compared.

0 Quasi-periodic behaviour occurs when the solution is a function of two or more periodic signals and there is an irrational relation between these periods. Therefore a Poincaré section of a quasi-periodic solution does not contain one single point, but a closed curve, which is the cross-section of a torus. This means that it is impossible to display quasi-periodic behaviour in a frequency response. Quasi-periodic solutions are due to Neimark bifurcations.

o Chaotic behaviour is neither a periodic nor a quasi-periodic solution.

The stability of the steady-state behaviour can be classified as follows:

o A solution is locally stable if the solution is asymptotically stable for a small perturbation of the steady-state. This can be determined by linearizing the equations of motion around the periodic solution and investigating the evolution i n time of a perturbation.

o A solution is globally stable if the solution is both locally asymptotically stable and asymptotically stable for any initial value of a disturbance. In other words, an equilibrium point is


globally stable if any trajectory in the state-space tends to that equilibrium point over time. See also Szemplinska [6].

When in this or coming chapters stable harmonic solutions or stable sub- and superharmonic solutions are mentioned, then is always meant local s tabi l i ty .


node 51 1 o-1

. . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . :$ ; ; :&#e ; ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... .O o:. . . . . . . . . . .

. . . . . . . . . ...o. 0 .o.o..

U

r f 5 1 30 40 50 60 70 80


node 151

90 1 O0 .- 20

10-l : 1 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

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: :A: . . . . . . . < . . . . . . . . . . . . . . . . . . . . .

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::\:: . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : : : : : j/&J :$& i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I - . . . . . . . . . . . . . . . . . . .

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I I I I I I I I 30 40 50 60 70 80 90 1 O0

I o - ~ L 20


Figure 5.2: Stable (..) and unstable (o o) displacement response of eight dof model (ksnub = 9.45*104 N/m) for a lg . base excitation.


node 51 I o4

$ 1 cb - x -1 x‘ E

1

1

t Y

c j o cb

8 E

1

c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . I I I I I I I

30 40 50 60 70 80 90 1 O0 O0 20


. . . . . . . . . . . . . . . . . . i: i 9; i 2 i i I I :j i i i i i i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :.: :* : : : : ............... . . . . . . . . . . . . . ’ . * . . . . . x : : : . : : : : : : : :

node 151

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . .

. . . . . . . . . . . . . : .j J3 : : . . . i ; ; ; ; ; . isbfl . . . . . ^ . . . . . . . . . . . . . . . . . . . . . . .

2 . . . . . . . . . . . . . . . . . ? . . . . . . . . . . . . . . . . . I.’..”’ . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : : : : : : .... ::41;i;;i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ¶ . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ’1

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 5.3: Stable (..) and unstable (o o) acceleration response of eight dof model (ksnub = 9.45*104 N/m) for a lg. base excitation.


Figure 5.4: Frequency spectrum of node 151 for fe=71.94 Hz.

Figure 5.5: Frequency spectrum of node 151 for fe = 32.02 Hz.

C H A P T E R 5. NONLINEAR DYNAMIC ANALYSES

5.2 Estimation of the bilinear eigenfrequencies

22

The eigenfrequencies of the system will change due to a one-sided linear spI,.ig. According t o Shaw and Holmes [3] the bilinear eigenfrequency f b l of the single dof system shown in figure 5.6 is as follow, provided tha t the backlash is equal t o zero:

In where fi the eigenfrequency without snubber is. In this equation, CY = is defined as the ratio of the stiffness of the one-sided linear snubber and the modal stiffness of the first bending mode of the solar panel without dummy mass. This in contra.distinction to the definition of CY in chapter 4. The term bilinear eigenfrequency is used here for the frequency a t which the bilinear model resonates.

. backlash I ,

Figure 5.6: Bilinear model according to Shaw.

To estimate the bilinear eigenfrequencies of the idealised solar panels with a bilinear snubber, a modification to formula 5.4 has to be made. Assume that the backlash is equal to zero and that both beams pass their equilibrium point a t the same time. Then the bilinear eigenfrequency f b z

caE be ca!cü!ated as fû!!ûws:

1 2 The total period time T = -(Tnosnub + Tsnub) ( 5 . 5 ) 1 f 6 , - - = 2 f s n u b f n o s n u b

- T f n o s n u b + f s n u b

In where f (no)snub the eigenfrequency for the linear system with(out) snubber is. To use formula 5.6 only the eigenfrequencies of the model without a snubber ( f n o s n u b ) and with a linear snubber ( f s n u b ) must be known. These values can be obtained from a simple eigenvalue analysis in DIANA. Table 5.2 shows the eigenfrequencies obtained by an eigenvalue analysis in DIANA, the bilinear eigenfrequencies obtained by a nonlinear analysis in DIANA and the bilinear eigenfrequencies calculated according t o formula 5.6. Note that formula 5.6 gives an underestimation of f b l and an overestimation of f b 2 , since the panels are not moving through their equilibrium point at the same time. But it is not possible to predict whether it is a n under- or overestimation as this can only be determined after some nonlinear dynamic analyses have been done and the motion of both


Linear (DIANA)

23

1.6 35.07 65.92 9.45 36.65 115.3

panels is exactly known.

(DIANA)

The ratio (Y will still be used, as this is a measure for the degree of nonlinearity. According to Appendix B, the modal stiffness k of the outer panel is 2.43*104 N/m.

I, I

9.45 11 35.78 I 64.03 I Nonlinear (according t o 5.6)

Nonlinear

1.6 32.88 56.44 9.45 33.56 69.11

I 1.6 11-1


5.3 Damped, bilinear model of snubber

In this paragraph the dynamic behaviour of an eight dof model with a bilinear snubber will be examined. The bilinear snubber has been modelled as a one-sided linear spring. In this case, no backlash is considered, and the snubber forces are only acting on both panels in case of a compression of the snubber. In forma! terms the bilinear sniibber is defined as:

where x is the relative displacement between the mountings of the snubber, ksnub and dsnub are considered to be constant. As long as the backlash is zero, there is still a linear relationship between input and output, i.e. between excitation and response (see Appendix C), just like the linear case. The model is excited by a prescribed harmonic base acceleration, with an amplitude of 9.81 m/s2. Therefore the prescribed base displacement, velocity and acceleration of the restraint look like:

Ü = 9.81 cos ( 2 r f e t ) [m/s2], (5.10)

where fe is the excitation frequency. In DIANA the one-sided snubber is implemented in the user- supplied routine nlsptr. f, while the prescribed base excitation is implemented in the user-supplied routine usr1od.f. They are both listed in Appendix H. Figure E.8 and E.9 (Appendix F) show the frequency response of both panel-ends, for a modal damping of 0.5 % of every eigenmode of the reduced model, a snubber stiffness ksnub = 9.45*104 N/m, thus a = 3.9 [-) and a snubber damping dsnub = 2.0 Ns/m. The periodic solutions have been calculated by means of the finite difference method using a fourth order central difference method. Branches of periodic solutions have been calculated with a path-following method using a tangential prediction step. From figure E.8 and E.9 all phenomena mentioned in paragraph 5.1 can be noted. For instance, the two lowest bilinear eigenfrequencies ( f b l = 35.78 Hz; fb2 = 64.03 Hz) can be seen clearly. Because all observed phenomena wil! return for each investigated value of a, all these phenomena will be discussed in more detail i n the following section.

5.3.1 Harmonic solutions

The lowest bilinear eigenfrequency is found a t f e = 35.78 Hz. In figure 5.7 the phase-plane has been drawn for fe = 35.78 Hz, while figure 5.8 shows the time histories for one period of both panel ends. From these time histories can be concluded that both panels move in phase, comparable with the linear model with a linear snubber. But the response of node 151 contains higher frequency components. This can also be seen i n the frequency spectra of the response of both panels (figure 5.9). From Figure 5.8 it can also be concluded that the snubber is only loaded from t=17.15 ms till 22.15 ms. Around 20 ms the largest snubber compression of approximately 8 mm occurs. Considering an initial distance of 15 mm between the two panels, it means that the panels make no contact with


node 51 (-) and node 151 (--)

25

y-displacement [m]

Figure 5.7: Phase-plane a t fe=35.78 Hz.

each other.

The second bilinear eigenfrequency is found a t f e = 64.03 Hz. In figure 5.10 the phase-plane has been drawn for fe = 64.03 Hz, while figure 5.11 shows the time histories for one period. It can be seen that both panels are moving out-of-phase comparable wi th the linear case. It is remarkable tha t node 51 oscillates around a position under its initial position, i.e. the undeformed case. The motion of the panels can be described as follow: the panel with the tip-mass (node 51) does hardly move, while the panel without tip-mass (node 151) oscillates with a large amplitude. Node 151 is bouncing against the relative not moving node 51. The impact is mainly reduced by the snubber. The lower mass of the panel (node 151) is also the reason for the fact that the response of node 151 contains more high frequency components than the response of node 51. From figure 5.11 it can also be concluded that the snubber is only loaded from t= 2 ms till 6 ms. For the second eigenfrequency, the maximum snubber deformation is approximately 1.8 mm. Thus in case of reducing the chance of contact between the panels, the first bilinear eigenfrequency is the most critical one. Another design criterion is that the stress in the panels is not higher than a certain stress value. The stress in the panels is depending on the acceleration of the panels. So for this criterion, the highest acceleration peak is the most important one. These conclusions are only valid for this particular set of parameter values.


node 51 (-) and node 151 (--) 0.021 I I I I I

-0.0251 I O 0.005 0.01 0.01 5 0.02 0.025 I

time [SI

Figure 5.8: Periodic time histories for fe=35.78 Hz.

node 151 0.0151

O 20 40 60 80 100 120 140 160 180 200

0.015

P

le [Hz]

node 51

..... . . . . <..<..... ... ..... . . . . . <.. . . . . ..,._ ... <

O I I I A I I

O 20 40 60 80 100 120 140 160 180 fe [Hz]

26

i3

D

Figure 5.9: Frequency spectra at fe=35.7S Hz.


I I I I I I I :.-.-<---. : \:

I. . * . . c . 1

: \ : .. e:,/ :

I . : I :

node 51 I-) and node 151 I-)

y-displacement [m] X

Figure 5.10: Phase-plane at fe=64.03 Hz.

-3: O.iO2 O . i O 4 O.iO6 O . i O 8 0.bl O h 2 O h 4 0.016 time [SI

27

Figure 5.11: Periodic time histories for fe=64.03 Hz.

CHAPTER 5. NONLINEAR DYNAMIC ANALYSES

f [Hz]

f i f 2

f 3 f h

28

linear model bilinear model a=O 0 ~ 3 . 9 according to 5.6 DIANA

30.95 36.65 33.56 3.5.78 49.34 115.31 69.11 64.03

263.76 267.14 265.4 308.91 326.58 317.5 -

-

5.3.2 Superharmonic resonances

Superharmonic resonances are, among others, found for f e = 32.02, 44.03, 52.17 and 78.56 Hz.


Figure 5.12: Stable (..) and unstable (o o) acceleration response (node 151) of eight dof model (ksnub = 9.45*104 N/m) for a lg . base excitation.

Table 5.3: Estimated bilinear eigenfrequencies.

Table 5.3 shows the approximate bilinear eigenfrequencies according to equation 5.6. As we have seen in the previous paragraph, the n-th superharmonic resonance peaks can be found near excitation frequencies fe = fb;/n in a weakly nonlinear system. The found superharmonic resonances are listed in table 5.4. The first column indicates which figure the frequency spectrum of that resonance shows (Appendix E). The third column indicates which other dominant frequency occurs in the frequency spectrum. Combining this information with table 5.3 leads to the conclusions in the last column of table 5.4.


figure f e [Hz] fba [Hz] 5.5 32.02 64.04

29

conclusion second superh. of f h 2

node 51

L E.6 -

..... ..<.: ......... : ................... : ......... : ....... <..: ...... ..: . . . . . . . . : ......... :... . . . .

" - - 62.83 314.2 fifth superh. of fb4 78.56 314.2 fourth superh. Of .fb4

.................. :.... ..... :,. ...... : ......... :.." ..... .:. ....... :... ..... .........

''!O $2 & i 6 excitation frequency [Hz]

node 151

lo:\ i 2 & 46 48 &I i 2 i4 i 6 i8 6'0 exciîation frequency [Hz]

Figure 5.13: Stable (..) 9.45*104 N/m) for a lg. base excitation.

and unstable (o o) acceleration response of eight dof model (ksn& =

Table 5.4: Superharmonic resonances.


5.3.3 Sub harmonic solutions

A branch of a 1/2-subharmonic solution has been found in the frequency interval 66.0-73.0 Hz, which bifurcate from the harmonic branch via flip bifurcations at the boundaries of the interval. The resonance peak (figure 5.14) around 71.5 Hz is a 1/2-subliarmonic solution of fbl. Figure 5.15 and figure 5.16 show the time histories and the phase-plane at fe = 71.94 Hz for both panels. These time-histories are, except for a scaling factor, t.he same as the time-histwies d the first !dinear eigenfrequency. From combining figure 5.12 and 5.14 it can be concluded tha t there also exist a l/Zsubharmonic of the second superharmonic of fb2. In fact the part of the response between 30 and 40 Hz has been copied to the response between 65 and 7 5 Hz.

node 51

=lo-

3 g 10' E

exutation frequency [Hz]

1 o"

1 0.2

- E ü zloJ

4 8 10' E

1 o* 60 62 64 66 E8 70 72 74 76 78 80

exutation frequency [Hz]

Figure 5.14: Subharmonic solutions.

The 1/2-subharmonic branch shows the same dip as the harmonic response at 32 Hz does. In fact it is possibie tha t there exist more subharmonic solutions originating from superharmonic resonances, although no flip or cyclic fold bifurcation point is found on the harmonic branch. Between 93.2 and 96.0 Hz a stable 1/3-subharmonic solution of the second superharmonic resonance of fb2 has been found. This 1/3-subharmonic solution is an island above the harmonic branch. For 93.2 and 96.0 Hz, cyclic fold bifurcation points ha.ve been found. Only a part of the branch has been found as it was impossible t o pass these bifurcation points. Figure E.7 shows the frequency spectrum of node 151 at fe = 94.8 Hz. This spectrum shows dominant frequencies for 31.6, 63.2 and 94.8 Hz. Thus i t is a 1/3-subharmonic resonance of 31.6 Hz, which is a second superharmonic resonance of the second bilinear eigenfrequency. More subharmonics of superharmonic resonances have not been found thus far. When passing through the harmonic branch after 100 Hz, mostly flip-bifurcations were found. This suggests more subharmonic solutions. For fe z 107 Hz, a 1/3-subharmonic of fbl has been found and for f e = N 127 Hz a 1/2-subharmonic of f b 2 has been found.


I

-0. -1 -0 5 O 0.5

y-dsplacement [m] x 1(

Figure 5.15: Phase-plane aft fe=71.94 Hz.

-1.5' I I I I I

O 0.005 0.01 0.015 0.02 0.025 0.03 time [SI

31

Figure 5.16: T i m e histories at .fe=71.94 Hz.


5.3.4 Other points of interest

So far the most important resonance peaks have been identified. But there are some areas that need to be examined more accurate.

a Frequencies below 30 Hz: In this area a lot of unstable solutions have been found, indicating subharmonic solutions and quasi-periodic motion. For fe=24. 1 Hz a stable superharmonic resonance is found. Out of the frequency-spectrum it appeared that this is a third superharmonic resonance of the 1/2-subharmonic of f b i , because the spectrum showed dominant peaks for 24.1 and 72.3 Hz. It appeared that it is very difficult and time consuming to calculate stable solutions in this area. For instance a t fe=20.30 Hz an unstable harmonic solution has been found. Numerical integration lead to a stable 1/2-subharmonic. But the phase-plane was equal to the harmonic solution. And for instance the solution obtained by numerical integration over 10,000 periods at fe=16.0 Hz had still not converged. So after 10,000 periods the transient had not damped out. This is probably due t o the low damping ratio and the fact that most solutions are marginally stable. A 1/2-subharmonic has been found at fe=28.3 Hz after 1500 periods. This is a 1/2-subharmonic of a superharmonic resonance of 14 Hz.

But as the response of the panels in this frequency-range is very small, this area is of no interest anymore.

a In the frequency interval from 37.59 Hz till 43.44 Hz unstable harmonic solutions have been found. This is an area enclosed by two flip bifurcations. Thus a subharmonic solution is expected. Although the unstable solutions were not marginally stable, it was difficult to find stable solutions. For instance a numerical integration a t f e = 41.0 Hz did not converge for 0.5% modal damping. But for 6% modal damping, the transient was damped out after 10,000 periods. Path-following from this solution was very difficult. Most times it was even not possible to find a periodic solution after the transient had damped out, although one period is divided in 6400 discretization points. Trying to find a solution for 5% modal damping out of the solution with 6% modal damping was also not possible. The phase-plane of a stable 1/2-subharmonic solution looks almost the same as the unstable harmonic solution. And there is only a small difference between the response for 0.5% (Iylmaz (node 51) = 0.877 mm) and 6% (lylmaz (node 51) = 0.83 mm) modal damping. This is obvious because low modal damping only influences the resonance peaks and not the rest of the response. Also there is practically no difference between the motions of the beams just before and just after the flip bifurcation at 43.44 Hz. The response of the system does not drastically change when passing this bifurcation point. As it is difficult to find the 1/2-subharmonic branch and of the subharmonic is almost equal t o IyJmaz of the harmonic solution, no more effort is put in the 1/2-subharmonic branch.

a For 49.7 Hz an unstable harmonic solution via a Neimark bifurcation has been found. Fig- ure 5.17 shows the Poincaré section at f e ~ 4 9 . 7 Hz. This figure shows that the free frequency is somewhere between fe/6 and fe/7. Thus this is a quasi-periodic solution.

o The area between 86 and 100 Hz contains only unstable harmonic solutions. In this interval mainly quasi-periodic behaviour occurs. Thus two frequencies or more are incommensurable: they relate irrational.

o More quasi-periodic behaviour has been found for f,=7*5-76 and 80.7-81.7 Hz.


-o.02-..+tb...( . . . . . . . . . ;.. . . . . . ; ._ ...... ;. ........ ( . . . . . ;

-0.02%;6

. . . . . . . . . . . + + i

i + + + + . i + ;

0.8 1 1.2 1.4 1.6 1.8 2 2.2

x i a y-displacement [m] 4

Figure 5.17: Poincaré section at fe=49.7 Hz.

In the 1/2-subharmonic branch of f 6 i also quasi-periodic behaviour has been found, namely for fe=69.0-69.2 Hz and for 66.6-67.2 Hz.

5.3.5

Comparing the response of the six and eight dof model leads to the following conclusions (see figure 5.18):

The six and eight dof model compared

o The displacement response is almost the same for both models.

0 The acceleration response shows some differences. The major difference is the fact that the fourth superharmonic resonance of fb4 a!ters tg another rxcitôtim frequency ( si:: dof model: fe = 79.10 Hz, eight dof model: f e = 78.56 Hz). This is due to the fact that f64 is higher for the six dof model (see table 5.1). The height of the resonance peak however remains the same. Further some small differences occur between 40 and 60 Hz.

o No attention has been paid to excitation frequencies lower than 30 Hz. This is mainly because of the calculation difficulties mentioned earlier and the minor importance of this branch. But it is expected that in this area too, some new superharmonics will occur.

Although it is not shown in figure 5.18, also a 1/3-subharmonic solution of the second superharmonic resonance of f62 has been found.

Because the difference between the six and eight dof model is very small, it is advised to use the six dof model in order to have lower calculation times.


node 51 1 o4

t : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,..,., . . , . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...... . . . . . ftj2;; i i ; : : : : : : ; ; ; : i ; 2 : : :

: ::: : : : :-: : : : : : :li/2iSUb:fbjifi i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. .

. . . . . . . . . . . . i .< . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . ;::i$ . . . . . . . . . . . . . ..... .. . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

: : : : : j . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 : : > : : i . . . . . . . . . . . . . . . . . . . . . . . .

;@,:.: . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : :.i :

. . . . . . . . . . . . . . . . . . . . . .

. . . . : : I : . . . . . . . :.: :&? :

. . . . . . . . . . . . . . . . ... . . . . . . . . . . .

- $ l o i p

. . . . . .

................................ E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . : : . . . . . . . . . . .: . . . . . ........................... ........................... ...........................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . : : . . . . . . . . . :?QQti j j j j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . : : : ;o ;.: ::. . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

: : : : : i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I I I I I I I I 30 40 50 60 70 80 90 1 O0

1 oo' 20


node 151

. . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .:.*.@e: . . . '* 0 0: 0: j i j j i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.a:. . . . . . . . . . . . : . . . . o

. . . . . . . . . s : : : : 4 ; : i * . . . . . . . .y . . : . : : .... .*. . . . . . . . . . . . . .

I : j i ; ; i ;q . . . . . . . . . . . .

. . . . . . i ' i : : : : : : : : . .

. . . .:. . . . . . . . . .*.(&*: :

. , . . . . . . : . . : i . . . . . . . . . . . . . . . . . . . . . . . . . . , ,o : 1.: : . I ' . . ' . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -_ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .: . . . . . . . . .&W. . . . . . . .

I I I I I I I O 0 0 30 40 50 60 ?O 80 90 1 O0


. . . . . . . . . .

Figure 5.18: Stable (..) and unstable (o o) acceleration response of six dof model (ksnub = 9.45*104 N/m) for a lg. base excitation.

Experimental verification

To verify the numerical results, an experimental setup has been developed. In this chapter it will be shown that the numerically found phenomena also show up in experiments. As starting-point, the numerical model has been taken. But some modification had to be made due to experimental limitations. A shaker table with an eccentric drive mechanism is used to prescribe a harmonic excitation of the model. In figure 6.1 the experimental setup can be seen.

, eccentric shafi

Figure 6.1: Experimental setup.

Here a base displacement is prescribed with a constant amplitude. This is in contrast with the numerical model, where a prescribed base acceleration with a constant amplitude of 9.81 [m/s2] is used. Unfortunately the eccentric shaft is not completely harmonic, so also high frequency components do occur. The use of this eccentric shaft introduces two limitations. First the mass of the model must be low in order t o obtain a good performance of the eccentric table for higher frequencies. Therefore the model must be scaled. Second, the excitation of the shaker table is harmonic up to 50 Hz. Therefore the bilinear model must be designed so that the 1/2-subharmonic branch can still be measured. Consequently the first bilinear eigenfrequency must be lower than 25 Hz. In Appendix D it is described how the experimental model has been designed to satisfy these limitations. Some remarks can be made about the experimental model:

e The aluminium strips have been cut out of aluminium plates. By that the strips are curved, twisted and the cut side-walls are not perfectly straight. Due to the torsion of a beam also the gravity is included in the measurements of the accelerometers. Therefore the measured

35

CHAPTER 6. EXPERIMENTAL VERIFICATION 36

acceleration must be corrected for the gravity, see figure 6.2.

Figure 6.2: Accelerometer mounted on the twisted beam.

o Because the gap, in which the strips are clamped, is not smoothly milled, some foil has been used to improve the fixation.

o The eccentric shaft, a crank-connecting rod mechanism, is not perfectly harmonic. Also higher frequency components are introduced by this mechanism. This will give a distortion in the measured accelerations.

Before measurements of the dynamic behaviour of the model can be compared with numerical results, some parameters must be determined. These are the amplitude of the excitation, the modal damping of the beams, the stiffness of the beams and the stiffness and damping value of the snubber. Accelerations are measured with Bruël & Kjaer accelerometers. All used measuring-instruments are described in Appendix F. To cut-off high frequency components, analog high band-pass filters (1 and 10 kHz) and digital filters of 200 and 400 Hz (Difa) have been used. The analog filters are used in the Kistler amplifier. For these filters the cut-off frequency must be chosen at least ten times higher than the desired frequency range. Analog filters may cause a shift in the phase. It was not necessary to calibrate the acce!erometers again, because their output was correct (see Appendix F).

6.1 Determination of model parameters

The excitation is a prescribed base displacement of the restraint:

o As excitation the displacement of the restraint is prescribed:

9 = -Ae2 COS ( 2 ~ . f ~ t )

ij = A e 5 ( 2 r f J 2 COS ( 2 r f e t ) 3 log(amp1itude of y) = log(A,,) + 2 1 o g ( 2 ~ ) + 2 1 0 g ( f e )

C H A P T E R 6. EXPERIMENTAL VERIFICATION 37

...... .... .......................... ~ .............. . , . . . . . . <.<.,......<....I... . . . . . . . . . . . ............................. : ................ ; ......... ,.: ................................... . . . . . . . . . . . . . . . . . . . . 10' 1 o2


,lo2 r E I

.e 3 a l o ' E H ............_. .................................. . . . . . . . . . . . . . . . . . . . . . . . . . . .

I 15 20 25 30 35 40

excitaiion frequency [Hz]

Figure 6.3: Measured (*) and expected (-) base acceleration amplitude (top: double logarithmic scale, bottom: linear-logarithmic scale),

In figure 6.3 the measured and expected amplitude of the acceleration of the restraint is given. Also a line has been fitted through these values. From the fitted line could be derived that A,, = 0.51 mm. For this frequency range can be concluded that the input-signal is harmonic up to 36 Hz, as the measured values make a n angle of 2 on double logarithmic paper.

e The modal damping of the aluminium beams has been determined as follow:

The impulse-response h(t - t o ) of an one dof system is:

with : t = modal damping, (6.7) f o = undamped eigenfrequency and

f n = 27r foJ1 - t2 = damped eigenfrequency.

If the modal damping t is small, the impulse-response becomes:

h(t - t o ) N ~ 2 r f o e - ~ ~ ~ f o ( ' - ~ o ) sin ( 2 r f o ( t - t o ) ) (6.10) w In(k( t - t o ) ) = Zn(K27r f o ) - @7r f o ( t - t o ) + ln(sin (27r f o ( t - t o ) ) ) (6.11)

The modal damping can now be determined by fitting a line through the local maxima of the impulse-response. The mean value of the modal damping is 0.64 %.

e In preliminary measurements of the dynamic response of the beams, a difference between the calculated and measured eigenfrequencies was noticed. The measured values were lower than expected, although the fixation does clamp enough, the dimensions of the beam are


correct and the density of the aluminium beam is correct. This difference could also not be explained by the modelling of the point-masses. Therefore the only logical explanation is tha t the stiffness of the beam is not correct. This means that the Young-modulus E is considered too high. To verify this, it is tried t o estimate the correct stiffness value. Force- displacement measurements showed that the static stiffness kbeam = 1000 & 25 % N/m. While in the mmerieal model the static Stiffness kbeam = i433.6 N/m. So this is a prove that the Young-modulus is lower than the expected value of 70*109 N/rn2. The lower Young-modulus of the aluminium beams is not unique for this problem. Also the stiffness of a steel beam (with a better shape than the aluminium strips, because the steel beams are produced in strips of 4x20 mm) has been determined. This resulted in a 8% lower stiffness. The beams are modelled in DIANA. Appendix D gives all dimensions of the beam. For the numerical model, the Young-modulus is modified so that the lowest eigenfrequency of the numerical and experimental model are simular.

- -

3.516 EI f i = ~ d ; o . = 48.4 Hz (measured without tipmass a.nd accelerometer).

f i a r 2pAL4 * E = ( - - = 59 * 109N/m2 and

3.516) 1

(6.12)

(6.13)

= 1208.7N/m (6.14) 3 E I L3 kbeam = -

In the tables 6.1, 6.2 and 6.3 it can be seen how accurate the approach is by using an Young- modulus of 59.0*109 N/m2 instead of 70.0*109. The eigenfrequencies have been measured with 8000 samples and a Af N 0.05 Hz.

I measured I numerical I relative error [%I E [lo9 Nlm21 1 - I 70.0 I 59.0 I -

Table 6.1: Eigenfrequency of a n aluminium beam without a dummy mass.

I measured I numerical I relative error [%I I I . . I fl [Hzl I 29.88 1 29.76 I 0.43

Table 6.2: Eigenfrequency of an aluminium beam wi th a dummy mass (m = 27.7*10-3 kg, E = 59.0*109 N/rn2).

Next some snubber parameters will be determined.

To make sure tha t nonlinear phenomena can be measured, a snubber with a low damping value is chosen. Therefore instead of a snubber made of rubber, a spring is used. This spring is shown in figure 6.4. Both ends of the spring are necessary to fix the spring to the beams. The total mass of the spring is 30.0*10-3 kg. To determine the spring stiffness, one side of the spring was attached to the real world, while the other side was loaded with two different masses.

C H A P T E R 6. EXPERIMENTAL VERIFICATION

measured

fl [Hzl 25.59 f 2 [HZ1 34.57

E [lo9 N/rn2] -

39

numerical relative error [%]

27.46 25.22 1.47 36.96 33.93 1.85

- 70.0 59.0

r M4 _-

filled with epoxy

M4

Figure 6.4: The snubber used in experiments.

By measuring the impulse-response of both masses, the modal damping and the eigenfrequencies, and thus the stiffness could be derived. Each mass results in a different eigenfrequency. See figure 6.5.

gravity 1 Figure 6.5: Determination of the snubber stiffness.

0 - Lp fi = 'J" 2n m + m l

f - 2 n m (6.15)

(6.16)

The spring is initially loaded with a mass of 0.331.5 kg. The additional mass ml is exactly known. But the mass m is not known, because the mass also represents the mass of the


spring. This mass m however can be calculated as follow:

For:

f o = 16.602 HZ fi = 13.135 HZ ml = 188.1 * kg

(6.17)

(6.18)

(6.19)

(6.20)

* m = 314.8 * kg and k,,,b = 3425.4 N/m. (6.21)

The modal damping cl for the discrete mass-spring model wi th the additional mass ml is 1.05 %, while (without mass ml) is 1.47 %. The discrete viscous damping value for each model is:

In the model a mean value of dsnub = 0.92 Ns/m has been used.

Once the snubber parameters are known, measurements have been done for the model with snubber shown in Appendix D.

6.2 Linear model

In the linear case the snubber mass has been divided equally over both beams. Since the total snubber mass (including both ends) is 0.030 kg, this means a point mass of 0.015 kg attached to each beam. Table 6.4 gives the eigenfrequencies for ksnub = 3425.4 N/m and = 0.92 Ns/m. The base excitation has been modelled as follow:

y = A,, cos (2xfet) 5, = A,,27r f e sin ( 2 ~ . f & ) y = -A , , (2~f , t )~ cos (2r fe t ) in where A,, = 0.51 *

(6.24) (6.25)

(6.26) m.

Figure 6.6 shows the measured and calculated acceleration response of the model with linear snubber. The difference between the measured and calculated eigenfrequencies are quite large. Espe- cially the lowest one. The measured and calculated response of the linear system are only the same up to 20 Hz due to the difference in the measured and calculated eigenfrequencies.

CHAPTER 6. EXPERIMENTAL VERIFICATION

r >

fl b z l f 2 [Hzl

41

I

26.76 25.20 5.83 59.77 57.73 3.41

I I measured 1 nameïicai I relative error [%] I 1 E iiû' î i / rnz] i - I 59.0 i - I

Table 6.4: Eigenfrequencies for model with linear snubber (k,,,b = 3425.4 N/m).

node 51

10 15 20 25 30 35 40 45 50 55 excitation frequency [Hz

node 151 1 o*

&. .........,......... /'i,.\.. =..: ... <... ..: . . < < ..... .: .... . . . . . ;. . .<. ...,.. <<.. . . . : I ._ ... <. . ..

lo-'' I 5 10 15 20 25 30 35 40 45 50 55


Figure 6.6: Measured (*) and calculated (-) acceleration response of linear model for Asnub = 3425.4 N/m.

C H A P T E R 6. EXPERIMENTAL VERIFICATION

E [lo9 N/m2] f l [Hzl f 2 K " 1

42

measured numerical relative error [%I ' - 59.0 -

21.09 21.13 -0.16 34.38 33.93 1.30

6.3 Nonlinear model

frequency [Hz] f b i

f b 2 1/2-subharmonic

In the bilinear case, the snubber mass is applied t o only one beam, i.e. the beam with the dummy mass. Therefore the eigenfrequencies of both beams will change. On the contact-surface of the snubber a small piece of rubber has been stuck to reduce the noise level and to reduce high frequency components caused by the impact. Due to this piece of rubber there is a very smal! preload of the rntihher. T&!e 6.5 shows the eigefifreqüencies vf both beams sepaîaiely, wltet-i there is no contact between snubber and beam (800@ samples,Af N 0.05 EZ). The first eigenfrequency gives the smallest error. Therefore it is expected that in the bilinear case also the measured and calculated lowest bilinear eigenfrequency will agree the most. There is a major difference between the relative error in table 6.4 and in table 6.5. The model with snubber gives a larger relative error than the model without snubber. This difference is due to a not completely correct snubber stiffness value.

measured calculated 24.5 24.02 40.4 41.99

47.0-49.0 45.2-50.9

Table 6.5: Eigenfrequencies of bilinear model when there is no contact between snubber and beam.

Table 6.6 gives the measured and calculated bilinear eigenfrequencies, the 1/2-subharmonic and the second superharmonic resonance (8000 samples, Af N 0.05 Hz). During the measurements of the

2nd superharmonic 1 21.68 I 20.97 1 Table 6.6: Resonance peaks of the bilinear model.

response, a digitai fiiter of 200 Hz has been used ( j e < 36 Hz: 2000 samples, Af z 0.2 Hz; je 2 36 Hz: 4000 samples, Af N 0.1 Hz). A cut-off frequency of 200 Hz has been used to avoid a distortion of the measured excitation. This has as a consequence that â.lso high frequency Components in the measured response are filtered away. Figure 6.7 shows the measured, fitted and calculated base acceleration amplitude. It can be seen that the excitation is linear in the frequency range 22-50 Hz. There is however a small difference between the fitted and calculated base excitation. The fitted line through the measured values resulted in a n amplitude of 0.458 mm instead of A,, = 0.51 mm and an exponent of 2.05 instead of exactly 2. The excitation is thus influenced by the response. But the simulated base excitation according t o equation 6.26 is a very good approach. In practice this means that , for excitation frequencies greater than 25 Hz, the measured response will be a little higher than the calculated response. In figure 6.8 the measured and calculated acceleration response of the bilinear model is shown. The response has been calculated the same as has been done in the previous chapters. The numerical model has been reduced to six dof. The DIANA-input file and the user-supplied routines have been enclosed i n .4ppendix H. Regarding figure 6.8, some remarks can be made:


The measured harmonic resonance peaks of the bilinear eigenfrequencies are for both nodes lower than the calculated values. It is possible that the damping values in the numerical model are too low or that high frequency components are filtered away during the measurements. Another explanation is that in the resonances, energy is dissipated during the impact (like noise production), which has not been modelled. Therefor peaks might become lower. For the second hilinear eigenfrequency it C C U ! ~ he seen that the pafie! with the tip-inass does h a r d y move. i ne panel without tip-mass however, is bouncing against the other panel. This is conform the calculations done in chapter 5.

Between 45 and 50 Hz a 1/2-subharmonic resonance has been found. After 50 Hz, the measured response is harmonic again. The figures F.l and F.2 in Appendix F show the time histories for fe = 47.07 and fe = 49.02 Hz. Note that the response period is twice the excitation period. In figure F.3 the frequency spectra at f e = 49.02 Hz are shown. The spectrum of the response has not only a peak for fe = 49.02 Hz, but also for 24.5 Hz. The measured peaks of the 1/2-subharmonic branch are lower than the calculated values. This is probably due to a different damping ratio.

For fe = 21.68 Hz a second superharmonic resonance has been found, although this resonance was expected to occur around 20 Hz. Figure F.4 (Appendix F) gives the time histories for this frequency and figure F.5 gives the frequency spectra. The frequency spectra not only show a peak at 21.68 Hz, but also a peak at 43.4 Hz, which is the second bilinear eigenfrequency.

r n l

P

It can be concluded that there is a difference between the absolute values calculated response. But the global acceleration responses are comparable.

1 o' excitation frequency [Hz]

m : i }.........i ................... ......... :..H. ..... .:.. ...... ................ . - E

a .z 10'

E

I 10 15 20 25 30 35 40 45 50 55


of the measured and

Figure 6.7: Measured (**), fitted (- -) and calculated (-) base acceleration amplitude (top: double logarithmic scale, bottom: linear-logarithmic scale).


node 51 Io"

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I I I I I I I I I 1 i 5

. - 5 i o 20 25 30 35 40 45 50

excitation frequency [Hz].

node 151

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...................... -I . . . . . . . . . ; . . . . . . . . . ; . . . .......

I I I I I I I I I I 5 10 15 20 25 30 35 40 45 50 55

excitation frequency [Hz].

Figure 6.8: Measured (*) and calculated stable (..) and unstable (00) acceleration response of the bilinear model: ksnub = 3425.4 N/m, dsnub = 0.92 Ns/m, A,, = 0.51 mm, a modal damping of 0.64 %, no backlash and a static beam stiffness kbeam = 1205.7 N/m.

Chanter 1- - -- 7

Parameter variations

7.1 Snubber damping

In the linear case it appeared that the damping of the snubber has the largest influence on the second eigenresonance peak. This is due to the second eigenmode where both panels are moving out-of-phase. Therefore the difference of the velocity between the panels will be larger than when both panels are moving in phase, which results in higher damping forces.

damping [Ndm] damping [Ndm]

damping [Ndm] damping [Ndm]

Figure 7.1: Variation of clsnub (a = 3.9, modal damping of 0.5 %); fbl, node 51 (-) and node 151 (- -); fb2, node 51 (-.) and node 151 (*-*); 1/2-subharmonic resonance, node 51 (x-x) and node 151 (0-0). A logarithmic scale on the left side and a linear scale on the right side.

Figure 7.1 shows the results of a variation of dsnub for the six dof model, considering that the resonance peaks do not change to other frequencies due to a damping variation. Note that for the first bilinear eigenfrequency f b l , node 51 ha.s the largest displacement value. While node 151 has the largest acceleration value. This is in conformity with the remark made earlier that the response of node 151 contains more higher frequency components than the response of node 51. I t can be

4 5

CHAPTER 7. PARAMETER VARIATIONS 46

concluded tha t the damping has relatively the same influence on node 51 and node 151 for the first bilinear eigenfrequency, as both lines have the same angle on logarithmic paper. The lines are only shifted for a constant value. For the second bilinear eigenfrequency and the 1/2-subharmonic solution the same conclusions can be made. But there is a difference between the first and second bilinear eigenfrequency. The latter one has a steeper angle. This means tha t the damping has re!atiue!y a, larger infiüence on Jb2 than f b i . This is like predicted from the linear case. It can also be seen that the peak of node 51 for fb2 is higher than the 1/2-subharmonic peak of node í 5 í tili d,n,b=2.0 Ns/m. For damping values higher than 2.0 Ns/m, the 1/2-subharmonic peak of node 151 will become higher. Further the following remarks can be made:

o The unstable harmonic branch around 40 Hz will become smaller for larger dsnub values.

o The harmonic branch before the first bilinear eigenfrequency, with a lot of superharmonic resonances, will become stable.

7.2 Modal damping

In figure 7.2 a variation of the modal damping is shown on the peaks of the bilinear eigenfrequencies and the 1/2-subharmonic peak €or both node 51 and 151, considering that the resonance peaks do not change t o other frequencies due to a variation of the damping. The modal damping is the same for every eigenmode of the reduced model. It can be concluded that the 1/2-subharmonic branch is influenced the most by the modal damping. The first bilinear eigenmode is an in-phase motion, thus the influence of the snubber damping is small. This means that for this motion only the modal damping of the beam is important. Therefore, the amplification of both beams goes t o infinity for a modal damping value close to zero. However, the second bilinear eigenmode is an out-of-phase motion. This means that the amplification of both beams does not go to infinity for a modal damping value close to zero, due to the fact that the snubber is still active for this out-of-phase motion. The trivial conclusion can be made that the damping values of the snubber and the beams must be chosen as high as possible to reduce the resonance peaks. Sub- and superharmonic resonances will even disappear when the damping values are large enough. Figure 7.3 shows the acceleration response of an eight dof modal for a lg. base excitation with 2.0 % modal damping of each eigenmode of the reduced model (ksnub = 9.45*104 N/m, dsnob = 2.0 Ns/m). The resonance peaks decrease and become less sharp. The peaks do not alter to other frequencies. Note tha t the second superharmonic resonance decreases relatively less than the peak of the first bilinear eigenfrequency. Due to more modal damping, less superharmonic resonances occur. The intervals with unstable branches however remain the same.

CHAPTER 7. PARAMETER VARIATIONS

0.05 I

I 0.5 1 1.5 2 modal damping [%]

l oo o 5 1 115 L modal damping [%]

47

o o 5 1 1 5 2 modal damping [%]

Figure 7.2: Variation of the modal damping (dsnub = 2.0 Ns/m, cx = 3.9); f b i , node 51 (-) and node 151 (- -); fb2, node 51 (-.) and node 151 (*-*); 1/2-subharmonic resonance, node 51 (x-x) and node 151 (0-0). A logarithmic scale on the left side and a linear scale on the right side.

node 51 1 o’

- u>

< . < . < . < .... ”.. e u E lo2

o

I 40 50 60 70 80 90 1 O0

exciiaiion frequency [Hz]

node 151


Figure 7.3: Stable (..) and unstable (o o) acceleration response of eight dof model for a lg. base excitation with 2.0 % modal damping (ksnub = 9.4.5*104 N/m, dsnub = 2.0 Ns/m).


7.3 Snubber stiffness

It is mentioned before that a variation of the bilinear snubber stiffness leads t o a change of the bilinear eigenfrequencies, and thus a change of the sub- and superharmonic resonances. Figure 7.4 shows the acceleration response of the bilinear eight dof model with a snubber stiffness ksnub = l.6*104 N/m (a = 0.66 >, dsnub = 2.0 Ns/m, a rrioda! darnping of G.5 '76 2nd T, ?g base excitation. Note that fbl = m.34 n z and fb2 = 55.û5 Ez. H i jZsubharmonic branch is found in the frequency kiterval 64.3 - 68.3 Hz. A second superharmonic resonance of fb2 occurs for fe = 27.55 Hz. in contradistinction t o the model with a! =3.9, a fifth superharmonic resonance of fb3 for fe = 52.97 Hz arises in the branch of the second bilinear eigenfrequency of node 51. A variation of the bilinear snubber stiffness and thus a variation of a! means that there is also a change i n the degree of nonlinearity, as a! is a measure for the degree of nonlinearity. See Fey [ 5 ] . When the response of the model with a! = 0.66 and Q = 3.9 are compared, it can be concluded that a lower cy, thus a lower degree of nonlinearity, leads to lower harmonic, s u b- and superharmonic resonance peaks. The frequency intervals, in which subharmonic solutions occur, become smaller and other unstable harmonic branches become smaller too. Thus the height of resonance peaks decreases if Q becomes lower.

nn ! - A TT


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. . . . . . . . . . . . . . . . . . . . .

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~~


Figure 7.4: Stable (..) and unstable (o o) acceleration response of eight dof model for a Ig. base excitation with = 1.6*104 N/m and dsnzLb = 2.0 Ns/m.


7.4 Backlash

In Appendix C is shown tha t the equations of motion and therefore the system response, for a bilinear system with a nonzero backlash, are excitation and frequency dependant. A dimensionless backlash is defined as S = v. From this definition it can be seen that 6 becomes smaller when the excitation amplitude A is increased. Put a m t h r r w2y b remains the same when both b and A are ch-mgr'! with afi e q ~ a l ïatio. Tv;litli the Dackiash in the biiinear system is meant that the undeforrned state of the h e a a s does not correspond with the point of application of the one-sided snubber, Two situations can be distinguished:

e There is no contact between the one-sided snubber and the beam in the undeformed position. A certain beam displacement is necessary to make contact. This situation is called the 'free backlash'.

e There is contact between beam and the one-sided snubber in the initial state. The snubber causes a certain deflection of both beams. This situation is defined as the 'preload'.

In the following the bilinear six dof model has been analysed for different 6. Note that for all these models ksnub = 9.45*104 N/m, dsnub = 2.0 Ns/m, the modal damping is 0.5 % and a base acceleration of 1 g.

7.4.1 Free backlash

Free backlash means tha t (compared to the model without backlash) the share of the eigenmodes, of the model without snubber, in the bilinear eigenfrequencies will increase. This is because there will be less contact due t o the increased backlash. As a consequence the bilinear eigenfrequencies, especially the first and the second one, will alter to lower values. Further, resonance peaks will bend, due t o the frequency dependence of the equations of motion. The bending of the peaks results in jump phenomena when a frequency sweep is done. In figure 7.5 the acceleration response for a model with a free backlash of 5 mm, t h u s 6 = 0.51*10-3(2~fe)2, is shown. It can be seen clearly that the resonance peaks are curved and that the base of the bilinear eigenfrequencies has shifted to lower frequencies. The base of . fb i star ts a t 31 Hz instea,d of 35.8 Hz and the base of f b 2 star ts a t 50 Hz instead of 64 Hz. In fact the response is becoming more simular to the response of the iinear model without a snubber. The jump phenomena of the resonance peaks are due to cyclic fold bifurcations. For an increasing frequency starting a t 30 Hz, around 34 Hz two periodic solutions coexist; one of them is stable, the other is unstable. At the bifurcation point (fe = 34.98 Hz) the two solutions merge into one marginally stable periodic solution. Just after the bifurcation point locally no periodic solution exists anymore, which results in a jumping behaviour. Cyclic fold bifurcation points are also called turning points. It is expected that the height of the resonance peaks does not change compared to the bilinear model without backlash. The height of a resonance peak is only depending on the damping values. But it can be noted that for node 151 the acceleration peak of the second bilinear eigenfrequency has increased with at least a factor 2 due t o the backlash. For this large backlash value no subharmonic resonances have been found so far. But no prove can be given that the subharmonic resonances are disappeared due to the large backlash value. Also the second superharmonic resonance of the second bilinear eigenfrequency is disappeared.

i


node 51

10 ::::::.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ ........................ ........................ 4r ........................ . . . . . . . . . . . . . . . . . 1: : : : : : : . . . . . . . . .:. . . . . . . 0 . i i . . . . . . . . . . . . . . . . . . . . . ..................... ........................ ....................... . . . . . . . . . .

........................ 3l ....................... .............. 1: : : : : : : . . . . . . . . .I. . . .

. . . . . . . . . . . . . . . . . . . . ~ : 1.: : : : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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excitation frequency [Hz] node 151

. . . . . . . . . . . . . . . . . . . . . . . . a." . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . i , . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . ... . . . . . .:.

. . . .

......

. . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . < . < . . . . . . . . . . . . . . . . . . . . . . . I ." . . . . . . ... ..: . . . . . . . . . . . . . . . . . . . . . . . . . . . -:"'-.-.-. .. . . . . . . Lrt:. :.: . . . . . . . . . . . . . . . . ~: . . . . . . . . . . . . . . . . . . . . . . . * . .

I I I I I 30 40 50 60 70


Figure 7.5: Acceleration response of bilinear model for a free backlash of 5 mm and a base acceleration of 1 g (6 = 0.51*10-3(2~fe)2).


7.4.2 Preload

Preload means tha t in the initial state there is contact between the beam and the one-sided snubber. Compared t o the bilinear model without backlash this means that the share of the eigenmodes, of the linear model with snubber, in the bilinear eigenfrequencies will increase. This is because there will be more contact between the beam and the one-sided snubber due to the preload. As a consequence the bilinear eigenÎrequencies wiii alter to higher values. Jiist like the m d e ! with fïee bacitiash, resonance peaks will curve, but now in the opposite direction. The reason for this wil! be expiained later.

node 51 1 o3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .:. . . . . . . . . i , . . . . . . .I

...........................................

. . . . . . . . . . . . . . . . . . .

I I I excitation frequency [Hz].

1 o4

Y E lo2

m 10'

&j 10

3 CI 10 .(?

0:

v>

- x - O

E 1 O'ho -

30 40

....... . . . . . . ....... ....... ....... ! ! ! !.! ! ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! ! i !:! ! . . . . . . ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

node 151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..................... . . . . . . . . . . . . . . . . . . . ..................... ..................... ..................... ! ! I ! j !.!!i j ! ! ! I / / 8 I . . . . . . . . . . . . . . . . < _ : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! ! ! ! i : . ! ! ! j : 1 i ! j i,!!! . . . . . . . . . . . . . . . . . . . ..................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! ! ! i ! ! - I : i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, , , : , , : a , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . ....i

. . . . . . .

I '1 * : ; . A '!.:.; . . . . . . . . . . . : : ; ; ~ ; : : : . . : : : : , : : ; . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................

1 O0

i i : 1:: i : . . . . . . . . ......... . . . . . . . . ........ ......... ......... ! ! ! ? ! ! ! ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! ! ! i.! ! i ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

: : i . i : : i :.i i : i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....: . . . . . . . . . . . . . . < . , : : : ; . / j j j 7 : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

110 120

i : i : : : : : : i . . . . . . . . . . . ............ . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . ! ! r ! ! ! ! ! ! ! ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ : i % 3 ; j !'ii j i i j

........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .:. .... . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . .. . . . . . . . . . . . . . . * . . . . . . . . . . . . : : : : i > : i :

. . . . . . . . . I I I I I I I

50 60 70 80 90 100 110 excitation frequency [Hz].

! i : : . . i : : : . . . . . . . . . . . . . . . . . . . . .J . . . . . . . < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

120

Figure 7.6: Acceleration response of bilinear model with a preload: backlash = -2.0 mm and a base acceleration of 1 g, thus 6 = -0.20*10-3(2.xfe)2.

Figure 7.6 shows the acceleration response of a model with a preload. The backlash = -2.0 mm, thus 6 = -0.20*10-3(2~fe)2. The preload in the deformed state of the beams is 21 N (see Appendix G). The response is very simular to the response of the linear model for a = 3.9. The first bilinear eigenfrequency has altered towards 35.9 Hz, while the second bilinear eigenfrequency has shifted towards 115 Hz. The l/Zsubharmonic resonance peak has probably disappeared. This is due to the fact (for the model without backlash) the maximum distance between both panels in the 1/2-subharmonic resonance is smaller than the backlash that causes the preload. So the beam and the snubber will no longer lose contact. Therefore no 1/2-subharmonic resonance will occur.


But no prove can be given that the 1/2-subharmonic solution does not exist, CLS it is numerically seen difficult to find a stable subharmonic solution around 71 Hz. Numerical integration of the equations of motion for fe = 71 Hz will converge to the stable harmonic solution instead of the stable subharmonic solution. The height of the resonance peaks of the model with preload are lower than the nonlinear model without backlash. In table 7.1 the ratio of the height of the resonance peaks of this mede! with prdoad, accûïding to the nonlinear model without backiash and according io the h e a r model are listed. For ail models, ksnub = 9.45*104 N/m and dsnub = 2.0 Ns/m. For the valiies of these peaks, the maximum stable values of the acceleration have been taken. Note that

nonlinear reference linear reference node 51 node 151 node 51 node 151

f h 2 0.3 0.5 1.3 3.0

Table 7.1: Ratio of the height of the peaks of the nonlinear model with a preload of 21 N.

all peaks are lower than the response of the nonlinear model without backlash. But most peaks are higher than the linear model. Figure 7.7 shows the enlargement of the response between 100 and 120 Hz for the model with a backlash of -2.0 mm. A cyclic fold bifurcation point has been found a t f e = 108.99 Hz. Figure 7.8 shows an enlargement of the response between 30 and 40 Hz. This is the response of the model for a backlash of -20 mm and a base acceleration of 10 g. As 6 = +, the dimensionless backlash S remains the same. Therefore the response is similar to the response in figure 7.6, except for a scaling factor of 10 as the base excitation is 10 times higher. Calculations of the response demonstrated tha t this is true. Following the harmonic branch from left to right, between 30 and 34.5 Hz stable harmonic solutions have been found. Cyclic fold bifurcations are found at 34.52 Hz, 33.42 Hz and again for 34.97 Hz. Also a cyclic fold bifurcation point has been found at 35.44 Hz. So far no stable or unstable solution has been found between 34.52 and 35.44 Hz. A numerical integration over 10.000 periods at fe = 35.0 Hz did not result i n a convergent solution. Note that for example at 34.5 Hz various harmonic solutions have been found. Figure 7.9 shows the time histories for stable solutions a t 34.49 (A) and 34.46 (C) Hz and for unstable solutions at 34.49 (B) and 34.44 (0) Hz. Also the distance (ylsl - y51) between both panel ends has been drawn. As long as this distance is smaller than the backlash of 20 mm, the snubber still makes contact with both beams. The preload is not overruled. In point A there is always contact between the beams and the snubber during the whole period. In point B the preload is slightly overruled. But in point C and D the snubber sometimes loses contact. For point C this happens around 5 ms and 25 ms. Note tha t the motion of both beams contains higher frequency components. In point D the preload is overruled during the majority of one period. So figure 7.9 gives an idea of the different kind of motions tha t might occur at one and the same excitation frequency.

As a preload results in lower resonance peaks and a free backlash even gives higher peaks, it is advisable to use a preload instead of a free backlash when designing solar panels with a bilinear snubber. The preload also causes higher bilinear eigenfrequencies. For so far it can be concluded tha t if the preload is large enough, the response becomes close to the response of the linear model with snubber. As the height of the resonance peaks of the nonlinear model are in general higher than the response of the linear model, it is not advisable to rely on a cheaper linear calculation solely. The bilinear eigenfrequencies however can be estimated accurately by linear calculations. But to determine the minimum required preload, so that linear dynamic analyses give reliable re-

27rf ) 2 b


node 151

...................

.................. 1 : . . . . . . . . . . . .

............

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............

..................................................................................... i

.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! \ ... - . 1

._ . o . . ...... ....... - .................................

................................................... ..................... .- .......................... .@ ::: : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ .:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .

. . . . . . . . . . . . . . .

. . . . . - . . . . . . . . ._ . .

. . . . . . . . . . . . . . . . . . . . . . . - , ........ - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- p :. :. ;. .:. .;. ; ; <.: ,;

Y E I'.' .... . . . . . . . . . . . . . . . : . . . . . . . . . . : . . . . . . . . . . : . . . .

. . . . . . . 4

. . . . . . . I . . . . . . . . . . . . .

. . . . . .

. . . . . . . :::.:.I . . . . . . o o - > - i! 101

'$00 102 104 106 108 110 112 114 116 118 120 excitation frequency [Hz].

Figure 7.7: Acceleration response of bilinear model with a preload: backlash = -2.0 mm and a base acceleration of 1 g, thus 6 = -0.20*10-3(2nfe)2.

sults, nonlinear calculations still have to be done.

Further it can be noted that , due to a backlash, the equations of motion become dependant on the base excitation. An increase of the base acceleration results in a decrease of the dimensionless backlash 6, which leads to a completely different response. This will be shown in figure 7.10 and figure G.1 for the following six dof model: ks,,b = 9.45* lo4 N/m, &.nub = 2.0 Ns/m and a modal damping of 0.5 %. The backlash = -2.0 mm (preload = 21 N) and the amplitude of the base excitation is 10 g. Therefor S = -0.20*10-4(27rfe)2. When the response of this model is compared to the model with a backlash of -2.0 mm and a base excitation of 1 g (figure 7.6), the following remarks can be made:

o When the dimensionless backlash decreases towards zero, the response will become closer t o the response of the model without backlash. This will happen first for lower excitation frequencies, as S is dependant on the excitation frequency and the fact that both panels are moving in-phase for the lowest bilinear eigenfrequency. When both panels are moving out- of-phase, there is a higher chance that the snubber and the panel will lose contact than for the in-phase motion. Therefor the first bilinear eigenfrequency at 35.84 Hz does not curve

55

io4: - . v)

v) . E L

m x - . - x 2 io3:

node 151

. . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,. . . . . . . . , . . . . . . . . ._

I I I I I I I I I

1 o5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....................................................................................................... . . . ; . . . . . . -1

;i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I I I

...

. .

. . . .

. .

. <

. .

.~ . . . . . < . . . . . .

....

..... ..... . . . ....

. . . .

......................................... . . . . . . .1

...... . . . . . . . . . . . . . . ....... ....... 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . % . . . . . . . . . . . . . . . . . . . . . o . . . . . . . . . . . .

.. .:. . & ;.&34:97. . . . . . . . . . . . . . . . . ’ . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . ............. t i

. .

Figure 7.8: Acceleration response of bilinear model with a preload: backlash = -20.0 mm and a base acceleration of 10 g, t,hus 6 = -0.’20*10-3(‘2nfe)2.

di l \ i i iorc.

o The second bilinear eigeiifrequeiicy s t a r t s around 100 Hz and the maximum of the peak occurs for 64.77 1 1 ~ . 1 he curve oí t l i i s Iesonaiicc peak can be explained as follow. For 100 Hz, the amplification of both beams is small. ‘Thus the backlash is relatively large according to the beam displacements. Which means that the response is close to the response of the linear model with snubber. When the amplification increases, the backlash relatively decreases. Thus the response is becoming closer to the response of the bilinear model without backlash. Thi5 i 5 tlic explanation for thc hciiding of t lit’ resonance peaks to the left for a preload. Put another Mak. for a free Ixìcklasli. tiit> resonaiice peaks will curve to the right. In figure 7.11 thc hcndiiig of 1 IIc secoiid bilinear resoriaiice peak can be seen for several backlash values.

The base of this resonance will move towards 64 Hz if the dimensionless backlash is decreased further. ‘The second superharmonic resonance is not curved, like the second bilinear eigen- frequciicy, because of the lower excitation frequency (the dimensionless backlash is closer to zero).

o Due to tliis lower dimensionless backlash, a 1/2-subharmonic branch has been found between


point A: stable, 34.49 Hz point 8: unstable, 34.49 Hz 0.041 I I

-0.04' I -0.04' I O 0.01 0.02 0.03 O 0.01 0.02 0.03

time [SI point C: stable, 34.46 Hz

1

-0.04' I O 0.01 0.02 0.03

time [SI

time [SI point D: unstable, 34.44 Hz

0.1

. . . . . . .

. .

-0.1 1 O 0.01 0.02 O

time [SI 13

Figure 7.9: Time histories for various harmonic solutions at fe = 34.5 Hz: backlash = -20 mm and a base acceleration of 10 g. Node 51 (-), node 151 (- -) and y151 - y51 ( - . -).

68.34 and 73.03 Hz. Compared to the bilinear model without backlash it can be noted that all resonance peaks have the same height.

It can be concluded tha t the response of the bilinea,r model with a preload becomes closer to the response of the bilinear model without backlash (including sub- and superharmonics), if the base excitation is increased. Therefor the minimum required preload to be aIlowed to use linear dynamic analyses, must be determined a t the largest occurring base excitation. Also can be noted that a higher dimensionless backlash (providing a preload) results in lower resonance peaks and subharmonic resonances may disappear. Therefore it is advisable to chose the preload as high as admissible. 1

CHAPTER 7. PARAMETER VARIATIONS

node 51

..........

.........

-3 10 E::::::: - ........ ......... . . . . . . . . . ......... . . . . . . . . . . ..........

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' . ? ' i

57

. . . . . . . . . . ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .".] t 1 I I I I I J 30 40 50 60 70 80 90 1 O0 110


Figure 7.10: Displacement response of bilinear model wi th a preload: backlash = -2.0 mm and a base acceleration of 10 g, thus ó = -O.2O*lQ-*((2~f,)~.


node 151 1 o4

. . . . . . . . " . . < .......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t ...................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 > . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .aeita: cI . . . .-'o:2o : : ............................... . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

*lo1 x 1: : : . . . . .... . . . . E . . . .

'4 . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -1 . . . . . . . . . . . . ./ :.

r : 4

. . . . . . . . . . . . . . . . . .L': T%&. . . . . . . . . . . I.. . ";-;,*f ; : :.: : : . . . . . . . . . . . . . . . . . . . . . . . . . . n.**.. . . . . . . . . . . . . . . . . -

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .:... . . . .<.;:::... ::: : : : LE : jtngat

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. . . . . . . . . . . . . . . . . . . . . . . . . . /. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ -.<. . . . . . . . . . . . . . . . : . . . . . . . h.<. . . . . . . . . . . . . . . . . . . . . . . . %.<. . . . . . : . . .

. . . . . . . . . . ........... . . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . .

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. . . . . . .

. . . . . . . . . . . . . . . . . . . . : . x : : . . . . . . . :...

.L. . . . . . . . . . .:. . . . . . . . . . . . . . . - . . . .....................................

;. . . . . . . . . . . .:. . . . . . . Y Y ' . . . . . . . . ~. .:. . . . . . . . . . . *:. . . . . . . . . . . ' . . . .

. . . . . . . . . . . .

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. . . . . . . . . . . . . . . . . . . . . . . .:. . . . . .

. . . . . . . . . . . . . . . . . . .................... .................... .................... .................... loo 1 ....................

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . . 1. . . . . . . . . . . .:. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I

'I


Figure 7.11: Acceleration response of bilinear models with a dimensionless backlash S = O ( f b 2 = 64 Hz), S = -0.20*10-3(2~fe)2 (fb2 N 115 Hz), S = 0.51*10-3(2n.fe)2 (fbz N 50 Hz) and of linear models with (- -) and without (-) snubber.

CHAPTER 7. PA RAME TER VA RIATIONS 59

7.5 Linear and nonlinear model compared

Former measurements at Fokker Space & Systems demonstrated that local nonlinearities, introduced by a bilinear snubber, can be seen as some sort of damping. To answer t h i s question, the response of various models have been compared.

node 51

node 151

I I I I I I I I I I I

Figure 7.12: Displacement response of bilinear model without backlash: stable (...) , unstable (0-0)

and of the linear model (- -) for a 1 g base excitation: o = 3.9 and dsnub = 2.0 Ns/m.

Figure 7.12 and 7.13 show the response of the bilinear model without backlash and the response of the linear model. Both for cv = 3.9 and dsnub = 2.0 Ns/m. Note that for the linear model in this chapter, dsnub = 2.0 Ns/m. This in contradistinction to chapter 4, where no discrete damping was modelled. Thus the resonance peaks of the linear model in this chapter are lower than the resonance peaks of the linear model in chapter 4. For the displacement response the height of the first bilinear eigenfrequency is almost the same as the height of the first eigenfrequency of the linear model. This is as expected. But the acceleration response gives another notion. The peaks of the bilinear system are much higher than the linear system because of the higher frequency components. Thus in this case it is not possible to use linear calculation techniques as the peaks are much higher. Therefore the introduction of local nonlinearities can not be seen as some sort of damping.

_ _ excitation frequency [Hz]

CHAPTER 7. PA RA ME TER VA RIA TIONS

node 51

60

E Y

.. - 2 2 1 -

E ::: x 10 5 ; .. . m I: :

I I I I I I I I I I

10 20 30 40 50 60 70 80 90 100 excitation frequency [Hz]

node 151

90 100 excitation frequency [Hz]

Figure 7.13: Acceleration response of bilinear model without backlash: stable (...), unstable (0-0)

and of the linear model (- -) for a 1 g base excitation: a = 3.9 and dsnub = 2.0 Ns/m.

In figure 7.14 the response of a bilinear model with a preload of 21 N is compared to the response of a linear model. Far both models Q = 3.9, &nub = 2 . Û Ns /m and the modal damping is 0.5 %. I t can be seen that the bilinear eigenfrequencies are estimated very good by the linear model. Further can be noted tha t for the acceleration response, the height of the resonance peaks of the nonlinear model are higher than the peaks of the linear model. So in case of a preload it can be concluded that local nonlinearities introduced by a bilinear snubber can not be seen as some sort of damping.


node 51 I o4

. . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . .

. . . . . . . . . . .

.......... ...................... . . . . . . . . . . . i . . . . . . . . . I . . . . . ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-.. . . . . . . . . " , .

c4?.:.: . . . . . . . . . 1 . . . . . ' i r

. . . . . . . . . . . . . . . . . . . . . . ...................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 O"' I I I I I I I I I I 9n 30 40 50 60 70 80

excitation frequency [Hz] 90

m . . . . -: . ..... . . . . . ..... ! !'! ! . . . . . . . . . ..... . . . . . . . . . . . . ! !.f ! . . . . . . . . . . . . ..... . . . . . . . . . . . .

1 O0 110 120 L U

node 151 I O 4 p ...

. . . . -1 . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . .

r; . . . . . . . . . . . . ! ! . . . . . . . . . . . . ! ! . . . . . . . . . . . ,

?? . . . . . . . . . . . . ! ! . . . . , . . . . . . . ! ! . . . . . . . . . . . . . .

- . . . . . . . . . . . . . . . . . . . . . . . . ! ! ! ! . . . . . . . . . . . . . < . . . . . . . . . . ! ! ! ! . . . . . . . . . . . . . . . . . . . . . . . .

r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " . . . : < . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ._. ... !'! ... . . ... . . . . . . !.! . . ... . . ... ... ...

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i 3

Y

! ! ! ! ! ! !'! ! ! ! ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . !? ! ! ! ! . . . . . . . . . . . . . ....... . . . . . . . . . . . . . . . . . .

! ! ! ' ! ! ! ! ! ! ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ! ! ! ! ! ! Y ! ! ! ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

! ! ! ! ! ! !.! ! ! ! ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~. . . . . . . . . . . . . !.! ! ! ! ! . . . . . . . . . . . . . . . . . . . . . . . . . .

j j f ; j j . ; f f j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . j f f . f f f f j j f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c . . . . . . . . . .

c.4 O' 1; 1 f f ; f j ;: f f f j j ; f j j:; j f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

'.&. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i i : : i ! ! : : : . . . . . . . . . . . . . . . . .

110 120

L . .................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0-1 20 30 40

1 5Q 60 70 80 9c 1 O0


Figure 7.14: Acceleration response of bilinear model with a preload of 21 N (...) and of the linear model (- -) for a 1 g base excitation: cy = 3.9 and dsnub = 2.0 Ns/m.

Chapter * 8

Conclusions and recommendat ions

From the dynamic analyses of idealised solar panels with bilinear snubbers, the following conclusions can be drawn. Also some recommendations for further research are given. The application of bilinear snubbers results in a completely different system response. Nonlinear phenomena like subharmonic and superharmonic resonances do occur. For the examined model in this report, it is accurate enough to reduce the system to six dof, as the system response of the six dof model is comparable with the response of the eight dof model. It appears that the response of the outer solar panel (this is the panel without dummy mass) shows more higher frequency components than the response of the inner panel. This is due to the additional mass of the yoke. From the experimental verification can be concluded that the global acceleration response of the numerical and experimental model are comparable. Parameter variations showed that sub- and superharmonic resonances can be suppressed by the addition of damping. Bilinear eigenfrequencies can be increased by applying stiffer snubbers. On the other hand, for higher snubber stiffness values, the degree of nonlinearity increases too. This leads to higher harmonic, sub- and superharmonic resonance peaks and unstable harmonic branches become larger. Because of the excitation and frequency dependence of the system response, resonance peaks of models with a backlash are curved. This results in jumping behaviour when a frequency sweep is done. For models with a free backlash the resonance peaks curve to higher frequencies, while for models with a preload, the peaks bend in opposite direction. A snubber under preload is preferred to a free back!ash becaüse application of bilinear snubbers under preioad results in iower resonance peaks and higher bilinear eigenfrequencies. The response of the nonlinear and the linear model are comparable if the preload of the snubber is large enough. But i t is not advisable to rely on a cheaper linear calculation solely, because the resonance peaks of the nonlinear model are still higher than the response of the linear model. The minimum required preload to be allowed to use linear dynamic analyses must be determined at the largest occurring base excitation. Nevertheless, the bilinear eigenfrequencies of the model with a bilinear snubber can be estimated accurately by linear calculations only. Next superharmonic and subharmonic resonances can be estimated. From a nonlinear dynamic analysis of the motions of the solar panels can be estimated how large the preload must he to avoid jumping phenomena. It appeared that local nonlinearities introduced by a bilinear snubber can not be seen as some sort of damping. Summarized, the height of the resonance peaks is dependant on the snubber stiffness, the damping values, the base excitation and the backlash. It is very good possible t o perform nonlinear dynamic analyses with the tools developed a t the TNO Centre for Mechanical Engineering and implemented in DIANA. But a high knowledge of nonlinear dynamics is necessary, as these techniques are no straight on calculations like linear

62

C H A P T E R 8. CONCLUSIONS A N D RECOMMENDATIONS 63

dynamic analyses. For more accurate analysis, the model of the damping of the snubber should be improved. In this report a linear viscous damping model for the snubber is considered. But for rubber a structural damping model should be applied. Thus far only the snubber design has been investigated. Another way to increase the lowest eige~frerjueitcy is by changing the position of a hoid-down and reiease system. If this system is moved towards the end of a solar panel, the lowest frequency in the end of t ha t panel will increase. For the ñrst bilinear eigenfrequency, both panels are in an in-phase motion. Thus the damping of this motion is low due t o the small snubber deformation. An additional bilinear snubber mounted on the spacecraft (positioned between the inner panel and the spacecraft body) makes i t possible to damp this in-phase motion more. This means that also the 1/2-subharmonic resonance can be damped more. In DIANA it is possible to define more than one nonlinear element. A side effect of this additional snubber is the fact that the bilinear eigenfrequencies will s h i f t towards higher values. Modelling more than two idealised solar panels is also possible, but then reduced models with more than six dof need t o be analysed. In chapter 7 the response of a nonlinear model with a preload is compared with the response of a linear model without preload. But it is better to compare the nonlinear response with the response of a linear model with a preload of the linear snubber. The differences will be small as the eigenfrequencies of the linear model remain the same, because the snubber stiffness does not change due t o a preload. Note that the influence of a preload on a bilinear model is enormous. In this case the preload results in a longer contact between snubber and panels during one period, thus the bilinear eigenfrequencies will change drastically. It is recommended to examine the response of the nonlinear model for a frequency sweep. A frequency sweep of 2 octaves per minute appeared to be very fast for these slightly damped systems. To make sure tha t the transient damps out, a lower sweep can be done. But for example a frequency sweep of one octave per 70 minutes for an eight dof model leads to high calculation times and large output-files. The output for one dof in the frequency interval 5 - 11 Hz resulted in a output-file of 2 megabyte. Further, it is a disadvantage that a back transformation of the reduced set of dof is impossible. This means that it is impossible to get information of nodes that are not in the reduced set of degrees of freedom.

Dimensioning the linear model

The following conditions of the idealised wing are important:

o The lowest eigenfrequency of a panel without yoke N 50 Hz.

o The lowest eigenfrequency of a panel due to the yoke N 30 Hz.

o Length of the hang over = 0.5 m.

o Mass of the hang over N 2 kg, t h u s mass one panel N 1 kg.

o Effective yoke-mass N 1 kg, thus mass of one leg of the yoke N 0.5 kg.

To fulfil these conditions, the numerical model must have the following dimensions:

L = 0.50 m, m = ph6L = 1.0 kg and

EaI = 70.0 * lo9 N / m 2

h = 0.015m and ai b = 0.049 m.

In DIANA the following model has been used:

width 6 = 0.050 m, height h = 0.015 m, length L = 0.50 m,

3 density pal = 2.7 * IO3 kglm , Young-modulus E,/ = 70.0 * lo9 N / m 2

inertia 1 = 1.406 * lo-' m'.

and

64

APPENDIX A. DIMENSIONING THE LINEAR MODEL

Figure A.1: Dimensions of the beam.

heam = 9 = 2.36 * lo4 N / m f = 49.34 Hz

65

The yoke has been modelled i1s a aluminium block (80*.50*4.5 mm) w i t h a mass of 0.486 kg.

(A.lO)

,

>. U U Q

a

66

Figure A.2: Retrx ted wing of a.n idealised configuration of a solar wing.

67

Figtir(1 :1.:3: Ilcploycd Mirig of i t t i idcalised configuration of a solar wing.

APPENDIX A. DIMENSIONING THE LINEAR MODEL 68

Figure A.4: Snubber mounted on the yoke.

APPENDIX A. DIMENSIONING THE LINEAR MODEL 69

I

Figure A.5: Snu bhcr mountTed on a deployment mechanism.

Eigenfrequencies depending on a

In this Appendix the eigenfrequencies will be calculated for the following equations of motion:

w i t h M = [ ml o ] : mass-ma.trix; 0 m2

: stiffness-matrix, 1 kl + ksnub -ksnub K = - [ -ksnub ksnub -k k2

N y = [ y: ] : matrix with dof and (B.4)

y is the prescribed base displacement. (B.5)

Ir, which fi the Uisp!acement 8f f i d e 51 aad y2 the disp!acement of nûde 151, with respect to the initial position of each mass, represents. Defining:

mi = m,

m2 = Bm, kl = k2 = k and

ksnub = a k

leads to:

70

APPENDIX B. EIGENFREQUENCIES DEPENDING ON o( 71

Considering harmonic displacements:

u( t ) = %{iieJwt}

With w real and .U, the complex amplitude of the excitation. Subs ituting a response y(t) = N

82(@eJwf), with ij the complex amplitude column of the response, in equation 3.1 gives: N N

This equations counts for every t , therefore:

-u2m + k + a k -u2pm - a k + k + a k ] [ i: ] = [ i::] =A[ ] (B.lO) [ -ak

With A-' is the inverse matrix A. For this inverse yields:

1 -w2/3m + k + a k a k

a k -w2m + k + a k

A-' = $ [ -

(B. l l )

(B.12)

With determinant D of A: D = w4,Bm2 - w2(mk + m a k + k p m + f l m a k ) + ( 1 + a ) k 2 (B.13)

Resonance will occur for D=o, w2 = A.

(B.14) k k2 m

=$ À 2 p - &(i + O + p + .P) + (1 + O)? = o

(B.15) cy + /3 + a@) f LJi + 20 + - 2/3 - 4a.P + P2 + a2P2 + 2aP2 + 2a2P

2B -(i + m +- x1,2 =

For B = 1, i.e. ml = m2 = m

'"(2 + 2a) f $vi- 2 X l , 2 = (B.16)

k * A 1 , 2 = - ( l+a&a. ) 112 (B.17)

APPENDIX B. EIGENFREQ UENCIES DEPENDING ON Q

/ / / / / / / / / / / / / / / / / /

For a = 1, i.e. k, = k :

Figure B.l: Discrete two dof model.

U f i = i/- and 2x m

f2 = ““E. 2x

f i = G and

72

(B.18)

(B.19)

(B.20)

(B.21)

This means for the lowest eigenmode both masses are moving in-phase and there is no deformation of the snubber. See figure B.2. For the second eigenmode both masses are moving out-of-phase. This can be seen as a single dof system with 2 springs with stiffness k and 2k (figure B.3).

Figure B.2: Lowest eigenmode of two dof model for cv = 1 and ,B = i.

APPENDIX B. EIGENFREQ UENCIES DEPENDING ON a

{

73

'ml = 0.727 kg = m

m2 = 0.246 kg

k = 2.36 * l o4 Nlm k,,,b = @k + ksnub = 1.6 i; lo4 j a = 0.677 [-]

+- ,B = 2 = 0.338 [-]

~ ksnub = 9.45 i: lo4 =+ Q = 4.0 [-]

Figure B.3: Second eigenmode of two dof mociei for a = 1 and 0 = 1

k,,,b [N/m]

0.0

For:

a [-I analytic (two dof) numerical (DIANA) f l [Hzl I f 2 [Hzl fl [Hzl f 2 [Hzl

0.0 28.69 1 49.34 28.69 49.34 _ _

1.6*104 9.45*104

equation B.15 leads to the results listed in table B.1.

0.678 32.79 66.25 35.07 65.92 4.0 34.57 122.87 36.65 115.3

I I I I

Ou I C 0 I 35.08 I 00 I 37.08 I 213.4 I I I I 1 I 2

(B.22)

~ ~ _ _ _

Table B.l: Analytic and numerical calculated eigenfrequencies depending on a.

Next the difference between a static and a dynamic reduction is explained. The equations of motion of a finite element model of a beam are as follow:

I

(R.23)

o Static reduction: The flexibility mode y, of a static reduction to a single dof (node 151 at the end of the clamped outer panel) is:

(B.24)

Substituting this in the equations of motion B.23 finally gives:

msY151 + ksy151 = fl51 (B.25)

In where m, is the reduced static mass and k , is the reduced static beam stiffness.

APPENDIX B. EIGENFREQUENCIES DEPENDING ON cv

( Y S ) t M F S m, = ( p ~ , 1 5 1 ) ~

( 9 ~ , 1 5 1 ) ~

And y s , 1 5 1 is the element of q, corresponding to the displacement of node i51.

tIQ9 k , = (”) - = 2.36 * lo4 N/m

74

(B.26)

(B.27)

e Dynamic reduction The lowest eigenmode y d of a dynamic reduction to a single dof (node 151 at the end of the clamped outer panel) is derived from the equation:

( -w2M + i)pd = o (B.28)

Uncoupling the equations of motion B.23 and substituting pd finally gives:

mdYii51 + kdY151 = f 1 5 1 (B.29)

In where m d is the modal mass and kd is the modal stiffness of the first bending mode.

(B.30)

(B.31)

And (3d,151 is the element of pd corresponding to the displacement of node 151. The lowest

eigenfrequency is thus fi = &

The difference between the modal stiffness and the static stiffness is due to the fact that there is a difference, although small, between the eigenmode y d and the static flexibility mode y,.

The dimensionless bilinear two dof model

The equations of motion for the bilinear two dof model in figure C.1 are as follow:

In which y1 is the displacement of node 51 and y2 the displacement of node 151, with respect t o the initial position of each mass, represents. In these equations of motion 11 quantities are involved. Two quantities can be eliminated by making equation C.1 and C.2 dimensionless. Heretu the dimensionless displacement q1 and q2 and the dimensionless time 0 are introduced:

; ; backlash b / I , / / -

/ k l k2 / ‘snub ;

/ m 1 +

/ _I

/ - u - y 2 dsnub - Y 1 -u

Figure C.1: Bilinea,r two dof model.

7 5

APPENDIX C. THE DIMENSIONLESS BILINEAR TWO DOF MODEL

t = $ - = - E w”,

76

Substituting equation C.8 and C.9 in C.l and C.2 followed by division by (Lh;el2 results in the dimensionless equation of motion ( I = 8/88):

si’ + “(!li, q 2 ) [ Q l ( q 1 - q2 - 6) + Cds: - q1)1 + 91 = cos (ne) 92 + 4 q 1 , !72) [ r (q2 - 41 + 6) + C 2 ( d - d)1+ F l 2 = P C O S (ne)

(C.10) (C.11)

The 7 dimensionless parameters are:

Q1 =

Cl =

6 =

Y =

C2 =

P =

f l =

(C.12)

(C.13)

(C.14)

(C.15)

(C.16)

(C.17)

(C.18)

(C.19)

The ratio Q (chapter 5) is defined as cv = y. The dimensionless equations of motion C.10 and C.11 are linear in the input if the dimensionless backlash 6 = O. This is because the transition takes

APPENDIX C. THE DIMENSIONLESS BILINEAR TWO DOF MODEL 77

place for (41 - q 2 ) = O. But at that moment the contribution of the snubber to the equations of motion C.10 and C.11 are also zero. So there is no discontinuity for (yi - 42) = O. If the backlash b # O, it can be seen that the dimensionless backlash S is depending on the excitation frequency fe and the acceleration amplitude of the excitation A. Thus for b # O, the equations of motion C.10 and C.ll are s ta te dependant.

Design of the experimental model

Both aluminium beams are clamped in an aluminium block as shown i n figure D. l . The deflection of the bottom of the block is due to the impressed forces of two bolts. This deflection causes a moment. This moment results in a deformation of both side-walls of the block and the beams are clamped. The advantages of this construction are a low mass and the fact that different beams can be exchanged. Further no holes have to be drilled in the beams, so no notch effect. In Appendix A can be seen which parameters influence the eigenfrequency:

h f - 3

Thus a reduction of twice the lenght L and a reduction of four times the thickness h, gives the same eigenfrequency. These dimensions are the starting point for the experimental model. Due to manufacturing limitations the thickness h is chosen 4 mm instead of h/4 = 15/4 = 3.75 mm. The dimensions of the model are shown in figure D.2.

Thickness h = 4.0 mm. Width b = 20.0 mm. Length L = 250.0 mm. pal = 2.7*103 N/m3. ,Tal = 7û.û*iû9 N/m2. Inertia I = &bh3 = 1.067*10-10 m4. mdummy = 27.7*10-3 kg = m47.

macceleromete = 13.0*10-3 kg. + m50 = 14..5*10-3 kg = 1i2150. Modal damping = 0.64 %.

mst i ck = 1.5*10-3 kg.

In this model (in contrast with the model used in chapter 3, 4 and 5 ) the dummy mass has been modelled as one single point mass. This because one single point mass can be easily modified if necessary. This model without snubber gives the following numerical results:

f i = 27.26 HZ f:! = 36.96 HZ

When the snubber (with its stiffness and mass) is taken into account, f1 will become (due to the snubber mass) even lower. So then is satisfied to the criterion that the 1/2-subharmonic branch is found around 50 Hz.

78

APPENDIX D. DESIGN OF THE EXPERIMENTAL MODEL 79

Figure D.l: Experimental model.

, accelerometer L C I

gravity I @ ,. A-- node 151

* - - -

-- h 145

snubber _- - - - - - - < 150

4’ ,50 4 5 = h

“ V c‘e--- node 51

’4 u 5mm excitation J. * :

20 mm 30 mm

Figure D.2: Dimensioi-is of the experimental model.

APPENDIX D. DESIGN OF T H E EXPERIMENTAL MODEL 80

This experimental model is not similar to the idealised solar panel anymore. One reason is that scaling down the model also requests a scaling down of the accelerometers. But tha t is not possible. Because of the relative low mass of the beams, the eigenmodes are very sensitive for additional masses of the accelerometers. A model with high beam mass is more robust for additional masses, but this is not possible due to the limitations of the eccentric shaft.

I

Annendix E - -rr ----*_-

Response of bilinear eight dof model

This Appendix contains some frequency spectra of the response of the bilinear eight dof model discussed in chapter 5 . Also the displarement and acceleration response of this model is printed on A3-format.

81

APPENDIX E. RESPONSE OF BILLINEAR EIGHT DOF MODEL.

0.9 'I

0.4

0.3

0.2

o. 1

O

fe=34.70 HZ

L 50

frequency [Hz]

82

O

Figure E.l : Frequency spectrum at fe = 34.70 Hz (node 151).

1 o' - I I

fee44.03 Hz

O 100 150 200 250 300 350 4 frequency [Hz]

D

Figure E.2: Frequency spectrum at fe = 44.03 Hz (node 51).

APPENDIX E. RESPONSE OF BILINEAR EIGHT DOF MODEL

0.7

U. i 0.5 - ü ::

0.4 -

0.3 -

0.2-

k 4 4 . 8 5 Hz

irequency [Hz]

Figure E.3: Frequency spectrum at fe = 44.85 Hz (node 151).

6 7l 5 -

t 6 4

- la ü

3-

2 -

ied2.17 Hz

1 -

O 1 I O 50 100 150 200 250 300 350 400

frequency [Hz]

83

Figure E.4: Frequency spectrum a t f e = 52.17 Hz (node 151).


0.9

,

- feA3.12 Hz

0.7

0.6

t 6 0 . 5

ü 0.4

W

0.3

o. o.2 1 t

- -

-

-

-

O

Figure E.5: Frequency spectrum at f e = 53.12 Hz (node 51).

1 li

1 od T

I 50

I

fed2.83 Hz

100 150 200 250 300 350 I

frequency [Hz]

84

Figure E.6: Frequency spectrum a t fe = 62.83 Hz (node 151)

1

APPENDIX E. RESPONSE OF BILINEAR EIGHT DOF MODEL 85

fe=94.80 Hz

D frequency [Hz]

Figure E.7: Frequency spectrum at f e = 94.80 Hz (node 151).


node 51 1 O-' . ...................................................................................................... . . . . . . . . . . . . . . . . ................................. 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... :;>: IftjZ i j i i . . . . . . . . . . . . . .......... .:.=. . . . . . . . 1 . . . . . . . . . . . . . . . ............... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...............

. . . . . < . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . .

. . . . i$; i

. . . . . .

. . . . . . , . . . . . . . . . . .

. . . . . . . . . . .. ..

...... . . . . . . ...... . . . . . . ...... . . . . . . . . . . . . . . . . . . . . . . . . i : : : : : . . . . . . . . . . . . . . . . . . . . . . . . ......

. . . . . . . . . . . . . ............... *i .. . . . . . . . . . . . . . . . . . . . . . . . .......... . . .- . . . . . . . . . . . . . . . . . . . . ............... .......... . . . . . . . i .. . . . . . . . . . . . . . . . .i:-.:::: i . . . . . . : : 5 :

. . . . . . . ". .................... . . i 3133jjtji I I . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -

..... . . . .

.... . . . . . . . . ..... .....

..... .....

. . . .

. . . . I 40

. . . . . . . . . . . . *.i .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ l o p i . . . . . . . . . . . . . . . . . i ; ; i ; ; ; i:;; ;; ;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : . : : : : : : : : : : : y $ i i ; i i ; i i ; : : - : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o c : . . . . : : . : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n:..: ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .?:-:ho:; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -5 t . . . . . . . . . . . .I. . . . .

1020 30

. . 1 I I I

50 60 70 80 excitation frequency [Hz]

90 1 O0

node 151 1 o"

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .fb2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2nd super. - 10 : i i i i ; i j i i i i :

. . . . . . . . . . . . . . ;g; . . . . . . . . . . . . . . . . . . . . . . . . . . . u E -*I . . . . . . . . . . . . . .

. .

. . . . . . . . . . . .

. i .. ..

. . . . < . * . < .

. . . .

I ......................... : . : :: :::: ::::. . . . . :i:::::: i i i i 1 $ij j i t ' : : : : : : : : : : :< : : : : : : . . . . . ....................... ........................ c . . . . . . . . . . . . . f . . ............ .......... . . . . . . . . . . .................... ..................... I .@ - 10-310;

-u . . . . x

. . . . . . . . - - . . . .

. . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . .'. .a... . . . I . . d e 9 ;OL:; ; i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. - : : : : ; :i: : : . . . . . . . . . . . . . . ̂ . . . . . . . . . . . . ..........

...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . : < : * . . . : : : : 2 j : f l + $ ~ ~ : ; J . . . . . . . . . . . . . . . . . .......... . . . . . "... . . . . . . . . . . . . . .

. . . . . . . . . . . . * .i . .* . . . . . - . . . . . . . . . .................... . . . . . . . %;Q; . . . . . . . . . . . . . . . .......... . . . . . . . . . . . . . . . . . . . .

4 x 10 1::: i ; ; ;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 . . . . . . . . . . . . . . . . . . . . .

1 I I I I 1 I I 30 40 50 60 70 130 98 1 O0

1 O-=' 20


Figiire E.S: Stablc I..) a n d iinstablc (o o) clispiacemcnt response of eight dof model (ICsnub = 9.4.5* 10" N /m) for a Ig. basc cxcit2atiori.

.4 PPENDIX E. RESPONSE OF BILINEAR EIGHT DOF MODEL 87

A node 51

. . . . . . . . . .

...............................


node 151

................................. . . ..................... . _ . ..........

. . . . . . . . . .


Figure E.9: Sta.ble (..) a.nd unsta.blc (o o) a.cceiera.tion response of eight dof model (Ic,,,b = 9.45*104 N/m) for a 10. basc excitation.

Measuring-inst rument s

instrument accelerometer 77

77

charge amdifier

This Appendix contains a list of the used measuring-instruments, some measured time histories and some measured frequency spectra.

brand t y Pe serial no. Brüel & Kjaer 4367 1074096

1074097 1074098

77 77

77 > 7

Kistler 5007 43579

serial no. 1074096 1074097 1074098

77

acceleration [m/s2] calibration frequency [Hz] 0.9936 79.49 1 .O047 79.49 1.0066 79.49

Table F.l: Measuring-instruments.

Table F.2: Calibration of the accelerometers for 1.0 m/s2 .

88

APPENDIX F. MEAS URING-INSTR UMENTS 89

time [s]

-200' I I O 0.05 0.1 0.15 0.2 0.25 time [SI

Figure F.l: Time histories for f e = 47.07 Hz: escitat,ion (-.). node .51 (-) and node 151 (- -).

-100' 1 O 0.05 0.1 0.15 0.2 0.25 time [s]

time [SI

Figure F.2: Time histories for f e = 49.0'2 Hz: escitíìt,ion (-.). iiode .51 (-) and node 151 (- -).

APPENDIX F. ILlEASURING-1iVSTRUiL.iENT.S

1 o2

8 10'

t m L

0 l oo

1 O-'

1 o.2

lo30 20 40 60 80 100 120 140 160 180 200 frequency [Hz]

frequency [Hz]

Figure F.3: Frequency spectra at j e = 49.07 Hz: excitation (-.), node 51 (-)

0.05 0.1 0.15 0.2 C time [SI

90

and node 151 (- -).

!5

I 0.05 0.1 0.15 0.2 0.25

time [cl

Figure F.4: Time histories for j e = '21.68 Hz: escitatioii (-.); node 51 (-) aiid node 151 (- -).

APPENDIX F. A/TEASUR7NG-lNSTRUMENTS

10‘

1 oo

10.’

1 o,*

1 o”

lo4

t

O 20 40 60 80 100 120 140 160 180 200 frequency [Hz]

1 o*

1 o’

~ loo LL

10’’

1 o.2

“ 0 20 40 60 80 100 120 140 160 180 200 frequency [Hz]

91

Figure F..5: Frcclu(~iic~. spectra a t = 21.08 í l ~ : cxcitaliori (-.), node 51 (-) and node 151 (- -),

Response of bilinear model with a preload

In this Appendix the preload of the snubber in the static situation is calculated. Consider: the deflection of the inner panel 201 = y, the deflection of the outer panel w:! = u and b is the backlash in the undeformed state. In the static situation, the snubber deformation is then: k?

dx = b - 201 - ~ 2 .

Thus the snubber force is Fsnub = ksnub * dx.

For b = -2.0 mm, ksnub = 9.45*104 N/m and kl = k:! = 2.36*104 N/m, Fsnub = -21.0 N. Thus a preload of 21.0 N.

92

APPENDIX G. RESPONSE OF BILINEAR MODEL WITH A PRELOAD 93

r node 51 1 O”

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~

i i i ip i i i i i i i i i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i\: . . . . . . : : :

i i : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . w; . . . . . . . . I . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..‘,

. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. .

. , . . . . . . . .

. . . . . < . z . . . . . . . . . .

. . . . . . . ... .. i .

i :

,. . . .. . . _ . . . . . .. . . i : ... .. . .

. I .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . i . .

: : : : : : ; ! i ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . . . . . . : : : : i ; : . . . . . . . . . . . . . . . . . . . . . . . I . . , . . . . . . ::!?;o . . . . . . . . . . . . . . . . . . . : : : . . . . i : : . . . . i . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : : : : : : : . . : i : : : : : . . . : . . . . . . . . . . . . . . . . . . . . . . . * . . . . . . . . . .

. . . . . . . . . . 0 ....... i . .

. . . . . . . . . : : : ; i ; ; *. . . . . . . . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . e . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : i ; : & : : ::ai: . . . . . 6 : : : . . .............. E t:::::: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L::] . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1n’I I I I I I I I I I ’ “30 40 50 60 70 80 90

excitation frequency [Hz] 1 O0 110 120

node 151 IO^^,^ i i : I : : : : : : : : : : I : : : : : : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . : . . . . . . . . . . . . . . . . . . . : : : : . . . . ( . . . . . . . i . . .

> : : :

Bi! . .e

-1 . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

po .....

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . j . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . I . I . . . I

. . . . . . : ! . . . I

. I

. i

i ! . , . . . < . . . . . .

. . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . I . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . i;;;;;kii;;j . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . . . .

30 40 50 60 70 80 98 110 120 excitation frequency [Hz]

. . . . . . . . . . . . . . . . . . . . . . . . . . 1 . i . . . . . . . . . . $ 1 m

. . . . . . . . . . > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C . l . . . . . . . . : : B : : i ; . . .

. . . :;. : .:, . . . . . . . . . . . . . .

x ’ L

1

Figure G.1: Acceleration response of bilinear model with a preload: backlash = -2.0 mm and a base acceleration of 10 g, thus S = -0.20*íO-4(2~fe)2.

Appendix H

DIANA-input files and user-supplied routines

This Appendix contains the DIANA-input files and the user-supplied routines nlsptr. f and usrlod. f for the numerical model of an idealised solar panel shown in figure H. l and for t he experimental model in chapter 6.

node 101 I ,

I I element 101 A -<'

element 150 I

' , e' ' ,' // I

/&I I I I 1 I I I V/I I I I I I I I I I I I I I I I I I I A J . / ,

- - - - - - - - snubber (sp2tr) node 150 (

I element 58 I

dummy mass excitation

Figure H.l: Numerical model of a n idealised solar panel implemented in DIANA.

94

P

. . . . ! , ..

.:. . . . . . . . .

< \ . . . - .

. .

SUBROUTINE NLSPTR( TASK, ELMNR, N, DESNR, DESVL, TIME, DISL, $ FORL, DFDI, DFDII, STIFF, DAMP, DSTFI, DDMPI, $ DSTFII, DDMPII, DFLDS )

C C.................................................COPYRIGHT (C) TNO-BOUW c.. . c.. . c.. . c...

c... c... c.. . c... c.. . c.. . c.. . c.. . c.. . c... c.. . c.. *

I

L. . *

DIANA/SD/XC30/USRSUB/NLSPTR

CALLED FROM: 'NLSWIT'

PIUmKPûSES f FOR THE 3-DIM. 2-NODE SPRING ELEMENT 'SP2TR': CALCULATE INTERNAL FORCES... etc.

OF ONE-SIDED (BILINEAR) SPRING AND DAMPER

desvl(3) := backlash in [m] decvl(6):= damping in [Nc/m] desvl(7):= stiffness in [N/m]

PROGRAMMED BY EVDV. TNO-BOUW.

(SNUBBER).

c........................................................................ C C c.. . ARGUMENTS

LOGICAL INTEGER DOUBLE PRECISION

$ $

C DOUBLE PRECISION INTEGER LOGICAL

TASK ( * ) ELMNR, N, DESNR(*) TIME, DESVL(*) , DISL(*) , FORL(*) , DFDI(*) , DFDII ( * ) , STIFF ( * ) , DAMP ( * ) , DSTFI ( * ) , DDMPI ( * ) , DSTFII ( * ) , DDMPII (*) , DFLDS ( * )

RPAR, TWOPI IPAR LPAR

COMMON /SDPARM/ RPAR(10), IPAR(2O), LPAR(20) EQUIVALENCE ( TWOPI, RPAR(1) )

C C... local

C C... calculate snubber forces C

DOUBLE PRECISION TDELTA, TVELTA, TFORCE, SPSTIF, SPDAMP

TFORCE=O.DO SPSTIF=O.DO SPDAMP=O.DO TDELTA = ( DISL(2) - DISL(1) ) TVELTA = ( DISL(4) - DISL(3) ) IF ( TDELTA.LT.-DESVL(3) ) THEN

TFORCE = TFORCE + DESVL(7) * (TDELTA+DESVL(3)) + $ DESVL(6) * TVELTA

SPSTIF = SPSTIF + DESVL(7) SPDAMP = SPDAMP + DESVL(6)

END IF C

C C c.. .

C 100

IF( .NOT. TASK(1)

SETUP < FOR > FORL(1) = -TFORCE FORL(2) = +TFORCE

CONTINUE IF( .NOT. TASK(2)

) GOTO 100

.- .- < -TFORCE +TFORCE >

) GOTO 200

r

C c.. . Setup Jacobian:

STIFF(1) = +SPSTIF STIFF(2) = -SPSTIF STIFF(3) = -SPSTIF STIFF(4) = +SPSTIF DAMP(1) = +SPDAMP DAMP(2) = -SPDAMP DAMP(3) = -SPEAMP f)L"I;0(4) = +spuLyp

C 200 CONTINUE

IF( DESNR(l).EQ.O ) GOTO 400 C c.. . Calculate deriv. force of design variable C

TFORCE = O.DO IF( DESNR(1) .EQ.7 ) THEN

IF (TDELTA.LT.-DESVL(3)) THEN TFORCE = TDELTA + DESVL(3)

ELSE TFORCE = O.DO

END IF END IF IF( DESNR(1) .EQ.6 ) THEN

IF (TDELTA.LT.-DESVL(3)) THEN

ELSE TFORCE = TVELTA

TFORCE = O.DO END IF

END IF DFDI(1) = -TFORCE DFDI(2) = TFORCE

IF( DESNR(2).EQ.O

TFORCE = O.DO IF( DESNR(2) .EQ.7

C

C ) GOTO 400

) THEN IF (TDELTA. LT. -DESVL ( 3 ) ) THEN

TFORCE = TDELTA + DESVL(3) ELSE

TFORCE = O.DO END IF

END IF IF( DESNR(2).EQ.6 ) THEN

IF (TDELTA.LT.-DESVL(3)) THEN

ELSE

END IF

TFORCE = TVELTA

TFORCE = O.DO

END IF DFDII(1) = -TFORCE DFDII(2) = TFORCE

IF( .NOT. TASK(3) ) GOTO 400 C

C C 400 CONTINUE

RETURN END

i

DIANA-input file (model chapter 5)

P

IDEALISED SOLAR PANELS WITH BILINEAR SNUBBER UNITS: m, kg, s, N 'COORDI' DI=2 : ELEMENT LENGTH IS 10 mm. : eight degrees of freedom

: NUMERICAL MODEL (chapter 5)

: inner panel 1 0.0 0.0 2 0.010 0.0 3 0.020 0.0 4 0.030 0.0 5 0.040 0.0 6 0.050 0 . 0 7 0.060 0.0 8 0.070 0.0 9 0.080 0.0 10 0.090 0.0 11 0.100 0.0 12 0.110 0.0 13 0.120 0.0 14 0.130 0.0 15 0.140 0.0 16 0.150 0.0 17 0.160 0.0 18 0.170 0.0 19 0.180 0.0 20 0.190 0.0 21 0.200 0.0 22 0.210 0.0 23 0.220 0.0 24 0.230 0.0 25 0.240 0.0 26 0.250 0.0 27 0.260 0.0 28 0.270 0.0 29 0.28û 0.û 30 0.290 0.0 31 0.300 0.0 32 0.310 0.0 33 0.320 0.0 34 0.330 0.0 35 0.340 0.0 36 0.350 0.0 37 0.360 0.0 38 0.370 0.0 39 0.380 0.0 40 0.390 0.0 41 0.400 0.0 42 0.410 0.0 43 0.420 0.0 44 0.430 0.0 45 0.440 0.0 46 0.450 0.0 47 0.460 0.0 48 0.470 0.0 49 0.480 0.0 50 0.490 0.0

51 0.500 0.0 : outer panel 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 12 1 122 12 3 124 12 5 126 127 128 129 13 O 13 1 132 13 3 134 13 5 136 137 138 139 14 O i4 i 142 14 3 144 14 5 14 6 147 148 149 150 151

o: o o. o11 0.010 0.011 0.020 0.011 0.030 0.011 0.040 0.011 0.050 0.011 0.060 0.011 0.070 0.011 0.080 0.011 0.090 0.011 0.100 0.011 0.110 0.011 0.120 0.011 0.130 0.011 0.140 0.011 0.150 0.011 0.160 0.011 0.170 0.011 0.180 0.011 0.190 0.011 0.200 0.011 0.210 0.011 0.220 0.011 0.230 0.011 0.240 0.011 0.250 0.011 0.260 0.011 0.270 0.011 0.280 0.011 0.290 0.011 0.300 0.011 0.310 0.011 0.320 0.011 0.330 0.011 0.340 0.011 0.350 0.011 0.360 0.011 0.370 0.011 0.380 0.011 0.390 0.011 û.4ûû û.011 0.410 0.011 0.420 0.011 0.430 0.011 0.440 0.011 0.450 0.011 0.460 0.011 0.470 0.011 0.480 0.011 0.490 0.011 0.500 0.011

DIRECT' 1 1.0 0.0 0.0 2 0.0 1.0 0.0 3 0.0 0.0 1.0

: supports SUPPOR'

1 TR 1 TR 2 RO 3 1 0 1 TR 1 TR 2 RO 3 51 TR 3 151 TR 3

/ 2-51 / TR 1

/ 1 0 2 - 1 5 1 / TR 1

: aluminium MATER1

1 YOUNG 70 .OE09 DENSIT 2 . 7 0 E 0 3

: l i n e a r spr ing/snubber 1001 SPRING 0 . 0

DAMP 0 . 0 GEOMET ' 1 CROSSE 7 .50E-4 : panel

: dummy mass INERTIA 1 . 4 0 6 3 E - 8

2 CROSSE 2 .25E-3 INERTIA 3 .80E-7

GENELM

1 TYPE panel1 : C o m p o n e n t Mode S y n t h e s i s

ELEMEN 1-58 i f cnod / 51 / tr 2

CMS FREERB NOFLEX FREQUB 1 0 0 . 0 : cut-off frequency RELNOD / 2 3 / TR 2

ELEMEN 1 0 1 - 1 5 0 i f cnod / 151 / t r 2

/ 1 0 1 / t r 2 CMS FREERB NOFLEX FREQUB 1 0 0 . 0 : cut -of f f requency RELNOD / 1 2 3 / TR 2

/ 1 / t r 2

2 TYPE p a n e l 2

' ELEMEN' CONNEC : i n n e r pane l 1 2 3 4 5 6 7 8 9 1 0 11 1 2 1 3 1 4 15 1 6 17 18 1 9 2 0 2 1 22 2 3 24 2 5 2 6 27 2 8 2 9 3 0

LGBËN 1 2 LGBEN 2 3 LGBEN 3 4 LGBEN 4 5 LGBEN 5 6 LGBEN 6 7 LGBEN 7 8 LGBEN 8 9 L6BEN 9 10 LGBEN 1 0 11 LGBEN 11 1 2 LGBEN 1 2 1 3 LGBEN 1 3 1 4 LGBEN 1 4 15 LGBEN 15 1 6 LGBEN 1 6 1 7 L6BEN 17 18 LGBEN 18 1 9 LGBEN 1 9 2 0 L6BEN 2 0 2 1 LGBEN 2 1 22 LGBEN 2 2 2 3 LGBEN 2 3 2 4 LGBEN 2 4 2 5 LGBEN 2 5 2 6 LGBEN 2 6 2 7 LGBEN 2 7 2 8 LGBEN 2 8 2 9 LGBEN 2 9 3 0 LGBEN 3 0 3 1

31 32 33 34 35 36 37 38 39 4 0 41 42 43 44 45 46 47 48 49 50

LGBEN 31 32 LGBEN 32 33 LGBEN 33 34 LGBEN 34 35 LGBEN 35 36 LGBEN 36 37 LGBEN 37 38 LGBEN 38 39 LGBEN 39 40 j-6BEN- 40 41 LGBEN 41 42 L6BEN 42 43 L63EN 43 44 L6BEN 44 45 LGBEN 45 46 LGBEN 4 6 47 L6BEN 47 48 LGBEN 48 49 LGBEN 49 50 LGBEN 50 51

: dummy mass 51 LGBEN 43 44 52 L6BEN 44 45 53 LGBEN 45 46 54 LGBEN 46 47 55 LGBEN 47 48 56 LGBEN 48 49 57 LGBEN 49 50 58 LGBEN 50 51 : outer panel 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 12 1 122 12 3 124 125 12 6 12 7 128 12 9 130 131 132 133 134

LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN EGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN

101 102 102 103 103 104 104 105 105 106 106 107 107 108 108 109 109 110 110 111 111 112 112 113 113 3.14 114 115 115 116 116 117 117 118 118 119 119 120 120 121 121 122 122 123 123 124 124 125 125 126 126 127 127 128 128 129 129 130 130 131 131 132 132 133 133 134 134 135

135 LGBEN 135 136 136 LGBEN 136 137 137 LGBEN 137 138 138 LGBEN 138 139 139 LGBEN 139 140 140 LGBEN 140 141 1 4 1 LGBEN 141 142 142 LGBEN 142 143 143 LGBEN 143 144 144 LGBEN 144 145 145 L6BEN 145 146 146 LGBEN 146 147 147 LGBEN 147 148 148 LGBEN 148 149 149 LGBEN 149 150 150 LGBEN 150 151 : linear spring 1001 SP2TR 51 151 MATER1

1

1

1001 GEOMET 1 1-50 1

1 1 51-58 1

2 1 101-150 1

1 1-58 1

101-150 1

1 1001 1

I ' LOADS

CASE 1 DEFORM 1 1 1 0 1 1 TR 2 1 . 0 DESIGN : specification of the design values : for the user-supplied routines

: excitation frequency [Hz]

: backlash [m;

1 0 . 0

2 50.0

3 0 . 0

4 0.0

5 9.81 : amplitude of base acceleration [m/s**2]

: BILINEAR SNUBBER : one-sided damping [ N s l m ]

: one-sided stiffness [ N l m ] 6 2 . 0

7 9.45E4 10 0.0 20 0 . 0 END

... - .... ,..- .... ,,-e SUBROUTINE USRLOD( TASK, FNR, DESVL, T, DESNR, RLOD , DDES )

C C.................................................COPYRIGHT (C) TNO-BOUW C... DIANA/SD/XC30/USRSUB/USRLOD c. * .

C... c.. . C... CALLED FROM: EXTLOD, FINTNL, FNLSHO, FNLBIF, FDERIV, FNLTRA c... c.. . TASK(1) = T externai ioad F-ext c.. * TASK(2) = T d F-ext! d desnar1 c. * * TASK(3) = T d F-ext/ d desnr2 c.. . c... FNR = function number c. * . T = time c.. . DESVL = design variables c.. . DESVL(2) = excitation frequency [Hz] c.. . DESVL(3) = backlash [m] c.. . DESVL(5) = amplitude of base acceleration [m/s**2] c.. . DESVL(6) = one-sided damping [Ns/m] c.. . DESVL(7) = one-sided stiffness [ N / m ] c. * . DESNR(2) = design variables numbers which are varied c. * .

C... COMMON /SDPARM/ c. * . RPAR(1) = TWOPI = 2 * PI c... C.. . OUTPUT:

EXTENDED WITH A prescribed base acceleration

c.. . RLOD(1) = external load factor c.. . RLOD(2) = prescribed displacement c.. . RLOD(3) = prescribed velocity c.. . RLOD(4) = prescribed acceleration c.. . DDES(1-4) = d F-ext/ d desnrl c.. . DDES(5-8) = d F-ext/ d desnr2 c.. . C... PROGRAMMED BY EVDV. c........ ............................................................... c.. . c.. . ARGUMENTS

LOGICAL TASK ( * ) INTEGER DESNR( *) , FNR DOUBLE PRECISION RLOD(4), T, DDES(8), DESVL(*)

DOTTBLE PRECISIIN RPAli, TWOPI INTEGER IPAR LOGICAL LPAR COMMON /SDPARM/ RPAR(101, IPAR(20), LPAR(20) EQUIVALENCE ( TWOPI, RPAR(1) )

C

C C.. . local

C DOUBLE PRECISION PHI, OMEGA

IF ( DESVL(g).NE.O.DO ) THEN

ELSE

END IF CALL RSET( O.DO, RLOD, 4 ) CALL RSET( O.DO, DDES, 8 )

PHI = DESVL(9)/36O.DO*TWOPI

PHI = O.DO

C C

C

C

IF ( .NOT. TASK(1) ) GOT0 300

IF ( FNR.EQ.l ) THEN

c... C c.. . C

C c... C

c.. . C

C c... C

Calculate RLOD

load RLOD ( 1) = DESVL (4) +DESVL (5) *COS ( TWOPI*DESVL (2) *T + PHI )

ELSE IF ( FNR.EQ.2 ) THEN

prescribed displacement

RLOD (2) = -DESVL (5) * COS ( TWÖPI*DESVL (2 j *'l'+PHI j $ ( TWOPI*TW@PI*DESVL(2)*DESVL(2) )

prescribed velocity

RLOD(3) = DESVL(5) *SIN (TWOPI*DESVL(2) *T+PHI ) / $ ( TW@PI*DESVL(2) )

prescribed base acceleration

RLOD (4) = DESVL (5) *COS ( TWOPI*DESVL (2) *T+PHI )

END IF C 300 CONTINUE

IF ( .NOT. TASK(2) ) GOTO 500 C C

C CALL RSET( O.DO, DDES, 4 )

IF ( DESNR(1) .EQ. 2 ) THEN

DDES(1) = O.DO

DDES (2)

DDES(3) = -DESVL(5) *SIN(TWOPI*DESVL(2) *T+PHI) /

DDES(4) = O.DO

IF ( FNR .EQ. 1 ) THEN

ELSE IF ( FNR.EQ.2 ) THEN = 2. DO*DESVL (5) *COS (TWOPI*DESVL (2) *T+PHI) /

$ ( TWOPI*TWOPI*DESVL(2)*DESVL(2)*DESVL(2) )

$ ( TWOPI*DESVL(2)*DESVL(2) )

END IF

IF ( FNR.EQ.l ) THEN DDES(1) = 1.DO

END IF


ELSE IF( FNR.EQ.2 THEN

ELSE IF ( DESNR(1) .EQ.4 ) THEN

ELSE IF( DESNR(1) .EQ.5 ) TEEN

DDES (1) = COS ( TWOPI*DESVL (2) *T+PHI )

DDES(2) = -COS( TWOPI*DESVL(2) *T+PHI ) /

DDES(3) = SIN (TWOPI*DESVL(2)*T+PHI ) / ( TWOPI*DESVL(2) ) DDES(4) = COS( TWOPI*DESVL(2)*T+PHI )

$ ( TWOPI*TWOPI*DESVL(2)*DESVL(2) )

END IF END IF

C

C 400 IF ( .NOT. TASK(3) ) GOTO 500

CALL RSET( O.DO, DDES(5) , 4 ) IF ( DESNR(2) .EQ. 2 ) THEN

IF ( FNR .EQ. 1 ) THEN DDES(5) = O.DO

ELSE IF( FNR.EQ.2 ) THEN DDES(6) = 2.DO*DESVL(5)*COS(TWOPI*DESVL(2)*T+PHI)/

$ ( TWOPI*TWOPI*DESVL(2)*DESVL(2)*DESVL(2) )

$ ( TWOPI*DESVL(2)*DESVL(2) ) DDES (7) = -DESVL (5) *SIN (TWOPI*DESVL (2) *T+PHI) /

DDES(8) = O.DO END IF


END IF


ELSE IF( FNR.EQ.2 ) THEN

ELSE IF ( DESNR(2).EQ.4 ) THEN

ELSE IF( DESNR(2).EQ.5 ) THEN

BBES(5) = COS( TWBPï*DECVL(2J*T+PHI 1

DDES(6) = -COS( TWO?I*DESVL(2)*T+PHI I / $ ( TWOPI*TWOPI*DESVL(2)*DESVL(2) )

DDES(7) = SIN (TWOPI*DESVL(2)*T+PHI ) / ( TWOPI*DESVL(2) ) DDES(8) = COS( TWOPI*DESVL(2)*T+PHI )

END IF END IF

C 500 CONTINUE

DIANA-input file (experimental model)

IDEALISED SOLAR PANELS WITH BILINEAR SNUBBER UNITS: m, kg, s, N 'COORDI' DI=2 : ELEMENT LENGTH IS 5 ìaí. : six aegrees of freetiom

: Experimental Model

: inner panel 1 0.0 0.0 2 0.005 0.0 3 0.010 0.0 4 0.015 0.0 5 0.020 0.0 6 0.025 0.0 7 0.030 0.0 8 0.035 0.0 9 0.040 0.0 1 0 0.045 0.0 11 0.050 0.0 12 0.055 0.0 1 3 0.060 0.0 1 4 0.065 0.0 15 0.070 0.0 16 0.075 0.0 17 0 . 0 8 0 0.0 1 8 0.085 0.0 1 9 0.090 0.0 20 0.095 0.0 2 1 0.100 0.0 22 0.105 0.0 23 0.110 0.0 24 0.115 0.0 25 0.120 0.0 26 0.125 0.0 27 0.130 0.0 28 0.135 0.0 29 0 .140 0.0 30 0.145 0.0 3 1 0 .150 0.0 32 0.155 0.0 3 3 0.160 0.0 3 4 0 .165 0.0 35 0.170 0.0 36 0.175 0.0 37 0.180 0.0 38 0.185 0.0 39 0.190 0.0 40 0.195 0.0 4 1 0.200 0.0 42 0.205 0.0 4 3 0.210 0.0 4 4 0.215 0.0 45 0.220 0.0 46 0.225 0.0 47 0.230 0.0 48 0.235 0.0 49 0.240 0.0 50 0.245 0.0

51 0.250 0.0 : o u t e r pane l 1 0 1 0.0 o . o11 102 0.005 0.011 103 0.010 0.011 104 0.015 0.011 105 0.020 0.011 106 0.025 0.011 107 0.030 0.011 108 0.035 0.011

110 0.045 0.011 111 0.050 0.011 112 0.055 0.011 113 0.060 0.011 114 0.065 0.011 115 0.070 0.011 116 0.075 0.011 117 0.080 0.011 118 0.085 0.011 119 0.090 0.011 120 0.095 0.011 12 1 0.100 0.011 122 0.105 0.011 123 0.110 0.011 124 0.115 0.011 125 0.120 0.011 12 6 0.125 0.011 127 0.130 0.011 128 0.135 0.011 129 0.140 0.011 13 O 0.145 0.011 13 1 0.150 0.011 132 0.155 0.011 133 0.160 0.011 134 0.165 0.011 135 0.170 0.011 13 6 0.175 0.011 137 0.180 0.011 138 0.185 0.011 139 0.190 0.011 140 0.195 0.011 141 0.200 9.011 142 0.205 0.011 14 3 0.210 0.011 144 0.215 0.011 14 5 0.220 0.011 14 6 0.225 0.011 147 0.230 0.011 14 8 0.235 0.011 149 0.240 0.011 150 0.245 0.011 151 0.250 0.011

1 1.0 0.0 0.0 2 0.0 1.0 0.0 3 0.0 0.0 1.0 ' SUPPOR' : suppor t s

109 0.040 0.011

'DIRECT' :SPECIFICEREN VAN VECTOREN

1 TR 1 TR 2 RO 3 101 TR 1 TR 2 RO 3 150 TR 3 50 TR 3 47 TR 3

4 5 TR 3 1 4 5 TR 3

1 2 - 5 1 / TR 1 1 1 0 2 - 1 5 1 / TR 1 ‘ MATER1 ‘ : modif ied aluminium 1 YOUNG 5 9 . 0 E 0 9

DENSIT 2 . 7 0 E 0 3 : linear spring;snUbber

DAMP 0.0 : accelerometer masses

~ n n i C ~ D T X T P n n LVUL cazn~i’lu V . W

1500 MASS 14.5E-3 5 0 0 MASS 1 4 . 5 3 - 3

4 7 0 MASS 2 7 . 7 0 3 - 3

450 MASS 30 .00E-3 1 4 5 0 MASS 0.00E-9

: dummy mass

: snubber mass

I GEOMET I : panel 1 CROSSE 80.OE-6

INERTIA 1 . 0 6 6 7 3 - 1 0 I GENELM’ : C o m p o n e n t Mode S y n t h e s i s 1 TYPE panel1

ELEMEN 1-50 4 5 0 4 7 0 5 0 0 i f cnod / 4 5 1 tr 2

/ 1 / t r 2 CMS FREERB FREQUB 3 0 . 0 : RELNOD 1 2 3 1

2 TYPE p a n e l 2 ELEMEN 1 0 1 - 1 5 0 i f cnod 1 1 4 5 1

CMS FREERB FREQUB 30 .0 : RELNOD 1 1 2 3 1

I 101 I

’ ELEMEN ’ CONNEC : i n n e r pane l 1 2 3 4 5 6 7 8 9 10 11 1 2 13 1 4 15 16 17 18 19 20 2 1 2 2

LGBËN 1 2 LGBEN 2 3 LGBEN 3 4 LGBEN 4 5 LGBEN 5 6 LGBEN 6 7 L6BEN 7 8 L6BEN 8 9 LGBEN 9 1 0 LGBEN 1 0 11 LGBEN 11 1 2 LGBEN 1 2 1 3 LGBEN 1 3 1 4 LGBEN 1 4 15 LGBEN 15 1 6 LGBEN 16 1 7 LGBEN 17 18 LGBEN 18 1 9 LGBEN 19 2 0 LGBEN 2 0 2 1 LGBEN 2 1 2 2 LGBEN 2 2 2 3

NOFLEX cut-off frequency

TR 2

1 4 5 0 1 5 0 0 t r 2 t r 2 NOFLEX c u t - o f f f requency

TR 2

23 24 25 26 27 28 29 30 31 32 33 34 35 3 6 37 38 39 4 0 41 42 43 44 45 46 47 48 49 50

LGBEN 23 24 LGBEN 24 25 LGBEN 25 26 LGBEN 26 27 LGBEN 27 28 LGBEN 28 29 LGBEN 29 30 LGBEN 30 31 LGBEN 31 32 LGBEN 32 33 LGBEN 3 3 34 LGBEN 34 35 LGBEN 35 3 6 LGBEN 36 37 LGBEN 37 38 L6BEN 38 39 L6BEN 39 40 LGBEN 40 41 LGBEN 41 42 LGBEN 42 43 LGBEN 43 44 LGBEN 44 45 LGBEN 45 46 L6BEN 46 47 L6BEN 47 48 LGBEN 48 49 LGBEN 49 50 LGBEN 50 51

: outer panel 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 12 o 12 1 12 2 123 124 125 12 6 127 12 8 12 9 13 O 13 1 132 13 3 134 135

LGkEN LGBEN LGBEN LGBEN LGBEN L6BEN LGBEN L6BEN LGBEN LGBEN LGBEN LGBEN LGBEN L6BEN LGBEN LGBEN LGBEN L6BEN LGBEN LGBEN LGBEN LGBEN L6BEN LGBEN L6BEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN LGBEN

101 102 102 103 103 104 104 105 105 106 1 0 6 107 107 108 108 109 109 110 110 111 111 112 112 113 113 114 114 115 115 116 116 117 117 118 118 119 119 120 120 121 121 122 122 123 123 124 124 125 125 126 126 127 127 128 128 129 129 130 130 131 131 132 132 133 133 134 134 135 135 136

P

136 LGBEN 136 137 LGBEN 137 138 LGBEN 138 139 LGBEN 139 140 LGBEN 140 141 LGBEN 141 142 LGBEN 142 143 LGBEN 143 144 LGBEN 144 145 L6BEN 145 145 56BEN 145 147 LGBEN 147 148 LGBEN 148 149 LGBEN 149 150 LGBEN 150 : point-masses 1500 PTST 150 500 PT3T 50 470 PT3T 47 450 PT3T 45 1450 PT3T 145 : linear spring 1001 SP2TR 45 MATER1

1 1-50

1 101-150 1 1

1001 1 1500 1

1500

470

450 1 1450 /

1450

500

1001 1

I 470 /

I 450 /

I 500 /

GEOMET 1-50 1

137 138 139 14 O 14 1 14 2 14 3 144 14 5 i4 6 147 148 14 9 150 15 1

14 5

1 101-150 1 1

I LOADS CASE 1 DEFORM

DESIGN’ 1 1 0 1 TR 2 1 . 0

: specification of the design values : for the user-supplied routines

: excitation frequency [Hz]

: backlash [m]

1 0.0

2 47.0

3 0.0

4 0.0

5 0.510E-3 : amplitude of base displacement [m]

: BILINEAR SNUBBER : one-sided damping [Ns/m] 6 0.92

: one-sided stiffness [ N / m ] 7 3425.4 10 0.0 20 0.0 ' END'

SUBROUTINE USRLOD( TASK, FNR, DESVL, T, DESNR, RLOD , DDES ) C C.................................................COPYRIGHT (C) TNO-BOUW C... DIANA/SD/XC30/USRSUB/USRLOD c.. * C... c.. . C... CALLED FROM: EXTLOD, FINTNL, FNLSHO, FNLBIF, FDERIV, FNLTRA c.. . L . . . TASKíjJ = rl; external loa6 F-ext

EXTENDED WITH A prescribed base displacement

I

c.. . TASK(2) = T d F ext/ d desnrá

c.. . c.. . FNR = function number c... T = time c.. . DESVL = design variables c. . . DESVL(2) = excitation frequency [Hz] c.. . DESVL(3) = backlash [m] c.. . DESVL(5) = amplitude of base displacement [m] c.. . DESVL(6) = one-sided damping [Ns/m] c.. . DESVL(7) = one-sided stiffness [N/m] c.. . DESNR(2) = design variables numbers which are varied

c.. . TASK(3) = T d FIext/ d desnr2

c.. . C... COMMON /SDPARM/ c.. * RPAR(1) = TWOPI = 2 * PI c.. . C.. . OUTPUT:

a

c.. . RLOD(1) = external load factor c.. . RLOD(2) = prescribed displacement c.. . RLOD(3) = prescribed velocity c.. . RLOD(4) = prescribed acceleration c.. . DDES(1-4) = d F ext/ d desnrl c.. . DDES(5-8) = d FIext/ d desnr2

c.. . Prescribed displacement c... c.. . C.. . PROGRAMMED BY EVDV. c....................................................................... c.. . c... ARGUMENTS

LOGICAL TASK ( * ) INTEGER DESNR ( * ) , FNR DOUBLE PRECISION RLOB(4); T, BDES(8); BESVL(*)

DOUBLE PRECISION RPAR, TWOPI INTEGER IPAR LOGICAL LPAR COMMON /SDPARM/ RPAR(10), IPAR(20), LPAR(20) EQUIVALENCE ( TWOPI, RPAR(1) )

C

C C... local

C

C

DOUBLE PRECISION PHI, OMEGA

IF( TASK(4) ) CALL ERRMSG( ’SDXC30’, 17, O, O, O, O, ‘:::USRLOD’ )

IF ( DESVL(g).NE.O.DO ) THEN

ELSE

END IF CALL RSET( O.DO, RLOD, 4 ) CALL RSET( O.DO, DDES, 8 )

PHI = DESVL(9)/36O.DO*TWOPI

PHI = O.DO

C C

IF ( .NOT. TASK(1) ) GOTO 300

IF ( FNR.EQ.l ) THEN C

C C... Calculate RLOD c c.. . load

C

C c.. . C c...

C c.. . C

C 300

C C

C

RLOD (1) = DESVL (4) +DESVL (5) *COS ( TWOPI*DESVL (2) *T + PHI )

IF ( ~::a.rq.2 m r T - x T I n r i i v

prescribed displacement

prescribed velocity

prescribed acceleration

RLOD (2) = DESVL (5) *COS ( TWOPI*DESVL (2) *T+PHI )

RLOD(3) = -TWOPI*DESVL(2) *DESVL(5) *SIN( TWOPI*DESVL(2) *T+PHI )

RLOD(4) = -TWOPI*DESVL(2) *DESVL(2) *DESVL(5) *TWOPI* $ COS( TWOPI*DESVL(2)*T+PHI )

ELSE IF ( FNR.EQ.3 ) THEN OMEGA = TWOPI*DESVL(23) RLOD(1) = DESVL(24)*OMEGA*OMEGA*COS(OMEGA*T+PHI)

OMEGA = TWOPI*DESVL(23) RLOD(1) = DESVL(24)*OMEGA*OMEGA*SIN(OMEGA*T+PHI)

IF( T.GT.DESVL(23) ) THEN



DESVL(20) = DESVL(21) + ( T - DESVL(23) ) * $ ( DESVL(22) -DESVL(21) ) / ( DESVL(24) -DESVL(23) )

ELSE

END IF RLOD(1) = DESVL(4)+DESVL(5)*COS( TWOPI*DESVL(20)*T + PHI )

ELSE IF( FNR.EQ.6 ) THEN RLOD(2) = DESVL(50)

END IF

DESVL(2O) = DESVL(21)

CONTINUE IF ( .NOT. TASK(2) ) GOTO 500

CALL RSET( O.DO, DDES, 4 )

IF ( DESNR(1) .EQ. 2 ) THEN IF ( FNR .EQ. 1 ) THEN

ELSE IF ( FNR.EQ.2 ) THEN DDES(1) = O.DO

DDES(2) = 0.ODO DDES(3) = -DESVL(5)*TWOPI*SIN(TWOPI*DESVL(2)*T+PHI) DDES(4) = -TWOPI*2.DO*DESVL(2) *DESVL(5) *TWOPI*

$ COS(TWOPI*DESVL(2)*T+PHI) END IF

ELSE IF ( DESNR(l).EQ.4 ) THEN IF ( FNR.EQ.l ) THEN

DDES(1) = 1.DO END IF

ELSE IF( DESNR(l).EQ.5 ) THEN IF ( FNR.EQ.l ) THEN

ELSE IF( FNR.EQ.2 ) THEN DDES (1) = COS ( TWOPI*DESVL (2) *T+PHI )

DDES(2) = COS( TWOPI*DESVL(2)*T+PHI ) DDES(3) = -DESVL(2) *TWOPI*SIN( TWOPI*DESVL(2) *T+PHI )

DDES (4) = -DESVL( 2) *TWOPI*DESVL (2) *TWOPI* $ COS ( TWOPI*DESVL (2) *T+PHI )

END IF ELSE IF( DESNR(1) .EQ.23 ) THEN

IF ( FNR.EQ.3 ) THEN OMEGA = TWOPI*DESVL(23) DDES(1) = 2.DO*OMEGA*TWOPI*DESVL(24)*COS(OMEGA*T+PHI)

OMEGA = TWOII*DESVL(23j ELSE IF ( FNR.EQ.4 ) THEN

ijijEs (ij = 2 .iju*ûì"íCZAKTWûPI*DES-"-~~~4j *SIN(mEGA+..+pnIj END IF

IF ( FNR.EQ.3 ) THEN ELSE IF( DESNR(l).EQ.24 ) THEN

OMEGA = TWOPI*DESVL(23) DDES(1) = OMEGA*OMEGA*COS(OMEGA*T+PHI)

OMEGA = TWOPI*DESVL(23) DDES(1) = OMEGA*OMEGA*SIN(OMEGA*T+PHI)


END IF

IF ( FNR.EQ.6 ) THEN DDES(2) = 1.DO

END IF

ELSE IF( DESNR(l).EQ.SO ) THEN

END IF C

C 400 IF ( .NOT. TASK(3) ) GOT0 500

CALL RSET( O.DO, DDES(5), 4 ) IF ( DESNR(2) .EQ.4 ) THEN


END IF


ELSE IF( FNR.EQ.2 ) THEN

ELSE IF( DESNR(2).EQ.5 ) THEN

DDES (5) = COS ( TWOPI*DESVL( 2) *T+PHI )

DDES (6) = COS ( TWOPI*DESVL (2) *T+PHI ) DDES(7) = -DESVL(2)*TWOPI*SIN( TWOPI*DESVL(2)*T+PHI ) DDES(8) = -DESVL(2) *TWOPI*DESVL(2) *TWOPI*

$ COS( TWOPI*DESVL(2)*T+PHI ) END IF

XF ( FNR.EQ.2 1 THEN ELSE IF( DESNR(2).EQ.2 ) THEN

DDES(7) = -DESVL(5)*TWOPI*SIN(TWOPI*DESVL(2)*T+PHI) DDES(8) = -TWOPI*2.DO*DESVL(2) *DESVL(5) *TWOPI*

$ COS(TWOPI*DESVL(2)*T+PHI) END IF

IF ( FNR.EQ.3 ) THEN ELSE IF( DESNR(2).EQ.23 ) THEN

OMEGA = TWOPI*DESVE(23j DDES(5) = 2.DO*OMEGA*TWOPI*DESVL(24)*COS(OMEGA*T+PHI)

OMEGA = TWOPI*DESVL(23) DDES(5) = 2.DO*OMEGA*TWOPI*DESVL(24)*SIN(OMEGA*T+PHI)


END IF

IF ( FNR.EQ.3 ) THEN ELSE IF( DESNR(2).EQ.24 ) THEN

OMEGA = TWOPI*DESVL(23) DDES(5) = OMEGA*OMEGA*COS(OMEGA*T+PHI

OMEGA = TWOPI*DESVL(23) DDES(5) = OMEGA*OMEGA*SIN(OMEGA*T+PHI


END IF ELSE IF( DESNR(2) .EQ.50 ) THEN

IF ( FNR.EQ.6 ) THEN DDES(6) = 1.DO

END IF END IF

C

C 500 CONTINUE

RETURN END

[i] TNO Building and Construction Research, DIANA User’s Manual, Volume 1, 1992.

[2] J.P. den Hartog, Mechanical vibrations, 1956.

[3] S.W. Shaw and P.J. Holmes, A periodically forced piecewise linear oscillator. Journal of Sound and Vibration, 90(1):129-155, 1983.

[4] A. de Kraker, R. Fey, D. van Campen and C.J. Langeveld, Some aspects of the analysis of systems with local nonlinearities. In W. Schiehlen, editor, Nonlinear Dynamics in Engineering Systems, pages 165-172, IUTAM, Springer-Verlag, August 1989.

[5] R.H.B. Fey, Steady-state behaviour of reduced dynamic systems with local nonlinearities. Ph.D. thesis, Eindhoven University of Technology, 1992.

[6] W. Szemplinska-Stupnicka, The Behavior of Nonlinear Vibrating Systems, Volume 1+2, 1990.

[7] R.M. Evan-Iwanowski, Resonance Oscillations in Mechanical Systems, Elsevier Scientific Pub- lishing Co., 1976.

[8] A.H. Nayfeh and D.T. Mook, Nonlinear oscillations, John Wiley and Sons Inc., 1979.

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dynamic analyses of idealised solar panels with a bilinear snubber

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