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U04522: Engineering Dynamics 1

A Study of Single Degree of Freedom Vibrating

Systems

Written by:

Kilian Mayr 06012549

[email protected]

For the attention of:

Dr. Anand Thite

WORD COUNT:

(Excluding: Headings, Reference, Appendix)

1496
mailto:[email protected]:[email protected]


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1 Introduction

1.1 Abstract

This report covers a experiment carried out to evaluate the response of a single degree of

freedom (SDOF) Spring/Mass model with grounded damping which is excited by a base

motion. Thereby this report is split into two sections.

Part 1 (Theory) reports how the response has been expected to be and how this has been

evaluated.

Part 2 ( Procedure) reports how the experimental procedure has been carried out.

After these topics, the results are going to be critically compared in the discussion chapter.

All the outcome derived from the discussion chapter will be summarised in the conclusion

chapter.

1.2 Objective

Following objectives have to be met in order to be able to compare the results.

Part 1

1. Deriving Equation Of Motion (EOM)

2. Transforming EOM (differential equation) to exponential form

3. Obtaining Graphs of: amplitude ratio Vs. Frequency Ratio & Phase Shift Vs.

Frequency Ratio

4. Establishing a graph of Amplitude Vs. Frequency & Phase Shift Vs. Frequency by

using model parameter of the experimental equipment

Part 2

1. Estimating damping coefficient

2. Measuring and establishing a graph of amplitude Ratio Vs. Frequency Ratio & Phase

shift Vs. Frequency Ratio

2 Part 1 (Theory)

2.1 Equation Of Motion

The first step is to establish the EOM, which describes the forces acting on the vibrating mass.

In order to do so, the system configuration has to be known. The following illustration show the

schematic system configuration:

Written by: Kilian Mayr 06012549 2 / 17 Vibration Lab Report


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From this illustration it can be seen, how forces are fed into the mass. Therefore it is possible

to draw a Free Body Diagram (FBD).


Figure 1: Schematic Model Configuration

Figure 2: Free Body Diagram


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From this Free body diagram following equations can be derived to achieve a force

equilibrium:

F=2kyxc x -Equation 1

As F=m x , EOM is: m xc x2k x=2 k y -Equation 2

2.2 Complex expression

Using Euler's expression and complex algebra is the simplest way of obtaining the desired

steady state amplitude ratio and phase shift.

In order to do so, following Euler's equations have to be substituted into the EOM (Equation 2):

x=Xe jt

x= jXej t

x=

2

Xe

j t

y=Yej t

Hence, m2Xe

j tcjXe

jt2 kXe

j t=2kYe

j t -Equation 3

By cancelling out the ejt term, substituting amplitude ratio r=

n

& damping ratio

=c

2mn, the EOM can be rearranged to the amplitude ratio:

X

Y =1

1r22rj -Equation 4

As the amplitude ratio contains a complex component, it can be expressed in the polar from.

The magnitude thereby represents the amplitude ratio and the angle the phase shift.

Amplitude Ratio=X

Y=

1

1r222r2 -Equation 5

Phase shift==0atan2r

1r2

-Equation 6

Using these formulae, the two charts 1 & 2 shown in the Results chapter has been obtained.

The absolute values of the amplitude of the mass can be obtained by re arranging equation 5

for X only, and substituting the amplitude ratio back. The natural frequency thereby can be

obtained by substituting c with 0 in equation 3 n=2km

.

Therefore the response of the mass is determined by:



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Amplitude Ratio=X

Y=

Y

1

2km

2

2

2

2km

2 -Equation 7

Phase shift==0atan

2

2km

1

2km

2

-Equation 8

3 Procedure(Part 2)

The experiment has been carried out to verify the theoretical calculations. Therefore the firststep was to establish the damping ratio. This has been achieved by dropping the damped

mass from its highest position. This system then behaves as a free damped vibration. Using

the theory of logarithmic decrement, the damping ratio was able to be determined by the

following equation:

=

lnx

1

x2

2cot2 ln x

1

x2

2-Equation 9

Once the damping ratio is obtained, the theoretical calculations has been carried out. At 9different speeds ranging from 110 to 413 RPM with two different damper settings has been

recorded. Thereby the input frequency has been measured by a optical measuring device on

the fly wheel. The Amplitudes of the input and out put has been recorded by potentiometers

and displayed on a oscilloscope. The phase shift has been measured and displayed by a

phase meter, which is fed by the same source as the oscilloscope. All raw data has been

attached in the appendix.

4 Results

Please Note: the tables of the results are attached at the end of this report.

Please note that these tables are simplified, as the tables used to plot the graphs uses beyond

100 rows to achieve a high resolution of the graph.



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Chart 1: Amplitude Ratio Vs. Frequency Ratio


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Chart 2: Phase Shift Vs. Frequency Ratio


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Chart 3: Amplitude Vs. Frequency


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Chart 4: Phase Shift Vs. Frequency


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Chart 5: Amplitude Comparison


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Chart 6: Phase Shift Comparison


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5 Analysis/Discussion

5.1 Part 1

5.1.1 Chart 1

From the Chart 1 it can be seen that at a frequency ratio close to 1 is the resonance

frequency. From calculation and the graph it can be seen, that the damping ratio has not just

an effect on the amplitude. It also shifts the point where the maximum amplitude (resonance

point) is reached. For example for a damping coefficient of 0.01 the max. amplitude is at

rmax=0.99980.9998999949995 . To contrast, at a damping coefficient of 0.4, the

resonance is reached at rmax=0.680.824621125123532 . This phenomenon can be

observed until the damping coefficient reaches a value of =1

2. From damping

coefficients above this value, there will be no peak or amplification any more. To visualisethis behaviour, this graph has been inserted as the yellow dotted line. A further information

that could be gained from the table is the various regions where each component influences

the amplitude ration. At low frequency ratios ( 0.2 ), the amplification is close to one.This region called the stiffness controlled region, as the spring rate of the system contribute

the response. Surprisingly the damping coefficient has a impact on the range of the stiffness

controlled region. Extreme damping values (either high or low) will reduce the stiffness

controlled region, while a damping ratio around =1

2seems to have the longest region

where the amplitude ratio amount to one. At frequency ratio close to one, the system response

is mainly determined by the damping ratio, as it has been mentioned earlier, hence this regionis called the damping controlled region. The point onwards where all amplitude ratios merges

together( 2.5 ) is called the mass controlled region, as the inertia of the mass does notallow the system to vibrate at high amplitudes.

5.1.2 Chart 2

This chart shows the phase shift for different damping ratios across the frequency ratio range.

The most eye catching detail is that at a Frequency Ratio of 1 all plots crosses 90 phase shift.

5.1.3 Chart 3

From this chart it can be seen that the resonance frequency of the experiment is going to beexpected at a frequency ratio of 0.992387740068976 (22.6 rad/s = 3.61 Hz) for =0.09

and 0.969880991002984 (22.14 rad/s = 33.52Hz) for =0.17 . Thereby the amplitude is

expected to be 5.76 units for =0.09 and 2.95 units for =0.17 at a 1 unit input.

5.1.4 Chart 4

The phase shift is expected to occur early and stays virtually constant (@0) throughout the

rest of the region. The level 1 setting is expected to have a very sudden phase shift, while the

level 8 setting starts to phase shift earlier but remains a higher phase shift after resonance

frequency until it merges with the level 1 plot.



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5.2 Part 2

The damping ratio has been determined to be =0.09 for the level 1 setting and

=0.17 for the level 8 setting. This will equal to a damping coefficient of

c=16.78Nsm1 and c=33.19Nsm

1 .

5.2.1 Amplitude Ratio (Chart 5)

Level 1

Actual max. Amplitude ratio reached earlier then predicted.

Estimated Amplitude ratio lower then actual Amplitude Ratio (14.29% in respect to

measured value)

Over estimated Amplitude Ratio in the stiffness controlled region.

Very high compliance between the results in the mass controlled region .

Level 2

max. Amplitude Ratio reached at the same point

Theoretical results over estimate amplitude ratio throughout the region.

Results merges closer at elevated amplitude ratio

5.2.2 Phase Shift (Chart6)

Level 1

Very high compliance of results up to a Frequency Ratio of 1.1

Almost constant deviation of 20 between the results from a frequency ratio above 1.4

Phase shift very close to 90 even at unstable resonance frequency

Level 2

Very high compliance of results up to a Frequency Ratio of 1

High non constant deviation from frequency Ratio 1 onwards

Measured result has a outstanding step between 1.2 &1.4

From the measured values it can be seen, that the increased damping coefficient (hence,

damping ratio) has a positive effect on limiting the amplification of the input at and near the

resonance frequency.

The Level 8 has a higher amplification then the Level 1 Setting in the lower frequencies. The

increased damping coefficient of Level 8 successfully reduces the peak amplification but for

the expense of the stiffness controlled region.



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6 Conclusion

The theoretical part (Part1) can concluded to be satisfactory. It was able to visualise the

amplification and how it is affected by the damping ratio (max amplitude shift & amplitude

decrease).

Similar, the experiment (Part2) can be considered to be successful. Some deviation was

present, especially at resonance frequencies. As stated earlier, at resonance a very high

amplitude amplification is to be expected. The variance can be explained by the very violent

oscillation which will be disturbed by the friction in the rail, imperfect spring and other effects,

which are not taken into account in the theoretical calculation. Also the system is very

sensitive at resonance and small disturbance can have a significant impact. Taken the

imperfect testing condition into account and the fact that the the measuring equipment is also

subjected to inaccuracies, the results is satisfactory. Therefore the expected behaviour

(Amplification & Phase Shift) are close enough to the measured results to be able to explain

the difference in result by the measuring and testing limitations.

7 Reference

1. Thomson, W. T. and Dillon Dahleh, M. (1998). Theory of Vibration with Applications 5th

ed. Upper Saddle River: Prentice Hall.

2. Tongue, B. H. (2002). Principles of Vibration 2th ed. Oxford: Oxford University Press.

8 Appendix

8.1 Sample Calculations

For a Damping Ratio of 0.4 at a frequency ratio of 2

Amplitude Ratio=X

Y=

1

1r222r2 -Equation 5

X

Y=

1

122220.422=0.29

Phase shift==0atan 2r1r

2

-Equation 6

=0atan 20.42

122=28.07



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8.2 Tables


Appendix 1: Measured Data

Appendix 2: Amplitude Ratio / Used for Chart 1


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Appendix 3: Phase Shift / Used for Chart 2

Appendix 4: Amplitude / Used

for Chart 3Appendix 5: Phase Shift / Used

for Chart 4


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Appendix 6: Comparison Table Level 1/ Used for Chart 5 & 6

Appendix 7: Comparison Table Level 8 / Used for Chart 5 & 6

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