vibration!duetorotation!unbalance! -...

Prepared by: Keivan Anbarani Abstract

In this experiment four eccentric masses are used in conjunction with four springs and one damper to simulate the vibration. Masses are aligned in different orders to simulate in-‐phase and out-‐phase situation. Acceleration, frequency and phase shift of the system is measure, which then are used to do calculations and generating graphs to describe the simple harmonic motion. For in-‐phase situation natural frequency is measured to be 9.16 Hz and calculated to be 9.6 Hz. This is a fairly good result since in real life nothing is perfect and it is not possible to generate theoretical results. As expected the calculated natural frequency is slightly larger due to the fact that damping is assumed to be zero using theoretical calculation, where the system really is slightly damped. For out-‐phase situation natural frequency happens at frequency of 12.81 Hz and also at 6.13 Hz, however at 12.81 Hz the amplitude is much larger.

Fall 08

VIBRATION DUE TO ROTATION UNBALANCE

Table of Contents

1.0 INTRODUCTION......................................................................................................................................1

2.0 METHODS ..............................................................................................................................................1

3.0 RESULTS.................................................................................................................................................2 Sample Phase Shift Calculation ...............................................................................................................................2 Pre-‐filtered and Post-‐filtered Signals.......................................................................................................................3 In Phase Results ......................................................................................................................................................3

Experimental versus Theoretical comparison............................................................................................................. 5 Damping ratio calculation:.......................................................................................................................................... 6

Out of Phase ...........................................................................................................................................................7 Damping ratio calculation:.......................................................................................................................................... 8

4.0 DISCUSSION AND CONCLUSION..............................................................................................................9

5.0 LEARNING OUTCOMES .........................................................................................................................10


1.0 INTRODUCTION Vibration due to rotating eccentric masses is a common phenomenal that one sees in everyday life

such as vibration of automobile engine or the washing machine. This phenomenal is really important because if not dealt with promptly one could have catastrophic failures due to the fatigue caused by the vibration. As observed in this lab at natural frequency vibration force could get really big which could lead into an unstable system and causing failure.

2.0 METHODS

As shown in figure 1, this experiment consists of:

Engine Model: The box that contains the shaft and essentric masses

Optical Encoder: used to generate an optical pulse per rotation of essentric mass

Speed Control: used to control the speed of rotation

Accelerometer: a piezoelectric element that produces an electric charge when subject to a load, it is used to measure the vertical acceleration.

There is one rectangular box which could model a cars engine. The front and back of the system contains two rotating shaft which have two essentric masses attached to them. In total there are four essentric masses that rotate. For in-‐phase all the masses are pointing in the same direction to start with, for our-‐phase masses point opposite of the masses on the other side. Figure 2 is the front view of the experiment which is really simular to the back view. Figure 3 shows the arrangments of the 4 springs and the damper from the side.

The values that are measured with hand in this lab are the

radious of the eccentric mass, the distance of the masses from the center, which is where the damper is located. The values that are measured with computer are frequency, acceleration and the phase shift.

Figure 1 -‐ Experiment Apparatus

Figure 2 -‐ Front view of the system

Figure 3 -‐ Side view of the system


3.0 RESULTS

This section illustrates how Matlab measures frequency, amplitude and phase shift to determine how the system behaves for given frequencies. Graphs will also be included in this section to help understand these calculations.

Sample Phase Shift Calculation

In order to determine the phase shift, snapshot of the figure 4 is taken and the following formula is used.

φ = 360 − τω2π

× 360 , where τ is the difference

between the leading edge of the pulse signal and the upward zero crossing of the sign wave which the system is oscillating. The sample phase shift calculation below is done at 19.65Hz.

φ = 360 − (6.132 − 6.108)× (19.65)2π

× 360 = 332.98

The sample phase shift is calculated to be 332.98 , however Matlab calculates the value to be 186.0 . The calculated error in terms of time is only 0.021 seconds, which is acceptable since the data was taken from the graph.

t = φ

360 × f

t = (332.98 −186)

360 × (19.65)= .021seconds

Figure 4 -‐ Graph used to measure phase shift


Pre-‐filtered and Post-‐filtered Signals

There are so many noises involved in this experiment. Therefore, a filter is being used in order to get a clear signal. A band pass filtration system is used for this specific experiment. The figures below is taken before and after filtration is used so that it is clear why filtration is needed.

In Phase Results

An experiment is conducted to determine the system’s behavior for varies of frequencies. The measured values are Acceleration, Frequency and Phase shift. However acceleration is measured in Volts and later converted to m/s2 using the conversion factor of 9.81 m/s2 = 99 mV. Also rotation speed is measured in rotation per minute and one rotation is equivalent to 2π . These values are then used in the following formula to calculate systems excitation vertical displacement.

x =xω2 , where x is the vertical displacement, x is the vertical acceleration of the system and ω is the

rotation speed.

The system showed maximum displacement at 9.16 Hz. Resonance frequency is a term used to define the tendency of a system to oscillate at larger amplitude at some frequencies than at others, therefore in this experiment the experimental resonance frequency is 9.16 Hz.

Theoretical value for un-‐damped resonance frequency ωn , is calculated using the following formula

ωn =kM

where k is the spring constant and M is mass of the whole system.

ωn =

4 ×14365.3N /m15.674kg2πrad

= 9.6s−1

Figure 5 -‐ Pre-‐filtered signal Figure 6 – Post-‐filtered signal


Based on our experimental graph the maximum displacement is measured to be at 0.13 cm, however the theoretical value for the maximum displacement is calculated to be 0.14 cm. There is a slight difference and that is because there are external forces that was not taken into account when calculating the theoretical value such as friction between the bearings, air friction and damping of the system.

Figure 7 – Experimental displacement versus rotation speed graph

Figure 8 -‐ Experimental Phase shift versus rotation speed graph


Experimental versus Theoretical comparison

In the shaky table the system’s motion can be described as a second order differential equation. Which is:

M d 2xdt 2

+C dxdt+ kx = 4meω2 sin(ωt)

the solution for this second order differential equation is

x(t) = e−ξωnt [Acos(ωdt)+ Bsin(ωdt)+Y sin(ωt − Φ)

where the red part of the equation is the homogeneous solution and the blue part is particular solution which is what we are interested in this lab. Frequency response function is a dimensionless function which is the ratio of the output to input expressed as a function of excitation frequency.

xu=

r2

(1− r2 )2 + (2ζr)2, where r = ω

ωn

and u = 4meM

e is the essentric radious, m is the essentric mass and M is the mass of the system. Above equation was used to plot the theoretical displacement versus rotation speed graph.

Figure 9 – Displacement comparison graph

As showen in figure 9 the maximum theoritical displacement is 0.14 cm, but the experimental value is 0.13 cm.


Figure 11 reproduces the curve in figure 6 for the damping value.

Damping ratio calculation:

In order to estimate the damping coefficient, the decrement in amplitude is estimated by taking two consecutive amplitudes and measuring their difference. We considered a few different cycles to get the most accurate value. The following equation and graph is used to determine the damping coefficient of the system.

δ = ln( AiAi+r

)× 1r, ζ ≅

δ

2π

where r is the number of cycles and Ai and Ai+r .

Ai r Ai+r δ ζ 0.0204 1 0.0159 0.24921 0.03966

2 0.0124 0.24892 0.03961 3 0.009509 0.25443 0.0405

Taking the average = 0.03992

The damping coefficient ζ turns out to be 0.04.

Figure 11 – Frequency Response curves equation illustration

Figure 10 – Frequency response curve

Figure 12 – In-‐phase damping


Out of Phase

The same methods as in phase section is used to calculate and plot the following graphs.

Figure 12 -‐ Phase shift

Figure 13 – Frequency response


Damping ratio calculation:

In order to estimate the damping coefficient, the decrement in amplitude is estimated by taking two consecutive amplitudes and measuring their difference. We considered a few different cycles to get the most accurate value. The following equation and graph is used to determine the damping coefficient of the system.

δ = ln( AiAi+r

)× 1r, ζ ≅

δ

2π

where r is the number of cycles and Ai and Ai+r .

Ai r Ai+r δ ζ 0.082 1 0.0763 0.005 0.018

The damping coefficient ζ turns out to be 0.018.

Figure 14 – Out-‐phase damping


4.0 DISCUSSION AND CONCLUSION

Quantity Formula In-‐phase Out-‐of-‐phase

Resonant frequency ωn =

4km

9.6 Hz 12.81 Hz

Maximum Acceleration Level

98mV9.81m / s2

440.4 cms2 7507.65 cm

s2

!" #$%

Maximum Displacement x =xω2

0.14 cm 1.16 cm

!#$%

Modal stiffness, k

k = F2ζX

max

87143N

m

21695Nm

Modal Mass, m m =kω2

n

24 kg 2.2 kg

Modal damping, c c = FωnXmax

115.6 kg/s 9.7 kg/s

Resonant frequency is the point that the system has maximum oscillation and in out of phase case we have two natural frequencies, one around 7 Hz and other around 11 Hz. The damping ratio at 7 Hz is much smaller because r is equal to 1.

The source of error in our theoretical versus experimental values includes friction, air drag and the damper since we assume zero damping for the theoretical calculations.

Channel two didn't show a clear harmonic acceleration response because there are other external forces that could affect the harmonic motion such as, air friction, internal friction for the motor and the mass of the spring.

There is a big phase shift close to the resonance. Right before resonance the phase shift was 209 and at resonance it jumps to 311

We excited the system as close to resonance as possible so we can observe the maximum amplitude respond.


5.0 LEARNING OUTCOMES

In this experiment I learned the importance of excitation frequency on vibration response and how one can use computer measurements and graphs to calculate important values such as damping coefficient. My team being the first group to do this experiment had extremely hard time preparing this report due to lack of understanding of the equations and concepts of the experiment. However thankfully with the help of the TA and Dr. Srikanth we were able to understand everything and finish the lab with excellent results.

vibration!duetorotation!unbalance! -...

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