lab 9_undamped vibration absorber_v3

8/18/2019 Lab 9_undamped Vibration Absorber_v3

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ME 413: System Dynamics & ControlME 413: System Dynamics & ControlME 413: System Dynamics & ControlME 413: System Dynamics & Control

Undamped ViUndamped ViUndamped ViUndamped Vibration Absorberbration Absorberbration Absorberbration Absorber

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UNDAMPED VIBRATION

ABSORBER

OBJECTIVES

1. To show the discomfort and dangers that result from the resonancephenomenon.

2. To study the effect of adding an absorber to a vibratory resonant system.

Part 1: THEORY

If a single or a multi-degree of freedom system is excited into resonance (theexcitation frequency nearly coincides with the natural frequency of the system), large

amplitudes of vibration result with accompanying high dynamic stresses and noiseand fatigue problems. Excessive vibrations in engineering systems are generallyundesirable and therefore must be avoided for the sake of safety and comfort. Ifneither the excitation frequency nor the natural frequency can conveniently be

altered, this resonance condition can often be successfully controlled. It is possible toreduce the unwanted vibrations by extracting the energy that causes thesevibrations. The extraction of this energy can be established by attaching to the main

vibrating system a dynamic vibration absorber , which is simply a spring-masssystem. The dynamic vibration absorber is designed such that the naturalfrequencies of the resulting system are away from the excitation frequency.

1m

)(1 t x

2 / 1k 2 / 1k

t F ο sin

Figure 1 Idealization of a machine


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When we attach an auxiliary mass2m to a machine of mass 1m through a spring of

stiffness2k , the resulting two degrees of freedom system will look as shown in

Figure 2. The equations of motion of the masses1

m and2

m are

( )

( )

1 1 1 1 2 1 2

2 2 2 2 1

sin

0

m x k x k x x F t

m x k x x

ο ω + + − =

+ − =

(1)

By assuming a harmonic solution,

t X t x j j ω sin)( = , j=1, 2 (2)

We can obtain the steady-state amplitude of the masses1

m and2

m as we can

obtain

2

2

2

22

2

121

2

22

1k mk mk k

F mk X

−−−+

−=

))((

)(

ω ω

ω ο

(3)

2

2

2

22

2

121

2

2k mk mk k

F k X

−−−+=

))(( ω ω

ο (4)

Machine 1m

)(1 t x

2 / 1k 2 / 1k

t F ο sin

)(2 t x

2m

2k

vibration absorber

Figure 2 Dynamic vibration absorber


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We are primarily interested in reducing the amplitude of the machine1

X . In order to

make the amplitude of1

m zero, the numerator of Eq. (3) should be set equal to zero.

This gives

2

22

mk =ω (5)

if the machine, before the addition of the dynamic vibration absorber, operates near

its resonance,11

2

1

2 / mk =≈ ω ω . Thus if the absorber is designed such that

1

1

2

22

m

k

m

k ==ω (6)

The amplitude of vibration of the machine, while operating at its original resonant

frequency, will be zero. By defining

,1

k

F st

ο δ =

1

1

1m

k =ω

as the natural frequency of the machine or main system, and

2

2

2m

k =ω (7)

as the natural frequency of the absorber or auxiliary system, equations. (3) and (4)can be rewritten as

1

2

2

2

2

11

2

21

11

1

k

k

k

k

X

st

−

−

−+

−

=

ω

ω

ω

ω

ω

ω

δ (8)

And

1

2

2

2

2

11

2

2

11

1

k

k

k

k

X

st

−

−

−+

=

ω

ω

ω

ω δ (9)

Figure (3) shows the variation of the amplitude of vibration of the machine st X δ / 1

with the machine speed1

/ ω ω . The two peaks correspond to the two natural

frequencies of the composite system. As seen before, 01

= X at1

ω ω = . At this

frequency, equation (9) gives


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1

2

2 2

st

F k X

k k

ο δ = − = − (10)

This shows that the force exerted by the auxiliary spring is opposite to the impressed

force ( )ο F X k −=22 and neutralizes it, thus reducing 1 X to zero. The size of thedynamic vibration absorber can be found from equations (10) and (6):

2

2 2 2 2k X m X F

ο ω = = − (11)

Thus the values of2

k and2

m depend on the allowable value of2

X . It can be seen

from Figure 3 that the dynamic vibration absorber, while eliminating vibration at the

known impressed frequency ,ω introduces two resonant frequencies1

Ω and2

Ω at

which the amplitude of the machine is infinite. In practice, the operating frequency

ω must therefore be kept away from the frequencies1

Ω and2

Ω .

2Ω1Ω

21 ω ω =

20

1

1

2=

m

m

st X δ / 1

1 / ω ω

Figure 3 Effect of undamped vibration absorber on the response of machine


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NOTES

1. The primary system possess now the characteristics of a two-degrees of

freedom, it has two natural frequencies1

Ω and2

Ω . The new natural

frequencies lie in the neighborhood of the natural frequency1

ω of the primary

system alone as shown in Figure 3. It can be seen from Figure 3 that1

Ω ≤

1ω ≤

2Ω . Thus the machine must pass through

1Ω during start-up and

stopping leading to large amplitude vibrations during these transient periods.

2. Since the dynamic absorber is tuned to one excitation frequency ω , the

steady-state amplitude of the machine is zero only at that frequency. If themachine operates at other frequencies or if the force acting on the machinehas several frequencies, then the amplitude of vibration of the machine may

become large.

3 The preceding analysis is valid only for an undamped system. If damping ispresent in the absorber it is not possible to eliminate steady state vibrations

of the original mass. The amplitude of vibration can only be reduced.

Part 1: EXPERIMENT

The above theory is applied to a simply supported beam carrying a motor with mass

unbalance at its mid-span as shown in Figure 4. In this figure, a simply supportedbeam carrying a motor with mass unbalance at its mid-span is shown. The motor is

connected to a speed control unit through which the speed of rotation can be varied.In order to measure the amplitude of vibration an accelerometer can be attached at

the beam mid-span. The output of the accelerometer is connected to a vibrationmeter that will provide reading of the amplitude of vibration. Underneath the motorassembly, the vibration absorber can be fixed.

Figure 4 Setup of the experiment (No vibration absorber attached).


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Apparatus

1. The absorber

Figure 5 shows the vibration absorber clamped below the motor. It comprises two

bodies of equal mass fixed equidistant from the midpoint of the horizontal cantilever.The distance apart of the bodies varies until the system is ‘tuned’.

l

Figure 5 The vibration absorber clamped below the motor.

2.

Speed Control Unit and Exciter Motor

Figure 6 shows the speed control unit that is used in this experiment. A d.c. motor is

used for all forced vibrations experiments powered by a control unit. Thiscombination comprises of a control box and d.c. motor, which provides high precisionspeed control of the motor up to 3000 rev/min, irrespective of the normal loadfluctuations of the motor.

The front panel of the unit contains a speed control, a fully calibrated speed meterincorporating an automatic range switching device (there being two ranges: 0 – 1500

and 1500 – 3000 rev/min), and a power socket for:

1. Mains inputs2. d.c. motor3.

Auxiliary output (either to stroboscope or chart recorder), sometimesmarked drum supply.


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Figure 6 Speed control unit front panel layout.

3. Vibration Meter Type 2511

The Vibration Meter Type 2511 shown in Figure 7 is a wide range instrument that isused in conjunction with piezo-electric vibration pick-up to measure mechanical

vibration and shock in terms of acceleration, velocity and displacement. It iscompletely portable and is built to a high standard of accuracy and stability making itsuitable for laboratory and field conditions. The instrument is fully calibrated in bothmetric and British units and has a charge amplifier input that allows the use of long

cables between the pick-up and the meter without any reduction in sensitivity.

Figure 7 Front panel of the 2511 Type Vibration Meter.


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Procedure

For a given frequency, the masses of the vibration absorber are adjusted along their

cantilever leaf spring so that the energy of vibration transmits to the absorber and

the amplitude of the main (primary) system, i.e. the motor and beam, is reduced tozero.

The aim is to determine the length l , the distance of the center of each of thebodies from the midpoint of the cantilever so that the natural frequency of transversevibration of this sub-system corresponds to the running speed of the main (primary)system, i.e. the motor and beam.

IMPORTANT

For your safety and the safety of the equipment

WHEN THE SYSTEM VIBRATES AT RESONANCE CONDITIONS

DO NOT LET IT VIBRATE FOR A LONG PERIOD OF TIME

1. No Absorber

1.

Turn the upper right switch of the Vibration Meter Type 2511 on displacement,velocity or acceleration depending on what you are intending to measure.

2.

Turn the lower right switch of the Vibration Meter Type 2511 for theappropriate range (Range I: metric units and Range II: British units). Notice

that you are reading the amplitude of the quantity to be measured as shownin Table 1.

Table 1 Reading of the Vibration Meter Type 2511

Quantity Equation of Motion Reading Vibration Meter Type2511

Displacement ( ) sint X t ω = Displacement Amplitude: X

Velocity ( ) cost X t ω ω = Velocity Amplitude: X ω

Acceleration2

( ) sint X t ω ω = − Acceleration Amplitude:2 X ω −

3. Vary the motor speed and read the vibration amplitudes from the VibrationMeter Type 2511.

4.

For each speed, record your reading (vibration amplitude) in Table 2.


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5. Increase the speed gradually until you get the resonance phenomenon thatresults in large amplitudes of excessive vibrations. From the tabulated dataplot the response curve and determine the resonant frequency of the system.

2. With Absorber

1. With the auxiliary system (vibration absorber) attached, vary the motor speedand record the corresponding frequency and the resulting amplitude ofvibration.

2.

Repeat the above (1 to 4) steps of the previous case.

3. With the aid of the experimentally defined resonant frequency, the dynamic

vibration absorber is to be designed such that the frequency of oscillations is

equal to

3

1 3

2 2

EI

ml

ω

π π = = (12)

where is natural frequency of the auxiliary system, m is the mass of eachof the two bodies, and EI is the flexural rigidity of the double cantileverbeam. The mass m is a given constant and l is to be found from the aboveformula.

4. One can easily conclude, that any three parameters of equation (12) can befixed, in order to determine the fourth parameter. In this experiment we willdetermine the position of the mass m , at which the absorber effect isverified. Experimentally, one can vary the position of the mass m , and excitethe system at the required excitation frequency until no vibrations of the

primary system are observed, or the position of the mass l can bedetermined from equation (12), adjusted accordingly and the absorbingeffect can be verified.


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REQUIREMENTS

The report should include the following:

1. Organize your measurements of the vibration amplitudes versus the rotationalspeed as shown in Table 2.

2.

Plot on the same graph the velocity amplitude X ω versus the rotationalspeed for the cases with and without the dynamic vibration absorber. (UseMATLAB or Excel).

3. Plot on the same graph the displacement amplitude X versus therotational speed for the cases with and without the dynamic vibration

absorber. (Use MATLAB or Excel)

4. Indicate resonant frequencies and the frequency at which the primary system

(no absorber included) has zero vibration

5. Measure the length l for which the amplitude of vibration is zero when theabsorber is used. Compare this by solving Equation (12) for the length l .

Use 200 GPaE = and31

12I bh = .

6. Give a brief discussion of your findings.

7.

Give conclusions on the effect of the vibration absorber.


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References

[1] Experiments in Vibration Using The TM 16 Universal Vibration, TQ Ltd[2] K. Ogata, System Dynamics, Fourth Edition, Pearson Prentice Hall, 2004.

[3] S. S. Rao, Mechanical vibrations, SI Edition, Pearson Prentice Hall, 2005.[4] http://ta.twi.tudelft.nl/nw/users/vuik/information/tacoma_eng.html

[5] http://abel.math.harvard.edu/archive/21b_fall_03/tacoma/ [6] http://perso.wanadoo.fr/olivier.granier/meca/accueil.htm [7] http://www.kettering.edu/~drussell/Demos/absorber/DynamicAbsorber.html [8] http://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/absorb.html [9] http://www.kettering.edu/~drussell/Demos/absorber/DynamicAbsorber.html

lab 9_undamped vibration absorber_v3

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