ME 413: System Dynamics & Control
Undamped Vibration
Vibration Absorber
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ME 413: System Dynamics and Control Lab Manual
UNDAMPED VIBRATION
ABSORBER
OBJECTIVES
1. To show the discomfort and dangers that result from the resonance
phenomenon.
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 (the
excitation frequency nearly coincides with the natural frequency of the system), large
amplitudes of vibration result with accompanying high dynamic stresses and noise
and fatigue problems. Excessive vibrations in engineering systems are generally
undesirable and therefore must be avoided for the sake of safety and comfort. If
neither the excitation frequency nor the natural frequency can conveniently be
altered, this resonance condition can often be successfully controlled. It is possible to
reduce the unwanted vibrations by extracting the energy that causes these
vibrations. The extraction of this energy can be established by attaching to the main
vibrating system a dynamic vibration absorber, which is simply a spring-mass
system. The dynamic vibration absorber is designed such that the natural
frequencies of the resulting system are away from the excitation frequency.
Fο sin ωt
m1
x1 (t )
k1 / 2
Figure 1
Undamped Vibration Absorber
k1 / 2
Idealization of a machine
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ME 413: System Dynamics and Control Lab Manual
When we attach an auxiliary mass
m2 to a machine of mass m1 through a spring of
k 2 , the resulting two degrees of freedom system will look as shown in
Figure 2. The equations of motion of the masses m1 and m 2 are
stiffness
m1x1 + k 1x 1 + k 2 ( x 1 − x 2 ) = Fο sin ωt
(1)
m 2 x2 + k 2 ( x 2 − x 1 ) = 0
By assuming a harmonic solution,
x j (t ) = X j sin ωt ,
(2)
j=1, 2
We can obtain the steady-state amplitude of the masses
m1 and m2 as we can
obtain
X1 =
(k 2 − m2ω 2 ) Fο
(k1 + k 2 − m1ω 2 )(k 2 − m2ω 2 ) − k 22
(3)
X2 =
k 2 Fο
(k1 + k 2 − m1ω 2 )( k 2 − m2ω 2 ) − k 22
(4)
Fο sin ωt
Machine m1
x1 (t )
vibration absorber
k2
k1 / 2
k1 / 2
m2
x2 (t )
Figure 2
Undamped Vibration Absorber
Dynamic vibration absorber
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ME 413: System Dynamics and Control Lab Manual
We are primarily interested in reducing the amplitude of the machine
make the amplitude of
X 1 . In order to
m1 zero, the numerator of Eq. (3) should be set equal to zero.
This gives
ω2 =
k2
m2
(5)
if the machine, before the addition of the dynamic vibration absorber, operates near
its resonance,
ω 2 ≈ ω 12 = k1 / m1 . Thus if the absorber is designed such that
ω2 =
k2
k
= 1
m2 m1
(6)
The amplitude of vibration of the machine, while operating at its original resonant
frequency, will be zero. By defining
δ st =
Fο
,
k1
ω1 =
k1
m1
as the natural frequency of the machine or main system, and
ω2 =
k2
m2
(7)
as the natural frequency of the absorber or auxiliary system, equations. (3) and (4)
can be rewritten as
ω 
1 −  
X1
 ω2 
=
2
2
δ st  k


1 + 2 −  ω  1 −  ω   − k 2

k1  ω 1    ω 2   k1



(8)
And
X2
δ st
=
 k ω 
1 + 2 −  

k1  ω1 

2
1
  ω  2  k
1 −    − 2
  ω 2   k1


Figure (3) shows the variation of the amplitude of vibration of the machine
with the machine speed
ω / ω1 .
(9)
X 1 / δ st
The two peaks correspond to the two natural
frequencies of the composite system. As seen before,
X 1 = 0 at ω = ω1 . At this
frequency, equation (9) gives
Undamped Vibration Absorber
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ME 413: System Dynamics and Control Lab Manual
X2 =−
F
k1
δ st = − ο
k2
k2
(10)
This shows that the force exerted by the auxiliary spring is opposite to the impressed
force (k 2 X 2 = − Fο ) and neutralizes it, thus reducing X 1 to zero. The size of the
dynamic vibration absorber can be found from equations (10) and (6):
k 2 X 2 = m 2ω 2 X 2 = −Fο
Thus the values of
(11)
k 2 and m2 depend on the allowable value of X 2 . It can be seen
from Figure 3 that the dynamic vibration absorber, while eliminating vibration at the
known impressed frequency ω , introduces two resonant frequencies Ω1 and Ω 2 at
which the amplitude of the machine is infinite. In practice, the operating frequency
ω must therefore be kept away from the frequencies Ω1 and Ω 2 .
ω1 = ω 2
m2
1
=
m1 20
X 1 / δ st
Ω1
Ω2
ω / ω1
Figure 3
Effect of undamped vibration absorber on the response of machine
Undamped Vibration Absorber
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ME 413: System Dynamics and Control Lab Manual
NOTES
1. The primary system possess now the characteristics of a two-degrees of
freedom, it has two natural frequencies
Ω1 and Ω 2 . The new natural
frequencies lie in the neighborhood of the natural frequency
ω1
of the primary
system alone as shown in Figure 3. It can be seen from Figure 3 that
ω1
≤
Ω1 ≤
Ω 2 . Thus the machine must pass through Ω1 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 the
machine operates at other frequencies or if the force acting on the machine
has 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 is
present 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 supported
beam 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 vibration
meter that will provide reading of the amplitude of vibration. Underneath the motor
assembly, the vibration absorber can be fixed.
Figure 4
Setup of the experiment (No vibration absorber attached).
Undamped Vibration Absorber
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ME 413: System Dynamics and Control Lab Manual
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
2.
The vibration absorber clamped below the motor.
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. This
combination comprises of a control box and d.c. motor, which provides high precision
speed control of the motor up to 3000 rev/min, irrespective of the normal load
fluctuations of the motor.
The front panel of the unit contains a speed control, a fully calibrated speed meter
incorporating an automatic range switching device (there being two ranges: 0 – 1500
and 1500 – 3000 rev/min), and a power socket for:
1. Mains inputs
2. d.c. motor
3. Auxiliary output (either to stroboscope or chart recorder), sometimes
marked drum supply.
Undamped Vibration Absorber
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ME 413: System Dynamics and Control Lab Manual
Figure 6
3.
Speed control unit front panel layout.
Vibration Meter Type 2511
The Vibration Meter Type 2511 shown in Figure 7 is a wide range instrument that is
used in conjunction with piezo-electric vibration pick-up to measure mechanical
vibration and shock in terms of acceleration, velocity and displacement. It is
completely portable and is built to a high standard of accuracy and stability making it
suitable for laboratory and field conditions. The instrument is fully calibrated in both
metric 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
Undamped Vibration Absorber
Front panel of the 2511 Type Vibration Meter.
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ME 413: System Dynamics and Control Lab Manual
Figure 8
4.
Rear panel of the 2511 Type Vibration Meter.
Transducer (Accelerometer)
In general, the transducers employed in vibration analyses convert mechanical
energy into electrical energy; that is, they produce an electrical signal which is a
function of mechanical vibration. See Figure 5.
Figure 9
Undamped Vibration Absorber
Accelerometers.
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ME 413: System Dynamics and Control Lab Manual
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 to
zero.
The aim is to determine the length l , the distance of the center of each of the
bodies from the midpoint of the cantilever so that the natural frequency of transverse
vibration 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 the
appropriate range (Range I: metric units and Range II: British units). Notice
that you are reading the amplitude of the quantity to be measured as shown
in Table 1.
Table 1
Reading of the Vibration Meter Type 2511
Quantity
Displacement
Velocity
Acceleration
Equation of Motion
x (t ) = X sin ωt
x (t ) = X ω cos ωt
x(t ) = − X ω 2 sin ωt
Reading Vibration Meter Type
2511
Displacement Amplitude:
X
Xω
2
Acceleration Amplitude: − X ω
Velocity Amplitude:
3. Vary the motor speed and read the vibration amplitudes from the Vibration
Meter Type 2511.
4. For each speed, record your reading (vibration amplitude) in Table 2.
Undamped Vibration Absorber
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ME 413: System Dynamics and Control Lab Manual
5. Increase the speed gradually until you get the resonance phenomenon that
results in large amplitudes of excessive vibrations. From the tabulated data
plot 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 speed
and record the corresponding frequency and the resulting amplitude of
vibration.
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
f =
where
f
ω
1
=
2π 2π
3EI
ml 3
(12)
is natural frequency of the auxiliary system,
of the two bodies, and
beam. The mass
formula.
m is
EI
m
is the mass of each
is the flexural rigidity of the double cantilever
a given constant and
l is
to be found from the above
4. One can easily conclude, that any three parameters of equation (12) can be
fixed, in order to determine the fourth parameter. In this experiment we will
determine the position of the mass m , at which the absorber effect is
verified. Experimentally, one can vary the position of the mass m , and excite
the system at the required excitation frequency until no vibrations of the
primary system are observed, or the position of the mass l can be
determined from equation (12), adjusted accordingly and the absorbing
effect can be verified.
Undamped Vibration Absorber
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ME 413: System Dynamics and Control Lab Manual
REQUIREMENTS
The report should include the following:
1. Organize your measurements of the vibration amplitudes versus the rotational
speed as shown in Table 2.
2. Plot on the same graph the velocity amplitude X ω versus the rotational
speed for the cases with and without the dynamic vibration absorber. (Use
MATLAB or Excel).
3. Plot on the same graph the displacement amplitude X
versus the
rotational 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 the
absorber is used. Compare this by solving Equation (12) for the length
Use
E = 200 GPa and I =
l.
1 3
bh .
12
6. Give a brief discussion of your findings.
7. Give conclusions on the effect of the vibration absorber.
Undamped Vibration Absorber
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ME 413: System Dynamics and Control Lab Manual
Table 2
Readings
With no Absorber
N
(rpm)
Disp.
Ampl.
(mm)
Vel.
Ampl.
(mm/s)
Undamped Vibration Absorber
Acc.
Ampl.
(mm/s2)
With Absorber
N
(rpm)
Disp.
Ampl.
(mm)
Vel.
Ampl.
(mm/s)
Acc.
Ampl.
(mm/s2)
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ME 413: System Dynamics and Control Lab Manual
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
Experiments in Vibration Using The TM 16 Universal Vibration, TQ Ltd
K. Ogata, System Dynamics, Fourth Edition, Pearson Prentice Hall, 2004.
S. S. Rao, Mechanical vibrations, SI Edition, Pearson Prentice Hall, 2005.
http://ta.twi.tudelft.nl/nw/users/vuik/information/tacoma_eng.html
http://abel.math.harvard.edu/archive/21b_fall_03/tacoma/
http://perso.wanadoo.fr/olivier.granier/meca/accueil.htm
http://www.kettering.edu/~drussell/Demos/absorber/DynamicAbsorber.html
http://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/absorb.html
http://www.kettering.edu/~drussell/Demos/absorber/DynamicAbsorber.html
Undamped Vibration Absorber
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