1. Mechanical Engineering at Virginia Tech
Chapter 5 Design
Acceptable vibration levels (ISO)
Vibration isolation
Vibration absorbers
Effects of damping in absorbers
Optimization
Viscoelastic damping treatments
Critical Speeds
Design for vibration suppression
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2. 5.1 Acceptable levels of vibration
Each part or system in a dynamic setting is required to pass “vibration” muster
Military and ISO provide a regulation and standards
Individual companies provide their own standards
Usually stated in terms of amplitude, frequency and duration of test
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3. Example 5.1.2 Dissimilar devices with the same frequency
m=1 kg
k=400 N/m
c=8 Ns/m
m=1000 kg
k=400,000 N/m
c=8000 Ns/m
car
CD drive
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4. But: response magnitudes different
Magnitude plot will have the same shape
Time responses will have the same form for similar (but scaled) disturbances BUT WITH DIFFERENT MAGNITUDES
different
Fig 5.3
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5. Mechanical Engineering at Virginia Tech
Section 5.2 Isolation
A major job of vibration engineers is to isolate systems from vibration disturbances or visa versa.
Uses heavily material from Sections 2.4 on Base Excitation
Important class of vibration analysis
Preventing excitations from passing from a vibrating base through its mount into a structure
Vibration isolation
Shocks on your car
Satellite launch and operation
Disk drives
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6. Recall from Section 2.4 that the FBD of SDOF for base excitation is
x(t) m m k c
y(t)
base
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7. SDOF Base Excitation assumes the input motion at the base has the form
The steady-state solution is just the superposition of the two individual particular solutions
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8. Particular Solutions (sine input)
With a sine for the forcing function,
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9. Particular Solutions (cosine input)
With a cosine for the forcing function, we showed
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10. Mechanical Engineering at Virginia Tech
Magnitude X/Y
Magnitude of the full particular solution
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11. The magnitude plot of X/Y
0.5 1
1.5 2
2.5 3
-20
-10 10 20 30 40
Frequency ratio r
X/Y (dB) z
=0.01
=0.1
=0.3
=0.7
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12. Notes on Displacement Transmissibility
Potentially severe amplification at resonance
Attenuation only for r > sqrt(2)
If r< sqrt(2) transmissibility decreases with damping ratio
If r>>1 then transmissibility increases with damping ratio Xp=2Yz/r
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13. It is also important to look at the Force Transmissibility:
x(t) m FT k c
y(t)
base
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14. Plot of Force Transmissibility
0.5 1
1.5 2
2.5 3
-20
-10 10 20 30 40
Frequency ratio r
F/kY (dB) z
=0.01
=0.1
=0.3
=0.7
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15. Isolation is a sdof concept;
Two types: moving base and fixed base
Three magnitude plots to consider TR= transmissibility ratio
Moving base displacement
Moving base force
Fixed base force
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16. Mechanical Engineering at Virginia Tech
For displacement transmissibility, isolation occurs as a function of stiffness
For stiffness such that the frequency ration is larger the root 2, isolation occurs, but increasing damping reduces the effect
For less then root 2, increased damping reduces the magnitude.
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17. Example 5.2.1 Design an isolation mount
Fig 5.6
Design an isolator (chose k, c) to hold a 3 kg electronics module to less then m deflection if the base is moving at y(t)=(0.01)sin(35t)
Calculate the force transmitted through the isolator
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18. T.R. Plot for moving base displacement
1.5
For T.R. =0.5 z
0.01
0.05
0.1
0.2
0.5
1.2 z r
1.73
1.74
1.76
1.84
2.35
4.41 1
T.R. z
=0.01 z
=0.05 z
0.5
=0.1 z
=0.2 z
=0.5 z
=1.2
0.5 1
1.5 2
2.5 3
Frequency ratio r
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19. Mechanical Engineering at Virginia Tech
From the plot, note that
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20. Mechanical Engineering at Virginia Tech
Rattle Space
Choice of k and c must also be reasonable
As must force transmitted:
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21. The transmitted force is
Transmitted force, T.R., static deflection,
damping and stiffness values must all be
reasonable for the application.
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22. Mechanical Engineering at Virginia Tech
Shock Isolation
Shock pulse
Pulse duration
Increased isolation with increasing k
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23. Mechanical Engineering at Virginia Tech
Figure 5.8 Shock Response
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24. Shock versus Vibration Isolation
In figure 5.8 for  = 0.5 requires
for shock isolation to occur.
Thus shock isolation requires a bound on the stiffness
Also from Figure 5, high damping is desirable for shock attenuation.
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25. Mechanical Engineering at Virginia Tech
Example Design a system that is good for both shock and vibration isolation.
The design constraints are that we have the choice of 3 off the shelve isolation mounts:
5 Hz, 6 Hz and 7 Hz each with 8% damping
The shock input is a 15 g half sine at 40 ms
The vibration source is a sine at 15 Hz
The response should be limited to 15 g’s and 76.2 mm, and 20 dB of vibration isolation
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26. Mechanical Engineering at Virginia Tech
Simulation of the response to the shock input for all three mounts choices
From these numerical simulations, only the 7 Hz mount satisfies all of the shock isolation goals:
Less then 15 g’s
Less then 3 in deflections
Fig 5.11
Fig 5.12
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27. Mechanical Engineering at Virginia Tech
Now consider the vibration isolation by plotting shock isolator design’s transmissibility:
For the 7 Hz shock isolator design, the reduction in Transmissibility is only 9.4 dB.
From this plot, and recalling Fig 5.7 less damping is required.
However, less damping is not possible
Fig 5.13
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28. Mechanical Engineering at Virginia Tech
5.3 Vibration Absorbers
Consider a harmonic disturbance to a single-degree-of freedom system
Suppose the disturbance causes large amplitude vibration of the mass in steady state
A vibration absorber is a second spring mass system added to this “primary” mass, designed to absorb the input disturbance by shifting the motion to the new added mass (called the absorber mass).
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29. Mechanical Engineering at Virginia Tech
Absorber concept
F(t) = F0sinwt x
Primary mass (optical table) m ka xa
k /2
absorber
k/2 ma
Primary system experiences resonance
Add absorber system as indicated
Look at equations of motion (now 2 dof)
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30. The equations of motion become:
To solve assume a harmonic displacement:
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31. Mechanical Engineering at Virginia Tech
The form of the response magnitude suggests a
design condition allowing the motion of the primary
mass to become zero:
pick ma and ka
to make zero
All the system motion goes into the absorber motion
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32. Mechanical Engineering at Virginia Tech
Choose the absorber mass and stiffness from:
This causes the primary mass to be fixed
and the absorber mass to oscillate at:
As in the case of the isolator, static deflection, rattle space and
force magnitudes need to be checked in each design
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33. Other pitfalls in absorber design
Depends on knowing w exactly
Single frequency device
If w shifts it could end up exciting a system natural frequency (resonance)
Damping, which always exists to some degree, spoils the absorption let’s examine these:
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34. Avoiding resonance (robustness)
Mass ratio
Original natural frequency of primary system before
absorber is attached
Natural frequency of absorber before it is attached
to primary mass
Stiffness ratio
frequency ratio
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35. Define a dimensionless amplitude of the primary mass
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36. Normalized Magnitude of Primary
Fig 5.15
Absorber
zone
w/w1
w/w2
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37. Robustness to driving frequency shifts
If w hits w1 or w2 resonance occurs
Using |Xk/F0|<1, defines useful operating range of absorber
In this range some absorption still occurs
The characteristic equation is:
= Frequency dependence on mass and frequency ratio
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38. Mass ratio versus frequency
Referring to fig. 5.16, as m increases, frequencies split farther apart for fixed b
thus if m is too small, system will not tolerate much fluctuation in driving frequency indicating a poor design
Rule of thumb 0.05< m <0.25
Fig 5.16
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39. Normalized Magnitude of the primary mass with and without the absorber
m=0.1 and b=0.71
0.5 1
1.5 2
-40
-20 20 40 60
Frequency ratio (ra)
Amplitude (Xk/F)
No vibration absorber
With vibration absorber
Adding absorber increase the number of resonances (or modes) from one to two.
Smaller response of primary structure is at absorber natural frequency
Effective over limited bandwidth
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40. What happens to the mass of the vibration absorber?
m=0.1 and b=0.71
In the operational range of the vibration absorber the absorber mass has relatively large motion
Beware of deflection limits!!
0.5 1
1.5 2
-40
-20 20 40 60
Frequency ratio (ra)
Amplitude (Xak/F)
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41. Mechanical Engineering at Virginia Tech
Example 5.3.1: design an absorber given F0= 13 N m=73.16 kg, k=2600 N/m, w = 180 cpm, xa< 0.02 m
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42. Example 5.3.2 Compute the bandwidth of the absorber design in 5.3.1
3 roots
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43. Mechanical Engineering at Virginia Tech
Comparing these 3 roots to the plot yields that:
This is the range that the driving frequency can
safely lie in and the absorber will still reduce the
vibration of the primary mass.
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44. 5.4 Damped absorber system
Undamped primary
Cannot be zero!
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45. Mechanical Engineering at Virginia Tech
Magnitude of primary mass for 3 levels of damping
As damping increases,
the absorber fails, but
the resonance goes away 1
Region of absorption
Go to mathcad example 5.16
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46. Effect of damping on performance
In the operational range of the vibration absorber decreases with damping
The bandwidth increases with damping
Resonances are decreased i.e. could be used to reduce resonance problems during run up
See Fig 5.19
m=0.25 and b=0.8
0.5 1
1.5 2
-30
-20
-10 10 20 30 40 50
Frequency ratio (ra)
Amplitude (Xk/F)
No VA
ca=0.01
ca=0.1
ca=1
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47. Three parameters effect making the amplitude small
This curves show that just increasing the damping does not result in the smallest amplitude.
The mass ratio and also matter
This brings us to the question of optimization
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48. With damping in the absorber:
Undamped absorber has poor bandwidth
Small damping extends bandwidth
But, ruins complete absorption of motion
Becomes a design problem to pick the most favorable m, b, z.
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49. Viscous Vibration Absorber
Rotating machine applications
Rotational inertia, shaft stiffness and a fluid damper
Often called a Houdaille damper illustrated in the following:
Primary system
Viscous absorber
x(t) xa k ca ma m
Fig 5.22
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50. Mechanical Engineering at Virginia Tech
Houdaille Damper
Fig 5.23
Equation of motion:
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51. Frequency response of primary mass
Figure 5.24
Design on m and z
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