MODULE 09
Inman chapter 5
1
DISPLACEMENT TRANSMISSIBILITY FROM BASE TO MASS
IN BASE EXCITATION
10
9
8
Z

7
6
X
0.01
0.1
5
0.5
4
3
2
1
0
0
1
2
3
4
r
Relative / base
Absolute / base
Amplification of displacement amplitude (transmissibility of displacement amplitude)
2
FORCE TRANSMISSIBILITY FROM BASE TO MASS
IN BASE EXCITATION
3
FORCE TRANSMISSIBILITY FROM MASS TO BASE
IN FORCE EXCITATION
F0 - amplitude of
excitation force
Excitation force is transferred to ground through damper and spring.
Therefore, the base “feels” the effect of vibration through the
combination of damping of stiffness forces.
TTransmissibility 
FT
F0
1  (2 r ) 2
T
(1  r 2 ) 2  (2 r ) 2
FT - amplitude of force
transmitted to ground
4
FORCE TRANSMISSIBILITY FROM MASS TO BASE
IN FORCE EXCITATION
T
T<1
1  (2 r ) 2
(1  r 2 ) 2  (2 r ) 2
when

r 2
r 2
T = Tmax when
  n 1  2 2
Force transmissibility to the base.
Isolation (reduction of force transmitted to base) occurs only for
r
r 2
5
Displacement
transmissibility
Force transmitted
to the base
Force transmitted
to the mass
Inman p 398
6
VIBRATION ABSORBING
7
VIBRATION ABSORBING
Vibration absorber is a spring-mass system added to a device to protect it from steady –state
harmonic disturbance.
The major effect of adding the second mass-spring system is to change from single degree of
freedom system to a two degree of freedom system.
The new system has two natural frequencies. The value of the absorber mass and stiffness are
chosen such that the motion of the original mass is a minimum. This is accompanied by
substantial motion of the added absorber system.
Inman p. 410
8
VIBRATION ABSORBING
A vibration absorber is used to protect the primary system from steady-state harmonic disturbance. By
attaching the absorber to the primary system which is modeled as a SDOF system, the new system
becomes a two DOF system as shown in the model below. Depending on the driving frequency of the
original system, the absorber needs to be carefully tuned, that is, to choose adequate values of absorber
mass and stiffness, so that the motion of the original mass is a minimum.
http://www.mfg.mtu.edu/cyberman/machtool/machtool/vibration/absorb.html
9
VIBRATION ABSORBING
We are trying to minimize the amplitude of vibration of mass m .
Mass m is subjected to a harmonic force excitation.
Mass ma is subjected to harmonic base excitation by mass m.
Inman p. 410
10
VIBRATION ABSORBING
Using the method of undetermined coefficients we guess solution in the form:
Amplitude of vibration of mass m:
Amplitude of vibration of mass ma
Inman p. 411
11
VIBRATION ABSORBING
Inman p. 411
12
VIBRATION ABSORBING
Amplitude of vibration of mass m:
Amplitude of vibration of mass ma
Inman p. 411
13
VIBRATION ABSORBING
X can be made zero by making the natural frequency of absorber (before it is added to the mass
m) equal to the excitation frequency:
ka  ma 2  0
The motion of mass m is “absorbed” by
the motion of the ma
Then the amplitude of vibration of absorber Xa will be:
Xa  
F0
ka
14
VIBRATION ABSORBING
The addition of vibration absorber adds 1DOF to the system and shifts
the natural frequencies away from the excitation frequency. The lower of
the new system’s natural frequency is less than the natural frequency of
the primary system while the higher natural frequency is greater than the
natural frequency of the primary system.
15
VIBRATION ABSORBER EXERCISE
n 
4.68kg
k2 = 1872N/m
F = 100cos20t
6.24kg
Absorber added
Configuration 01
Configuration 02
One natural frequency
Two natural frequencies:
1872
 17.3rad/s
6.24
Vibration absorber is tuned to
20rad/s by setting k1=k2 and
making mass equal to 4.68kg
k1 = 1872N/m
No absorber
n 
  20rad/s
F = 100cos20t
6.24kg
k1 = 1872N/m
1872
 20rad/s
4.68
ω1 = 12rad/s
ω2 = 30rad/s
Confirm this using
eigenvalue solver
vib_absorber.SLDASM
16
VIBRATION ABSORBER EXERCISE - PLAN
F = 100cos20t
F = 100cos20t
F = 100cos20t
Study 01
Study 02
Study 03
Time
response
Time
response
Frequency
response
17
VIBRATION ABSORBER EXERCISE - STUDY 01
Study 01
Maximum displacement amplitude is 340mm
Time
5s
Time step
0.025s
18
VIBRATION ABSORBER EXERCISE - STUDY 02
Note that the main
mass was supposed to
remain motionless but is
moving
Absorber
Main mass
Study 02
Displacement amplitude main mass is 50mm
Displacement amplitude of absorber is ~100mm
Time
5s
Time step
0.025s
19
VIBRATION ABSORBER EXERCISE - STUDY 03
Absorber
Main
mass
Results of frequency sweep between the two natural frequencies demonstrate that
main mass is stationary for excitation 20rad/s
20
Why the full effect of vibration absorbing could not be
demonstrated in time response study 02?
21
VIBRATION ISOLATION
TWO FORMS OF VIBRATION ISOLATION
Isolating a device from source of vibration:
isolating displacements transmitted through the base
Isolating the source of vibration from its surroundings:
reducing force transmitted by the source through its mounting points
23
VIBRATION ISOLATION
Isolation problem 1
We need to isolate the fan from surroundings so that no more than 19% of imbalance force is
transmitted to base. Fan has mass 200kg and operates at 1000RPM (104.7rad/s)
Damping can be ignored.
What is the maximum stiffness of an undamped support (isolator)?
Force transmitted to the base:
FT
1
1


F0
1 r2
(1  r 2 ) 2
1
 0.19
2
1 r
1
 0.19
r 2 1
r  2.5
104.7
n
No solution
This is required to have 81% isolation
 2.5
n  41.9
(k / m) 2  41.9 2
k  351200 N / m
So the support must be “soft enough”
24
VIBRATION ISOLATION
Isolation problem 2
What is the minimum static deflection of an undamped isolator to provide at least 75% isolation to a
pump that operates at speed between 1500 and 2000RPM (157-209.4 rad/s).
Isolation is better for higher speeds, therefore if we ensure 75% isolation at 1500RPM it will be only
better at 2000RPM
FT
1
1


F0
1 r2
(1  r 2 ) 2
1
 0.25
2
1 r
1
 0.25
r 2 1
r  2.24
n 

r
 70
No solution
Force transmitted to the base. Isolation (reduction of
force transmitted to base) occurs only for r > √2
Relation between natural frequency and operating
frequency ω = 157rad/s (1500RPM). This is the maximum
natural frequency if we want 75% isolation.
r
k  n m
2
d static 
mg
g
9.81


 0.002
2
2
4900
n m n
25
VIBRATION ISOLATION
Isolation problem 3
A 20kg lab experiment is to be mounted to a table that is bolted to the floor in a laboratory. Measurements indicate that
due to the operation of a nearby pump that runs at 2000RPM (209.4 rad/s) the table has steady-state displacement
0.25mm. What is the maximum stiffness of an undamped oscillator, placed between the experiment and the table such
that the experiment’s acceleration amplitude is less that 4m/s^2?
Transmissibility ratio:
Transmissibility ratio of acceleration is the same as
transmissibility ratio of displacement :
26