1. MODULE 09
Inman chapter 5

2. DISPLACEMENT TRANSMISSIBILITY FROM BASE TO MASS
IN BASE EXCITATION Z X r
Relative / base
Absolute / base
Amplification of displacement amplitude (transmissibility of displacement amplitude)

3. FORCE TRANSMISSIBILITY FROM BASE TO MASS
IN BASE EXCITATION

4. 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.
FT - amplitude of force transmitted to ground

5. FORCE TRANSMISSIBILITY FROM MASS TO BASE IN FORCE EXCITATION
T < 1 when
T = Tmax when r
Force transmissibility to the base.
Isolation (reduction of force transmitted to base) occurs only for

6. Inman p 398 Displacement transmissibility
Force transmitted to the base
Force transmitted to the mass
Inman p 398

7. VIBRATION ABSORBING

8. 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

9. 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.

10. 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

11. 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

12. VIBRATION ABSORBING
Inman p. 411

13. VIBRATION ABSORBING Amplitude of vibration of mass m:
Amplitude of vibration of mass ma
Inman p. 411

14. 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:
The motion of mass m is “absorbed” by the motion of the ma
Then the amplitude of vibration of absorber Xa will be:

15. 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.

16. VIBRATION ABSORBER EXERCISE
4.68kg
k2 = 1872N/m
F = 100cos20t
F = 100cos20t
Vibration absorber is tuned to 20rad/s by setting k1=k2 and making mass equal to 4.68kg
6.24kg
6.24kg
k1 = 1872N/m
k1 = 1872N/m
No absorber
Configuration 01
Absorber added
Configuration 02
Two natural frequencies:
ω1 = 12rad/s
ω2 = 30rad/s
Confirm this using eigenvalue solver
vib_absorber.SLDASM

17. VIBRATION ABSORBER EXERCISE - PLAN
F = 100cos20t
F = 100cos20t
F = 100cos20t
Study 01
Time response
Study 02
Time response
Study 03
Frequency response

18. VIBRATION ABSORBER EXERCISE - STUDY 01
Maximum displacement amplitude is 340mm
Time 5s
Time step s

19. 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 s

20. VIBRATION ABSORBER EXERCISE - STUDY 03
Main mass
Results of frequency sweep between the two natural frequencies demonstrate that main mass is stationary for excitation 20rad/s

21. Why the full effect of vibration absorbing could not be demonstrated in time response study 02?

22. VIBRATION ISOLATION

23. 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

24. 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:
No solution
This is required to have 81% isolation
So the support must be “soft enough”

25. 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 ( rad/s).
Isolation is better for higher speeds, therefore if we ensure 75% isolation at 1500RPM it will be only better at 2000RPM
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

26. 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 :