1

MODULE 09

Inman chapter 5

2Amplification of displacement amplitude (transmissibility of displacement amplitude)

DISPLACEMENT TRANSMISSIBILITY FROM BASE TO MASS

IN BASE EXCITATION

Relative / base

Absolute / base

X

Z

r

3

FORCE TRANSMISSIBILITY FROM BASE TO MASS

IN BASE EXCITATION

4

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

Transmissibility0

2

2 2 2

1 (2 )

(1 ) (2 )

TFTF

rT

r r

FORCE TRANSMISSIBILITY FROM MASS TO BASE

IN FORCE EXCITATION

5

2

2

1 2

n

rT < 1 when

T = Tmax when

2

2 2 2

1 (2 )

(1 ) (2 )

rT

r r

Force transmissibility to the base.

2r

Isolation (reduction of force transmitted to base) occurs only for 2r

r

FORCE TRANSMISSIBILITY FROM MASS TO BASE

IN FORCE EXCITATION

6

Inman p 398

Displacement transmissibility

Force transmitted to the mass

Force transmitted to the base

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

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

VIBRATION ABSORBING

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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

VIBRATION ABSORBING

11

Using the method of undetermined coefficients we guess solution in the form:

Inman p. 411

Amplitude of vibration of mass m:

Amplitude of vibration of mass ma

VIBRATION ABSORBING

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Inman p. 411

VIBRATION ABSORBING

13

Inman p. 411

Amplitude of vibration of mass m:

Amplitude of vibration of mass ma

VIBRATION ABSORBING

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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:

2 0a ak m

Then the amplitude of vibration of absorber Xa will be:

0aa

FX

k

The motion of mass m is “absorbed” by

the motion of the ma

VIBRATION ABSORBING

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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.

VIBRATION ABSORBING

16vib_absorber.SLDASM

k1 = 1872N/m

6.24kg 6.24kg

4.68kg

F = 100cos20t F = 100cos20t

No absorber

Configuration 01

VIBRATION ABSORBER EXERCISE

k1 = 1872N/m

Absorber added

Configuration 02

k2 = 1872N/m

187220rad/s

4.68

20rad/s

n

One natural frequency

187217.3rad/s

6.24 n

Vibration absorber is tuned to

20rad/s by setting k1=k2 and

making mass equal to 4.68kg

Two natural frequencies:

ω1 = 12rad/s

ω2 = 30rad/s

Confirm this using eigenvalue solver

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Study 01

Time

response

Study 02

Time

response

Study 03

Frequency

response

VIBRATION ABSORBER EXERCISE - PLAN

F = 100cos20tF = 100cos20t F = 100cos20t

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Study 01

Maximum displacement amplitude is 340mm

Time 5s

Time step 0.025s

VIBRATION ABSORBER EXERCISE - STUDY 01

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VIBRATION ABSORBER EXERCISE - STUDY 02

Absorber Main mass

Study 02

Displacement amplitude main mass is 50mm

Displacement amplitude of absorber is ~100mm

Time 5s

Time step 0.025s

Note that the main mass was supposed to remain motionless but is moving

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Main mass

Absorber

Results of frequency sweep between the two natural frequencies demonstrate that main mass is stationary for excitation 20rad/s

VIBRATION ABSORBER EXERCISE - STUDY 03

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

VIBRATION ISOLATION

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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

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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)?

22 20

2

2

2 2

1 1

1(1 )

10.19

11

0.191

2.5

104.72.5

41.9

( / ) 41.9

351200 /

T

n

n

F

F rr

r

rr

k m

k N m

VIBRATION ISOLATION

Force transmitted to the base:

This is required to have 81% isolation

So the support must be “soft enough”

No solution

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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

22 20

2

2

2

2 2

1 1

1(1 )

10.25

11

0.251

2.24

70

9.810.002

4900

T

n

n

static

n n

F

F rr

r

rr

r

k m

mg gd

m

Force transmitted to the base. Isolation (reduction of force transmitted to base) occurs only for r > √2

VIBRATION ISOLATION

No solution

rRelation between natural frequency and operating frequency ω = 157rad/s (1500RPM). This is the maximum natural frequency if we want 75% isolation.

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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 of acceleration is the same as transmissibility ratio of displacement :

Transmissibility ratio:

VIBRATION ISOLATION