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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 . 5.1 Acceptable levels of vibration.

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Chapter 5 design l.jpg
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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5 1 acceptable levels of vibration l.jpg
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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Example 5 1 2 dissimilar devices with the same frequency l.jpg
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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But response magnitudes different l.jpg
But: response magnitudes different

  • Magnitude plot will have the same shape

  • Time responses will have the same form for similar (but scaled) disturbancesBUT WITH DIFFERENT MAGNITUDES

different

Fig 5.3

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Section 5 2 isolation l.jpg

Section 5.2 Isolation

  • 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

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

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Recall from section 2 4 that the fbd of sdof for base excitation is l.jpg

m

m

Recall from Section 2.4 that the FBD of SDOF for base excitation is

x(t)

k

c

y(t)

base

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Sdof base excitation assumes the input motion at the base has the form l.jpg
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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Particular solutions sine input l.jpg
Particular Solutions (sine input) has the form

With a sine for the forcing function,

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Particular solutions cosine input l.jpg
Particular Solutions (cosine input) has the form

With a cosine for the forcing function, we showed

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Magnitude x y l.jpg
Magnitude X/Y has the form

Magnitude of the full particular solution

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The magnitude plot of x y l.jpg

40 has the form

z

=0.01

z

=0.1

30

z

=0.3

z

=0.7

20

X/Y (dB)

10

0

-10

-20

0

0.5

1

1.5

2

2.5

3

Frequency ratio r

The magnitude plot of X/Y

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Notes on displacement transmissibility l.jpg
Notes on Displacement Transmissibility has the form

  • 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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It is also important to look at the force transmissibility l.jpg

m has the form

It is also important to look at the Force Transmissibility:

x(t)

FT

k

c

y(t)

base

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Plot of force transmissibility l.jpg

40 has the form

z

=0.01

z

=0.1

30

z

=0.3

z

=0.7

20

F/kY (dB)

10

0

-10

-20

0

0.5

1

1.5

2

2.5

3

Frequency ratio r

Plot of Force Transmissibility

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Isolation is a sdof concept l.jpg
Isolation is a sdof concept; has the form

  • 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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For displacement transmissibility isolation occurs as a function of stiffness l.jpg
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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Example 5 2 1 design an isolation mount l.jpg
Example 5.2.1 Design an isolation mount function of stiffness

Fig 5.6

  • Design an isolator (chose k, c) to hold a 3 kg electronics module to less then 0.005 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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T r plot for moving base displacement l.jpg

z function of stiffness

r

1.73

1.74

1.76

1.84

2.35

4.41

0.01

0.05

0.1

0.2

0.5

1.2

T.R. Plot for moving base displacement

1.5

For T.R. =0.5

z

1

z

=0.01

T.R.

z

=0.05

z

=0.1

0.5

z

=0.2

z

=0.5

z

=1.2

0

0

0.5

1

1.5

2

2.5

3

Frequency ratio r

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From the plot note that l.jpg
From the plot, note that function of stiffness

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Rattle space l.jpg
Rattle Space function of stiffness

Choice of k and c must also be reasonable

As must force transmitted:

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The transmitted force is l.jpg
The transmitted force is function of stiffness

Transmitted force, T.R., static deflection,

damping and stiffness values must all be

reasonable for the application.

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Shock isolation l.jpg
Shock Isolation function of stiffness

Shock pulse

Pulse duration

Increased isolation with increasing k

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Figure 5 8 shock response l.jpg
Figure 5.8 Shock Response function of stiffness

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Shock versus vibration isolation l.jpg
Shock versus Vibration Isolation function of stiffness

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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Example 5 2 3 design a system that is good for both shock and vibration isolation l.jpg
Example 5.2.3 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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Simulation of the response to the shock input for all three mounts choices l.jpg
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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Now consider the vibration isolation by plotting shock isolator design s transmissibility l.jpg
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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5 3 vibration absorbers l.jpg
5.3 Vibration Absorbers isolator design’s transmissibility:

  • 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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Absorber concept l.jpg
Absorber concept isolator design’s transmissibility:

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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The equations of motion become l.jpg
The equations of motion become: isolator design’s transmissibility:

To solve assume a harmonic displacement:

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Slide31 l.jpg

The form of the response magnitude suggests a isolator design’s transmissibility:

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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Slide32 l.jpg

Choose the absorber mass and stiffness from: isolator design’s transmissibility:

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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Other pitfalls in absorber design l.jpg
Other pitfalls in absorber design isolator design’s transmissibility:

  • 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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Avoiding resonance robustness l.jpg
Avoiding resonance (robustness) isolator design’s transmissibility:

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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Define a dimensionless amplitude of the primary mass l.jpg
Define a dimensionless amplitude of the primary mass isolator design’s transmissibility:

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Normalized magnitude of primary l.jpg
Normalized Magnitude of Primary isolator design’s transmissibility:

Fig 5.15

Absorber

zone

w/w2

w/w1

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Robustness to driving frequency shifts l.jpg
Robustness to driving frequency shifts isolator design’s transmissibility:

  • 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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Mass ratio versus frequency l.jpg
Mass ratio versus frequency isolator design’s transmissibility:

  • 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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Normalized magnitude of the primary mass with and without the absorber l.jpg

60 isolator design’s transmissibility:

No vibration absorber

With vibration absorber

40

20

Amplitude (Xk/F)

0

-20

-40

0

0.5

1

1.5

2

Frequency ratio (ra)

Normalized Magnitude of the primary mass with and without the absorber

m=0.1 and b=0.71

  • 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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What happens to the mass of the vibration absorber l.jpg

60 isolator design’s transmissibility:

40

20

Amplitude (Xak/F)

0

-20

-40

0

0.5

1

1.5

2

Frequency ratio (ra)

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!!

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Example 5 3 1 design an absorber given f 0 13 n m 73 16 kg k 2600 n m w 180 cpm x a 0 02 m l.jpg
Example 5.3.1: design an absorber given isolator design’s transmissibility:F0= 13 Nm=73.16 kg, k=2600 N/m, w = 180 cpm, xa< 0.02 m

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Example 5 3 2 compute the bandwidth of the absorber design in 5 3 1 l.jpg
Example 5.3.2 Compute the bandwidth of the absorber design in 5.3.1

3 roots

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Slide43 l.jpg

Comparing these 3 roots to the plot yields that: in 5.3.1

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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5 4 damped absorber system l.jpg
5.4 Damped absorber system in 5.3.1

Undamped primary

Cannot be zero!

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Slide45 l.jpg

Magnitude of primary mass for 3 levels of damping in 5.3.1

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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Effect of damping on performance l.jpg

50 in 5.3.1

No VA

40

ca=0.01

30

ca=0.1

ca=1

20

10

Amplitude (Xk/F)

0

-10

-20

-30

0

0.5

1

1.5

2

Frequency ratio (ra)

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

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Three parameters effect making the amplitude small l.jpg
Three parameters effect making the amplitude small in 5.3.1

  • 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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With damping in the absorber l.jpg
With damping in the absorber: in 5.3.1

  • 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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Viscous vibration absorber l.jpg
Viscous Vibration Absorber in 5.3.1

  • 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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Houdaille damper l.jpg
Houdaille in 5.3.1Damper

Fig 5.23

Equation of motion:

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Frequency response of primary mass l.jpg
Frequency response of primary mass in 5.3.1

Figure 5.24

Design on m and z

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