30079_23a.pdf

CHAPTER 23

VIBRATION AND SHOCK

Wayne T\istin

Equipment Reliability Institute

Santa Barbara, California

23.1 VIBRATION

661

23.5 SHOCK MEASUREMENT AND

ANALYSIS

692

23.2 ROTATIONAL IMBALANCE

668

23.6 SHOCKTESTING

695

673

23.3 VIBRATION MEASUREMENT

23.7 SHAKE TESTS FOR

ELECTRONIC ASSEMBLIES

23.4

ACCELERATION

MEASUREMENT

705

681

23.1 VIBRATION

In any structure or assembly, certain whole-body motions and certain deformations are more common

than others; the most likely (easiest to excite) motions will occur at certain natural frequencies. Certain

exciting or forcing frequencies may coincide with the natural frequencies (resonance) and give rel-

atively severe vibration responses.

We will now discuss the much-simplified system shown in Fig. 23.1. It includes a weight W (it

is technically preferred to use mass M here, but weight W is what people tend to think about), a

spring of stiffness K, and a viscous damper of damping constant C. K is usually called the spring

rate; a static force of K newtons will statically deflect the spring by 8 mm, so that spring length /

becomes d + L (In "English" units, a force of K Ib will statically deflect the spring by 1 in.) This

simplified system is constrained to just one motion—vertical translation of the mass. Such single-

degree-of-freedom (SDF) systems are not found in the real world, but the dynamic behavior of many

real systems approximate the behavior of SDF systems over small ranges of frequency.

Suppose that we pull weight W down a short distance further and then let it go. The system will

oscillate with W moving up-and-down at natural frequency f Nt expressed in cycles per second (cps)

or in hertz (Hz); this condition is called "free vibration." Let us here ignore the effect of the damper,

which acts like the "shock absorbers" or dampers on your automobile's suspension—using up vi-

bratory energy so that oscillations die out. f N may be calculated by

J_ [Kg

fN ~ 2ir V~^

(

}

It is often convenient to relate f N to the static deflection d due to the force caused by earth's gravity,

F=W = Mg, where g = 386 in./sec 2 = 9807 mm/sec 2 , opposed by spring stiffness K expressed

in either Ib/in. or N/mm. On the moon, both g and W would be considerably less (about one-sixth

as large as on earth). Yet f N will be the same. Classical texts show Eq. (23.1) as

f =- I^

277 \M

/-»

,/ TL*

In the "English" System:

s- F - w

S ~K'J

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

Fig. 23.1 Single-degree-of-freedom system.

Then

= JL /£ = -L I™

* N

27T^l 8

2TrV 5

( 23.2a)

= 19.7 = 3.13

~ 2TrVs "" V5

In the International System:

* = * = **

Then

= JL ^ = J- /^Z

/A r

2TrV^

277 V 8

(23.2b)

_ 99.1 _ 15.76

~ 27r\/6 ~ Vd

Relationships (23.2a) and (23.2b) often appear on specialized "vibration calculators." As increasingly

large mass is supported by a spring; 8 becomes larger and f N drops.

Let K = 1000 Ib/in. and vary W:

W (Ib)

S = ^ (in.)

f N (Hz)

0.001

0.000 001

3 130

0.01

0.000 Ol

990

0.1

0.000 1

313

0.001

10.

0.01

31.3

100.

0.1

9.9

1 000.

3.13

10 000.

10.

0.99

Let K = 1000 N/mm and vary M:

8 = 9 -^-

f N (Hz)

M (kg)

0.00102

10 nm

4 980

0.0102

100 nm

1 576

0.102

1 /mi

498

1.02

10 ptm

157.6

10.2

100 fjan.

49.8

102.

1 mm

15.76

1 020.

10 mm

4.98

102 000.

1 m

0.498

Note, from Eqs. (23.2a) and (23.2b), that f N depends on 6, and thus on both M and K (or W and

K). As long as both load and stiffness change proportionately, f N does not change.

The peak-to-peak or double displacement amplitude D will remain constant if there is no damping

to use up energy. The potential energy we put into the spring becomes zero each time the mass

passes through the original position and becomes maximum at each extreme. Kinetic energy becomes

maximum as the mass passes through zero (greatest velocity) and becomes zero at each extreme

(zero velocity). Without damping, energy is continually transferred back and forth from potential to

kinetic energy. But with damping, motion gradually decreases; energy is converted to heat. A vibration

pickup on the weight would give oscilloscope time history patterns like Fig. 23.2; more damping

was present for the lower pattern and motion decreased more rapidly.

Assume that the "support" at the top of Fig. 23.1 is vibrating with a constant D of, say, 1 in. Its

frequency may be varied. How much vibration will occur at weight W? The answer will depend on

1. The frequency of "input" vibration.

2. The natural frequency and damping of the system.

Let us assume that this system has an f N of 1 Hz while the forcing frequency is 0.1 Hz, one-tenth

the natural frequency, Fig. 23.3. We will find that weight W has about the same motion as does the

input, around 1 in D. Find this at the left edge of Fig. 23.3; transmissibility, the ratio of response

vibration divided by input vibration, is 1/1 = 1. As we increase the forcing frequency, we find that

the response increases. How much? It depends on the amount of damping in the system. Let us

assume that our system is lightly damped, that it has a ratio C/C 0 of 0.05 (ratio of actual damping

to "critical" damping is 0.05). When our forcing frequency reaches 1 Hz (exactly /^), weight W has

a response D of about 10 in., 10 times as great as the input D. At this "maximum response" frequency,

we have the condition of "resonance"; the exciting frequency is the same as the f N of the load. As

we further increase the forcing frequency (see Fig. 23.3), we find that response decreases. At 1,414

times f N , the response has dropped so that D is again 1 in. As we further increase the forcing

frequency, the response decreases further. At a forcing frequency of 2 Hz, the response D will be

about 0.3 in. and at 3 Hz it will be about 0.1 in.

Note that the abscissa of Fig. 23.3 is "normalized"; that is, the transmissibility values of the

preceding paragraph would be found for another system whose natural frequency is 10 Hz, when the

forcing frequency is, respectively, 1, 10, 14.14, 20, and 30 Hz. Note also that the vertical scale of

Fig. 23.3 can represent (in addition to ratios of motion) ratios of force, where force can be measured

in pounds or newtons.

The region above 1.414 times f N (where transmissibility is less than 1) is called the region of

"isolation." That is, weight W has less vibration than the input; it is isolated. This illustrates the use

of vibration isolators—rubber elements or springs that reduce the vibration input to delicate units in

aircraft, missiles, ships, and other vehicles, and on certain machines. We normally try to set f N (by

selecting isolators) considerably below the expected forcing frequency of vibration. Thus, if our

Fig. 23.2 Oscilloscope time history patterns of damped vibration.

Fig. 23.3 Transmissibility of a lightly damped system.

support vibrates at 50 Hz, we might select isolators whose K makes f N 25 Hz or less. According to

Fig. 23.3, we will have best isolation at 50 Hz if f N is as low as possible. (However, we should not

use too-soft isolators because of instabilities that could arise from too-large static deflections, and

because of need for excessive clearance to any nearby structures.)

Imagine a system with a weight supported by a spring whose stiffness K is sufficient that f N =

10 Hz. At an exciting frequency of 50 Hz, the frequency ratio will be 50/10 or 5, and we can read

transmissibility = 0.042 from Fig. 23.3. The weight would "feel" only 4.2% as much vibration as

if it were rigidly mounted to the support. We might also read the "isolation efficiency" as being

96%. However, as the source of 50-Hz vibration comes up to speed (passing slowly through 10 Hz),

the isolated item will "feel" about 10 times as much vibration as if it were rigidly attached, without

any isolators. Here is where damping is helpful: to limit the "g" or "mechanical buildup" at reso-

nance. Observe Fig. 23.4, plotted for several different values of damping. With little damping present,

there is much resonant magnification of the input vibration. With more damping, maximum trans-

missibility is not so high. For instance, when C/C C is 0.01, "g" is about 40. Even higher Q values

are found with certain structures having little damping; <2's over 1000 are sometimes found.

Most structures (ships, aircraft, missiles, etc.) have g's ranging from 10 to 40. Bonded rubber

vibration isolating systems often have <2's around 10; if additional damping is needed (to keep Q

lower), dashpots or rubbing elements may be used. Note that there is less buildup at resonance, but

that isolation is not as effective when damping is present.

Figure 23.1 shows guides or constraints that restrict the motion to up-and-down translation. A

single measurement on an SDF system will describe the arrangement of its parts at any instant.

Another SDF system is a wheel attached to a shaft. If the wheel is given an initial twist, that system

will also oscillate at a certain f N , which is determined by shaft stiffness and wheel inertia. This

imagined rotational system is an exact counterpart of the SDF system shown in Fig. 23.1.

If weight W in Fig. 23.1 did not have the guides shown, it would be possible for weight W to

move in five other motions—five additional degrees of freedom. Visualize the six possibilities:

Fig. 23.4 Transmissibility for several different values of damping.

Vertical translation.

North-south translation.

East-west translation.

Rotation about the vertical axis.

Rotation about the north-south axis.

Rotation about the east-west axis.

Now this solid body has six degrees of freedom—six measurements would be required in order to

describe the various whole-body motions that may be occurring.

Suppose now that the system of Fig. 23.1 were attached to another mass, which in turn is sup-

ported by another spring and damper, as shown in Fig. 23.5. The reader will recognize that this is

more typical of many actual systems than is Fig. 23.1. Machine tools, for example, are seldom

attached directly to bedrock, but rather to other structures that have their own vibration characteristics.

Weight W 2 will introduce six additional degrees of freedom, making a total of 12 for the system of

Fig. 23.5. That is, in order to describe all of the possible solid-body motions of W 1 and W 2 , it would

be necessary to consider all 12 motions and to describe the instantaneous positions of the two masses.

The reader can extend that reasoning to include additional masses, springs, and dampers, and

additional degrees of freedom—possible motions. Finally, consider a continuous beam or plate, where

mass, spring, and damping are distributed rather than being concentrated as in Fig. 23.1 and 23.5.

Now we have an infinite number of possible motions, depending on the exciting frequency, the

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