Harris C.M., Piersol A.G. Harris Shock and vibration handbook

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The amplitude of the oscillating force F

in the X direction and the amplitude of

the oscillating moment M

about the Y axis transmitted to the foundation by the

combination of resilient supports are

= 4k

c 0

− a

)

= 4k

(3.69)

where F

and M

have the same meaning as in Eqs. (3.65). Low vibration transmis-

sion of force and moment to the foundation is decreased at the higher frequencies in

a manner similar to that shown in Fig. 3.29.

FOUNDATION-INDUCED VELOCITY SHOCK

A velocity shock is an instantaneous change in the velocity of one portion of a sys-

tem relative to another portion. In this section, the system is a rigid body supported

by orthogonal resilient elements within a rigid container; the container experiences

a velocity shock.The system has one plane of symmetry; modal coupling and natural

frequencies for such a system are considered above. Two types of velocity shock are

analyzed: (1) a sudden change in the translational velocity of the container and (2) a

sudden change in the rotational velocity of the container. In both instances the

change in velocity is from an initial velocity to zero. No forces or moments are

VIBRATION OF A RESILIENTLY SUPPORTED RIGID BODY 3.51

FIGURE 3.29 Force and moment amplitudes transmitted to the foundation for the system shown

in Fig. 3.27 when excited by a rotating force F

acting about the Y axis. The parameters of the system

are k

= 1.16, a

/ρ

=±1.32, a

/ρ

=−1.62. The amplitudes of the oscillating forces in the X and Z

directions transmitted to the foundation are F

and F

, respectively. The amplitude of the total oscil-

lating moment about the Y axis transmitted to the foundation is M

8434_Harris_03_b.qxd 09/20/2001 11:32 AM Page 3.51

applied directly to the resiliently supported body; i.e., only the forces transmitted by

the resilient supports act. Thus, in the equations of motion, Eqs. (3.31):

= F

= M

= 0 (3.70)

The modal coupling and natural frequencies for this system have been deter-

mined when the YZ plane is a plane of symmetry and the conditions of symmetry

of Eqs. (3.33) to (3.35) apply. It is assumed that the velocity components of the

body ( ˙x

,˙y

,˙z

, ˙α,

β,˙γ) and the velocity components of the supporting container

(˙u, ˙v, ˙w, ˙a,

b,˙g) are respectively equal at time t < 0.At t = 0, all velocity components

of the supporting container are brought to zero instantaneously. To determine the

subsequent motion of the resiliently supported body, the natural frequencies f

the coupled modes of response are first calculated using Eq. (3.36). Then the

response motion of the resiliently supported body to the two types of velocity

shock can be found by the analyses which follow.

One Plane of Symmetry with Orthogonal Resilient Supports Excited by a

Translational Velocity Shock. Figure 3.30 shows a rigid body supported within a

rigid container by resilient supports in such a manner that the YZ plane is a plane of

symmetry. The entire system moves with constant velocity ˙v

and without relative

motion.At time t = 0, the container impacts inelastically against the rigid wall shown

at the right.The following initial conditions of displacement and velocity apply at the

instant of impact (t = 0):

˙y

(0) = ˙v

(0) = y

(0) = z

(0) =α(0) =β(0) =γ(0) = 0 (3.71)

˙x

(0) = ˙z

(0) = ˙α(0) =

β(0) = ˙γ(0) = 0

As a result of the impact, the velocity of the supported rigid body tends to continue

and is responsible for excitation of the system in the coupled mode of the y

, α

motions. The maximum displacements of the center-of-gravity of the supported

body are



n = 1





n = 1



(3.72)



n = 1



The maximum accelerations of the center-of-gravity of the supported body are



n = 1







n = 1





(3.73)



n = 1







¨α



2πf

˙v

/ρ



¨z



2πf

˙v



2πf

˙v





2π˙v

/ρ



2π

˙v



2π

˙v

3.52 CHAPTER THREE

8434_Harris_03_b.qxd 09/20/2001 11:32 AM Page 3.52

where the subscript m denotes maximum value and



−







−





(3.74)

= M

n + 1

n + 2

− M

n + 2

n + 1

B =





n = 1

n + 1

− N

n + 2

)



The fictitious natural frequency f

is defined for mathematical purposes by Eq.

(3.37).The numerical values of the subscript numbers n, n + 1, n + 2 denote the three

natural frequencies in the coupled mode of the y

, α motions determined from

Eq. (3.36). These natural frequencies are arbitrarily assigned the values n = 1, 2, 3.

When n + 1 or n + 2 equals 4, use 1 instead; when n + 2 equals 5, use 2 instead. Max-

imum displacements and accelerations may be calculated for other points in the sup-

ported rigid body by using Eqs. (3.1) except that each of the terms must be made

numerically additive. For example, the maximum value of the y displacement at the

point b having the Z coordinate b

= y

+ |b

|α

(3.75)

since γ=0.

Σk



Σk



Σk



Σk



Σk



Σk



Σk



1 − (f

)

VIBRATION OF A RESILIENTLY SUPPORTED RIGID BODY 3.53

FIGURE 3.30 Example of a rigid body supported within a rigid container by resilient elements

with YZ a plane of symmetry. Excitation is by a translational velocity shock in the Y direction. Prior

to impact the entire system moves with constant velocity ˙v

and without relative motion. The rigid

container impacts inelastically against the wall shown at the right, and y

, α motions of the inter-

nally supported body result, as described mathematically by Eqs. (3.72) and (3.73).

8434_Harris_03_b.qxd 09/20/2001 11:32 AM Page 3.53

Since the system is assumed undamped, the response of the suspended body in

terms of displacement or acceleration consists of a superposition of three sinusoidal

components at the three natural frequencies in the coupled y

, α mode. The

absolute values of terms appear in Eq. (3.75) because the maximum response is the

sum of the amplitudes of the three component vibrations which make up the over-all

response. In general, the maximum response occurs when the three component vibra-

tions reach their maximum positive or negative values at the same instant. Thus, the

maximum values of response apply both in positive and negative directions.

One Plane of Symmetry with Orthogonal Resilient Supports Excited by a

Rotational Velocity Shock. Alternative to the type of impact illustrated in Fig.

3.30, the system may be excited by imparting a rotational velocity shock (e.g., by lift-

ing and dropping one end of the container), as illustrated in Fig. 3.31. It is assumed

that the container impacts inelastically. The system has the same form of symmetry

as that shown in Fig. 3.30, and only the y

, α modes are excited. The initial condi-

tions at the instant of impact (t = 0), based upon the angular velocity ˙a

of the rigid

container about point A in Fig. 3.31, are

˙y

(0) =−d

˙a

˙z

(0) = d

˙a

˙a(0) = ˙a

(0) = y

(0) = z

(0) =α(0) =β(0) =γ(0) = 0 (3.76)

˙x

(0) =

β(0) = ˙γ(0) = 0

Note that d

and d

are negative quantities. The initial conditions in Eqs. (3.76) are

based on the assumption that motion of the rigid body relative to the container dur-

ing the fall is negligible compared to that which occurs after the impact. The maxi-

mum displacements of the center-of-gravity of the supported body are



n = 1



+ (N

n + 1

− N

n + 2

) + (M

n + 2

− M

n + 1

)





n = 1





+ (N

n + 1

− N

n + 2

) + (M

n + 2

− M

n + 1

 

(3.77)



n = 1





+ (N

n + 1

− N

n + 2

) + (M

n + 2

− M

n + 1

)

 

The maximum accelerations of the center-of-gravity of the supported body are



n = 1



+ (N

n + 1

− N

n + 2

) + (M

n + 2

− M

n + 1

)





n = 1





+ (N

n + 1

− N

n + 2

) + (M

n + 2

− M

n + 1

)







(3.78)



n = 1





+ (N

n + 1

− N

n + 2

) + (M

n + 2

− M

n + 1

)

 

where d

and d

are the Z and Y coordinates, respectively, of the edges of the con-

tainer, as shown in Fig. 3.31, and the other quantities are the same as those appear-



¨α



2πf

˙a



¨z



2πρ

˙a





2πρ

˙a



2π ˙a





2πρ

˙a





2πρ

˙a

3.54 CHAPTER THREE

8434_Harris_03_b.qxd 09/20/2001 11:32 AM Page 3.54

ing in Eqs. (3.72) and (3.74). The maximum response at any point in the suspended

body can be found in the manner of Eq. (3.75).

The rotational velocity ˙a

of the container about the corner A in Fig. 3.31 may be

induced by lifting the opposite end to a height h and dropping it.The resulting veloc-

ity ˙a

˙a



′+d

′



1 −



− d

′



1/2

(3.79)

where g is the acceleration of gravity, ρ

is the radius of gyration of the rigid body

plus container about the corner A, h is the initial elevation of the raised end of the

container, l is the length of the container, and d

′ and d

′ are the Y and Z coordinates,

respectively, of the edges of the container with respect to the center-of-gravity of the

assembly of rigid body plus container (see Fig. 3.31).

Example 3.3. The rigid body shown in Fig. 3.31 weighs 1,500 lb and has a radius

of gyration ρ

= 42 in. with respect to the X axis. The resilient supporting elements

apply forces parallel to their longitudinal axes only. Each element with its longitudi-

nal axis in the X or Y direction has a stiffness of k

= k

= 500 lb/in. Each element

whose longitudinal axis extends in the Z direction has a stiffness k

= 1,000 lb/in.The

resilient elements are positioned as shown in Fig. 3.30, and l = 168 in., d

= d

′=

−84 in., d

= d

′=−21 in., ρ

= 308 in. The rotational velocity shock results from a

height of drop h = 36 in.

The fictitious natural frequency f

is obtained from Eq. (3.37), yielding f

7.22 Hz. From Eq. (3.36) or Fig. 3.14, the natural frequencies in the y

, α mode are

= 3.58 Hz, f

= 6.02 Hz, and f

= 9.75 Hz. From Eqs. (3.74), it is determined that

 0, M

= 11.7, M

=−15.3, N

=−0.1, N

= 7.1, N

= 25.1, A

= 402, A

= 2, A

= 1,

B = 405. Sample calculations for M

and A

are



−





=−0.04

= M

− M

= (11.7)(25.1) − (−15.3)(7.1) = 402

4(1,000)(68 − 26)



4(500)(−10.5)

3.58



7.22

4(500)



8(1,000)



1 − (3.58/7.22)



VIBRATION OF A RESILIENTLY SUPPORTED RIGID BODY 3.55

FIGURE 3.31 System shown in Fig. 3.30 excited by a rotational velocity

shock about the X axis. The shock is induced by lifting and dropping one end

of the rigid container to make inelastic impact with the foundation. If the

height of drop is h, the rotational velocity of the system about the corner A at

the instant of impact is given by Eq. (3.79). The response of the resiliently sup-

ported body is described mathematically by Eqs. (3.77) and (3.78).

8434_Harris_03_b.qxd 09/20/2001 11:32 AM Page 3.55

From Eq. (3.79), ˙a

= 0.38 rad/sec. Then Eqs. (3.78) give the maximum acceleration

of the center-of-gravity in the Y direction of the supported body as follows:



+ (N

− N

) + (M

− M

)



+ (N

− N

) + (M

− M

)



+ (N

− N

) + (M

− M

)



(402) + (7.1 − 25.1) + (−15.3 − 11.7)



(2) + (25.1 + 0.1) + (0 + 15.3)



(1) + (−0.1 − 7.1) + (11.7 − 0)



= 286 in./sec

= 0.74g

In a similar manner:

= 1,580 in./sec

= 4.09g

¨α

= 45.9 rad/sec

REFERENCES

1. Avallone, E. A., and T. Baumeister, III (eds.): “Marks’ Standard Handbook for Mechanical

Engineers,” 10th ed., The McGraw-Hill Companies, Inc., New York, 1996.

2. Housner, G. W., and D. E. Hudson: “Applied Mechanics-Dynamics,” 2d ed., D. Van Nos-

trand Company, Inc., Princeton, N.J., 1959.

3. Beer, F. P., and E. R. Johnston: “Vector Mechanics for Engineers: Dynamics,” 5th ed.,

McGraw-Hill Book Company, Inc., New York, 1988.

4. Boucher, R. W., D. A. Rich, H. L. Crane, and C. E. Matheny: NACA Tech. Note 3084, 1954.

5. Woodward, C. R.: “Handbook of Instructions for Experimentally Determining the

Moments and Products of Inertia of Aircraft by the Spring Oscillation Method,” WADC

Tech. Rept. 55–415, June 1955.

6. Vane, F. F.: “A Guide for the Selection and Application of Resilient Mountings to Ship-

board Equipment—Revised,” David W. Taylor Model Basin Rept. 880, February 1958.

7. Smollen, L. E.: J.Acoust. Soc. Amer., 40:195 (1966).

8. Lewis, R. C., and K. Unholtz: Trans.ASME, 69:813 (1947).

9. Taylor, E. S., and K. A. Browne: J.Aeronaut. Sci., 6:43 (1938).

10. Browne, K.A.: Trans. SAE, 44:185 (1939).

11. Derby,T. F.:“Decoupling the Three Translational Modes from the Three Rotational Modes

of a Rigid Body Supported by Four Corner-Located Isolators,” Shock and Vibration Bull.,

43, pt. 4, June 1973, pp. 91–108.

9.75



7.22

−84



−21



6.02



7.22

−84



−21



724 in./sec



405

3.58



7.22

−84



−21



2πρ

˙a





3.56 CHAPTER THREE

8434_Harris_03_b.qxd 09/20/2001 11:32 AM Page 3.56

CHAPTER 4

NONLINEAR VIBRATION

Fredric Ehrich

H. Norman Abramson

INTRODUCTION

A vast body of scientific knowledge has been developed over a long period of time

devoted to a description of natural phenomena. In the field of mechanics, rapid

progress in the past two centuries has occurred, due in large measure to the ability

of investigators to represent physical laws in terms of rather simple equations. In

many cases the governing equations were not so simple; therefore, certain assump-

tions, more or less consistent with the physical situation, were employed to reduce

the equations to types more easily soluble. Thus, the process of linearization has

become an intrinsic part of the rational analysis of physical problems. An analysis

based on linearized equations, then, may be thought of as an analysis of a corre-

sponding but idealized problem.

In many instances the linear analysis is insufficient to describe the behavior of

the physical system adequately. In fact, one of the most fascinating features of a

study of nonlinear problems is the occurrence of new and totally unsuspected phe-

nomena; i.e., new in the sense that the phenomena are not predicted, or even hinted

at, by the linear theory. On the other hand, certain phenomena observed physically

are unexplainable except by giving due consideration to nonlinearities present in

the system.

The branch of mechanics that has been subjected to the most intensive attack

from the nonlinear viewpoint is the theory of vibration of mechanical and electrical

systems. Other branches of mechanics, such as incompressible and compressible

fluid flow, elasticity, plasticity, wave propagation, etc., also have been studied as non-

linear problems, but the greatest progress has been made in treating vibration of

nonlinear systems.The systems treated in this chapter are systems with a finite num-

ber of degrees-of-freedom which can be defined by a finite number of simultaneous

ordinary differential equations; on the other hand, the mechanics of continua

involves partial differential equations. Nonlinear ordinary differential equations are

easier to handle than nonlinear partial differential equations. An interesting survey

of the entire realm of nonlinear mechanics is given in Ref. 1.

This chapter provides information concerning features of nonlinear vibration

theory likely to be encountered in practice and methods of nonlinear vibration

analysis which find ready application.

4.1

8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.1

EXAMPLES OF SYSTEMS POSSESSING

NONLINEAR CHARACTERISTICS

SIMPLE PENDULUM

As a first example of a system possessing nonlinear characteristics, consider a simple

pendulum of length l having a bob of mass m, as shown in Fig. 4.1. The well-known

differential equation governing free vibration is

θ+mglθ=0 (4.1)

This equation holds only for small oscillations about the position of equilibrium since

the actual restoring moment is characterized by the quantity sin θ. Equation (4.1) thus

employs the assumption sin θ  θ. The exact, but nonlinear, equation of motion is

θ+mgl sin θ=0 (4.2)

SIMPLE SPRING-MASS SYSTEM

A simple spring-mass system, as shown in Fig. 4.2, is characterized by the equation

m ¨x + kx = 0

This equation is based on the assumption that the elastic spring obeys Hooke’s law;

i.e., the characteristic curve of restoring force versus displacement is a straight line.

However, many materials do not exhibit such a linear characteristic. Further, in the

case of a simple coil spring, a deviation from linearity occurs at large compression as

the coils begin to close up, or conversely, when the extension becomes so great that

the coils begin to lose their individual identity. In either case, the spring exhibits a

characteristic such that the restoring force increases more rapidly than the displace-

ment. Such a characteristic is called hardening. In a similar manner, certain systems

(e.g., a simple pendulum) exhibit a softening characteristic. Both types of character-

istic are shown in Fig. 4.3.A simple system with either softening or hardening restor-

ing force may be described approximately by an equation of the form

m ¨x + k(x ±µ

) = 0

where the upper sign refers to the hardening characteristic and the lower to the soft-

ening characteristic.

4.2 CHAPTER FOUR

FIGURE 4.1 Simple pendulum. FIGURE 4.2 Simple spring-mass system.

8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.2

It is possible for a system with only

linear components to exhibit nonlinear

characteristics, by snubber action for

example, as shown in Fig. 4.4. A system

undergoing vibration of small amplitude

also may exhibit nonlinear characteris-

tics; for example,in the pendulum shown

in Fig. 4.5, the length depends on the

amplitude.

STRETCHED STRING WITH

CONCENTRATED MASS

The large amplitude vibration of a

stretched string with a concentrated

mass, as shown in Fig. 4.6, offers another

example of a nonlinear system. The governing nonlinear differential equation is,

approximately,

NONLINEAR VIBRATION 4.3

FIGURE 4.3 Restoring force characteristic

curves for linear, hardening, and softening vibra-

tion systems.

FIGURE 4.4 Nonlinear mechanical system with snubber

action showing piecewise linear restoring force characteristic

curve.

FIGURE 4.5 Pendulum with nonlinear char-

acteristic resulting from dependence of length

on vibration amplitude.

FIGURE 4.6 Vibration of a weighted string as

an example of a nonlinear system.

8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.3

m¨w + F



w + (SE − F

)



= 0

where F

is the initial tension, S is the cross-sectional area, and E is the elastic mod-

ulus of the string. Consider now the special case of a = b and denote the unstretched

length of the half string by l

. Then the initial tension and the restoring force

become

= SE



 SE





− 1

 



2 −

 



An interesting feature of this system is that it exhibits a wide variety of either harden-

ing or softening characteristics, depending upon the value of a/l

, as shown in Fig. 4.7.

SYSTEM WITH VISCOUS

DAMPING

The foregoing examples all involve non-

linearities in the elastic components,

either as a result of appreciable ampli-

tudes of vibration or as a result of pecu-

liarities of the elastic element. Consider

a simple spring-mass system which also

includes a dashpot. The usual assump-

tions pertaining to this system are that

the spring is linear and that the motion is

sufficiently slow that the viscous resist-

ance provided by the dashpot is pro-

portional to the velocity; therefore, the

governing equation of motion is linear.

Frequently, the dashpot resistance is

more correctly expressed by a term proportional to the square of the velocity. Fur-

ther, the resistance is always such as to oppose the motion; therefore, the nonlinear

equation of motion may be written

m¨x + c|˙x|˙x + kx = 0

BELT FRICTION SYSTEM

The system shown in Fig. 4.8A involves a nonlinearity depending upon the dry fric-

tion between the mass and the moving belt.The belt has a constant speed v

, and the

applicable equation of motion is

m¨x + F(˙x) + kx = 0

where the friction force F(˙x) is shown in Fig. 4.8B. For large values of displacement,

the damping term is positive, has positive slope, and removes energy from the sys-



a − l



+ b



4.4 CHAPTER FOUR

FIGURE 4.7 Restoring force characteristics

for the weighted string shown in Fig. 4.6.

8434_Harris_04_b.qxd 09/20/2001 11:30 AM Page 4.4