Tài liệu Handbook of Machine Design P45 doc

CHAPTER 38

VIBRATION AND CONTROL

OF VIBRATION

T. S. Sankar, Ph.D., Eng.

Professor and Chairman

Department of Mechanical Engineering

Concordia University

Montreal, Quebec, Canada

R. B. Bhat, Ph.D.

Associate Professor

Department of Mechanical Engineering

Concordia University

Montreal, Quebec, Canada

38.1 INTRODUCTION / 38.1

38.2 SINGLE-DEGREE-OF-FREEDOM SYSTEMS / 38.1

38.3 SYSTEMS WITH SEVERAL DEGREES OF FREEDOM /38.19

38.4 VIBRATION ISOLATION / 38.28

REFERENCES / 38.30

38.1 INTRODUCTION

Vibration analysis and control of vibrations are important and integral aspects of

every machine design procedure. Establishing an appropriate mathematical model,

its analysis, interpretation of the solutions, and incorporation of these results in the

design, testing, evaluation, maintenance, and troubleshooting require a sound under￾standing of the principles of vibration. All the essential materials dealing with vari￾ous aspects of machine vibrations are presented here in a form suitable for most

design applications. Readers are encouraged to consult the references for more

details.

38.2 SINGLE-DEGREE-OF-FREEDOMSYSTEMS

38.2.1 Free Vibration

A single-degree-of-freedom system is shown in Fig. 38.1. It consists of a mass m con￾strained by a spring of stiffness k, and a damper with viscous damping coefficient c.

The stiffness coefficient k is defined as the spring force per unit deflection. The coef-

FIGURE 38.1 Representation of a single-degree￾of-freedom system.

ficient of viscous damping c is the force provided by the damper opposing the

motion per unit velocity.

If the mass is given an initial displacement, it will start vibrating about its equi￾librium position. The equation of motion is given by

mx + cjc + kx = O (38.1)

where x is measured from the equilibrium position and dots above variables repre￾sent differentiation with respect to time. By substituting a solution of the form x = e

81

into Eq. (38.1), the characteristic equation is obtained:

ms

2

+ cs + k = Q (38.2)

The two roots of the characteristic equation are

S = ^tZCOn(I-CT2

(38.3)

where O)n = (klm)m

is undamped natural frequency

£ = clcc is damping ratio

cc = 2/TtCQn is critical damping coefficient

«=v-i

Depending on the value of £, four cases arise.

Undamped System (£= O). In this case, the two roots of the characteristic equation

are

s = ±mn = ±i(klm)m

(38.4)

and the corresponding solution is

x = A cos GV + B sin GV (38.5)

where A and B are arbitrary constants depending on the initial conditions of the

motion. If the initial displacement is Jt0 and the initial velocity is V0, by substituting

these values in Eq. (38.5) it is possible to solve for constants A and B. Accordingly,

the solution is

VQ

x = Jt0 cos GV + — sin GV (38.6)

03«

Here, G)n is the natural frequency of the system in radians per second (rad/s), which

is the frequency at which the system executes free vibrations. The natural frequency

is

/.=£ (38.7)

where fn is in cycles per second, or hertz (Hz). The period for one oscillation is

T=fn|

CO = -

n

W

The solution given in Eq. (38.6) can also be expressed in the form

jt = ^cos(con-6) (38.9)

where

X= \xl+( — }2

]

112 9 = tan-1

^- (38.10)

L \ con / J COn^0

The motion is harmonic with a phase angle 0 as given in Eq. (38.9) and is shown

graphically in Fig. 38.4.

UnderdampedSystem (O <£< 1). When the system damping is less than the criti￾cal damping, the solution is

x = [exp(-^GV)] (A cos GV + B sin GV) (38.11)

where

co, = con(l-C2

)"2

(38.12)

is the damped natural frequency and A and B are arbitrary constants to be deter￾mined from the initial conditions. For an initial amplitude of Jt0 and initial velocity V0,

x = [exp (-CcOnOl I *o cos co/ + —— sin GV (38.13)

V co, /

which can be written in the form

x = [exp (-^OV)] X cos (co/ - 6)

x L-^ftm

'*o+v°Yr

(3814)

^T0+I co, JJ

and

e.ta^itetZo

CO,

An underdamped system will execute exponentially decaying oscillations, as shown

graphically in Fig. 38.2.