Foundations for Machines-Analysis & Design_By Shamsher Prakash.pdf

Foundations for Machines

Foundations for Machines: Analysis and Design Shamsher Prakash University of Missq!)ri-Rolla

Vijay K. Puri Southern Illinois University, Carbondale

A Wiley-lnterscience Publication JOHN WILEY AND SONS

New York

Chichester

Brisbane

Toronto

Singapore

To our friend the enlightened saint, humble philosopher, and friend of all mankind who speaks the language of the heart; whose religion is Jove; who always aspires to fill lives of one and all with spiritual bliss.

Copyright

© 1988 by John Wiley

& Sons, Inc.

All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to ,., .:.t~~-·~,ermissions Department, John Wiley & Sons, Inc. {ibt-ary of'Congress Cataloging in Publication Data: .- ~. ,' · J?rakash~ Sham_s~~r. : Fo~ndations for machines. (Wiley series in geotechnical engineering) "A Wiley-Interscience publication." Includes bibliographies and index. 1. Machinery-Foundations. I. Pori, Vijay. II. Title. III. Series. TJ249.P618 1987 621.8 87-21678 ISBN 0-471-84686·4 Printed in the United State of America 10 9 8 7 6 5 4 3 2 1

Preface

The design of machine foundations involves a systematic application of the principles of soil eljgineering, soil dynarn,,i~s, and theory of vibrations-a fact that has been well recognised during tlie last three decades. Since the classical work by Lamb in 1904 and the paper on "Foundation Vibrations" by Richart in 1962, the subject of vibratory response of foundations has attracted the attention of several researchers. The state of art on the subject has since made significant strides. Methods are now available not only for computing the response of machine foundations resting on the surface of the elastic half space but also for embedded foundations and foundations on piles. Elastic half space analogs have further simplified the computation process and are a convenient tool for the designer. The linear spring approach of Barkan, which could previously be used only for surface footings, has also been extended to account for the embedment effects. Recent advances dealing with the determination of the dynamic soil prop- erties and rational interpretation of the test data are of direct application to the design of machine foundations. Information on several aspects of machine foundation design such as design of embedded foundations and pile supported machine foundations is either unavailable or only inadequately treated in the presently available texts. This text has been developed with the object of providing state-of-the-art ,,information on the analysis .,and design of machine foundations and is intended to cater to the interests of graduate students, senior undergraduates, and practicing engineers. Both authors have offered graduatelevel courses on the subject in the United States and India. They also organized many short courses for practicing engineers, including four by the senior author at University of Missouri, Rolla. The authors have also been engaged in the design and performance evaluation of machine foundations.

The feedback from the classroom and the professionals in the field has been of immense help in the planning and preparation of this text. vii

viii

PREFACE

The special features of this book are: (1) analysis of surface and embedded foundations by both the elastic half space method and the linear spring method; (2) analysis of pile supported machine foundations; (3) detailed discussion of the dynamic soil properties, methods for their determination, and evaluation of the test data; ( 4) detailed design procedure followed by examples; and (5) discussion of design of machine foundations on absorbers and vibration isolation. Knowledge of soil mechanics and elementary mathematics or mechanics is needed to follow the text. The reader is introduced to the problem of machine foundation and its special requirements in Chapter 1. In Chapter 2, the elementary theory of vibrations is discussed. Chapter 3 deals with the wave propagation in an elastic medium that provides an important basis for determination of dynamic soil properties as discussed in Chapter 4. Needless to say, soil properties play a critical role in the design of machine foundations. Chapter 4 thus forms a very important component of the text. Also included in this chapter is the procedure for rational selection of soil parameters for a given machine foundation problem. The determination of unbalanced forces and moments occasioned by the operation of a machine is reviewed in Chapter 5. The principal subject of the book, the analysis and design of machine foundations is introduced in Chapter 6, that deals with the design of rigid-block-type foundations for reciprocating machines. In this chapter the reader is made familiar with the concepts of elastic half space method and linear spring method for computing the vibratory response of surface footings. Foundations for impact-type machines such as hammers are discussed in Chapter 7. Foundations for high-speed rotary machines are discussed in Chapter 8 and for miscellaneous machines in Chapter 9. The principles of vibration isolation and absorption are considered in Chapter 10. The design of embedded block foundations for machines is described in Chapter 11 followed by pile supported machine foundations in Chapter 12. A few case histories are discussed in Chapter 13 and construction aspects in Chapter 14. Computer program for design of a block foundation based on principles discussed in Chapter 6 has been included in Appendix I, aud for design of a hammer foundation as in Chapter 7 has been included in Appendix II. A brief description of the commercially available programs PILAY for solution of piles and STRUDL for analysis of turbo-generator foundations is included in Appendix III. The subject matter has been developed in a logical progression from one chapter to the next. Every effort has been made to make the text selfcontained as far as possible. A comprehensive bibliography is included at the end of each chapter so that an interested reader may obtain additional information from published sources. Development in certain areas, particularly the determination of dynamic soil properties and analysis of embedded foundations and piles under

PREFACE

dynamic loads, is taking place at a very rapid rate. Analysis and design procedures may therefore undergo modifications. This fact has also been brought to the attention of the reader"' at appropriate places in the text. Thanks are due the American Society of Civil Engineers and National Research Council of Canada for permitting the use of materials from thefr publication. Acknowledgment to other copyrighted material is given at appropriate places in the text and figures. In preparing this text, several of our colleagues and graduate students have helped in a variety of ways. The authors wish to express their sincere thanks to them. Special mention must be made of Dr. Krishen Kumar, who read the entire manuscript and made useful suggestions, particularly on Chapter 12, and Dr. A Syed for his useful comments and suggestions and of Mr. Murat Hazinedarogulu for assistance in writing the computer programs. The manuscript was typed by Janet Pearson, Charlena Ousley, Allison Holdaway, and Mary Reynolds. The authors are most thankful to them for their care, painstaking efforts, and patience. John W. Koeing, technical editor at the University of Missouri, Rolla, provided editorial assistance and deserves our sincer~ !hanks. ->', en

N=O

. When r is equal to 1, resonance occurs, and the amplitude tends to be mfimte. Introduction of damping reduces the resonant amplitudes to finite values. . The phase angle is zero if r is less than 1; the displacement z is in phase With the exc1tmg force, F0 , and is equal to 1800 if r is greater than 1. Effect of Damping

The corresponding vector diagram is shown in Fig. 2.11. The solution given by Eq. (2.44a) is a steady-state solution, which is important in most practical problems. However, there are transient vibrations initially that correspond to the solution given by Eq. (2.35). These vibrations, of course, die out in the first few cycles.

2.6

t

As the dampi~g increases, the peak of the magnification factor shifts slightly to the left. This IS due to the fact that maximum amplitudes occur in damped VIbratiOns when the forcing frequency w equals the system's damped natural frequency, wnd [Eq. (2.34)], which is slightly smaller than the undamped natural frequency, wn.

FREQUENCY-DEPENDENT EXCITATION

In many probl~Iils of machine fouii'iflitions, the exciting force depends upon the machine's operating frequency. Figure 2.12 shows such a system mounted on elastic supports with m 0 representing the unbalanced mass placed at eccentricity e from the center of the rotating shaft. The unbalanced force is F = (m 0 ew 2 ) sin wt. Therefore, the equation of motion may be written as follows: d 2z (M- m 0 ) - 2 dt

d2 (z dt 2

dz dt

.

+m0 -

+ e sm wt) = - kz- c -

0

'

Ia I

wt z

Fo M

9 "" goo

cwZo

Figure 2.11. Vector diagram at resonance in a damped system under forced vibrations. (a) Displacement, velocity, and acceleration. (b) Forces.

(b)

=

Fo

~k/2 0!

!'

~k/2

0!

//,/#lW##/mrw####J#m/,7/ Figure 2.12.

Force of excitation due to rotating unbalance.

(2.48a)

30

THEORY OF VIBRATIONS

SYSTEMS UNDER TRANSIENT lOADS

By rearranging the terms in the preceding equation, we get

Mi + ci + kz

m 0 ew 2 sin wt

=

h 3.0

(2.48b)

z=

Y(k- Mw

2 2 )

.

+ (cw ) 2

sm wt

(2.49a)

0.25

z0 =

m 0 ew

'"'

:;:

" 0

2 2

+(cw)

2

ro

•ro

0.15

90" t---

1.0

h

~

~

~

~

0

1.0

2.0

3.0

4.0

5.0

Frequency ratio~

(b)

""

~

k-:::0.50

.o

(2.49b)

and 0

cw tan¢= k-Mw 2

0

tv

K, ~

M

~

0.375

2

v(k-Mw)

0.50

•

~~

01 .

0.05 0.25

~

0.

1L

Therefore, the maximum amplitude Z 0 is given by

1 o.~5 t-

2. 0 2

180"

t-0.10

In this system, M also includes m 0 . Equations (2.48b) and (2.37) are 2 similar, except that m 0 ew appears in Eq. (2.48b) in place of F in Eq. 0 (2.37). The solution of this equation may therefore be witten as, m 0 ew

31

~

/Y v

~ 1--\,1

I

~~~1.0

1.0

3.0

2.0

4.0

5. 0

-~}:ic}~quency ratio~

(2.50)

Wn

-t,.

(a)

In nondimensional form, these equations can be arranged as follows:

me

. mZ, Figure 2.13. Response of a system with rotating unbalance. (a) versus 1requency ratiO wlwn. (b) Phase angle tfJ versus frequency ratio wlwn. (After Thoms~n, 1972, p. 50. Reprinted with permission of PrenticeRHall, Englewood Cliffs, New Jersey.)

or

2.7 SYSTEMS UNDER TRANSIENT LOADS Zo

,z

moe! M

Y(1- r')' +Ag'rz

(2.51)

and 2gr tan 1> = - -2 1-r

(2.52)

The value of MZ0 1m 0 e and 1> are plotted in Figs. 2.13a and 2.13b, respectively. These curves are similar in shape to those in Fig. 2.10 except that the peak amplitudes occur at (2.53)

Transient loads may be caused by hammers, earthquakes, blasts, and the sudden dropping of weights. In several such cases, the maximum motion may occur within a relatively short time after the application of the force. For this reason, damping may be of secondary importance m transient loads. CASE I. SUDDENLY APPLIED LOAD

Consider a spring-mass system (Fig. 2.14a) that is subjected to suddenly applied force represented by the forcing function F = F0 (Fig. 2.14b). The equation of motion of mass, m, is given by

..,...

mi + kz

=

F0

(2.55)

The solution for displacement, z, is

and the value of the ordinate when r is equal to 1 in Eq. (2.51) is (2.56) Z0

1

m 0 el M

21;

(2.54)

Initially, if at t = 0, z and i are equal to zero, then A is equal to- Fofk,

THEORY OF VIBRATIONS

33

SYSTEMS UNDER TRANSIENT LOADS F(t)

F(t)

?fLk Sboo

Fof-------

Fof---,

0

'

0

(b)

(a)

Ia I

(b)

dT (c)

Figure 2.15.

Dynamic amplification due to a square pulse. (a) One degree of freedom system.

(b) Square pulse,~.Jc) Magnification. f," .•

lcl Figure 2.14. Dynamic amplification due to suddenly applied load. (a) Single degree of freedom system. (b) Suddenly applied load. (c) Magnification factor versus time.

Wben t is equal to r, Z

and B is equal to 0. Thus, Eq. (2.56) becomes

z=

Fa

k

(1- cos w"t)

z

=

F lk

=

1- COS wnt

i

(2.57)

(2.58)

0

Magnification N versus time is plotted in Fig. 2.14c. The magnification has a maxtmum value of 2, wh1ch occurs when cos w"t is equal to minus 1. The first peak ts reached when wnt is equal to 7T or tis equal to Tl2, in which Tis the natural period of vibration of the system. CASE II. SQUARE PULSE OF FINITE DURATION

= -Fo

k

(1

)

(2.59)

)

(2.60)

-COS W" T

and

If the force F0 is applied gradually, the static deflection is F 0 I k. Thus the magnification N of the deflection z is '

N

'

Co.nsider. the system, shown in Fig. 2.15a, that is s~bjected to a pulse of umform mtenstty F(t) for a given duration r (Fig. 2.15b). . When tis less than r, the motion is governed by Eq. (2.55). The solution IS given by Eq. (2.57).

When t is greater than

T,

7

=

F0 w" (Slll . -k-

(t)n

T

the equation of motion is (2.61)

mi+kz=O The solution for displacement z is

(2.62) in which t' = t - r. The values of A and B in Eq. (2.62) are determined from the initial conditions when tis eqOO.l to r. By equating z, and i, from Eq. (2.62), with those in Eqs. (2.59) and (2.60), respectively, we get A= (F,/k)(l- cos w"r) and B = (F0 1k) sin w" r. Therefore, Eq. (2.62) becomes Z =

or

kFo

(1

)

-COS Wn T COS Wnt

'+k Fo Sill .

.

W 11 T Sill Wn

t'

THEORY OF VIBRATIONS RAYLEIGH'S METHOD

(2.63a)

2.8

in which cjJ =tan -t

1- cos w11 r sin w 11 'T

_

(2.63b)

F = ; Y2(1- cos wn r) sin( wnt' - cf>)

or F0 z=k

:t

T)

w • 2smz sin(wnt'-q,)

(

RAYLEIGH'S METHOD

According to Rayleigh's method, the fundamental natural frequency, i.e., frequency in the first mode of vibrations, of a continuous elastic system with infinite degrees of freedom can be determined accurately by assuming a reasonable deflected curve for the elastic system. If the true deflected shape of the vibrating system is not known, the use of tbe static deflection curve of the elastic system gives a fairly accurate fundamental frequency. In illustrating the application of this method, the energy principle is used. Expressions are developed for the kinetic energy KE, and potential energy PE. Because the total energy is constant, the sum of KE and PE is constant. Thus

Therefore,

Z

(2.63c)

Fa k

maximum KE = maximum PE .

(z sm. 2wn T)-- 2 kF, sm. T

7TT

Zmax _

.

Wn

'T

.

(2.64a)

Solution Let the displacement of the mass from the equilibrium position be Yo and y = y 0 cos w,t. If the extension of the spring is assumed to be linear, the displacement of the element of the spring at a distance z from the fixed support is y = (z I L )y 0 cos wnt, and the velocity of element dz is y = -(zl L )wnYo sin w,t. The maximum KE of the element with mass (wig) dz is then d(KE)m, = (w/2g) dz ((z!L)wnYof

'TTT

(2.64b)

The maximum value of N is 2 when r I Tn is equal to ~. (Fig. 2.15c.) . Constd~r the bmttmg case in Eq. (2.64a), if r!Tn is very small so that sm 7TT I Tn ts approximately equal to 7TT 1T n> then

z

= 2F0 7TT m"

k

Tn

(2.66b)

EXAMPLE 2.8.1

Hence, the dynamic magnification N is N- F/k-2sm-=2smo 2 Tn

(2.66a)

In Fig. 2.16, the weight of the spring of length L is w per unit length. Determine the natural frequency of the spring-mass system.

n

_

(KE + PE) = 0

or

The maximum value of z is

(zm" ) =

35

(2.64c)

Now,_k is equal to mw;, and Tis equal to 27T/w"" By substituting these quanl!hes mto Eq. (2.64c), we obtain

I I

f Now, F0 r is equal to the impulse I. Therefore,

Figure 2.16.

System with spring having weight.

36

THEORY OF VIBRATIONS

37

RAYLEIGH'S METHOD

By integrating this expression, we obtain the maximum kinetic energy of the spring

(KE)m,

=

W

g 2

(L )21L WnYo

2

Z

0

dz

or Figure 2.17.

(2.67)

Fundamental frequency determination of cantilever beam.

The maximum kinetic energy of the rigid mass m is

Solution: d The deflected curve of the cantilever beam may be assumed to cor~espon to that of a weightless beam with the concentrated load P actmg at 1ts end. Then and the total maximum kinetic energy is 1 (W+~wL)

2

g

PL 3

(2.71)

Yo= 3EI 2

2

(2.68)

WnYo

The maximum potential energy of the spring can be computed as follows: Maximum potential energy =

P;

-;,;i.f

in which EI 'is the flexural stiffnesii~of the beam. The stiffness k of beam at the free end is

P 3El k=-=-,Yo L

LYo ky dy 1

(2.72)

The expression for the deflected shape of the cantilever is 2

= 2 kyo

(2.69)

(2. 73) In a conservative system, the maximum kinetic energy equals the maximum potential energy. By equating the values of the two energies, we obtain

and Maximum potential energy=

(2.70a)

3 EI

1

2 2 kyo= 2 L'

2

Yo

(2.74)

If the weight of the beam is w per unit length and if a harmonic motion is

assumed,

Therefore, the natural frequency wn is given by

Maximum KE of (2.70b)

The effect of the spring's weight can thus be accounted for by lumping one-third of its mass with the concentrated mass of the system.

system=~ LL (wnYl

~~

=

~

(

W;Yo )'

2

dx

r Hf)'- (f)T

_~ (33wL)w' 140g nYo

- 2

2

dx

(2.75)

EXAMPLE 2.8.2

By using Rayleigh's method, determine the fundamental frequency of the cantilever beam shown in Fig. 2.17.

By equating the two energies from Eqs. (2.74) and (2.75), we obtain the fundamental frequency of vibration of the cantliever beam as

THEORY OF VIBRATIONS

38

Wn

=

PEl -

L3

g /gEl ) = 3.56 \I ~-wL wL 4 140

( 33

(2.76a)

The exact solution is

(2.81b) Therefore,

Wn

2.9

/iEi

=3.515 \I~ wL

1

2.10

LOGARITHMIC DECREMENT

zl

/5 =log -

' z,

(2. 77)

in which z 1 and z 2 are two successive peak amplitudes. If z 1 and z 2 are determined at times t1 and (1 1 + 27T) from Eq. (2.35) and substituted into Eq. (2.77), we obtain

27Ti;

(2.82)

DETERMINATION OF VISCOUS DAMPING

Viscous damping may be determined from either a free-vibration or a forced-vibration test on a system. In a free-vibration test, the system is displaced from its position of equilibrium, and a record of the amplitude of displacement is made. Then, from Eqs. (2.77) and (2.80b) /5 1 z r

MULTIDEGREE FREEDOM SYSTEMS

55

g, uncouples the n degree of freedom system into n of single degree of freedom systems. The t;'s are termed as normal coordinates and this approach is known as normal mode method, i.e., the total solution is thus expressed as a sum of contribution of individual modes. For determination of f,(t), multiply both sides of Eq. (2.130) by A~') (=A)'lT, the transpose of A)'l) and summing up for all the masses, we get It is seen that the coordinate

n

(2.127)

n

"' L.J FAC'l I I

r=l

n

n

n

reel

i=l

="'LJ AC'l "'LJ mAc'lt, ="'L.J /,rL.i"' m.AC'l AC'l I

i=J

i=l

I

I

I

T

T"'l

l

I

(2.133)

Where r represents the rth mode. Then from Eq. (2.126), we get n

L r=l

n

mA(•)i: 1 1 br

Using the orthogonality relationship,

n

+"' L.J

"'k A(')c = F(t) L.J 1 Sr i IJ

r=t j=l

n

(2.128)

"' AC'l = 0 LJ m.AC'l I I l

for

r o;6 s

i=1

From Eq. (2.115), the right-hand side of Eq. (2.133) reduces to n

f,

L

i=l

m,(A)~);,

when

r=s

~·

Hence . Hence n

n

L r=

2 mA(') < +"' t = F(t) 1 1 ~r L-1 w nr mAC') t 1 Sr 1

1

(2.134)

'"' 1

and Using Eqs. (2.132) and (2.134), the complete solution from Eq. (2.127)

n

L

r=I

m,A~')( {, + w;,t;,) = F,(t)

is (2.129)

n

F,(t) =

L

m,A~') f,(t)

"

1

X,--~ A, -;;; -

Since the left-hand side is a summation involving different modes of vibration, the right-hand side should also be expressed as a summation of equivalent force contribution in the corresponding modes. Let F, be expanded for convenience as

r-1

(r)

r

l' ""

L.. j=

0

l::"

1

Fj (T )AC'l j • (A('l)' sm w.,(t- r) dr

J=lm;

1

EXAMPLE 2.14.1

Figure 2.24 shows a three degrees of freedom system. Determine the stiffness matrix.

(2.130)

r=l

in whichf,(t), the modal force is derived subsequently as Eq. (2.134). Then from Eqs. (2.129) and (2.130), where

r = 1, 2, ... , n

(2.131)

This is a single degree of freedom equation and its solution can be written as

t;, = - 1wnr

l' 0

J,(r) sin w,(t- r) dr

where 0 < r < t

(2.132)

Figure 2.24.

Computation of stiffness matrix.

56

THEORY OF VIBRATIONS

Solution Let x 1 = 1.0 and x 2 = x 3 = 0. The forces required at 1, 2, and 3, considering forces to the left as positive, are

VERTICAL DISPLACEMENT

0

ft = kl + k, = k11 / 2

=

-k 2

[, = 0 =

=

57

MUlTIDEGREE FREEDOM SYSTEMS

+

-

0.445

0

+

1.247

k 21

k,l 0.802

Repeat with x 2 = 1, and x 1 = x 3 = 0. The forces are now

!1

=

-k,

=

k1z

[, = k, + k, = k,,

1.0

[,=-k,=k,,

First mode

Repeat with x 3 = 1, and x 1 = x 2 = 0. The forces are

Ia I

!I= 0 = k13 h = -k, = k,,

+ k,)

[

-k,

(k 2 + k 3 )

k2

-

-k 3

0

-k 0 +

(k,

k.J

l

EXAMPLE 2.14.2

For the system shown in Fig. 2.25a, solve for natural frequencies and mode shapes.

kl2

k21 = -k,

k,, = 2k,

k,, = -k

k,, = -k,

k,,

= 0,

k13 =0 d," k

The equations of motion of the system, from Eq. (2.113) are m 1i

1

+ 2kx 1 -

(2k-m 1 w;) -k 0

0

-k (2k- m 2 w!)

-k

-k

(k- m,w;)

=0

The frequency equation can be obtained by expanding the determinant. Letting m 1 = m 2 = m 3 = m (for simplicity),

(2k- mw!)[(2k- mw;)(k- mw!)- (- k)( -k)] - ( -k)[(- k)(k- mw;)- ( -k)(O)J + (0)[( -k)( -k)- (2k- mw;)(O)] = 0

- A' 3 + 5A' 2

k 11 =2k,

k31

(b)

Example 2.14.2 (b) Mode shapes.

which, on simplification, gives

Solution The stiffness coefficients K,i for the system are = ~k,

sylt~dls for 1

The stiffness matrix can now be written as (kl

(a) Spring-mass

The corresponding frequency determinant from Eq. (2.116) is

[, = k, + k, = k,,

K =

Figur-?~:25.

Third mode

Second mode

kx 2 = 0

m 2i 2 - kx 1 + 2kx 2 - kx 3 = 0

-

6A' + 1 = 0

in which A' = mw ;; k (A' will be equal to 1.0 for a single degree of freedom). The roots of frequencfequation can be determined by any of the standard techniques. In this case, the trial-and-error method and the plotting of the graph of the function will be used. The roots are A;= 0.198, A;= 1.555, A;= 3.247, and since A'= mw;lk, 2 Will=

k 0.198 m '

2

w 112 =

1.sss

k m

(I)

2 n3

= 3.247

k m

The amplitude coefficients for the three modes of vibration can be obtained from Eq. (2.125), from which we get

58

THEORY OF VIBRATIONS

1

1

- -w-; A 1 + m k

A1+ m

1

k

A2 + m

1

k

A3 =0

PRACTICE PROBLEMS

59

2.3 For the system represented by Eq. (2.48b), show that the peak amplitude occurs at a frequency ratio of

1 1 2 2 --,A,+m-kA 1 +m-A +m-A =0 w, k 2 k 3

1

r = -Y-,=1,;;_=2t;""'

1 1 2 3 - - , A 3 +m k A, +m k A 2 +m -A 3 =0 k

(f)n

and

-

z

Letting A'= (mlk)w;,, the preceding equations can be rewritten as (A' -1)A 1 + A'A 2 + A'A 3 = 0

A'A 1 +(2A'-l)A 2 +2A'A 3 =0 A'A 1 + 2A'A 2 + (3A' -l)A 3 = 0 It is more convenient to work with particular numerical values rather than with ratios. Therefore, let us arbitrarily set A 3 (1) = A 3 (2) = A (3) = 1. In 3 th1s manner, we will obtain two simultaneous equations for A and A from 1 2 above:

(A' -1)A 1 + A'A 2 =-A' A'A 1 + (2A' -1)A 2 = -2A' Substituting numerical values of A' determined above, we get

First mode Second mode Thrid mode

AI 0.445 -1.247 1.802

A, 0.802 -0.555 -2.247

A, 1.0 1.0 1.0

=-~~= Z~

If t; is greater than 0.707, r is imaginary. Discuss the significance of these values with the help of a diagram. 2.4 An unknown weight W is attached to the end of an unknown spring k and the natural frequency of the system is 1.5 Hz. If 1 kg weight is added to W, the natural frequency is lowered to 75 cpm. Determine the weight W and the spring constant k. 2.5 A body weighing 60 kg is suspended from a spring, which deflects 1.2 em under the load. It is subjected to a damping effect adjusted to a value O.QO times that requir~ct:for critical damping. Find the natural frequency of the undamped and damped vibrations, and, in the latter case determine the ratio of successive amplitudes. If the body is subjected to a periodic disturbing force with a maximum value of 25 kg and a frequency equal to one-half its natural undamped frequency, determine the amplitude of forced vibrations and the phase difference with respect to the disturbing force.

a

"

b

•

.I m

I'

These mode shapes have been plotted in Fig. 2.25b.

~k

//~7/////.////Jm. .

PRACTICE PROBLEMS 2.1

1

max

(a)

Determine the numerical value of viscous damping from the free vibrations record in Fig. 2.8b.

2.2 Show that, in frequency-dependent excitation, the damping factor t; is given by the following expression:

t;=Ht,-J;) !,,

T

!

~k

l

J;

in which [2 and are frequencies at which the amplitude is 1/"1/2 times the amplitude at r = 1.

(b)

Figure 2.26.

r~a=3~~

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