A 3-N degree of freedom system with two natural frequencies

MECHANICS RESEARCH COMMUNICATIONS Vol.16(l), 59-63, 1989. Printed in the LISA. 0093-6413/89 $3.00 + .00 Copyright (c) 1989 Pergamon Press plc

A 3-N DEGREE OF FREEDOM SYSTEM WITH TWO NATURAL FREQUENCIES E.V. Wilms and H. Cohen Department of Civil Engineering University of Manitoba, Winnipeg, Canada

(Received 18 July 1988: accepted for print 22 September 1988) : :

INTRODUCTION The study of linear N degree of freedom vibrating mechanical systems sometimes leaves one with the impression that there are usually N natural frequencies corresponding to the N freedoms [1], and that the system is linear only for small deformations. Some authors (see for example [2], have pointed out that if the characteristic equation has multiple roots, then there are less than N distinct frequencies. The purpose of the present work is to investigate a system with 3N degrees of freedom which has only two natural frequencies regardless of the value of N and which remains linear for very large deformations. Previously [3] a vibrating system with N degrees of freedom was analyzed. With the appropriate choices of the masses and spring constants the system oscillated with a single natural frequency. Further [4] a system with two degrees of freedom was considered. It was shown that the system had a single natural frequency even when the particle was connected to N arbitrary springs. The effective lengths of the springs were zero, and the deformations were arbitrarily large. ANALYSIS The present work combines features of the systems analyzed in [3] and [4]. We consider n particles each of mass m = 1. A spring of constant k & 1- is connected from each particle to the origin of co-ordinates. In addition, each particle is connected to all the other particles by springs of constant k = 1. The unstretched lengths of the springs are zero. Three of the particles are shown in Fig. 1. -59-

60

E.V.

WILMS

a n d H. C O H E N

The kinetic energy of the system is:

2 2 (x~+yi +zt ).

T= [/2.

(I)

1=1

Xl ~Yl ~ Z! x2~Y 2 ~ Z 2

x31Y$ ~ Z 3

Fig. 1

Three of the Particles

The potential energy is:

1 1 ~ ~-, { xi)2+ (yj y~)2 1 n 2 2 (x j+ (zj- z i )2} + 2 ~ (xi2 + Y, + zi ). i=l j=l i=l

V = ~-. ~-

(2)

xi, Yi, and zi are the co-ordinates of particle i; hence the system has 3-n degrees of freedom. We neglect the effects of gravity. Lagrange's equations then yield the following equations of motion.

x,+ (n+ 1) x i - ~ x j = O , j=l

SYSTEM WITH TWO NATURAL FREQUENCIES

61

n

yi+ (n + l) Yi- ~ y j = O , j=l

zi+(n+l)z

i-~zj=0,i=

(3)

lton.

j=l

The solutions of Eqs. (3) are: Xi = (Xio- Xco ) COS (n + 1) 1/2 + XcoCOS t + (Xio- X c o )

1/2 +

,

(n+l) 1/2 sin (n+l)

Xco sin t

Yi = (Yi o- Yc o) cos (n + 1) l/2t + YcoCOS t + ())i°- ~'c°) 1/2 sin (n + 1)1/2t + Ycosin t (n +1) z i= (Zio- Zoo) cos (n + 1)l/2t + Zeocos t + (zi°- 7"¢°) sin (n + 1)1/2t+ Zcosin t , i = 1 to n. 1/2 (n+ 1) tl

n

(4) n

In Eqs. (4), Xco = (,~_, Xjo ) / n, Yco = ('~ y j o ) / n , and Zco= ()". Zjo)/n are the co-ordinates of j=l

j=l

j=l

the mass center of the system at t = O. Xjo, YJo, and Zjo are the initialvalues of the displacements n

n

n

of the individual particles.X Co = (~'~.xjo )/n, ~'Co= ( ~ yjo )/n, and 2:Co= (~-~~:jo )/n are the j=l

j=l

j-I

average values of the components of linear momentum of the particles at t = 0. Xjo, YJo' and zjo are the initial values of the velocities of the individual particles. Hence Eqs. (4) indicate that oscillation takes place with the two frequencies 1 and (n + 1)I/2 for the 3-n degree of freedom system. The position of the mass center of the entire system may be obtained from Eqs. (4). n

,Y__.,x i

Xc=

i=l n

= XcoCOS t + Xcosin t,

n

ZYi i=l

Yc- - -

=YcoCOS t + ~'Co sin t,

62

E.V.

WILMS

a n d H. C O H E N

n

~z, 1=1

Zc - - -

n

ZcoCOS t + 7.coSin t.

(5)

Hence the mass center oscillates with the frequency 1. We investigate two special cases. Oscillation of each oarticle with freauencv 1 Oscillation with frequency 1 requires; Eqs. (4): X,o- X c o = O, i = 1 to n.

(6-a)

Xio- Xco = 0 , i = 1 to n.

(6-b)

Since Eqs. (6-a) are identical for each variable Xio, we must have Xio, = x ° , i = 1 to n. The initial value of x i must be the same for each particle, in order for oscillation to take place at frequency 1. Similarly Eqs. (6-b) yield xio= x o , i = 1 to n. The solution for this case is then, from Eqs. (4): x , = x ocos t + x o sin t,

(7-a)

Yi = Yo cos t + Yo sin t,

(7-b)

z i = z o cos t + z o sin t,

(7-c)

fori = 1 ton. The particles coincide. Oscillation of each narticle with freouencv (n + 1) 1/2 Eqs• (4) then require: Xco = )(co = Y % = Yco = Zoo= Z % = 0, the mass center of the system of particles remains at the origin, and the total linear m o m e n t u m of the system is zero. The solution for this case is then:

xi= XioCOS(n+ 1)1/2+ y i = YioCOS (n + 1)l/2t +

Zi= ZioCOS(n + 1)1/2+

xl°_ s i n ( n + 1)l/2t (n + 1) 1/2

(8-a)

)'1° s i n ( n + 1)I/2t 1/2 ( n + 1)

(8-b)

zl° sin(n+ 1)lnt, 1/2 (n+ 1)

(8-c)

i= 1 ton. For this case, it is readily shown that the force acting on each particle passes through the origin• In order for the force on each particle to pass through the origin, we must have: Fx~

Fy i

Fz~

xi

y~

zi

(9)

SYSTEM WITH TWO NATURAL FREQUENCIES

63

Fxi= xi, Fyi= ~}i, Fzi= zi are the force components on particle i. Hence Eq. (9) can be replaced by: Xi_ Yi_ Zi Xi Yi Zi "

(10)

Eqs. (8) yield: Xi

Yi

Zi

. . . . . . . (n+ 1) xi Yi zi (11) Hence the force on each particle passes through the origin, so in effect we may regard each particle as acted upon by a single spring of stiffness (n + 1) attached to the origin.

ILW_KIIF./~J~,S [11 (21 [31 [41

W. T. Thomson, "Mechanical Vibrations" second edition, Prentice Hall, (1953) D.A. Wells, "Lagrangian Dynamics", Schaum Publishing Co. (1967) E. Wilms and H. Cohen, Am. J. Phys. 51, 1091 (1983) E. Wilms and H. Cohen, Am. J. Phys. 53, 942 (1985)