On the flexural vibrations of elastic manipulators with prismatic joints

Compurm

Pergamon PII: 10045-7949(%)00296-9

& S~rucrura Vol. 62. No. 5, pp. 897-908. 1997 Copyright 0 1996 Elxvier Science Ltd Printed in Great Britain.All rightsreserved cws-7949/97 $17.00 + 0.00

ON THE FLEXURAL VIBRATIONS OF ELASTIC MANIPULATORS WITH PRISMATIC JOINTS $. Yiikselt and M. Giirgiizez tDepartmcnt

of Mechanical Engineering, Faculty of Engineering and Architecture, Gazi University, 06570 Maltepe, Ankara, Turkey

SFaculty of Mechanical Engineering, Istanbul Technical University, 80191 Giimiigsuyu, fstanbul, Turkey (Received 5 April 1995)

Abstract-The aim of this paper is to investigate the flexural vibrations of an axially moving robotic arm sliding through a prismatic joint while the joint is undergoing both vertical translation and rotary motion. Considering not only Euler-Bernoulli beam assumption but also the effects of gravitation, rotary inertia and axial force, the equations of motion and associated boundary conditions of the dynamic model are derived by using the extended Hamilton’s principle. Since they form a complex boundary value problem which cannot be solved analytically, the assumed modes method is utilized to obtain an approximate solution. Simulation results are presented for some typical kinematical inputs. Copyright 0 1996 Elsevier Science Ltd.

1. INTRODUCTION

Finding applications in lightweight and high speed robotic manipulators, axially moving flexible beams have received considerable attention lately. The paper of Taborrok et al. [I] is one of the earliest works on this subject, which deals with a clamped-free beam whose length varies with time. They considered the beam in flexure, derived the equations of motion

via Newton’s law and presented certain properties of the mode shapes. In the literature, some of the following studies have sought direct analytical solutions for this vibration problem, like Bergamaschi and Sinopoli [2], while most of them have used approximate methods [3]-[l I]. Using the eigenfunctions of a clamped-free beam as trial functions of the desired series solution for a clamped-free beam whose length varies with time, Wang and Wei [3] and Tadikonda and Baruh [4] have derived the equations of motion as a system of second order ordinary differential equations, and given some sample results. Similarly, Buffinton and Kane [5] have investigated the dynamics of a free-free beam moving over supports by using assumed modes method associated! with the eigenfunctions of a clamped-free beam in flexure. On the other hand, Yuh and Young [6] and Banerjee and Kane [7] have studied an axially moving beam having not only translational motion but also rotational motion. Researchers in Ref. [6] have formulated the p.roblem via Newton’s law and Galerkin’s discretization method, whereas in Ref. [7},

they have proposed a new modeling method for the simulation of the motion of any continua that undergoes extrusion from or retraction into moving bases. As a more general problem, a robot with a flexible arm having a translational motion through an arbitrarily driven prismatic joint has been investigated by GiirgGze and Miiller [8]. They have used two approaches, continuum and multibody models, to investigate the vibration problems of an elastic robot arm in flexure and torsion. Chalhoub and Ulsoy [9] have investigated a particular robot arm which has two revolute joints and one prismatic joint. They have used assumed modes method in their simulation studies. In a different study, Wang and Wei [lo] have considered a robot with a long extendible robot arm which can undergo both vertical translation and rotary motion. They have used Newton’s second law to derive the equations of motion for the flexible arm and made use of a Galerkin-type approximation to solve them. In a following study, Krishnamurty [l l] has revised the latter by adding the vibration problem of the arm’s tail section. He has derived the equations of motion by using the extended Hamilton’s principle and applying Galerkin’s procedure. In almost all studies so far, Euler-Bernoulli beam assumption, that neglect the effects of rotary inertia and shear deformation, has been used to investigate the vibrations of axially moving flexible beams. In addition to this assumption, some researchers have included the axial force effect in their investigation,

897

$. Yiiksel and M. Giirgiize

898

while some have included the structural damping effect. Moreover, some researchers have attached a concentrated mass to one end of the beam, whereas, some attached an end-force and moment. Furthermore, the effect of the tail section of the moving beam has been investigated by Krishnamurty [ 111. In the present study, the flexural vibrations of a flexible robotic arm are investigated by considering the effects of gravitation and rotary inertia in addition to the effects of the axial force due to inertial forces, end-mass and the tail section of the beam. The arm is similar to those of Refs [lo] and [ 111. It moves axially through a prismatic joint while the joint is undergoing both vertical translation and rotary motion. In other words, the robotic arm under consideration has one revolute joint and two prismatic joints. The effects of shear deformation, axial deformation and structural damping are assumed to be negligible. Here, the equations of motion of the above described dynamic system are derived via extended Hamilton’s principle. Due to great difficulties in finding a direct analytical solution to this time dependent partial differential equation and its boundary conditions, it is discretized by means of a series solution, called also the assumed modes method, in order to obtain an approximate solution. In the rest of the paper, simulation results for various kinematic inputs of the prismatic joint and material values of the dynamic system are presented. 2. EQUATIONS

on the system. Here, however, the virtual work of the driving force is zero (6’A = 0) since the sliding motion u(t) is a prescribed motion and therefore its variation vanishes (6~ = 0). The kinetic energy consists of two parts which are due to translation and rotation of each element of the beam. Dividing the beam into two parts as the front and the tail sections, the rotational kinetic energy is

Trot=

$

s s -”

[I,ti;: + I,(~u;,)* + I:(? - LX,)‘] dy -L/2 L/2

@i: + h(Iju;,)’ + Z>(j -

ic:,)‘] dy

(2)

-u

where p shows mass density, I,, 1, and I, denote the moments of inertia with respect to moving coordinate system xEyEzEfor the uniform beam cross-section, U.&J, r), u.&, 0, a&, t) and u&, t) present the bending displacements corresponding to the two sections of the beam and y(t) denotes the rotation angle of the prismatic joint about zA axis. As usual, primes and dots in the above equation refer to partial derivatives with respect to the position coordinate y and time t, respectively. The second terms of both integrants in eqn (2) result from the transformation of the angular velocity j of the joint into moving coordinate system xEyEzE. Similarly, the translational kinetic energy can be obtained as

OF MOTION

s -u

The robotic arm to be considered is depicted in Fig. 1. It consists of an elastic beam in flexure in two axes, xA and zA.The arm has a concentrated end-mass M and is sliding through a prismatic joint according to u(t). The kinematics of the joint is given by the angle y(t) and the position r,(t) of its central point A in the direction of the axis 2. Besides, y shows the position of a typical element at point Eon the elastic beam with respect to its mass center S. As seen in Fig. 1, there are three coordinate systems which are to be used in the mathematical analysis of the dynamic system. One of them is an inertial coordinate system denoted by XYZ. The other two are moving coordinate systems denoted by xAyAzAattached at point A to the prismatic joint and xEyEzEattached to a typical element at point E on the elastic beam. In the system, the axes Z and zAcoincide permanently. In order to obtain the equations of motion, the extended Hamilton’s principle “[6(Ts 11

V)+6’A]dt=O

T

,rms =

:

-

iti:,

+

y’(u + y)’ - 2&J+ + Y)

-L/2

x [tit, + li’(u + y)’ - 2ti,,j(u + y)+C2

where A denotes the cross-sectional area of the beam, M denotes the concentrated end-mass and rz(t) denotes the position of the central point A of the prismatic joint. Additionally, a spatial Dirac delta function defined as

(1)

(4) will be applied, where T denotes the kinetic energy, V denotes the potential energy and 6’A represents the virtual work done by nonconservative forces acting

is utilized in the above equation.

Flexural vibrations of elastic manipulators On the other hand, the potential energy also consists of three parts. These are the strain energy due to the bending, the potential energy associated with the axial force arising due to inertial forces in connection with the so called axial foreshortening and the potential energy due to gravitation. At first, let the potential energy due to the bending be written in the form

v,, =;

J

-” (ELui:’ + El,&‘;) dy -L/2

J

899

Summing the kinetic and potential energy terms as T = T,,, + Tm,,,

v = v,, + VP, + v,

(10)

and then, making use of Hamilton’s principle given in eqn (l), one can obtain after lengthy calculations the equations of motion for the front section (-u Q y < L/2) in the following form

pZ& +

Jl

[21’ti- ii,, + y(u + y)

pA + M6 y - L

I

(

L/2

(Elru;’ + E&u::) dy

(5)

--u

where EL and EI; denote flexural rigidities. Secondly, the potential energy coming from the axial force associated with foreshortening can be written as

J -”

+fi-i’u)(y-~)-~(y’-~)ju”,

(11) 1 p[(-bu~ - w+a +m(,-$1 x,ii.,+~=+g~+~A[{-~(i +,.-~~.,-(y-~)-~(y~-~)} +(ii- j’(u +y)d, (12) 1+ {ii - i’(u + y)}u:,

p2b’.

t)(u::

+

14:)

-LIZ

J

dy

L/2

+;

PI(v, WC: + u,‘;) dy

(6)

--u

where P,(y, t) and P&J, t) are the axial forces which have different expressions for the front and tail sections of the beam. For the front section, the axial force can be shown to be

PIti,*)=

+,2u~~]+pA[j-~(ir-~2(u+~))

44[.r-y’(u+~)]

- El;u’l: = 0,

ELu;

+@[(ti-,i?u)(y-$)-f(Y’-y)]

(7)

and for the tail section (-L/2

= 0

< y < -u)

as

and for the tail section p1zti.C’~ + p-4 21ij - ii,, + j@ + y)+‘1’*u,, [

(8) Here, the terms associated with the elastic displacements are neglected by assuming that they are small and u(t) denotes the sliding motion of the rigid beam with respect to the prismatic joint. Finally, the gravitational potential energy can be expressed as

+{(.-y’u)(~+~)-~(y~-~)}~~~

1

+ [ii - y*(u + y)]u:, - EI:u:> = 0,

(13)

pA(~*z+ uz,) dy

where g denotes the gravitational

acceleration.

- (ii- j’(u +y)ju,‘, 1-El& = 0.

(14)

$. Yiiksel and M. GiirgGze

900

These four equations of motion are subject to the following boundary conditions, for the front section: t&,(-u,

t) = 0,

f&,(-u, 1) = 0,

d,(-u,

ul,(-u,

t) = 0,

U:(,(L/2, t) = 0,

UI’,(L/2, t) = 0,

t) = 0,

u,*(-u,

t) = 0,

Ul,(- u, t) = 0,

l&J-u,

t) = 0,

u.C,(-24,t) = 0,

U.3 - L/2, t) = 0,

Ul;(- L/2, t) = 0,

Z;(i; -

[

ti.:*)+$u:; II

= 0, v=

-L,Z

The differential eqns (1 lt(14) and the 16 boundary conditions given in (15) and (16) form a complex boundary value problem for which exact solution is not possible. Fortunately, one can solve this type of boundary value problems by using approximate methods.

+E&I’ 0 p I’ II F=L,2= and for the tail section

(15)

3. APPROXIMATE

SOLUTION

Since finding an exact solution is not possible, the partial differential equations of motion of the

Fig. 1. A flexible robotic arm with a prismatic joint.

Flexural vibrations of elastic manipulators 4,

Cm)

_.__~ 0

0.1

0.2

0.3

0.4

0.5

~

0

~ 0.1

0.2

0.3

0.4

0.5

time(s)

time(s) u,,(m)

u,(m)

0.004

0.002 0.0015 0.001 0.0005 0 -0.0005 -0.001

0.002 0 -0.002

0

0.1

0.2

0.3

0.4

1

-0.004 0.5 0

_\

0.1

0.2

0.3

0.4

0.5

time(s)

time(s)

Fig. 2. Tip deflections for UD= -0.35 m, U, = 0.7 m, YO= 0, 7, = n rad, rzO= 0, rz, = 0.5 m, T = 0.5 s, M= 0.15 kg.

dynamic system will be discretized by means of the assumed modes method in order to obtain an approximate solution. The method consists of assuming solutions for the bending displacements U&J, Z), u.&, t), u,,(!J, t) and a&, 1) in the form of infinite series as

U, = f: j=

ejd_Y,

t)pji(l),

I

i= 1,2

(17)

% = f, e,iCr, l)qj,(t), ,=I

where ejlb, t) and e,&, t) are properly selected functions which depend on the spatial coordinate y and time t and p,{(r) and qji(t) are the time-dependent generalized coordinates to be determined. Here, the eigenfunctions of a.clamped-free beam are chosen as the functions ejl@, t) and e,&, t) since they satisfy all of the geometric boundary conditions of the system. They are ej,(y, t) =

cosh(J %)-cost&

3)

where cash 1, + cos & D = sinh & + sin Ij

(19)

and the dimensionless frequency parameters 4 satisfy the well known characteristic equation of a clamped-free beam coshJcosL/=

-1,

j=

I,...,

m.

(20)

$. Yiiksel and M. Gtirgijze

902

Furthermore, it can be shown that these eigenfunctions satisfy the following conditions at the boundaries: q(-u,

t) =

e/q-u,1) =

e;;(L/2, t)

= eF(L/2. t) = 0, e,*(- 24,t) = e;( - u, t) = e;;( - L/2, t) i=l,2,

= e;;‘( - L/2, t) = 0.

It is seen that u,,(J, t), u&, t), u,,(y, t) and u&, t) expressed in eqn (17) satisfy all of the geometric boundary conditions in eqns (15) and (16). Additionally, they satisfy also the four dynamic (natural) boundary conditions related to the bending moment. However, they do not satisfy the remaining four natural boundary conditions related to the shearing force balance. Therefore, they can be used as admissible functions for this boundary value problem according to the assumed modes method [ 121. Now, substituting the infinite series given by eqn (17) into the kinetic and potential energy terms derived above, then using them in Lagrange’s equations

cc

(22)

one can obtain the equations of motion as a system of ordinary differential equations with time varying coefficients. Introducing the definitions, 1

U UI =7,

U2=t’ u+-

U--

u+z

2

u

uj=x,

z&=7,

2

U--

1

(23) 2

and making use of the boundary conditions given in eqn (21) and using the orthogonality property of the

ux*(m)

ux,04

0. 03 0.

k=l,...,

(21)

0.04,

02

0. 01 0 -0.

01

-0. 02 -0. 03, 0

V 0.1

-0.04 0.2

0.3

0.4

time (s)

time (s)

0.5

0

0.1

0.2

0.3

0.4

time (s)

time (s)

Fig. 3. Tip deflections for u0 = -0.35 m, u, = 0.7 m, 70 = 0, yI = a rad, rzo = 0, rz, = 0.5 m, T = 0.5 S, M = 0.30 kg.

0.5

Flexural vibrations of elastic manipulators

903

chosen admissible: functions, the equations of motion of the beam can be written after lengthy calculations. The resulting equ.ations are

L/2 e&dy =O, s -u I

k=l,__.,

co.

(24)

u, 04 0.03 0.02 0.01 0

-0.01

-0.01

-0.02,

-0.02

-0.03,

-0.03

-0.04 0.1

0

0.2

0.3

0.4

0.5

0

0.1

0.2

time (s)

-0.002

’

4,

‘1’

”

“V

u

0.5

time (s)

u,(m)

-0.003

0.4

0.3

Cm)

v

I 0

0.1

0.2

0.3

0.4

time (s)

0.5

0

0.1

0.2

0.3

0.4

time (s)

Fig. 4. Tip sdeflections for uo = 0.35 m, u, = -0.7 m, y0 = 0, y, = II rad, rzO= 0, r~, = 0.5 m, T = 0.5 s, A4 = 0.15 kg.

0.5

$. Yiiksel and M. Giirgijze

904

;,

{[s.’

{1+ 5

b(y - $)}

+~J‘I:‘e;,e;,dy]4;,+[zsf:2{~

xc5 y-3 (

L )I

e,‘,e;, + 2ti2@’ + ““2

ek,ej, dy

uz(ii’ + “y)

x e;;e;, + 01 + Gy)* u:

+s

ti, + z&y ___ e,:ekl dy u2

X

(?A;(, + g)ek!dy

.

II=o,

k=l,...,

--.--...

--

co.

(25)

-...-.

0.03) 0.02, 0.01 0, -0.01, -0.02 -0.03 -0.04)

0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

0.4

0.5

time (s)

time (s) uq (m)

u,(m)

0

0.3

0.1

0.2

0.3 time (s)

0.4

0.5

0

0.1

0.2

0.3

0.4

time (s)

Fig. 5. Tip deflections for uo = -0.35 m, u, = 0.7 m, y0 = 0, y, = n rad, rz, = 0, rz, = 2 m, T = 0.5 s, M = 0.15 kg.

0.5

Flexural vibrations of elastic manipulators for the front section

905

+ L (a3+ li4yy

and

A

u:

e$eh

dy >lp

,2

-u

z,

{pjk ($ - .I)+2

J:iI

-

e$& dy]tj,

f [I

- L,2 (i

+ 2tijekz dy >

L eh + (u + y)e k? _ A 1

=o, k=l,...,c (26) II

0. 0.

-0. -0. 0

0.1

0.2

0.3

0.4

0.5

0

0.1

0.2

time (s)

0.3

0.4

0.5

time (s)

u,(m)

4, (m) 0.004

0.002 0.0015

0.002

0.001 0.0005 0

0

-0.0005

-0.002-

-0.001

vvv

vi

-0.004~ 0

0.1

0.2

0.3

time (s)

0.4

0.5

0

0.1

0.2

0.3

0.4

time (s)

Fig. 6. Tip deflections for ua = -0.35 m, u, = 0.7 m, ya = 0, y, = 271rad, rzu = 0, 12,= 0.5 m, T= 0.5 s, M = 0.15 kg.

0.5

S. Yiiksel and M. Giirgijze

906

r,,,,=,.+!$[I-gsin(T)],

+

r, 2w3

+ ti4y)+u4@3+ i&y)

+ 3 @3+ hYj2 A d

-

e;eh

I

4

A

dy

e;‘e;*

J >

(?A:+ g) ek2dy

1

qi2

II

=O,

k=l,.

. . , co,

(27)

for the tail section. Here, ~5, denotes the Kronecker delta. The last four sets of ordinary differential equations with time varying coefficients completely govern the vibrational behaviour of the system. In these equations, the boundary conditions of the system have been taken into account and there is no remaining boundary condition term to be satisfied. Besides, the coefficients of the generalized coordinates in the equations consist of the various combinations of the integrals of the admissible functions over the section of the beam.

4. NUMERICAL

RESULTS

Simulation studies have been performed for typical cases. For this reason, one can take finite series instead of infinite ones given in eqn (17) in order to solve the approximate equations of motion of the system. In the simulation studies, it is assumed that the beam material is aluminum for which the density is p = 2700 kg m-’ and the modulus of elasticity is E = 7 x 10” N me2. Moreover, the following values are used for the beam geometry; the cross-sectional area A = 0.00075 m2, the moments of inertia 1, = 1.406 x 1O-9m4 and 1; = 1.563 x 10-“‘m4 and the length of the beam L = 1.4 m. Additionally, different end-masses (M = 0.15 kg and A4 = 0.30 kg) have been used to investigate end-mass effects on the vibrations. Similar to those in Refs [S]-[6], the following functions are used to generate the translational and rotational motions:

u(l)=h+F[t-&sin($!)],

(28)

where uo, rz, and y. denote the initial lengths and angle; u,, ri, and y, denote the total displacements and angle, respectively (Fig. l), and T denotes the operating period. Due to its importance, the tip deflections of the tail and front sections of the beam are calculated in the simulation studies by taking 1 to 3 terms in the series given by eqn (17). Since not much difference has been observed among the three results, Figs 2-7 illustrate the tip deflections by using two terms for various inputs which are presented in the legends. Here, fourth-order Runge-Kutta method is used to solve the differential equations of motion by considering the continuous system as a discrete one. The calculations are made by means of a suitable computer code written in Mathematics. As a first example, Fig. 2 shows the tip deflections of the beam whose front Section (1) has an extending motion through the prismatic joint. At the same time, the joint is in a rotation about and a translation on the axis Z. The beam is chosen such that it is stiffer for bending along zEaxis than xE axis (I2 < I,). Figure 2 confirms this phenomena, because the vibrations along zA axis have higher frequencies than the vibrations along xA axis. Furthermore, the tip deflections of the front section increase when its length increases. Similarly, the tip deflections of the tail section decrease when its length decreases. This demonstrates that decreasing length gives rise to the stiffness of the beam. As another example, Fig. 3 represents the tip deflections when all the inputs are the same as those in Fig. 2 with the exception that the magnitude of the end-mass is greater. It is shown in Fig. 3 that in comparison with Fig. 2, the larger end-mass causes larger tip deflections. Figure 4 illustrates the tip deflections for the case when the beam front section retracts into the joint. All the inputs of Fig. 4 are the same as those of Fig. 2; however, the beam front section retracts into the joint in the case of Fig. 4, whereas it extends from the joint in the case of Fig. 2. As expected, the deflections of the front section decrease and the deflections of the tail section increase as the beam slides. Figure 5 shows the changes in the tip deflections of the beam if one choose a faster vertical translation for the joint motion while the other inputs are the same as in Fig. 2. As seen from the two figures, faster joint translation results in different tip deflections in the zA direction, however, there is no change in the tip deflections with respect to axis XA. Figure 6 shows the tip deflections of the beam obtained from another simulation study where the

Flexural vibrations of elastic manipulators

907

-0 0

0.1

0.2

0.3

0.4

0

0.5

0.1

0.2

0.3

0.4

0.5

time(s)

time(s) u*,(m) o.oztT-_~n 0.015 0.01 0.005 0 -0.005 -0.01 -0.015

.....--....

.--

’

-a.ar\

time(s)

time (s)

Fig. 7. Tip deflections for uo= -0.35 m, tl, = 0.7 m, ye = 0, y, = n rad, rzg= 0, r2, = 0.5 m, T = 0.5 s, M = 0.15 kg and different cross-section. rotation of the joint about the Z axis is faster. All inputs are same as those of Fig. 2, except the joint rotation is faster. This input leads to larger tip deflections with respect to axis XA,whereas it brings slightly different tip deflections in the direction of ZA. Figure 7 represents the tip deflections of the beam obtained from a simulation in which the beam geometry is different from those of the preceding simulations. The new beam has a square cross-section with the following geometrical values, I, = Zr = 8.33 x lo-r0m4,

A = 0.0001

m.

Since the new beam has the same moments of inertia with respect to axes xA and ZA,such a choice indicates the differences between the magnitudes and mode shapes of the tip deflections in the two directions. As seen from Fig. 7, the vibrations of both axes have similar mode shapes and but slightly different magnitudes.

motion. The equations of the motion of the arm have been derived by using the extended Hamilton’s principle. In the derivation, the effects of rotary inertia, axial force, end-mass and gravitation have been considered in addition to Euler-Bernoulli beam assumption. Furthermore, the vibrations of the tail section of the beam have been taken into account. Since the equations of the motion and the associated boundary conditions form a very complex boundary value problem for which an exact solution is not possible, the assumed modes method is utilized to obtain an approximate solution. These approximate equations of motion have been solved to simulate the system. For some typical joint kinematics, corresponding results are presented in the form of plots. It is suggested that axial and torsional vibrations of the beam be investigated in the future studies in addition to the flexural vibrations. This way, more information about the vibrations of a robotic arm with an axially moving flexible beam can be obtained.

5. CONCLUSIONS The flexural vibrations of a robotic arm including an axially moving flexible beam have been investigated, The beam slides through a prismatic joint that is subject to a vertical translation and a rotary

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B. Tabarrok, C. M. Scott and Y. L. Kim, On the dynamics of an axially moving beam. J. Franklin Instit.

297, 201-220 (1974). 2. S. Bergamasehi and

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5. 6.

7.

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8. M. Giirgijze and P. C. Miiller, Modeling and control of elastic robot arm with prismatic joint. IUTAM-IFAC Symp. Dynamics of Controlled Mechanical

Systems,

Switzerland, pp. 235-245 (1989). 9. N. G. Chalhoub and A. G. Ulsoy, Dynamic simulation of a leadscrew driven flexible robot arm and controller. J. Dyn. Systems, Measmts Control loll, 119-126 (1986). 10. P. K. C. Wang and J. D. Wei, Feedback control of vibrations in a moving flexible robot arm with rotary and prismatic joints. IEEE Int. Conf. on Robotics and Automation, pp. 1683-1689 (1987). Il. K. Krishnamurthy, Dynamic modeling of a flexible cylindrical manipulator. J. Sound Vibr. 132, 143-154 (1989). 12. L. Meirovitch,

Analytical

Methods

Macmillan, New York (1967).

in

Vibrations.