Exact and direct analytical solutions to vibrating systems with discontinuities

Journal of Sound and Vibration (1976)

EXACT

AND

DIRECT

VIBRATING

44(l), 15-25

ANALYTICAL

SYSTEMS C.

WITH D.

SOLUTIONS

TO

DISCONTINUITIES

BAILEY

Department of Aeronautical and Astronautical Engineering,

The Ohio State University, Columbus, Ohio 43220, U.S.A. (Received 21 April 1975)

Direct analytical solutions, i.e., analytical solutions without any reference to the theory of differential equations, are given for the vibration frequencies and modes of beams with discontinuities in moment and shear. The calculated displacement, slope, moment and shear are given. In every case, these. data are compared to corresponding data from the rigorously exact solution to the differential equation subjected to the rigorously imposed boundary conditions. The results show that, contrary to the approximation that would be obtained from the Rayleigh-Ritz method when there are discontinuities in the higher derivatives, the direct solution converges to the exact solution as defined by the mathematician: i.e., the solution that satisfies the differential equation of equilibrium throughout the domain and the conditions on the boundary of the domain.

I. INTRODUCTION New insight into the application of Hamilton’s Law of Varying Action [l, 21 permits direct analytical solutions to conservative and non-conservative, linear and non-linear initial value systems whether stationary or non-stationary 131.This development has prompted the author to examine the direct solution of continuous stationary systems with discontinuities in the higher derivatives. To avoid any misunderstanding about interpretation of the direct solution, the results of the direct solution are compared to the results of the rigorously exact solution to the differential equation in each case. The nature of approximate solutions by the method of Rayleigh-Ritz is well known. If proper admissible functions can be found, this method will converge to an upper bound of the exact solution. It is the purpose of this paper to show that the theory of Ritz [4] applied to Hamilton’s Law of Varying Action [ 1] yields solutions that converge to the exact solution in the same sense that the mathematician defines the exact solution. The exact solution is an analytical solution that yields all of the information about a system that is available from mathematics; it is defined as that solution which satisfies the differential equations of equilibrium throughout the space-time domain of applicability and satisfies the conditions on the boundary or boundaries of the space-time domain. 2. THEORY Hamilton’s Law of Varying Action is the familiar equation which may be found in almost, but not all, texts in which Hamilton’s principle is discussed [2, 51, 6

f(T+W)dt-(aTiarii)6qiI:b=O.

Equation (l), applied with concepts that differ from those of Euler and Lagrange and the 15

16

C. D. BAILEY

variational calculus, provides an approach to the solution to the problems of mechanics that is devoid of the theory of solution to differential equations [3]. The observation of fact is made that the time-space path of any particle of matter is continuous with continuous first derivative in time-space regardless of the forces that may act upon it (infinite forces do not exist and cannot be postulated to act !) [3]. The condition of continuity of displacement and its first derivatives with respect to the independent space variable was recognized by Ritz [4] as a requirement for admissible functions. However, continuity of time-space, which is an observed fact and a fundamental requirement of the same “nature” that requires equilibrium of forces, i.e., F - ma = 0, permits the general application of Hamilton’s law through the theory of Ritz [3]. Whenever any system is properly defined, it is possible to choose, solely on the basis of the system definition and without a priori knowledge of the solution, both finite time intervals and finite space sub-domains such that the time-space path and aN derivatives are continuous within the time-space domain of the defined system [3]. This paper demonstrates that fact for the stationary motion of beams. The observation is now made, on the basis of the mathematics of algebra, geometry and the calculus, that the analytic function representing any known curve or surface that is continuous with continuous derivatives may be approximated, in principle, as closely as one pleases with power series. However, in the problems of mechanics, for the curve (say, beam deflection) or surface (say, plate or shell deflection) to be known implies that one has already achieved the solution or has apriori knowledge of the solution as was assumed by Raylergh [6]. No a priori knowledge of the solution is necessary when power series are properly employed in Hamilton’s law. Simply define the problem and generate from Hamilton’s law, and power series, the time-space path and/or configuration for whatever forces, both natural and applied, that may act on or within the system [33. 3. THE SYSTEM

This paper is restricted to systems that are assumed to obey the assumptions of “engineering” beam theory: i.e., “plane sections remain plane”, linear strain-displacement relations, and linear stress-strain equations, namely, v = -z(&@y), (2) E= -z(P G/+2), y = afi/laz+ aqay

= 0,

CJ= EC,

(3) (4) (5)

where, 5 is the displacement in the axial co-ordinate direction, y, @is the displacement in the lateral co-ordinate direction, z, E is the axial strain, &/ay, y is the shear strain, c is the stress and E is the modulus of elasticity of the material.

Figure 1. The co-ordinate system and the beam.

17

SYSTEMS WITH DISCONTINUITIES

A rigid body and two springs are assumed attached to a common point along the span of the beam. The rigid body is of mass, M, and mass moment of inertia, Z,. One of the springs exerts a linear restoring moment, -& &+?y, while the other spring exerts a restoring force, -Kg. The choice of co-ordinates is arbitrary. The system is defined as shown in Figure 1. The rigid body and springs are attached at a distance, 11,from the origin. y1 is the independent space variable in the interval, 0 < y, < I,. y, is the independent space variable over the remainder of the beam length, 0 < yz < 12. The total length is I= I, + 12. 4. KINETIC ENERGY When a system is defined, the kinetic energy may be written directly as the sum of the kinetic energies of the various elements of the system : T = MG,:(y, = 1,)/2 + Z,(c%5/+5)‘(y1 = 1,)/2 + (l/2) r m, $1:dy, 0

+

UP)

1Z,,,,(~~,?Q’

dy, + (l/2)

0

i”

m2 fi,:

dy, + (l/2) q I,Ja~,+2)2

0

dy,.

(6)

0

5. THE WORK The work of all forces, both natural and applied, may be written in two ways [3], either as functions of the displacement (i.e., negative of the potential), if that is the form in which the work of a force is known, or as the product of the defined force and its displacement regardless of the functional form of the force: i.e., II’= -K$(y,

= 1,)/2 - K,#tiI/~yI)2(yl

= 1,)/2

- (l/2) 1 Er,(a2 aJay:)2 dy, - (l/2) ]t E12(a2 i?2/+;)2 dy, 0

0

(Work of body forces and moments) + (Work of surface forces and moments).

+

(7)

6. HAMILTON’S LAW Substitute equations (6) and (7) into equation (1). Operate with 6 and integrate the kinetic energy by parts with respect to time to obtain Hamilton’s law in its second form [3] : fl --ME,, &(y,

= r,) - I&&#y,)

~(&P,,/i?_yl)(yl = f,) - 4 m, $I &ijii, dy,

‘( $0

0

- 1 EI,(3' ~l/i+~) &a' 0

wJay:>

dy, - r EI,(a2 g2/ZJ$)b(a2 P2/+:) dy2 0

dt = 0. (8) I Equation (8) applies to statics and dynamics, conservative or non-conservative, whether stationary or non-stationary [3]. This paper, for the sake of brevity, is constrained to free +

(applied forces) S( associated displacements)

18

C. D. BAILEY

vibrations. Thus, the applied forces and any other natural forces are assumed to be zero. For convenience, non-dimensionalize the independent space variables : (9)

Yl = VlL Y2 = v212.

(10)

Substitute equations (9) and (10) into equation (8). Multiply and divide where appropriate by reference structural property values, m, and El”. Assume simple harmonic motion : t) = w,(Y,) sin W

(11)

~2(y2, t) = w,(y,) sinot.

(12)

%(h

After some algebra, equation (8) (the rotary inertia terms, I,,,, and Z,,,,,are assumed negligibly small for the remainder of this paper) becomes

0

w21WV2

w2P2d)dv,

(13)

=O,

0

where I2 = (m, 14/EZo) co’,

l=I,+l,.

The search for admissible functions need go no further than power series for reasons cited earlier. Thus, the functional forms of the dependent displacement variables may be written as W,(R) = 5 Alg,(%)?f,

(14)

i=O

%(?2)

=

kIo

4cgz(q2)

(15)

G.

gl(ql) accounts for the specified constraint (boundary condition) on the displacement at q1 = 0. g2(q2) accounts for the specified constraint (boundary condition) on the displacement at q2 = 1.0. A thorough discussion of these functions is given in reference [7]. In the present paper, the boundary conditions at q1 = 0 and q2 = 1.0 will be restricted to free, pinned or clamped. Therefore [7], p = 0, 1, 2. (16) g1=& q = 0, 1, 2. (17) gz = (1 - r2)4, Note that nothing has been said about the force boundary conditions, i.e., moment and shear, which involve the second and third derivatives, respectively, of the displacement and which must be rigorously and explicitly satisfied in order to obtain the exact solution to the system through the differential equation. This is probably the most important difference between the direct solution through the energy equation and the exact solution through the differential equation for deformable bodies. In addition to the specified displacement boundary conditions, the displacements, wl and w2, must satisfy the natural continuity conditions, and nothing else, at q1 = 1.0, q2 = 0: i.e.,

w,(l) = w,(O), (~W,l?Y,)(Y, = Z1)= (%/?YJ

(Y, = 0).

(18) (19)

19

SYSTEMS WITH DISCONTINUITIES

From equation (18) and equations (14)-( 17), B,= 2 A,.

(20)

i=O

From equation (19) and equations (14)-( 17) and (20), B, = i: Al(l,/M(P + (Z,/Z,)q + i).

(21)

i=O

The deflection function for w1 remains unchanged; but, substituting equations (20) and (21) into equation (15) gives, for w2, w2 = Z

Ai(l

-

r1iJ4(1

+

(Ull)(P

+

(ll/lJq

+

i>1]2)

+

:

Bk(l

-

TJ”Vzk-

(22)

k=2

i=O

From equations (14) and (16), for thejth term, 6w, = v;+~ SAj,

(23)

6(aW,/aq,) = (j + p) vi+‘-’ 6Aj,

(24)

f?(a’Wl/+,)

=

(j

+

p)(j

+

p

-

1) pi+“-’ 6A

(25)

j.

From equations (22), for thejth term in the w1 function, and the 5th term in the w2 function, 6~2 W’

w2/Ml=

=

(1 a2{(l

-

y/2Nl -

r12)qU

+

(~2ll& +

U2/~,)(~

+

(lJl2)q +

V1/~2)q

+A

~2) t-j)

6Aj V2)

+ JAj

(1 +

(1

~21’~; -

r12Yvi

SBc, W/W

(26)

(27)

Substitution of equations (14) and (22)-(27) into equation (13) yields two sits of coupled linear homogeneous algebraic equations in the dependent variables Ai and Bk. The resulting set of equations may be expressed in matrix form:

(28) i,,j = 0, 1,2, . . ., N k, [ = 2,3,4, . . ., M

The matrix elements are given in Appendix I. It is not the purpose of this paper to discuss the solution to differential equations; but, because of the accuracy of the direct solution, it was necessary to obtain the exact solution with sufficient accuracy to provide a valid comparison to the direct solution. The matrix equation in Appendix II, from which the eigenvalues of the exact solution are obtained, applies only to a uniform cantilever beam with a discontinuity in moment and/or shear. Equation (28), however, with the elements given in Appendix I, is the direct analytical solution to the vibration frequencies and modes of a very large class of both uniform and non-uniform beams with or without a discontinuity in moment and/or shear and with various boundary conditions. Note that the parameters in the exact analytical solution are identical to those in the direct analytical solution, equation (28). The exact solution requires the rigorous satisfaction of both the prescribed force and displacement boundary conditions as well as the conditions on moment and shear, displacement and slope at the point of discontinuity. Ultimately, however, the exact solution requires finding the approximate transcendental numbers that are the roots of the transcendental algebraic equation that results from the expansion of the determinant of the matrix in Appendix II. Ultimately, the direct solution requires finding the approximate transcendental numbers that are the roots ofthe determinant ofthe matrix in equation (28). Not a single force boundary condition is satisfied in obtaining equation (28). Yet equation (28) will yield both the transcendental eigenvalues and the transcendental eigenfunctions that are the solution to the problem.

20

C.

D. BAILEY

In principle, both analytical solutions, exact and direct, can be carried to any desired degree of accuracy. In practice, the extent to which either solution, exact or direct, can be carried is dependent upon the computational equipment. The mathematical proof has been in the papers of Hamilton for more than 140 years. The practical proof is in the numerical results from the analytical solutions achieved. Satisfaction of the force conditions at the point of discontinuity for the exact solution requires, a priori, knowledge of the inertial forces and moments because these forces and moments must be written with the correct sign in the application of the force conditions. This is an elementary problem for the student of Newtonian and Lagrangian mechanics when the motion of the system is simple as that ofthe system considered in this paper. No knowledge, a priori, is required for the direct analytical solution, a matter of great importance in more complicated systems. Note that a solution, whether exact or direct, can be carried no further than Appendix II or equation (28) until the parameters of the system are specified as explicit numbers. Although I, is arbitrary within the restrictions imposed by the assumptions of beam theory, for this paper it is assigned the value, 1, = l/2: i.e., the springs and mass are attached at the center of the beam. 7. RESULTS

Three examples have been selected for this paper. The first two are for a cantilever beam for which p = 2 and q = 0. The third is a clamped-pinned beam for which p = 2 and q = 1. For the sake of brevity and to prevent cluttering the figures, only the derivatives of the first mode are presented as curves. The mode is not shown as a curve. Instead, ordinates of the normalized mode curves are shown in the tables for direct comparison of numerical values. Eigenvalues for the higher modes are also given in the tables. Ten terms were used for the direct solution. Case 1. Discontinuity in the third derivative due to a linear force srping, K13/Ei0

For this case, M = IM = KB = 0 and KZ3/EZ0= 100. Figure 2 shows the derivatives of I

I

1

I

I

I

I

I

I

_f

-0.2

-

-0.6

-

I

I

I.0

I.0

._

0 E

z z

-t.ot I

0.1

I 0.2

I 0.3

I 0.4

I 0.5

I 0.6

I 0.7

I 0.8

I 0.9

I I.0

77

Figure 2. Exact and direct solution; slope, moment and shear,

KP/El==

100.

SYSTEMS WITH DISCONTINUITIES

21

both the exact and the direct solution. Note the continuity of the first and second derivatives although the second derivative (moment) reflects the discontinuity that is seen in the third derivative (shear). These conditions on moment and shear must be satisfied rigorously in the process of achieving the exact solution; but, not so for the direct solution. TABLE 1

Normalizedfirst mode, Case 1 Deflection A

Q

fW(B) exact w(1)

Equilibrium , w(q) direct w(1)

0.0

0.0

0.0

0.2 0.4 0.5 O-5 0.6 0.8

0.019206 0.111094 0.20078 1 0.200781 0.325802 0.645671 1~000000

0.019206 0.111094 0.20078 1 0.200781 0.325802 0.645671 1~000000

1.0

r wvl) __ direct 2: w(1) -0~000119 0.019209 0.111096 0.200920 0.200770 0.325802 0.645670 1~000001

h

equation

w”(V) exact A: w(1)

, % error

0.0 0.019206 0.111094 0.20078 1 0.20078 1 0.325802 0.64567 1 1 *oooooo

0.016 0.002 0.069 -0.005 oGOo o+JOo 0.000

Eigenvalues, Case 1 A2= (m14/EI> w2 Mode number 1

2 3 4 5

A2 (exact)

I2 (direct)

43.556742 699.78875 3806.7097

43.556742 699.78875 3806.7097

14820~000 39943.832

14819.998 39943,864

% error

0~00000 0~00000 0~00000 0~0000 1 0.00008

The reason for showing the curves for both solutions on the same figure is seen in the numbers of Table 1. The direct solution is identical to the exact solution. In particular, note the degree to which the direct solution satisfies the fourth order differential equation of force equilibrium for this system. Accurate stresses may be calculated from the direct solution as well as from the exact solution. For the sake of brevity, curves for other modes are not shown; but, Table 1 gives the eigenvalues up to the highest calculated for the exact solution. When M = ZM= K = 0, but K, # 0, the first derivative reflects the discontinuity in the moment curve. However, the third derivative is smoothly continuous because there is no change in shear across the point of discontinuity. Case 2. Discontinuity in both derivatives due to an attached rigid body

A mass attached such that it will behave as a particle will cause only a discontinuity in the shear curve. However, an attached rigid body will cause discontinuities in both moment and shear. For this case, K = Ke = 0, M/m, 1= 1 .O and IM/mo13 = O-01.In Case 1, the frequencies are increased by the presence of the spring except for those modes for which the point of attachment is a nodal point. In the present case the frequencies will be decreased for all modes. The derivatives for both the exact and direct solutions are shown in Figure 3 where the discontinuities in both the second and third derivatives are seen to occur. The degree of agreement may be seen in Table 2.

22

C.D. BAILEY

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0-I

0.8

0.9

I.0

77

Figure 3. Exact and direct solution; slope, moment and shear, M/ml = 1, I&d3 = 0.01.

TABLET Normalizedfirst mode, Case 2 Deflection c , W(V) W(V) exact w(l) w(1) direct 0.0

0.0

0.0

0.2 0.4 0.5 0.5 0.6 0.8 1.0

0.069410 0.245104 0.357788 0.357788 0.479333 0.736326 1~000000

0.069410 0.245104 0.357788 0.357788 0.479333 0.736326 1*OOoooo

Equilibrium r WY?) A; w(l) direct 0~000077 0.069409 0.245107 0.357733 0.357827 0.479343 0.736342 1400000

equation A w”(V) exact n: w(1)

\ % error

0.0

0.069410 0.245104 0.357788 0.357788 0.479333 0.736326 1WOooo

-0-001 0.001

-0.015 0.011 0.002 0.002 0.000

Eigenvalues, Case 2 ?,2 = (m14/EI)w2 Mode number 1 2 3 4 5

A2 (exact)

A2 (direct)

%error

8.070108 196.5660 1352,978 9062.833 9729.336

8.070108 196.5660 1352.978 9062.833 9729.336

0.00 0.00 OGO O+lO O*OO

Case 3. A linear force spring, K13/E10, attached to a clamped-pinned

beam

This case is the same as Case 1, except that the boundary condition at qz = 1 has been changed from free to pinned. To calculate this case from equation (28), after Case 1, requires only that q be changed from q = 0 to q = 1.0 and nothing else. To obtain the exact solution, the change in force boundary conditions causes a bit more effort to modify the equation of Appendix II. The two solutions are shown in Figure 4. Note again the discontinuity in the third derivative; but, because of the different boundary conditions, the curves are entirely different from Case 1. The agreement between the exact solution and the direct solution can be seen in Table 3. In the first two cases, although the mass matrix was coupled in equation (28), the stiffness matrix was uncoupled due to the free end at qz = 1. But in this case and any other case with a constraint at both ends, both the stiffness matrix and the mass matrix are

SYSTEMS WITH DISCONTINUITIES

23

Figure 4. Exact and direct solution, clamped-pinned,

KP/Ef = 100.

TABLE 3

Normalized$rst mode, Case 3 Deflection

Equilibrium

vv -

I

w(d exact

w(O.5)

Wi’(?) ___

w(v)

-direct

w(O.5)

direct

n: w(O.5)

0.0

0.0

0.0

0.2 0.4 0.5 0.5 0.6 0.8 1.0

O-316202 O-826117 1 *oooOOo 1 *oOOOOO 1.080144 O-795228 0.0

0.316202 0.826117 1 *oOOOoo 1 wOO00 1.080144 0.795228 0.0

0~000001 0.316202 0.826117 0.999997 1~000000 1.080144 0.795228 0~000000

A

equation \

wvl)

n: w(0*5)exact

0,

/o error

0.0 0.316202 0.826117 1 GOOOOo 1 *oOOOOO 1.080144

0.000 o+lOo o*OOo 0.000 0.000

0.795228 0.0

o*ooo 0.000

Eigenvalues, Case 3 A2 = (m14,!EI) o2 Mode number 1 2 3 4 5

%

A2 (exact)

A2 (direct)

error

439.0827 2531.311 11039.72 31809.55 74172.43

439.0827 2531.311 11039.72 31809.55 74172.43

o*oo 0.00 0.00 0.00 0.00

coupled. Note that orthogonality has not been mentioned. from the direct solution are orthogonal; but, the property category of “nice, but not necessary”.

The eigenfunctions generated of orthogonality falls into the

24

C. D. BAILEY 8. CONCLUSION

It has been shown that the direct solution obtained from simple power series and the theory of Ritz applied to Hamilton’s Law of Varying Action yields results for systems with discontinuities that are, for all practical purposes, fully as accurate as the exact solution. The energy approach to mechanics, which was being pursued by Leibniz at the time that Newton made his great discovery, has been subordinated to differential equations, possibly because direct solutions to time dependent, non-conservative, non-stationary systems cannot be obtained from the theory as taught in the variational calculus. However, direct analytical solutions to these truly difficult time dependent problems have recently been obtained [3] and show that Hamilton’s law is not only the independent approach to mechanics that Osgood [5] claimed it to be, but, with the proper concepts, it can be a highly useful tool to the engineer for the solution to practical problems. A comparison of results for the two independent approaches to mechanics has been presented in this paper for a stationary system.

ACKNOWLEDGMENT

The author wishes to express his appreciation to his two Research Assistants, Mr Jon Ogg and Mr Anthony Straw, who not only did much of the analysis, but “supervised” the computer in making the calculations and prepared the tables and figures.

REFERENCES 1. W. R. HAMILTON1834Philosophical Transactions of the Royal Society, 247-308.

2. 3. 4. 5. 6.

On a general method in dynamics. W. YOURGRAUand S. MANDELSTAM1968 Variational Principles in Dynamics and Quantum Theory. Philadelphia: W. B. Saunders Company, third edition. C. D. BAILEY1975Foundations of Physics, issue 3. A new look at Hamilton’s principle. W. RITZ 1911Gesommelte Werke. Paris: Societe Suisse de Physique. W. F. OSGOOD1937 Mechanics. New York: The Macmillan Company. LORD RAYLEIGH1871Theory of Sound (two volumes). New York: Dover Publications, second

edition, 1945 re-issue. 7. C. D. BAILEY1973American Institute of Aeronautics and Astronautics Journal 11, 14-19.Vibration of thermally stressed plates with various boundary conditions.

APPENDIX

I: MATRIX

ELEMENTS

Ktj = (K13/EI,,)(1,/1)3 r/f+j+2p(qI = 1) + (K,

+ (i +p)(i +p - I)(j+p)(j+p

FOR DIRECT

SOLUTION

I/EZJ(l,/Z)(i + ~)(j +~)d+j+~~-~h

- 1) 5 (EZ~/Elo)rl’;+i+2P-“d~l 0

+

(h/O31(EI,IEl,W21~r13((1

- r2)Yl + U21Mp+ (4/12)q+ i)r12)x

0 x

&,=

W>"

(a'/%,z)((l

-

r2)'U

j W2l~~o>(~'lM)((~ 0

+

(~zll,)(p

-

+

0,/l&

~2)"4$P'/~$)((l

+j)&dqz, -

q2N1

+

V2/4)x

= 1)

2.5

SYSTEMS WITH DISCONTINUITIES

x (P + (4ll2)q + i)v,)dvz,

Mi j = (M/m0 1)(1,/1)3r/f+j+zp h =

1)+ VM/moI”)U,/O(i+ A(j + p) t$j+2P-2(q1= 1)

+ K/r)* j (mJmo) ?f:+j+zp drll +

(W3(~2/0

j!(m2lmo)(l

x

Mi:

=

(P

+

(b/12)q

(11/1)~(12/1)

-

~2)~Yl

+

(12/b)

x

0

0 +

i)v2)(1

S hzlmo)(l

+

U21h)(~

-

~2)~‘Yl

-

q2)2q&+c

+

+

(L/12)q

(12/l&p

+

+j)q2)dv2,

(ZJl2)q

+

i)q2)ddq2,

0

Mu

= (4/03(12/0

/(m,/mo)(l

dq,.

0

APPENDIX ~(cash X -

II: MATRIX

(sinh X sin X)

-I

cos X) (sinh X + sin X)

(cash X cos X)

0

(cash X + cos X)

(sinh X + sin X)

(sinh X sinX)X3

(cash X + cos X) X3

-(&IEI,)

EQUATION

FOR EXACT -1

0

A

0

0

-I

SOLUTION

-1

i

E

-1*(1/1,)3 x3 + &‘(~,I~)IX

,44*(//r,) X4 K*(l,/Q3

-(EI,/EI,)

X3

B

M*(l//,)

x4 -

x3

F

-sin (1,/l,) X

G

(EI,/EI,)

K*(W)’

0

0

cash (/Jr,) X

sinh (1,/1,) X

-cos (l*/l,)

0

0

sinh (&/l,) X

cash (1,/l,) X

sin (/2//1) X

x

H

-cos (h/l& x 1

I* = IM/m13, K$ = KoI/EIl, M* = M/ml, K* = K13/Ei,, X = (l,/l)(m14W21E1,)‘/4, w,(ql) = A (cash Xv, - cos XqJ + B(sinhXr], - sin X~J,

%(q2) = Ecosh (12/f,) Xv, + Fsinh (1,/l,) Xv2 + Gcos (f2/ll)Xq2 + H sin (1,/l,) Xv,.