By Barus C.

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**Sample text**

The resulting frequency is an upper bound solution, unless an exact eigenfunction of free vibration for the trial function is assumed. Ritz (1909) generalized the Rayleigh method by assuming a set of admissible trial functions, each having independent amplitude coefficients. He showed that a closer upper bound for the frequency could be achieved by minimizing the energy functional H with respect to each of the coefficients. Ritz demonstrated his method on a completely free square plate for which no exact solution is possible.

3 Formulation in Polar Coordinates II,SUM,NTAB,COEF) 60 70 S(II,I,J)=SUM CONTINUE CONTINUE DO 100 I=I,N CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 90 J= 1,N CALL FFXY(J,FXYSJ,IRSJ,ISSJ,NSJ) DO 80 II=6,10 CALL CALCULUS(FXYN,IRN,ISN,NN, FXYN,IRN,ISN,NN, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM 80 CONTINUE 90 CONTINUE 100 CONTINUE DO 130 I=I,N CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 120 J=I,L CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 110 II=l 1,13 CALL CALCULUS(FXYN,IRN,ISN,NN, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(II,I,J)=SUM 110 CONTINUE 120 CONTINUE 130 CONTINUE DO 160 I=I,L CALL FFXY(I,FXYSI,IRSI,ISSI,NSI) DO 150 J=I,L CALL FFXY(J,FXYSJ,IRSJ,IS SJ,NSJ) DO 140 II= 14,18 CALL CALCULUS (FXYL,IRL,ISL,NL, FXYL,IRL,ISL,NL, FXYSI,IRSI,ISSI,NSI, FXYSJ,IRSJ,ISSJ,NSJ, II,SUM,NTAB,COEF) S(n,U)=SUM 140 CONTINUE 150 CONTINUE 160 CONTINUE 53 See.

53) i=l where ~bi(x,y ) are the approximate functions which individually satisfies at least the geometric boundary conditions. 54) which yields a set of homogeneous equations expressed in terms of the coefficients q. 57) According to the Ritz method, the limit of the approximations in Eq. (x, y) are linearly independent. 9 Functions ~bi (x, y) form a complete system of functions. 9 Functions ~bi (x, y) satisfy the kinematic boundary conditions. It is not necessary for the chosen functions qki(x,y) to satisfy the static boundary conditions.

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