Finite Element Programs for Structural Vibrations by C. T. F. Ross BSc, PhD (auth.)

By C. T. F. Ross BSc, PhD (auth.)

Finite point courses for Structural Vibrations offers targeted descriptions of ways to take advantage of six machine courses (written in Fortran seventy seven) to figure out the resonant frequencies of 1, , and third-dimensional skeletal buildings throughout the finite aspect strategy. bankruptcy 1 is on "The Finite aspect procedure" and bankruptcy 2 demonstrates, via hand calculations, the finite point resolution of a few smaller constructions. bankruptcy three covers "The Modular Approach", and Chapters four to nine describe the six laptop courses, with quite a few labored examples. The six laptop courses are given in Appendices I via VI, and on a three 1/2'' disk integrated with the publication. The courses are compatible to be used on IBM (or suitable) laptop (640K or extra) or minicomputer.

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E. 1, for the case when node 2 has an added mass of 5 kg. 82 10 6 X _W 2[11. 3 Determine the resonant frequencies for the beam of Fig. 10) 3 1m Fig. 5. Encastre beam. 3, when node 2 has an additional mass attached to it. e. 5 Determine the resonant frequencies for the pin-jointed space truss shown in Fig. 5 m . . --. 8 m 1 1 1 ...... 5 m --~ Fig. 6. X",u o Pin-jointed space truss. a Plan. b Front elevation. 20), 11- 4 = V[(2 - O? 7 X lO lD 308 . 7 x 10 10 x 308 . e. 5 was a four-dimensional problem, and although the structure was of a very simple type, the eigenvalue solution involved the expansion of a third-order determinant.

1. Suppress zero node Calculate [KOj-l [M'j and determine eigenvalues Flow diagram for vibration programs. THE MODULAR APPROACH IN FINITE ELEMENT PROGRAMMING 39 FOR NELEM = 1 TO NELEMS Input I: REM the "I" node of the element Input J: REM the "J" node of the element Input other material and geometrical properties L = SQR «X(J) - X(I)) t 2 + (Y(J) - Y(I)) t 2): Rem length of element c = (X(J) - X(I))/L: Rem cos aS = (Y(J) - Y(IHjL: Rem sin a- Hence calculate the elemental [kO] and [rna]. To calculate the system [KO] and [M O], the following process is used: Rem to assemble [I<"] and [mOl into [KO] and [M'] 11 = NDF *I-NDF: REM NDF = number of degrees of freedom per node J1 = NDF *J-NDF FOR II = 1 To NDF FOR JJ = 1 To NDF MM = 11 + II MN = 11 + JJ NM = J1 + II NN = J1 + JJ K(MM,MN) == k(II,JJ) M(MM,MN) = m(II,JJ) K(MM,MN) = k(II,JJ + NDF) M(MM,MN) = m(II,JJ + NDF) K(NM,MN) = k(1I + NDF,JJ) M(NM,MN) = m(1I + NDF,JJ) K(NM,MN) = k(1I + NDF,JJ + NDF) M(NM,MN) = m(1I + NDF,JJ + NDF) NEXT JJ NEXT II NEXT ELEM After assembling the stiffness and mass matrices, it is necessary to feed in the positions of the zero displacements.

Plane pin-jointed truss. 22 FINITE ELEMENT PROGRAMS FOR STRUCTURAL VIBRATIONS AZ-4 = cross-sectional area of element 2-4 = 6 x 10-4 m Z Element 1-2 By defining the element as 1-2, as distinct from 2-1, it means that the element points from 1 to 2, as shown by Fig. 2. B. If this were defined as 2-1, then the element would point from 2 to 1, so that Cl' would equal 225°. 1) y", V" Y,v _ - - '_ _ _ _ _ _ _.... X",u" Fig. 2. Element 1-2. 2) Element 2-3 The element points from 2 (the start node) to 3 (the finish node), as shown in Fig.

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