Advanced Methods of Structural Analysis

14.3 Free Vibrations of Systems with Finite Number Degrees of Freedom 535 Equations (e) divide byA 1 .Let 2 DA 2 =A 1 ; 3 D A ...

536 14 Dynamics of Elastic Systems abm 1 =m, EI=• EI EI EI h h h 1 2 3 m 2 =2m, EI=• m 3 =2m, EI=• 1 2 3 M 1 r 31 r 21 r 11 Z 1 ...

14.3 Free Vibrations of Systems with Finite Number Degrees of Freedom 537 If eigenvalue is denoted as D m!^2 r 0 D m!^2 h^2 24i ...

538 14 Dynamics of Elastic Systems Equations (c) becomes .1 1 / 2 D0;  1 C2.1 1 / 2  3 D0;  2 C2.1 1 / 3 D0: Solu ...

14.4 Free Vibrations of One-Span Beams with Uniformly Distributed Mass 539 Table 14.2 Comparison of the force and displacement m ...

540 14 Dynamics of Elastic Systems 14.4.1 Differential Equation of Transversal Vibration of the Beam Differential equation of th ...

14.4 Free Vibrations of One-Span Beams with Uniformly Distributed Mass 541 Table 14.3 Classical boundary conditions Clamped end. ...

542 14 Dynamics of Elastic Systems The general solution of (14.18)is X.x/DAcoshkxCBsinhkxCCcoskxCDsinkx; (14.19) whereA, B, C,an ...

14.4 Free Vibrations of One-Span Beams with Uniformly Distributed Mass 543 The mode shape of vibration is Xi.x/DDsinkixDDsin i ...

544 14 Dynamics of Elastic Systems S(x) V(x) U(x) T(x) Fig. 14.14 Krylov-Duncan functions circular permutations Table 14.4 Krylo ...

14.4 Free Vibrations of One-Span Beams with Uniformly Distributed Mass 545 AtxD 0 the Krylov–Duncan functions and their second d ...

546 14 Dynamics of Elastic Systems Fundamental data for one-span uniform beams with classical boundary condi- tions are presente ...

Problems 547 14.4.Symmetrical statically indeterminate frame carried one lumped mass. Deter- mine the frequency of free horizont ...

548 14 Dynamics of Elastic Systems 14.7.A massless beam carried body of massMand moment of inertiaJxwith respect tox-axis (Fig.P ...

Problems 549 Ans.!i^2 D EI i0:7854MR^3 ;  1 D1:4195;  2 D0:03395 ˆD  11 0:6591:517 14.10.Symmetrical frame with absolutely ...

550 14 Dynamics of Elastic Systems Ans. (a)U^2 .kl /T.kl/V.kl/D 0 !coshklcosklD 1 !DklD4:7300I::: !iD ^2 i l^2 r EI m ;! 1 D ...

Appendix............................................................................ Ta b l e A. 1 Typical graphs, their area an ...

552 Appendix Ta b l e A. 2 Multiplication of two bending moment diagrams (EIDconst) l MA MB Rl 0 MMdx 1 MA MB l l 6  MAMBCMBMAC ...

Appendix 553 Tabulated data for standard uniform beams: Ta b l eA.3andA.4contain the reactions at supports for uniform fixed–pin ...

554 Appendix Ta b l e A. 3 (continued) No Loading conditions Reactions and bending moment diagrams Expressions for bending momen ...