, Volume 20, Issue 2–4, pp 441–450 | Cite as

Homology design of vibration mode shape under uncertain boundary conditions

  • Shigeru Nakagiri
  • Nobuhiro Yoshikawa
Advances in nonlinear structural dynamics


A methodology is proposed for homology design to realize such a vibration mode shape that satisfies a certain geometrical relation before and during vibration. The formulation is based on the finite element eigenvalue and sensitivity analyses so that a governing equation for the design variables is derived under the condition that the homology constraint holds while the eigenvalue problem is satisfied.

The nonlinear effect of uncertain boundary conditions on the homologous mode shape is examined through convex model of the uncertainties. The worst case of the disturbed mode shape due to the uncertainties is estimated employing the Lagrange multiplier method.

The numerical example of out-of-plane vibration of a planar lattice frame displays the validity of the proposed method for homology design. The worst case of the disturbed mode shape is discussed when rotational stiffness of the boundary fluctuates.


Vibration mode shape homology design finite element sensitivity analysis Moore-Penrose generalized inverse uncertain boundary condition convex model 


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  1. Ben-Haim Y, Elishakoff I 1990Convex models of uncertainty in applied mechanics (Amsterdam: Elsevier) chap. 2MATHGoogle Scholar
  2. Fox R L, Kapoor M P 1968 Rate of change of eigenvalues and eigenvectors.AIAA J. 6: 2426–2429MATHCrossRefGoogle Scholar
  3. Hangai Y, Guan F I 1989 Structural shape analysis with the constraint condition of homologous deformation.J. Struct. Constr. Eng., Trans. Arch. Inst. Jpn. 405: 97–102 (in Japanese)Google Scholar
  4. Hoerner S 1967 Homologous deformation of tiltable telescope.Proc. Am. Soc. Civ. Eng. 93 (ST3): 461–485Google Scholar
  5. Morimoto M, Kaifu N, Takizawa Y, Aoki K, Sakakibara O 1982 Homology design of large antenna. Mitsubishi Electric Corp., Technical Report, Vol. 56, No. 7: 495–502 (in Japanese)Google Scholar
  6. Rao C R, Mitra S K 1971Generalized inverse of matrices and its applications (New York: John Wiley & Sons) chap. 3MATHGoogle Scholar
  7. Yoshikawa N, Nakagiri S 1993 Homology design of frame structure by finite element method.Theoretical and Applied Mechanics, Proc. the 42nd Japan National Congress for Applied Mechanics (eds) M Hori, S Kobayashi (Tokyo: University Press) vol. 42, pp. 43–51Google Scholar

Copyright information

© Indian Academy of Sciences 1995

Authors and Affiliations

  • Shigeru Nakagiri
    • 1
  • Nobuhiro Yoshikawa
    • 1
  1. 1.Institute of Industrial ScienceUniversity of TokyoTokyoJapan

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