Qualitative Theory in Structural Mechanics pp 119-181 | Cite as

# Qualitative Properties of Vibration and Static Deformation Associated with Discrete Systems of Beams

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## Abstract

The focus of the first six sections of this chapter is on qualitative properties of the finite difference model or the related physical model of a beam. We will set up the governing equations of motion and boundary conditions associated with the finite difference system; derive various modal qualitative properties of the discrete model under different boundary constraints, by applying the theory of oscillatory matrices and the concept of conjugate beams; and establish qualitative properties in static deformation of the finite difference system of a well-constrained beam.

## References

- 1.Gantmacher FP, Krein MG (1961) Oscillation matrices and kernels and small vibrations of mechanical systems. U. S. Atomic Energy Commission, WashingtonGoogle Scholar
- 2.Gladwell GML (1985) Qualitative properties of vibrating systems. Proc Roy Soc Lond A 401:299–315CrossRefGoogle Scholar
- 3.Gladwell GML (2004) Inverse problems in vibration, 2nd edn. Springer, Dordrecht (1986, 1st edn, Martinus Nijhoff Publishers, Dordrecht)Google Scholar
- 4.Gladwell GML, England AH, Wang DJ (1987) Examples of reconstruction of an Euler-Bernoulli beam from spectral data. J Sound Vibr 19(1):81–94CrossRefGoogle Scholar
- 5.Gladwell GML (1991) Qualitative properties of finite-element models II: the Euler-Bernoulli beam. Q J Mech Appl Math 44(2):267–284CrossRefGoogle Scholar
- 6.He BC, Wang DJ, Low KH (1989) Inverse problem for a vibrating beam using a finite difference model. In: Proceedings international conference on noise & vibration, Singapore, B, pp 92–104Google Scholar
- 7.He BC, Wang DJ, Wang QS (1989) Inverse problem for the finite difference model of Euler beam in vibration. J Vibr Eng 2(2):1–9 (in Chinese)Google Scholar
- 8.He BC, Wang DJ, Wang QS (1991) The single mode inverse problem of Euler-Bernoulli beam. Acta Mech Solida Sin 12(1):85–89 (in Chinese)Google Scholar
- 9.Wang DJ, He BC, et al (1990) Inverse problems of the finite difference model of a vibrating Euler beam. In: Proceedings international conference on vibration problems in engineering, June 1990, China, vol 1(1), pp 21–26. International Academic Publishers, SingaporeGoogle Scholar
- 10.Wang DJ, He BC, Wang QS (1990) On the construction of the Euler-Bernoulli beam via two sets of modes and the corresponding frequencies. Acta Mech Sin 22(4):479–483 (in Chinese)Google Scholar
- 11.Wang QS, He BC, Wang DJ (1990) Some qualitative properties of frequencies and modes of Euler beams. J Vibr Eng 3(4):58–66 (in Chinese)Google Scholar
- 12.Wang QS, Wang DJ, He BC (1991) Construction of the difference discrete system for the simple-support beam from the spectra data. Eng Mech 8(4):10–19 (in Chinese)CrossRefGoogle Scholar
- 13.Wang QS, Wang DJ (2006) Difference discrete system of the Euler beam with arbitrary supports and sign oscillatory property of stiffness matrices. Appl Math & Mech 27(3):393–398CrossRefGoogle Scholar
- 14.Wang QS, Wu L, Wang DJ (2009) Some qualitative properties of frequency spectrum and modes of difference discrete system of multi-bearing beam. Chin J Theor Appl Mech 41(6):61–68 (in Chinese)Google Scholar
- 15.Wang QS, Wang DJ, Wu L et al (2009) Sign-oscillating property of stiffness matrix of difference discrete system and qualitative property of beam with overhang. J Vibr Shock 28(6):113–117 (in Chinese)Google Scholar
- 16.Wang QS, Wang DJ, Wu L et al (2011) Qualitative properties of frequencies and modes of vibrating multi-span beam. Q J Mech Appl Math 64(1):75–86CrossRefGoogle Scholar
- 17.Wang QS, Zhang LH, Wang DJ (2012) Some qualitative properties of modes of discrete system of beam with overhang. Chin J Theor Appl Mech 44(6):1071–1074 (in Chinese)Google Scholar
- 18.Wang QS, Wang DJ (2014) Supplementary proof of some oscillation property for discrete systems of rod and beam having rigid modes. Chin Q Mech 35(2):262–269 (in Chinese)Google Scholar
- 19.Zheng ZJ, Chen P, Wang DJ (2012) Oscillation property of modes for FE models of bars and beams. J Vibr Shock 31(20):79–83 (in Chinese)Google Scholar
- 20.Zheng ZJ, Chen P, Wang DJ (2013) Oscillation property of the vibrations for finite element models of Euler beam. Q J Mech Appl Math 66(4):587–608CrossRefGoogle Scholar
- 21.Zheng ZJ, Chen P, Wang DJ (2014) A unified proof to oscillation property of discrete beam models. Appl Math Mech 35(5):621–636CrossRefGoogle Scholar
- 22.Zheng ZJ (2014) The qualitative vibrational property and modal inverse problems of rods and Euler beams [D]. Department of Mechanics and Engineering Science, College of Engineering, Peking University (in Chinese)Google Scholar

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