Dynamic Response and Wave Propagation in Plane Trusses and Frames

Pao, Yih‐Hsing; Keh, Der-Ching; Howard, Samuel M.

doi:10.2514/2.778

Cited by 109 publications

(41 citation statements)

References 6 publications

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“…[25][26][27][28], the frequency equation of multi-span continuous Kirchhoff plates may be easily established. However, if the plate has a large number of spans, the unknown quantities involved in the solutions and hence the dimension of the reverberation-ray matrix increase rapidly, resulting in a low computational efficiency.…”

Section: A Recursive Formulation For Rmmmentioning

confidence: 99%

“…where R is the reverberation-ray matrix of dimension is the source vector due to external loads, which vanishes for the free vibration problem [25][26][27][28] . After IJ k d are obtained, IJ k a are determined from the scattering relations.…”

Section: Ji Ji Jimentioning

confidence: 99%

“…Although several techniques have been proposed to solve this problem [22][23][24] , the application is still limited because they are only valid or applicable for particular cases. Recently, an alternative approach, called the method of reverberation-ray matrix (MRRM), was proposed by Pao et al [25][26][27][28] to evaluate the transient response of planar trusses and wave propagation in layered media. A key idea in this method is to set up two local coordinate systems (dual coordinate system) for a single member/layer.…”

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confidence: 99%

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Recursive formulations for the method of reverberation-ray matrix and the application

Zhu

Chen

Gui-ru

et al. 2009

Sci. China Ser. G-Phys. Mech. Astron.

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A recursive formulation is proposed for the method of reverberation-ray matrix (MRRM) to exactly analyze the free vibration of a multi-span continuous rectangular Kirchhoff plate, which has two opposite simply-supported edges. In contrast to the traditional MRRM, numerical stability is achieved by using the present new formulation for high-order frequencies or/and for plates with large span-to-width ratios. The heavy computational cost of storage and memory are also cut down. An improved recursive formulation is further proposed by modifying the iterative formula to reduce the matrix inversion operations. Numerical examples are finally given to demonstrate the effectiveness and efficiency of the proposed recursive formulae. method of reverberation-ray matrix, recursive formulation, free vibration, continuous Kirchhoff plate Multi-span plates with orthogonal straight edges are frequently encountered in civil, aircraft/aerospace and marine engineering as basic structures or the key components of structures. For instance, the aeroplane skins, long-span roofs in building construction, steel bridge decks of oversea bridges and glass window panels may all be modeled as plates with multiple spans. As these plate systems are usually used as key bearing carriers in practice, it is essential to understand their dynamic characteristics. A vast number of researches of free vibration of Kirchhoff plates have been conducted by using different methods during the past several decades [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . One of the celebrated techniques is the method of transfer matrix (MTM), which was first formulated by Thomson for free vibration of inhomogeneous beams [21] . But this method was soon recognized to suffer from the inherent numerical instability, when the aspect ratio is large and/or the frequency is high. Although several techniques have been proposed to solve this problem [22][23][24] , the application is still limited because they are only valid or applicable for particular cases. Recently, an alternative approach, called the method of reverberation-ray matrix (MRRM), was proposed by Pao et al. [25][26][27][28] to evaluate the transient response of planar trusses and wave propagation in layered media. A key idea in this method is to set up two local coordinate systems (dual coordinate system) for a single member/layer. The equilibrium/compatibility conditions of all joints are considered to formulate a global scattering matrix, while a global phase matrix is obtained from the wave propagation mechanism in all members. The two matrices are combined to give the reverberation-ray matrix, which characterizes the wave motion of the whole structure. As we will show later, the use of dual coordinate system enables us to avoid the numerical instability during computations effectively. However, for complex structures, with the increase of members and joints, the dimension of the reverberation-ray matrix in MRRM becomes larger, and hence, leading to a heavy computational cost of stor...

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Section: A Recursive Formulation For Rmmmentioning

confidence: 99%

Section: Ji Ji Jimentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Recursive formulations for the method of reverberation-ray matrix and the application

Zhu

Chen

Gui-ru

et al. 2009

Sci. China Ser. G-Phys. Mech. Astron.

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“…Previous research by Pao et al (1999); Srikantha Phani et al (2006) and Yong and Lin (1992) describes models of discrete trusses under dynamic loading. Hutchinson and Fleck (2006) and Mohr (2005) develop equivalent continuum models of trusses under quasi-static loading, which are discussed in more detail subsequently.…”

Section: Introductionmentioning

confidence: 98%

Wave propagation in equivalent continuums representing truss lattice materials

Messner

Barham

Kumar

et al. 2015

International Journal of Solids and Structures

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Stiffness scales linearly with density in stretch-dominated lattice meta-materials offering the possibility of very light yet very stiff structures. Current additive manufacturing techniques can assemble structures from lattice materials, but the design of such structures will require accurate, efficient simulation techniques. Equivalent continuum models have several advantages over discrete truss models of stretch dominated lattices, including computational efficiency and ease of model construction. However, the development an equivalent model suitable for representing the dynamic response of a periodic truss in the small deformation regime is complicated by microinertial effects. This paper derives a dynamic equivalent continuum model for periodic truss structures suitable for representing long-wavelength wave propagation and verifies it against the full Bloch wave theory and detailed finite element simulations. The model must incorporate microinertial effects to accurately reproduce long wavelength characteristics of the response such as anisotropic elastic soundspeeds. The formulation presented here also improves upon previous work by preserving equilibrium at truss joints for simple lattices and by improving numerical stability by eliminating vertices in the effective yield surface.

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“…The method of reverberationray matrix (MRRM) is suitable for determining the natural frequencies and steady-state response of multi-span structures and space frame structures with complex geometry. MRRM was first proposed by Howard and Pao (1998) and Pao et al (1999) to analyze wave propagation in planar trusses. Subsequently, Pao and his associates extended the method to study waves propagating in multilayered media (Pao et al, 2000;Su et al, 2002;Tian and Xie, 2009).…”

Section: Introductionmentioning

confidence: 99%

Exact free vibration analysis of open circular cylindrical shells by the method of reverberation-ray matrix

Yao

Tang

Pang

et al. 2016

J. Zhejiang Univ. Sci. A

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Abstract:This paper is concerned with the free vibration analysis of open circular cylindrical shells with either the two straight edges or the two curved edges simply supported and the remaining two edges supported by arbitrary classical boundary conditions. Based on the Donnell-Mushtari-Vlasov thin shell theory, an analytical solution of the traveling wave form along the simply supported edges and the modal wave form along the remaining two edges is obtained. With such a unidirectional traveling wave form solution, the method of the reverberation-ray matrix is introduced to derive the equation of natural frequencies of the shell with different classical boundary conditions. The exact solutions for natural frequencies of the open circular cylindrical shell are obtained with the employment of a golden section search algorithm. The calculation results are compared with those obtained by the finite element method and the methods in the available literature. The influence of length, thickness, radius, included angle, and the boundary conditions of the open circular cylindrical shell on the natural frequencies is investigated. The exact calculation results can be used as benchmark values for researchers to check their numerical methods and for engineers to design structures with thin shell components.

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Dynamic Response and Wave Propagation in Plane Trusses and Frames

Cited by 109 publications

References 6 publications

Recursive formulations for the method of reverberation-ray matrix and the application

Recursive formulations for the method of reverberation-ray matrix and the application

Wave propagation in equivalent continuums representing truss lattice materials

Exact free vibration analysis of open circular cylindrical shells by the method of reverberation-ray matrix

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