A state space approach to elasticity

Bahar, Leon Y.

doi:10.1016/0016-0032(75)90082-4

Cited by 101 publications

(41 citation statements)

References 5 publications

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“…In the past few decades, the application of state-space approach has attracted the attention of a number of researchers [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] . This approach considers the transverse stresses and displacements synchronously.…”

Section: Introductionmentioning

confidence: 99%

Modeling vibration behavior of delaminated composite laminates using meshfree method in Hamilton system

Chen

Wang

Qing

2015

Appl. Math. Mech.-Engl. Ed.

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Free vibration analysis of composite laminates with delaminations is performed based on a three-dimensional semi-analytical model established by introducing the local radial point interpolation method (LRPIM) into a Hamilton system. The governing equation is derived with a transfer matrix technique and a spring layer model based on a local weak-form equivalent to the modified Hellinger-Reissner variational principle. Main superiority of the present model is that the scale of the governing equation involves only the so-called state variables at the top and bottom surfaces, and is insensitive to the thickness and the layer number of the composite laminates. Several numerical examples for analyzing the vibration frequencies and mode shapes of delaminated composite beams and plates are given to validate the model. The results are in good agreement with the pre-existing results.

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Section: Introductionmentioning

confidence: 99%

Modeling vibration behavior of delaminated composite laminates using meshfree method in Hamilton system

Chen

Wang

Qing

2015

Appl. Math. Mech.-Engl. Ed.

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“…Following the spatial state-space approach developed by Bahar (1975), by direct substitution of the kinematic relations (2b) into the constitutive relations (1), and subsequent incorporation of the results into the equations of motion/equilibrium (3), after some tedious manipulations, one advantageously arrives at…”

Section: The Spatial State Space Modelmentioning

confidence: 99%

Active vibration control of an arbitrary thick piezolaminated beam with imperfectly integrated sensor and actuator layers

Hasheminejad

Vahedi

Hoshangi

2014

Lat. Am. j. solids struct.

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A spatial state-space formulation based on the linear twodimensional piezoelasticity theory and involving local/global transfer matrices is applied to investigate the active vibration suppression of a simply supported, arbitrarily thick, orthotropic elastic beam, imperfectly integrated with spatially distributed piezoelectric actuator and sensor layers on its top and bottom surfaces, respectively. A linear spring-layer model is adopted to simulate the bonding imperfections between the host structure and the piezoelectric layers. To assist control system design, system identification is conducted by applying a frequency domain subspace approximation method with N4SID algorithm based on the first five structural modes of the system. The state space model is constructed from system identification and used for state estimation and development of control algorithm. A linear quadratic Gaussian (LQG) optimal controller is subsequently designed and simulated based on the identified model in order to actively control the response of the smart structure in both frequency and time domains.

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“…It was proved that all the Saint-Venant solutions may be obtained directly via the zero eigenvalue solutions and all their Jordan normal forms of the corresponding Hamiltonian operator matrix and the Saint-Venant principle correspond to neglect of all the nonzero eigenvalue solutions, in which the non-zero eigenvalue gives the decay rate [21][22][23][24][25][26][27][28][29][30] . Recently, some researches on symplectic solution have drawn attention to the advantage of the state variable method [31][32][33][34][35][36][37][38][39][40][41][42] . It is noted that classical elasticity is well suited to the state space approach, including important biorthogonality relations.…”

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confidence: 99%

A biorthogonality relationship for three-dimensional couple stress problem

Luo

Liu

2009

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

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The duality solution for elasticity and the biorthogonality relationship have been well researched. Now the couple stress theory becomes a new research spot but there is few research for the biorthogonality relationship for couple stress theory comparing to classical elasticity. A new state vector is presented for three dimensional couple stress problems of prismatic structures. A new biorthogonality relationship of couple stress is discovered. The dual partial differential equations of couple stress problem are derived by the new state vector. By two important identical equations the new biorthogonality relationship is proved based on the method of separation of variables. The symplectic orthogonality relationship to three dimensional couple stress theory may be decomposed into two independently and symmetrically orthogonality relationships. The new biorthogonality relationship includes the symplectic orthogonality relationship. The biorthogonality relationship of couple stress may also be degenerated into the theory of elasticity. The new state vector and biorthogonality relationship provide theoretic foundation for the research on the schemes of separation of variables and eigenfunction expansion of couple stress theory.couple stress, duality solution, biorthogonality relationship, Hamiltonian, state spaceThe solution of elasticity is a classical problem which has lasted for more than a century. Because of the complication of its partial differential equations, the semi-inverse solution became the classical solution method in elasticity [1] . By the analogy theory between computational structural mechanics and optimal control, the Hamiltonian system theory was introduced into the theory of elasticity [2][3][4][5][6] . Based on the separation of variables of the Hamiltonian system, it was proved that among its eigenfunction-vectors state space there existed a symplectic orthogonality relationship and the corresponding expansion theorem. The duality solution of elasticity has reached a new stage [7][8][9][10][11][12][13][14][15][16][17][18][19][20] .The Saint-Venant problem and the Saint-Venant principle were unified by the direct method. It was proved that all the Saint-Venant solutions may be obtained directly via the zero eigenvalue solutions and all their Jordan normal forms of the corresponding Hamiltonian operator matrix and the Saint-Venant principle correspond to neglect of all the nonzero eigenvalue solutions, in which the non-zero eigenvalue gives the decay rate [21][22][23][24][25][26][27][28][29][30] . Recently, some researches on symplectic solution have drawn attention to the advantage of the state variable method [31][32][33][34][35][36][37][38][39][40][41][42] . It is noted that classical elasticity is well suited to the state space approach, including important biorthogonality relations. Vasil and Lur [43] presented a method of homogeneous solutions and biorthogonal expansions in the plane problem of the theory of elasticity. It was proved that the biorthogonality condition was equivalent to a g...

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A state space approach to elasticity

Cited by 101 publications

References 5 publications

Modeling vibration behavior of delaminated composite laminates using meshfree method in Hamilton system

Modeling vibration behavior of delaminated composite laminates using meshfree method in Hamilton system

Active vibration control of an arbitrary thick piezolaminated beam with imperfectly integrated sensor and actuator layers

A biorthogonality relationship for three-dimensional couple stress problem

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