The Forced Vibration and Boundary Control of Pretwisted Timoshenko Beams With General Time Dependent Elastic Boundary Conditions

Lin, Shueei-Muh; Lee, Sen-Yung

doi:10.1006/jsvi.2001.4084

Cited by 11 publications

(5 citation statements)

References 27 publications

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“…To find the solution for the differential equation of the plate with variable coefficients and non-homogeneous elastic boundary conditions, one generalizes the method given by Lin and Lee (2002) for beam equation with variable coefficients and non-homogeneous elastic boundary conditions by taking…”

Section: Change Of Variablementioning

confidence: 99%

“…In general, by introducing a general change of the dependent variable with shifting functions, the original system with four non-homogeneous boundary conditions is transformed into a system composed of a timeindependent governing differential equation and a system with four homogeneous boundary conditions. Lee and Lin (1998) and Lin and Lee (2002) have investigated the dynamic analysis of non-uniform Timoshenko beams with time-dependent boundary conditions. In fact, they have generalized the method of Mindlin-Goodman to develop a solution procedure for studying the vibrations of an unpretwisted and pretwisted non-uniform Timoshenko beam with general time-dependent boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

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Bending vibration and buckling of non-uniform plate with time-dependent boundary conditions

Saeidifar

Ohadi

2010

Journal of Vibration and Control

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Current work presents an exact solution for investigating the buckling, free vibration and dynamic response of plates with variable parameters in one direction. The plate has two opposite simply-supported edges subjected to in-plane loads with desired functions, whereas other edges may have various general time-dependent boundary conditions. The solution procedure is obtained by developing and generalizing Mindlin-Goodman and Frobenius methods. First, by introducing a general change of dependent variable with shifting functions, the original system with four nonhomogeneous boundary conditions is transformed into a system composed of a time-independent governing differential equation and a system with four homogeneous boundary conditions. Then, assuming that the transverse displacement varies as a sinusoidal function, the governing partial differential equation of the plate is reduced to an ordinary differential equation in terms of y with variable coefficients. An exact solution is obtained as a power series to solve the resulting equation. Applying boundary conditions of two other edges yields the problem of finding eigenvalues of a fourth order characteristic determinant. Some examples are illustrated for validation with previous studies and the finite element method. Eventually, in other examples, the effect of some parameters on vibration behavior of the system is investigated.

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Section: Change Of Variablementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bending vibration and buckling of non-uniform plate with time-dependent boundary conditions

Saeidifar

Ohadi

2010

Journal of Vibration and Control

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“…In addition, Aravamudan and Murthy [25] combined the perturbation method, the Galerkin method, and the M-G method to study the nonlinear vibration of beams with time-dependent boundary conditions. In the 1990s, Lee and Lin [26][27][28] and Lin [29] proposed the shifting function method, which modified the M-G method to analyze Bernoulli-Euler beams, Timoshenko beams, and pre-twisted nonuniform beams with time-dependent boundary conditions. Lee et al [30] extended the prior studies and considered the transverse vibration problem to develop a set of shifting functions that consisted of fundamental solutions of nonuniform beams and to discuss the physical meaning of the shifting functions.…”

Section: Introductionmentioning

confidence: 99%

An Analytic Solution for the Dynamic Behavior of a Cantilever Beam with a Time-Dependent Spring-like Actuator

Chang¹,

Tu²,

Huang

2023

Axioms

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The purpose of this study is to derive an analytical solution for a cantilever beam with a novel spring-like actuator that behaves like a time-dependent spring and to study the dynamic behavior of the system. A time-dependent spring was set at the free end of the cantilever beam to model the novel spring-like actuator. First, the boundary conditions were transformed from being nonhomogeneous to being homogeneous using the shifting function method. The solution of the analytic series was then obtained by using the expansion theorem method. The correctness of the proposed analytical solution was verified by comparing the results with those obtained via the separation of variables in the special extreme case of a constant spring coefficient. We took the free end of a cantilever beam with harmonic spring stiffness and an external periodic unit load as an example. The influence of the actuator parameters, such as the effect of the magnitude and the frequency of the time-dependent spring stiffness on the resonance frequency, was investigated. An important new result was found, i.e., that the resonance frequency is clearly dependent on the magnitude and the frequency of the spring-like actuator in the first two modes, but not in the third and fourth modes. In practical engineering applications, system resonance can be avoided by adjusting the magnitude and frequency of the actuator.

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“…Hsueh (1999) obtained analytical and closed-form frequency responses for the longitudinal vibration of multi-stepped rods with or without lumped elements, having nonclassical boundary conditions. The bending–bending forced vibration behavior of pretwisted nonuniform Timoshenko beams was studied by Lin and Lee (2002), using Hamilton's principle. Using the shooting method, the geometric nonlinear large amplitude forced vibration of thin/thick beams was studied by Ribeiro (2004), based on p -version hierarchical finite element formulation.…”

Section: Introductionmentioning

confidence: 99%

A numerical analysis of large amplitude beam vibration under different boundary conditions and excitation patterns

Das

Sahoo

Saha

2011

Journal of Vibration and Control

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Large amplitude forced vibration behavior of thin beams under harmonic excitation is studied, incorporating the effect of geometric nonlinearity. Large amplitude free vibration analysis of the same system is also carried out to obtain the backbone curves. When a very small loading is considered, the system response becomes almost identical to the backbone curve, which confirms the validity of the large amplitude study. The sets of governing equations for both forced and free vibration problems are obtained using Hamilton’s principle, whereas the same for the static problem are obtained by applying the variational form of the energy principle. The solution of the static problem provides system stiffness as a function of deflection and it becomes an input to the corresponding free vibration problem. The results are presented in the nondimensional frequency–amplitude plane in the form of response curves, and the related backbone curves are shown to represent the complete system behavior. Results for different combinations of classical flexural boundary conditions and loading patterns are reported.

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The Forced Vibration and Boundary Control of Pretwisted Timoshenko Beams With General Time Dependent Elastic Boundary Conditions

Cited by 11 publications

References 27 publications

Bending vibration and buckling of non-uniform plate with time-dependent boundary conditions

Bending vibration and buckling of non-uniform plate with time-dependent boundary conditions

An Analytic Solution for the Dynamic Behavior of a Cantilever Beam with a Time-Dependent Spring-like Actuator

A numerical analysis of large amplitude beam vibration under different boundary conditions and excitation patterns

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