• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.15 no.4
• /
• pp.579-589
• /
• 2002

In this paper, a Wittrick-Williams algorithm is developed for the spectral element model of an elastic-piezoelectric two-layer active beam. This algorithm may help calculate all the required natural frequencies, which lie below any chosen frequency, without the possibility of missing any due to close grouping or due to the abrupt sign changes of the determinant of spectral element matrix via infinity instead of via zero. A uniform active beam and a partially patched active beam are considered as the illustrative examples to confirm the present algorithm.

• Structural Engineering and Mechanics
• /
• v.62 no.6
• /
• pp.759-769
• /
• 2017

The effect of central axial load on natural frequencies of various thin-walled beams, are investigated by some researchers using different methods such as finite element, transfer matrix and dynamic stiffness matrix methods. However, there are situations that the load will be off centre. This type of loading is called eccentric load. The effect of the eccentricity of axial load on the natural frequencies of asymmetric thin-walled beams is a subject that has not been investigated so far. In this paper, the mentioned effect is studied using exact dynamic stiffness matrix method. Flexure and torsion of the aforesaid thin-walled beam is based on the Bernoulli-Euler and Vlasov theories, respectively. Therefore, the intended thin-walled beam has flexural rigidity, saint-venant torsional rigidity and warping rigidity. In this paper, the Hamilton‟s principle is used for deriving governing partial differential equations of motion and force boundary conditions. Throughout the process, the uniform distribution of mass in the member is accounted for exactly and thus necessitates the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick-Williams algorithm. Finally, in order to verify the accuracy of the presented theory, the numerical solutions are given and compared with the results that are available in the literature and finite element solutions using ABAQUS software.

• Structural Engineering and Mechanics
• /
• v.24 no.2
• /
• pp.213-225
• /
• 2006

The propagation of non-uniformly modulated, evolutionary random waves in viscoelastic, transversely isotropic, stratified materials is investigated. The theory is developed in the context of a multi-layered soil medium overlying bedrock, where the material properties of the bedrock are considered to be much stiffer than those of the soil and the power spectral density of the random excitation is assumed to be known at the bedrock. The governing differential equations are first derived in the frequency/wave-number domain so that the displacement response of the ground may be computed. The eigen-solution expansion method is then used to solve for the responses of the layers. This utilizes the precise integration method, in combination with the extended Wittrick-Williams algorithm, to obtain all the eigen-solutions of the ordinary differential equation. The recently developed pseudo-excitation method for structural random vibration is then used to determine the solution of the layered soil responses.

• Structural Engineering and Mechanics
• /
• v.19 no.5
• /
• pp.551-566
• /
• 2005

Using two different, but related approaches, an exact dynamic stiffness matrix for a two-part beam-mass system is developed from the free vibration theory of a Bernoulli-Euler beam. The first approach is based on matrix transformation while the second one is a direct approach in which the kinematical conditions at the interfaces of the two-part beam-mass system are satisfied. Both procedures allow an exact free vibration analysis of structures such as a plane or a space frame, consisting of one or more two-part beam-mass systems. The two-part beam-mass system described in this paper is essentially a structural member consisting of two different beam segments between which there is a rigid mass element that may have rotatory inertia. Numerical checks to show that the two methods generate identical dynamic stiffness matrices were performed for a wide range of frequency values. Once the dynamic stiffness matrix is obtained using any of the two methods, the Wittrick-Williams algorithm is applied to compute the natural frequencies of some frameworks consisting of two-part beam-mass systems. Numerical results are discussed and the paper concludes with some remarks.

• Structural Engineering and Mechanics
• /
• v.38 no.2
• /
• pp.195-210
• /
• 2011

Bernoulli-Euler beam theory is used to develop an exact dynamic stiffness matrix for the flexural-torsional coupled motion of a three-dimensional, axially loaded, thin-walled beam of doubly asymmetric cross-section. This is achieved through solution of the differential equations governing the motion of the beam including warping stiffness. The uniform distribution of mass in the member is also accounted for exactly, thus necessitating the solution of a transcendental eigenvalue problem. This is accomplished using the Wittrick-Williams algorithm. Finally, examples are given to confirm the accuracy of the theory presented, together with an assessment of the effects of axial load and loading eccentricity.

• Advances in aircraft and spacecraft science
• /
• v.1 no.3
• /
• pp.291-310
• /
• 2014

An advanced model for the linear flutter analysis is introduced in this paper. Higher-order beam structural models are developed by using the Carrera Unified Formulation, which allows for the straightforward implementation of arbitrarily rich displacement fields without the need of a-priori kinematic assumptions. The strong form of the principle of virtual displacements is used to obtain the equations of motion and the natural boundary conditions for beams in free vibration. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The resulting dynamic stiffness matrix is used with particular reference to the Wittrick-Williams algorithm to carry out free vibration analyses. According to the doublet lattice method, the natural mode shapes are subsequently used as generalized motions for the generation of the unsteady aerodynamic generalized forces. Finally, the g-method is used to conduct flutter analyses of both isotropic and laminated composite lifting surfaces. The obtained results perfectly match those from 1D and 2D finite elements and those from experimental analyses. It can be stated that refined beam models are compulsory to deal with the flutter analysis of wing models whereas classical and lower-order models (up to the second-order) are not able to detect those flutter conditions that are characterized by bending-torsion couplings.

• Structural Engineering and Mechanics
• /
• v.58 no.5
• /
• pp.847-868
• /
• 2016

In this study, the free vibration analysis of axially moving beams is investigated according to Reddy-Bickford beam theory (RBT) by using dynamic stiffness method (DSM) and differential transform method (DTM). First of all, the governing differential equations of motion in free vibration are derived by using Hamilton's principle. The nondimensionalised multiplication factors for axial speed and axial tensile force are used to investigate their effects on natural frequencies. The natural frequencies are calculated by solving differential equations using analytical method (ANM). After the ANM solution, the governing equations of motion of axially moving Reddy-Bickford beams are solved by using DTM which is based on Finite Taylor Series. Besides DTM, DSM is used to obtain natural frequencies of moving Reddy-Bickford beams. DSM solution is performed via Wittrick-Williams algorithm. For different boundary conditions, the first three natural frequencies that calculated by using DTM and DSM are tabulated in tables and are compared with the results of ANM where a very good proximity is observed. The first three mode shapes and normalised bending moment diagrams are presented in figures.

• Advances in nano research
• /
• v.14 no.2
• /
• pp.127-139
• /
• 2023

This paper studies the free vibration behavior of bi-dimensional functionally graded (BFG) nanobeams subjected to arbitrary boundary conditions. According to Eringen's nonlocal theory and Hamilton's principle, the underlying equations of motion have been obtained for BFG nanobeams. Moreover, the variable substitution method is utilized to establish the structure's state-space differential equations, followed by forming the dynamic stiffness matrix based on state-space differential equations. In order to compute the natural frequencies, the current study utilizes the Wittrick-Williams algorithm as a solution technique. Moreover, the nonlinear vibration frequencies calculated by employing the proposed method are compared to the frequencies obtained in previous studies to evaluate the proposed method's performance. Some illustrative numerical examples are also given in order to study the impacts of the nonlocal parameters, material property gradient indices, nanobeam length, and boundary conditions on the BFG nanobeam's frequency. It is found that reducing the nonlocal parameter will usually result in increased vibration frequencies.

• Earthquakes and Structures
• /
• v.10 no.6
• /
• pp.1253-1272
• /
• 2016

For a nonlinear control system, there are many uncertainties, such as the structural model, controlled parameters and external loads. Although the significant progress has been achieved on the robust control of nonlinear systems through some researches on this issue, there are still some limitations, for instance, the complicated solving process, weak conservatism of system, involuted structures and high order of controllers. In this study, the computational structural mechanics and optimal control theory are adopted to address above problems. The induced norm is the eigenvalue problem in structural mechanics, i.e., the elastic stable Euler critical force or eigenfrequency of structural system. The segment mixed energy is introduced with a precise integration and an extended Wittrick-Williams (W-W) induced norm calculation method. This is then incorporated in the market-based control (MBC) theory and combined with the force analogy method (FAM) to solve the MBC robust strategy (R-MBC) of nonlinear systems. Finally, a single-degree-of-freedom (SDOF) system and a 9-stories steel frame structure are analyzed. The results are compared with those calculated by the $H{\infty}$-robust (R-$H{\infty}$) algorithm, and show the induced norm leads to the infinite control output as soon as it reaches the critical value. The R-MBC strategy has a better control effect than the R-$H{\infty}$ algorithm and has the advantage of strong strain capacity and short online computation time. Thus, it can be applied to large complex structures.