Exact Frequency Equation of a Linear Structure Carrying Lumped Elements Using the Assumed Modes Method

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
Philip D. Cha ◽  
Siyi Hu
Keyword(s):  
Linear Structure ◽  
Frequency Equation ◽  
Governing Equations ◽  
Linear Structures ◽  
Combined System ◽  
Digamma Function ◽  
Simply Supported ◽  
Lumped Elements ◽  
Finite Sum ◽  
Infinite Sum

Combined systems consisting of linear structures carrying lumped attachments have received considerable attention over the years. In this paper, the assumed modes method is first used to formulate the governing equations of the combined system, and the corresponding generalized eigenvalue problem is then manipulated into a frequency equation. As the number of modes used in the assumed modes method increases, the approximate eigenvalues converge to the exact solutions. Interestingly, under certain conditions, as the number of component modes goes to infinity, the infinite sum term in the frequency equation can be reduced to a finite sum using digamma function. The conditions that must be met in order to reduce an infinite sum to a finite sum are specified, and the closed-form expressions for the infinite sum are derived for certain linear structures. Knowing these expressions allows one to easily formulate the exact frequency equations of various combined systems, including a uniform fixed–fixed or fixed-free rod carrying lumped translational elements, a simply supported beam carrying any combination of lumped translational and torsional attachments, or a cantilever beam carrying lumped translational and/or torsional elements at the beam's tip. The scheme developed in this paper is easy to implement and simple to code. More importantly, numerical experiments show that the eigenvalues obtained using the proposed method match those found by solving a boundary value problem.

Applying Sherman-Morrison-Woodbury Formulas to Analyze the Free and Forced Responses of a Linear Structure Carrying Lumped Elements

2006 ◽  
Vol 129 (3) ◽  
pp. 307-316 ◽  
Author(s):  
Philip D. Cha ◽  
Nathanael C. Yoder
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Linear Structure ◽  
Frequency Equation ◽  
Continuous Linear ◽  
Computationally Efficient ◽  
Combined System ◽  
System A ◽  
Lumped Elements ◽  
Forced Responses

A simple approach is proposed that can be used to analyze the free and forced responses of a combined system, consisting of an arbitrarily supported continuous structure carrying any number of lumped attachments. The assumed modes method is utilized to formulate the equations of motion, which conveniently leads to a form that allows one to exploit the Sherman-Morrison or the Sherman-Morrison-Woodbury formulas to compute the natural frequencies and frequency response of the combined system. Rather than solving a generalized eigenvalue problem to obtain the natural frequencies of the system, a frequency equation is formulated whose solution can be easily solved either numerically or graphically. In order to determine the response of the structure to a harmonic input, a method is formulated that leads to a reduced matrix whose inverse yields the same result as the traditional method, which requires the inversion of a larger matrix. The proposed scheme is easy to code, computationally efficient, and can be easily modified to accommodate arbitrarily supported continuous linear structures that carry any number of miscellaneous lumped attachments.

The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

Post-Buckling of Piezoelectric Thin Plates

2015 ◽  
Vol 15 (07) ◽  
pp. 1540020 ◽  
Author(s):  
Michael Krommer ◽  
Hans Irschik
Keyword(s):  
Nonlinear Partial Differential Equation ◽  
Nonlinear Behavior ◽  
Dirichlet Boundary Conditions ◽  
Thin Plates ◽  
Governing Equations ◽  
Simply Supported ◽  
Buckling Behavior ◽  
Single Term ◽  
Post Buckling ◽  
Special Case

In the present paper, the geometrically nonlinear behavior of piezoelastic thin plates is studied. First, the governing equations for the electromechanically coupled problem are derived based on the von Karman–Tsien kinematic assumption. Here, the Berger approximation is extended to the coupled piezoelastic problem. The general equations are then reduced to a single nonlinear partial differential equation for the special case of simply supported polygonal edges. The nonlinear equations are approximated by using a problem-oriented Ritz Ansatz in combination with a Galerkin procedure. Based on the resulting equations the buckling and post-buckling behavior of a polygonal simply supported plate is studied in a nondimensional form, where the special geometry of the polygonal plate enters via the eigenvalues of a Helmholtz problem with Dirichlet boundary conditions. Single term as well as multi-term solutions are discussed including the effects of piezoelectric actuation and transverse force loadings upon the solution. Novel results concerning the buckling, snap through and snap buckling behavior are presented.

Free Vibration of a Circular Cylindrical Shell Elastically Restrained by Axially Spaced Springs

1983 ◽  
Vol 50 (3) ◽  
pp. 544-548 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Muramoto
Keyword(s):  
Cylindrical Shell ◽  
Free Vibration ◽  
Circular Cylindrical Shell ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Governing Equations ◽  
Structure Matrix ◽  
The Matrix ◽  
Circular Cylindrical Shells ◽  
Elastically Restrained

An analysis is presented for the free vibration of a circular cylindrical shell restrained by axially spaced elastic springs. The governing equations of vibration of a circular cylindrical shell are written as a coupled set of first-order differential equations by using the transfer matrix of the shell. Once the matrix has been determined, the entire structure matrix is obtained by the product of the transfer matrices and the point matrices at the springs, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method is applied to circular cylindrical shells supported by axially equispaced springs of the same stiffness, and the natural frequencies and the mode shapes of vibration are calculated numerically.

Multi-Pulse Chaotic Motion for a Non-Autonomous Buckled Plate by Using the Extended Melnikov Method

2007 ◽  
Author(s):  
Wei Zhang ◽  
Jun-Hua Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Thin Plate ◽  
Chaotic Dynamics ◽  
Melnikov Method ◽  
Type Equation ◽  
Rectangular Thin Plate ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Extended Melnikov Method ◽  
Two Degree Of Freedom

The multi-pulse Shilnikov orbits and chaotic dynamics for a parametrically excited, simply supported rectangular buckled thin plate are studied by using the extended Melnikov method. Based on von Karman type equation and the Galerkin’s approach, two-degree-of-freedom nonlinear system is obtained for the rectangular thin plate. The extended Melnikov method is directly applied to the non-autonomous governing equations of the thin plate. The results obtained here show that the multipulse chaotic motions can occur in the thin plate.

Impulsive Loading of a Simply Supported Circular Rigid Plastic Plate

1968 ◽  
Vol 35 (1) ◽  
pp. 59-65 ◽  
Author(s):  
Norman Jones
Keyword(s):  
Yield Condition ◽  
Impulsive Loading ◽  
Bending Moments ◽  
Circular Plates ◽  
Governing Equations ◽  
Rigid Plastic ◽  
Static Loads ◽  
Simply Supported ◽  
Experimental Values ◽  
Theoretical Procedure

It is clear from a survey of literature on the dynamic deformation of rigid-plastic plates that most work has been focused on plates in which either membrane forces or bending moments alone are considered important, while the combined effect of membrane forces and bending moments on the behavior of plates under static loads and beams under dynamic loads is fairly well established. This paper, therefore, is concerned with the behavior of circular plates loaded dynamically and with deflections in the range where both bending moments and membrane forces are important. A general theoretical procedure is developed from the equations for large deflections of plates and a simplified yield condition due to Hodge. The results obtained when solving the governing equations for the particular case of a simply supported circular plate loaded with a uniform impulsive velocity are found to compare favorably with the corresponding experimental values recorded by Florence.

Post-Buckling Analysis of FGM Beam Subjected to Non-Conservative Forces and In-Plane Thermal Loading

2012 ◽  
Vol 152-154 ◽  
pp. 474-479
Author(s):  
Feng Qun Zhao ◽  
Zhong Min Wang ◽  
Rui Ping Zhang
Keyword(s):  
Shooting Method ◽  
Thermal Loading ◽  
Governing Equations ◽  
Temperature Dependent ◽  
Tangential Load ◽  
Simply Supported ◽  
Runge Kutta Method ◽  
Buckling Behavior ◽  
Post Buckling ◽  
Fgm Beam

Based on the Kirchhoff large deformation theory, the post-buckling behavior of right movable simply supported FGM beam subjected to non-conservative forces and in-plane thermal loading was analyzed in this paper. The temperature-dependent and spatially dependent material properties of the FGM beam were assumed to vary through the thickness. The nonlinear governing equations of FGM beam subjected to a uniform distributed tangential load along the central axis and in-plane thermal loading were derived. Then, a shooting method and Runge-kutta method are employed to numerically solve the resulting equations. The post-buckling equilibrium paths of the FGM beam with different parameters were plotted, and the effects of non-conservative force, temperature, gradient index of FGM on the post-buckling behavior of right movable simply supported FGM beams were analyzed.

scholarly journals A quantum algorithm to approximate the linear structures of Boolean functions

2016 ◽  
Vol 28 (1) ◽  
pp. 1-13 ◽  
Author(s):  
HONGWEI LI ◽  
LI YANG
Keyword(s):  
Boolean Functions ◽  
High Probability ◽  
Success Probability ◽  
Quantum Algorithm ◽  
Linear Structure ◽  
Differential Uniformity ◽  
Linear Structures ◽  
High Success Probability ◽  
Time Required ◽  
Quantum Speedup

A quantum algorithm to determine approximations of linear structures of Boolean functions is presented and analysed. Similar results have already been published (see Simon's algorithm) but only for some promise versions of the problem, and it has been shown that no exponential quantum speedup can be obtained for the general (no promise) version of the problem. In this paper, no additional promise assumptions are made. The approach presented is based on the method used in the Bernstein–Vazirani algorithm to identify linear Boolean functions and on ideas from Simon's period finding algorithm. A proper combination of these two approaches results here to a polynomial-time approximation to the linear structures set. Specifically, we show how the accuracy of the approximation with high probability changes according to the running time of the algorithm. Moreover, we show that the time required for the linear structure determine problem with high success probability is related to so called relative differential uniformity δf of a Boolean function f. Smaller differential uniformity is, shorter time is needed.

Experimental Modal Analysis of a Fixed-Fixed Beam Under Axial Force

2017 ◽  
Author(s):  
Pezhman A. Hassanpour ◽  
Khaled Alghemlas ◽  
Adam Betancourt
Keyword(s):  
Analytical Model ◽  
Axial Force ◽  
Experimental Approach ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Governing Equations ◽  
Experimental Procedure ◽  
Resonance Frequencies ◽  
Fixed Beam

In this paper, an experimental procedure is proposed for determining the resonance frequencies and mode shapes of vibration of a fixed-fixed beam. Since it is fixed at both ends, the beam may sustain an axial force due to several factors including the fasteners and/or change of temperature. The analytical governing equations of motion, frequency equation, and mode shapes of vibration are presented. The analytical model is used to justify the experimental approach as well as interpretation of the experiment data. In this study, a hammer is used to excite the beam, and then the vibration of the beam is observed and recorded at two different points on the beam using two laser Doppler vibrometers. The data from the vibrometers are used to extract the resonance frequencies and mode shapes of vibrations. Using the analytical model, the axial force in the beam is estimated.

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