Vibration Analysis of Thin Plates Subject to Piezoelectric Actuation: A New Perspective in Modeling and Numerical Analysis

2011 ◽  
Author(s):  
Parikshit Mehta ◽  
Nader Jalili
Keyword(s):  
Piezoelectric Actuator ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Model Development ◽  
Thin Plates ◽  
Shape Functions ◽  
Assumed Mode Method ◽  
New Perspective ◽  
Forced Vibration Analysis

This paper undertakes model development and numerical simulations of vibration problem of piezoelectrically actuated thin plates with a holistic perspective. Constitutive laws governing piezoelectric actuator are integrated with the potential and kinetic energies of combined plate-actuator system. The equations of motions are derived using variational approach and verified with results obtained by Newton’s equilibrium approach. It is verified that the field coupled components associated with piezoelectric actuator appear as distributed moments over the area of the actuator. The equations of motion are solved using modal analysis deploying Raleigh Ritz method utilizing Boundary Characteristic Orthogonal Polynomials (BCOP). The shape functions generated using this method is used in Assumed Mode Method (AMM) to numerically simulate forced vibration analysis. Since Raleigh Ritz analysis with BCOP can be deployed with the plates of all the geometries, minor modifications in selecting the shape functions enables one to use the same method to calculate natural frequencies of annular plate as well.

scholarly journals FE-Meshfree QUAD4 Element with Modified Radial Point Interpolation Function for Structural Dynamic Analysis

2019 ◽  
Vol 2019 ◽  
pp. 1-23 ◽  
Author(s):  
Hongjun Zhang ◽  
Guangsong Chen ◽  
Linfang Qian ◽  
Jia Ma
Keyword(s):  
Forced Vibration ◽  
Vibration Analysis ◽  
Numerical Solutions ◽  
Integration Scheme ◽  
Interpolation Function ◽  
Shape Functions ◽  
Structural Dynamic ◽  
Radial Point Interpolation ◽  
Point Interpolation ◽  
Forced Vibration Analysis

The partition-of-unity method based on FE-Meshfree QUAD4 element synthesizes the respective advantages of meshfree and finite element methods by exploiting composite shape functions to obtain high-order global approximations. This method yields high accuracy and convergence rate without necessitating extra nodes or DOFs. In this study, the FE-Meshfree method is extended to the free and forced vibration analysis of two-dimensional solids. A modified radial point interpolation function without any supporting tuning parameters is applied to construct the composite shape functions. The governing equations of elastodynamic problem are transformed into a standard weak formulation and then discretized into time-dependent equations which are solved via Bathe time integration scheme to conduct the forced vibration analysis. Several numerical test problems are solved and compared against previously published numerical solutions. Results show that the proposed FE-Meshfree QUAD4 element owns greater tolerance for mesh distortion and provides more accurate solutions.

Elastodynamic Analysis of a Completely Elastic System

1977 ◽  
Vol 99 (3) ◽  
pp. 604-609 ◽  
Author(s):  
D. Kohli ◽  
D. Hunter ◽  
G. N. Sandor
Keyword(s):  
Vibration Analysis ◽  
Elastic System ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Lagrange Equations ◽  
Elastic Supports ◽  
Linear Ordinary Differential Equations ◽  
Elastic Links ◽  
Inertia Forces ◽  
Crank Mechanism

The completely elastic system considered for this vibration analysis consists of an offset slider-crank mechanism having (a) elastic supports and mountings of the mechanism permitting translational vibrations of the shafts and supports, (b) elastic shafts permitting torsional vibrations, (c) elastic links of the mechanism which deform due to external or internal body forces and allow flexural and axial vibrations. Both the effect of the deformations caused by the inertia forces in the mechanism links, shafts, and supports and the effect of change in the inertia forces due to these deformations are taken into account in constructing a general mathematical model for conducting elastodynamic analysis. The rigid displacements (finite and infinitesimal) of the mechanism links due to deformations in the support are evaluated using a truncated Taylor series approximation. Deformation in the links caused by the inertia forces is approximated by a finite number of terms in a Fourier series using the Raleigh-Ritz method. The Lagrange equations of motion are used to obtain coupled time varying linear ordinary differential equations of motion for the vibration analysis of the slider-crank mechanism. The method in general may be applied to any planar or spatial system consisting of elastic links, elastic shafts, and elastic supports. Numerical examples are presented for illustration.

scholarly journals Free and Forced Vibration Analysis of Airtight Cylindrical Vessels with Doubly Curved Shells of Revolution by Using Jacobi-Ritz Method

2017 ◽  
Vol 2017 ◽  
pp. 1-20 ◽  
Author(s):  
Fuzhen Pang ◽  
Haichao Li ◽  
Kwangnam Choe ◽  
Dongyan Shi ◽  
Kwanghun Kim
Keyword(s):  
Forced Vibration ◽  
Vibration Analysis ◽  
Jacobi Polynomials ◽  
Ritz Method ◽  
The Finite Element Method ◽  
Shells Of Revolution ◽  
Vessel Structure ◽  
Forced Vibration Analysis ◽  
Doubly Curved ◽  
Thin Shell Theory

This paper presents free and forced vibration analysis of airtight cylindrical vessels consisting of elliptical, paraboloidal, and cylindrical shells by using Jacobi-Ritz Method. In this research, the theoretical model for vibration analysis is formulated by Flügge’s thin shell theory and the solution is obtained by Rayleigh-Ritz method. The vessel structure is divided into shell components (i.e., ellipsoid, parabolic, and cylinder) and their segments, and each displacement field of shell segments is represented by the Jacobi polynomials and the standard Fourier series. The continuous conditions at the interface are modeled by using the spring stiffness technique. The reliability and the accuracy of the present method are verified by comparing the results of the proposed method with the results of the previous literature and the finite element method (FEM). Moreover, some numerical results for free and forced vibration of elliptical-cylindrical-elliptical vessel (ECE vessel) and paraboloidal-cylindrical-elliptical vessel (PCE vessel) are reported.

Free Vibration of Plates of Various Shapes with Intermediate Point Supports by the Hp-Cloud Method and Lagrange Multiplier

2016 ◽  
Vol 16 (09) ◽  
pp. 1550055 ◽  
Author(s):  
Sajad Jamshidi ◽  
Mojtaba Azhari ◽  
Hossein Amoushahi
Keyword(s):  
Lagrange Multiplier ◽  
Natural Frequencies ◽  
Cell Structure ◽  
Ritz Method ◽  
Thin Plates ◽  
Plate Boundaries ◽  
Shape Functions ◽  
Intermediate Point ◽  
Simply Supported ◽  
Kronecker Delta

The Hp-Cloud meshless method was developed to study the dynamic analysis of arbitrarily shaped thin plates with intermediate point supports. By proposing a special pattern for the influence radius of nodes and a polynomial type of enrichment function, the Hp-Cloud shape functions with Kronecker delta property were constructed. They can satisfy the zero deflection conditions for the field nodes at the point supports. The results obtained from these shape functions agree well with the previous ones, showing good accuracy and convergence. For plates with sharp corners, it is not possible to construct the Hp-Cloud shape function with Kronecker delta property. To this end, the Lagrange multiplier method was used for enforcing the boundary conditions. The computations were carried out by the Ritz method, and the cell structure method is refined to improve the speed and accuracy of numerical integration on the subscription surface of clouds intersecting with the plate boundaries. Using the algorithm developed, the natural frequencies of plates of various shapes and support patterns were computed. By increasing the number of point supports on the plate edges, the natural frequencies computed of the plate tend to those of the simply supported plate. Appropriate pattern of point supports distribution was presented for modeling the simply supported plates of various shapes by comparing the corresponding natural frequencies.

scholarly journals Forced Vibration Analysis of Composite Beams Reinforced by Carbon Nanotubes

Nanomaterials ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 571
Author(s):  
Ömer Civalek ◽  
Şeref D. Akbaş ◽  
Bekir Akgöz ◽  
Shahriar Dastjerdi
Keyword(s):  
Carbon Nanotubes ◽  
Carbon Nanotube ◽  
Forced Vibration ◽  
Vibration Analysis ◽  
Volume Fraction ◽  
Ritz Method ◽  
Time History ◽  
Composite Beams ◽  
Reinforced Composite ◽  
Forced Vibration Analysis

This paper presents forced vibration analysis of a simply supported beam made of carbon nanotube-reinforced composite material subjected to a harmonic point load at the midpoint of beam. The composite beam is made of a polymeric matrix and reinforced the single-walled carbon nanotubes with their various distributions. In the beam kinematics, the first-order shear deformation beam theory was used. The governing equations of problem were derived by using the Lagrange procedure. In the solution of the problem, the Ritz method was used, and algebraic polynomials were employed with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method was applied in the time history. In the numerical examples, the effects of carbon nanotube volume fraction, aspect ratio, and dynamic parameters on the forced vibration response of carbon nanotube-reinforced composite beams are investigated. In addition, some comparison studies were performed, with special results of published papers to validate the using formulations.

scholarly journals Free and forced vibration analysis of T-shaped plates with general elastic boundary supports

2018 ◽  
Vol 37 (2) ◽  
pp. 355-372 ◽  
Author(s):  
Xianjie Shi ◽  
Dongyan Shi
Keyword(s):  
Series Expansion ◽  
Forced Vibration ◽  
Vibration Analysis ◽  
Present Method ◽  
Ritz Method ◽  
Boundary Edge ◽  
Coupling Conditions ◽  
Solution Algorithms ◽  
Present Solution ◽  
Forced Vibration Analysis

This investigation proposes a series solution method for free and forced vibration analysis of T-shaped plate structure with general elastic supports, arbitrary coupling angles and elastically coupled conditions. The present solution framework can be readily utilized for various boundary or coupling conditions without modifying the solution algorithms or procedures. The general boundary and coupling conditions are considered with artificial spring method and can be easily achieved with assigning the restraining springs with specified values. In the current approach, the displacement functions for in-plane and bending vibration are expressed as a new form of trigonometric series expansion. The sine terms are introduced to eliminate the discontinuity along the boundary edge or coupling edge. Rayleigh–Ritz method is employed to determine the series expansion coefficients, which are treated as the generalized coordinates. Numerous numerical examples are carried out to verify the accuracy and reliability of the present method. The present method can be directly extended to more complicated structures with any number of plates.

Free and Forced Vibration Analysis of an Idealized Compliant Tower With Superstructure, due to Current Loads, Wave Excitation, and Earthquake Impulses

2014 ◽  
Author(s):  
Ankit ◽  
N. Datta
Keyword(s):  
Forced Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Wave Excitation ◽  
Uniform Beam ◽  
Compliant Tower ◽  
Forced Vibration Analysis ◽  
Beam Mode

A compliant tower (CT) is modeled as a partially dry, partially tapered, damped Timoshenko beam with the superstructure modeled as an eccentric tip mass, and a non-classical damped boundary at the base. The foundation is modeled as a combination of a linear spring and a torsional spring, along with linear and torsional dampers. The mean empty space factor due to the truss type structure of the tower is included. The effect of shear deformation and rotary inertia are included in the vibration analysis; with the non-uniform beam mode-shapes being a weighted sum of the uniform beam mode-shapes. The weights are evaluated by the Rayleigh-Ritz method, using the first ten modes and verified using Finite Element Method (FEM). The superstructure adds to the kinetic energy without affecting the stiffness of the beam, thereby reducing the natural frequencies. The weight of the superstructure acts as an axial compressive load on the beam, reducing its frequencies further. Kelvin-Voigt model of structural damping is included. A part of the structure being underwater, the virtual added inertia is included to calculate the wet natural frequencies. The CT is first subjected to steady current loads of a given velocity profile. The static deflection and overturning moment is estimated for current loads. The CT is then studied for wave excitation at various seas states. Morrison’s equation and Pierson-Moskowitz Spectrum are used to derive the forces for different sea states. The forced vibration analysis of the structure is done via Rayleigh-Ritz method and verified using FEM. The maximum horizontal deflection and shear stress of the base of the superstructure, and the normal/shear stresses at the foundation are analyzed. Finally, the CT is subjected to earthquake excitation, modeled as an arbitrary horizontal impact excitation at the base. The above forced vibration analysis is repeated.

scholarly journals Free Vibration Analysis of L-Shaped Folded Thin Plates

2019 ◽  
Vol 2 (1) ◽  
pp. 67-73
Author(s):  
Koji Sekine
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Energy Functional ◽  
Thin Plates ◽  
Mode Shapes ◽  
Width Ratio ◽  
Various Boundary Conditions

Free vibration analysis of L-shaped folded thin plates having various boundary conditions is presented. Vibration properties of the folded plates are analyzed by means of the Ritz method. Displacement functions satisfying the geometric boundary conditions are assumed in the form of double power series. The interconnection of plate elements of the folded plates is defined by translational and rotational coupling springs. The generalized eigenvalue problem, which is derived by means of minimizing the energy functional, is solved to determine the natural frequencies and mode shapes. The accuracy and validity of the present solutions are demonstrated through convergence studies and comparisons with the results from the literature and FEM (finite element method) analysis solutions. Numerical results are presented for different conditions, such as width ratio, length ratio and the four types of boundary condition.

scholarly journals Vibration Analysis of a New Type of Compliant Mechanism with Flexible-Link, Using Perturbation Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
N. S. Viliani ◽  
H. Zohoor ◽  
M. H. Kargarnovin
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Vibration Analysis ◽  
Compliant Mechanism ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Flexible Link ◽  
Assumed Mode Method ◽  
New Type ◽  
Assumed Mode

Vibration analysis of a new type of compliant parallel mechanism with flexible intermediate links is investigated. The application of the Timoshenko beam theory to the mathematical modeling of the intermediate flexible link is described, and the equations of motion of the flexible links are obtained by using Lagrange’s equation of motion. The equations of motion are obtained in the form of a set of ordinary differential equations by using assumed mode method theory. The governing differential equations of motion are solved using perturbation method. The assumed mode shapes and frequencies are to be obtained based on clamped-clamped boundary conditions. Comparing perturbation method with Runge-Kutta-Fehlberg 4, 5th leads to highly accurate solutions, and the results are performed and discussed.

scholarly journals Free Vibration Analysis of Piezoelectric Cylindrical Nanoshell: Nonlocal and Surface Elasticity Effects

2020 ◽  
Vol 15 ◽  
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Surface Elasticity ◽  
Free Vibration Analysis ◽  
Nonlocal Theory ◽  
Mode Shapes ◽  
Arbitrary Boundary ◽  
Assumed Mode Method ◽  
Mode Method

Vibration analysis of piezoelectric cylindrical nanoshell subjected to visco-Pasternak medium with arbitrary boundary conditions is investigated. In these analysis simultaneous effects of the nonlocal, surface elasticity and the different material scale parameter are considered. To this end, Eringen nonlocal theory and Gurtin–Murdoch surface/interface theory considering Donnell's shell theory are used. The governing equations and boundary conditions are derived using Hamilton’s principle and the assumed mode method combined with Euler–Lagrange method is used for discretizing the equations of motion. The viscoelastic nanoshell medium is modeled as Visco-Pasternak foundation. A variety of new vibration results including frequencies and mode shapes for piezoelectric cylindrical nano-shell with non-classical restraints as well as different material parameters are presented. The convergence, accuracy and reliability of the current formulation are validated by comparisons with existing experimental and numerical results. Also, the effects of nonlocality, surface energy, nanoshell radius, circumferential wavenumber, nanoshell damping coefficient, and foundation damping are accurately studied on frequencies and mode shapes of piezoelectric cylindrical nanoshell.

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