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.

A Unified Frequency Equation and the Motion Analysis of a Moving Flexible Beam Carrying a Concentrated Mass

1999 ◽  
Author(s):  
S. Park ◽  
J. W. Lee ◽  
Y. Youm ◽  
W. K. Chung
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Open Loop ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Lumped Mass ◽  
Concentrated Mass ◽  
Forcing Function ◽  
The Mathematical Model

Abstract In this paper, the mathematical model of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying an intermediate lumped mass is derived. The equations of motion of the beam-mass-cart system is analyzed utilizing unconstrained modal analysis, and a unified frequency equation which can be generally applied to this kind of system is obtained. The change of natural frequencies and mode shapes with respect to the change of the mass ratios of the beam, the lumped mass and the cart and to the position of the lumped mass is investigated. The open-loop responses of the system by arbitrary forcing function are also obtained through numerical simulations.

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.

Nonlinear Dynamic of Cantilever Functionally Graded Plates Under the Thermalmechanical Loads

2009 ◽  
Author(s):  
Yu-xin Hao ◽  
Wei Zhang ◽  
Jian-hua Wang
Keyword(s):  
Nonlinear System ◽  
Functionally Graded Materials ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Governing Equations ◽  
Graded Materials ◽  
Two Degree Of Freedom

An analysis on nonlinear dynamic of a cantilevered functionally graded materials (FGM) plate which subjected to the transverse excitation in the uniform thermal environment is presented for the first time. Materials properties of the constituents are graded in the thickness direction according to a power-law distribution and assumed to be temperature dependent. In the framework of the Third-order shear deformation plate theory, the nonlinear governing equations of motion for the functionally graded materials plate are derived by using the Hamilton’s principle. For cantilever rectangular plate, the first two vibration mode shapes that satisfy the boundary conditions is given. The Galerkin’s method is utilized to discretize the governing equations of motion to a two-degree-of-freedom nonlinear system under combined thermal and external excitations. By using the numerical method, the two-degree-of-freedom nonlinear system is analyzed to find the nonlinear responses of the cantilever FGMs plate. The influences of the thermal environments on the nonlinear dynamic response of the cantilevered FGM plate are discussed in detail through a parametric study.

Free in-plane vibration of cracked curved beams: Experimental, analytical, and numerical analyses

2018 ◽  
Vol 233 (3) ◽  
pp. 928-946
Author(s):  
M Zare
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Experimental Methods ◽  
Mode Shapes ◽  
Mode Transition ◽  
Curved Beams ◽  
Governing Equations ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Different Depths

In this study, free vibration of a cracked curved beam utilizing analytical, numerical, and experimental methods is investigated. The differential quadrature element method is used to solve the equations of motion numerically. The governing equations are also solved analytically. The crack, which is considered to be open, is modeled as a rotational spring. Furthermore, the effect of curvature on mode shapes is studied. To verify the validity of the proposed methods of determining frequencies and mode shapes, an experimental modal analysis test is conducted on a sample beam having crack with some different depths. This study revealed that the behavior of curved beams toward the mode transition phenomenon depends greatly on the boundary conditions of the beam. Also, both the location and depth of crack have considerable effects on natural frequencies.

Axisymmetrical Vibrations of Underwater Hemispherical Shells

1976 ◽  
Vol 98 (3) ◽  
pp. 941-947
Author(s):  
F. C. Chen ◽  
T. C. Huang
Keyword(s):  
Equations Of Motion ◽  
Hydrodynamic Pressure ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Legendre Functions ◽  
Water Field ◽  
Fluid Solid Interaction ◽  
Interaction Problem ◽  
Hemispherical Shells

Free vibrations of an underwater elastic hemispherical thin shell with fixed edge have been investigated based on the bending theory. The solution of this fluid-solid interaction problem involves the differential equations of motion of underwater spherical shells, the velocity potential of the water field, the hydrodynamic pressure, and the continuity and boundary conditions. A transcedental frequency equation in terms of Legendre functions is derived and the normal and tangential mode shapes are found. Examples are given and results are plotted for natural frequencies and modes shapes.

Analysis of Electromechanical Performance of Energy Harvesting Skin Based on the Kirchhoff Plate Theory

2014 ◽  
Author(s):  
Heonjun Yoon ◽  
Byeng D. Youn ◽  
Heung S. Kim
Keyword(s):  
Energy Harvesting ◽  
Differential Equations ◽  
Analytical Model ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Electrical Circuit ◽  
Kirchhoff Plate ◽  
Mode Shapes ◽  
Kirchhoff Plate Theory

As a compact and durable design concept, energy harvesting skin (EH skin), which consists of piezoelectric patches directly attached onto the surface of a vibrating structure as one embodiment, has been recently proposed. This study aims at developing an electromechanically-coupled analytical model of the EH skin so as to understand its electromechanical behavior and get physical insights about important design considerations. Based on the Kirchhoff plate theory, the Hamilton’s principle is used to derive the differential equations of motion. The Rayleigh-Ritz method is implemented to calculate the natural frequency and the corresponding mode shapes of the EH skin. The electrical circuit equation is derived by substituting the piezoelectric constitutive relation into Gauss’s law. Finally, the steady-state output voltage is obtained by solving the differential equations of motion and electrical circuit equation simultaneously. The results of the analytical model are verified by comparing those of the finite element analysis (FEA) in a hierarchical manner.

Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies

1995 ◽  
Vol 62 (1) ◽  
pp. 193-199 ◽  
Author(s):  
M. W. D. White ◽  
G. R. Heppler
Keyword(s):  
Boundary Conditions ◽  
Rigid Body ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Rigid Bodies ◽  
The Body ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
First Moment

The equations of motion and boundary conditions for a free-free Timoshenko beam with rigid bodies attached at the endpoints are derived. The natural boundary conditions, for an end that has an attached rigid body, that include the effects of the body mass, first moment of mass, and moment of inertia are included. The frequency equation for a free-free Timoshenko beam with rigid bodies attached at its ends which includes all the effects mentioned above is presented and given in terms of the fundamental frequency equations for Timoshenko beams that have no attached rigid bodies. It is shown how any support / rigid-body condition may be easily obtained by inspection from the reported frequency equation. The mode shapes and the orthogonality condition, which include the contribution of the rigid-body masses, first moments, and moments of inertia, are also developed. Finally, the effect of the first moment of the attached rigid bodies is considered in an illustrative example.

scholarly journals Simplified Vibration Model and analysis of a single-conductor transmission line with dampers

2016 ◽  
Vol 231 (22) ◽  
pp. 4150-4162 ◽  
Author(s):  
O Barry ◽  
R Long ◽  
DCD Oguamanam
Keyword(s):  
Transmission Line ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Viscous Damping ◽  
Equivalent Viscous Damping ◽  
Proposed Model ◽  
Vibration Model ◽  
Mass Spring ◽  
Discrete Mass

A novel model is developed for a vibrating single-conductor transmission line carrying Stockbridge dampers. Experiments are performed to determine the equivalent viscous damping of the damper. This damper is then reduced to an equivalent discrete mass-spring-mass and viscous damping system. The equations of motion of the model are derived using Hamilton’s principle and explicit expressions are determined for the frequency equation, and mode shapes. The proposed model is verified using experimental and finite element results from the literature. This proposed model excellently captures free vibration characteristics of the system and the vibration level of the conductor, but performs poorly in regard to the vibration of the counterweights.

Dynamic Analysis of Flexible Multi-Body Systems With Time-Variant Mode Shapes

1991 ◽  
Author(s):  
F. M. L. Amirouche ◽  
M. Xie
Keyword(s):  
Equations Of Motion ◽  
Rate Of Change ◽  
Mode Shapes ◽  
Flexible Body ◽  
Governing Equations ◽  
Dynamical Equations ◽  
Application Problem ◽  
Nodal Coordinates ◽  
Multi Body ◽  
Modal Coordinates

Abstract The paper extends the theory of flexible multi-body systems to include time-variant mode shapes to account for the changes of boundary conditions that a flexible body undergoes while in motion. The method presented herein makes use of finite element methods to discretize the elastic body, then the nodal coordinates are related to the modal coordinates through the mode shapes. The latter are computed separately and their rate of change is accounted for in the dynamical equations of motion. The resulting governing equations of motion are presented in a computer form and a possible application problem is proposed.

Free Vibration Analysis of a Beam Under Axial Load Carrying a Mass-Spring-Mass

2012 ◽  
Author(s):  
O. R. Barry ◽  
Y. Zhu ◽  
J. W. Zu ◽  
D. C. D. Oguamanam
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Axial Load ◽  
Equations Of Motion ◽  
Tensile Load ◽  
Free Vibration Analysis ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Mass System ◽  
Mass Spring

This paper deals with the free vibration analysis of a beam subjected to an axial tensile load with an attached in-span mass-spring-mass system. The equations of motion are derived by means of the Hamilton principle and an explicit expression of the frequency equation is presented. The formulation is validated with results in the literature and the finite element method. Parametric studies are done to investigate the effect of the axial load, the magnitude and location of the mass-spring-mass system on the lowest five natural frequencies and mode shapes. The results indicate that the fundamental mode is independent of the tension and the in-span mass. However, a significant change in all modes is observed when the position of the mass-spring-mass is varied.

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