Flexural–Torsional Free Vibration Analysis of a Double-Cantilever Structure

2021 ◽  
Vol 144 (3) ◽  
Author(s):  
Anahita Zargarani ◽  
S. Nima Mahmoodi
Keyword(s):  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Experimental Investigations ◽  
Torsional Vibrations ◽  
Cantilever Beams ◽  
Torsional Mode ◽  
Flexural Mode ◽  
Cantilever Structure ◽  
Flexural Vibrations

Abstract This paper aims to investigate the free coupled flexural–torsional vibrations of a double-cantilever structure. The structure consists of two identical Euler–Bernoulli cantilever beams with a piezoelectric layer on top. The beams are connected by a rigid tip connection at their free ends. The double-cantilever structure in this study vibrates in two distinct modes: flexural mode or coupled flexural–torsional mode. The flexural mode refers to the in-phase flexural vibrations of the two cantilever beams resulting in translation of the tip connection, while the coupled flexural–torsional mode refers to the coupled flexural–torsional vibrations of the cantilever beams resulting in rotation of the tip connection. The latter is the main interest of this research. The governing equations of motion and boundary conditions are developed using Hamilton’s principle. Two uncoupled equations are realized for each beam: one corresponding to the flexural vibrations and the other one corresponding to the torsional vibrations. The characteristic equations for both the flexural and the coupled flexural–torsional vibration modes are derived and solved to find the natural frequencies corresponding to each mode of vibration. The orthogonality condition among the mode shapes is derived and utilized to determine the modal coefficients corresponding to each mode of vibration. Moreover, the analytical and experimental investigations show that the coupled flexural–torsional fundamental frequency of the structure is dependent on dimensional parameters including the length of the cantilever beams and the length of the tip connection.

Free Vibration Analysis of a Carbon Nanotube Reinforced Composite Beam

2014 ◽  
Vol 592-594 ◽  
pp. 2041-2045 ◽  
Author(s):  
B. Naresh ◽  
A. Ananda Babu ◽  
P. Edwin Sudhagar ◽  
A. Anisa Thaslim ◽  
R. Vasudevan
Keyword(s):  
Carbon Nanotube ◽  
Free Vibration ◽  
Composite Beam ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Element Formulation ◽  
Reinforced Composite ◽  
Vibration Responses

In this study, free vibration responses of a carbon nanotube reinforced composite beam are investigated. The governing differential equations of motion of a carbon nanotube (CNT) reinforced composite beam are presented in finite element formulation. The validity of the developed formulation is demonstrated by comparing the natural frequencies evaluated using present FEM with those of available literature. Various parametric studies are also performed to investigate the effect of aspect ratio and percentage of CNT content and boundary conditions on natural frequencies and mode shapes of a carbon nanotube reinforced composite beam. It is shown that the addition of carbon nanotube in fiber reinforced composite beam increases the stiffness of the structure and consequently increases the natural frequencies and alter the mode shapes.

Flexural Vibration of Symmetrical Multi-Layer Beams with Viscoelastic Damping

1968 ◽  
Vol 10 (3) ◽  
pp. 269-281 ◽  
Author(s):  
J. A. Agbasiere ◽  
P. Grootenhuis
Keyword(s):  
Equations Of Motion ◽  
Finite Difference Methods ◽  
Flexural Vibration ◽  
Order Effect ◽  
Simultaneous Equations ◽  
Mode Shapes ◽  
Central Layer ◽  
Method Of Solution ◽  
Flexural Vibrations ◽  
Frequency Phase

The flexural vibrations of beams can be reduced by introducing sandwiched layers of energy-dissipating materials. The equations of motion are derived for flexural vibrations of symmetrical, multi-layer sandwich beams. Two types of beams are considered, depending on whether the central layer is energy dissipating when the number of layers will be n = 4 i − 1, where i = 1, 2,…, or whether the material of the central layer is perfectly elastic when n = 4 i + 1. The total number of equations of motion will be i + 1 for each type. These equations will be non-linear when the properties of the energy-dissipating materials are strain dependent. A numerical method of solution has been introduced by transforming the equations of motion into sets of non-dimensional simultaneous equations and using finite difference methods. The simplest type of beam has three layers but even this leads to a set of four simultaneous equations of the twelfth order. It is shown as an example that the strain dependence of a typical viscoelastic material has only a second order effect upon the computed response. Experimentally determined values of frequency, phase angle and mode shapes under conditions of steady state are compared with computed data and the agreements are such that the response to excitation in flexural motion of symmetrical, multi-layer damped beams can now be calculated with confidence even for the lower modes.

Modal analysis of cylindrical panels at elevated temperatures under nonuniform heating conditions: Experimental investigation

2020 ◽  
pp. 095440622093673
Author(s):  
CM Twinkle ◽  
C Nithun ◽  
Jeyaraj Pitchaimani ◽  
Vasudevan Rajamohan
Keyword(s):  
Free Vibration ◽  
Elevated Temperatures ◽  
Laminated Composite ◽  
Free Vibration Analysis ◽  
Cylindrical Panel ◽  
Experimental Results ◽  
Mode Shapes ◽  
Experimental Investigations ◽  
Nonuniform Heating ◽  
Cylindrical Panels

In this study, experimental investigations carried out to analyze the influences of different in-plane temperature variations on buckling and free vibration responses of metal and fiber-reinforced laminated composite cylindrical panels are presented. Initially, critical buckling temperature is calculated then free vibration analysis is performed as a function of the buckling temperature to analyze the changes in the natural frequencies and mode shapes. Experimental results revealed that the thermal buckling strength of the panel is significantly influenced by the nature of the heating condition. Similarly, significant changes in free vibration mode shapes are observed with the rise in temperature and also according to the heating conditions. It is also observed that, with the increase in temperature, nodal and anti-nodal lines of free vibration modes shifting towards the heating source. The experimental results are compared with the numerical simulation for the studies on the isotropic cylindrical panel and both the results are in good agreement.

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.

scholarly journals Nonlinear Flexural Vibrations of Thin Circular Rings

1966 ◽  
Vol 33 (3) ◽  
pp. 553-560 ◽  
Author(s):  
D. A. Evensen
Keyword(s):  
Equations Of Motion ◽  
Mode Shapes ◽  
Test Results ◽  
State Response ◽  
Coupled Mode ◽  
Bending Mode ◽  
Circular Rings ◽  
Flexural Vibrations ◽  
Bending Modes ◽  
Good Agreement

The nonlinear flexural vibrations of thin circular rings are analyzed by assuming two vibration modes and then applying Galerkin’s procedure on the equations of motion. The results show that vibrations involving either a single bending mode or two coupled bending modes can occur. Theory and experiment both indicate a nonlinearity of the softening type and the existence of these coupled-mode vibrations. Test results for the steady-state response are in good agreement with the calculated values, and the deflection modes used in the analysis agree with the experimental mode shapes. The analytical and experimental results exhibit several features that are characteristic of nonliner vibrations of axisymmetric systems in general and of circular cylindrial shells in particular.

Free Vibration Analysis of Planar Flexible Mechanisms

2003 ◽  
Vol 125 (4) ◽  
pp. 764-772 ◽  
Author(s):  
S. D. Yu ◽  
F. Xi
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Summation Method ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Modal Summation ◽  
Gyroscopic Effects ◽  
Flexible Mechanisms

This paper presents a methodology for accurate free vibration analysis of planar flexible mechanisms. Each flexible body is considered as a beam and modelled using higher-order beam elements for longitudinal and flexural deformations. The global equations of motion for a mechanism consisting of multiple flexible bodies are formulated using the augmented Lagrange equations. Free vibration analyses are conducted at desired fast Fourier configurations to determine instantaneous structural natural frequencies and structural mode shapes. Dynamical frequencies and dynamical mode shapes incorporating the gyroscopic effects and dynamic axial loads are obtained using the modal summation method. Numerical results and comparisons are given for a rotating beam and two four-bar crank-rocker mechanisms.

A rigid multibody method for free vibration analysis of beams with variable axial parameters

2016 ◽  
Vol 23 (1) ◽  
pp. 131-146 ◽  
Author(s):  
Aleksandar Nikolić ◽  
Slaviša Šalinić
Keyword(s):  
Multibody System ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Constant Thickness ◽  
Flexible Beam ◽  
Cantilever Beams ◽  
New Approach ◽  
Transverse Vibrations ◽  
Tip Mass ◽  
Rigid Multibody

This paper presents a new approach to the problem of determining the frequencies and mode shapes of Euler–Bernoulli tapered cantilever beams with a tip mass and a spring at the free end. The approach is based on the replacement of the flexible beam by a rigid multibody system. Beams with constant thickness and exponentially and linearly tapered width, as well as double-tapered cantilever beams are considered. The influence of the tip mass, stiffness of the spring, and taper on the frequencies of the free transverse vibrations of tapered cantilever beams are examined. Numerical examples with results confirming the convergence and accuracy of the approach are given.

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.

scholarly journals Transverse-Torsional Vibrations of the High-Speed Planetary Gear Transmission

2014 ◽  
Vol 8 (1) ◽  
pp. 396-401 ◽  
Author(s):  
He Zhaoxia ◽  
Liu Qingtao ◽  
Chang Lehao ◽  
Wu Jinsong
Keyword(s):  
High Speed ◽  
Equations Of Motion ◽  
Planetary Gear ◽  
Gear Transmission ◽  
Mode Shapes ◽  
Parameter Method ◽  
Analysis Model ◽  
Mesh Stiffness ◽  
Torsional Mode ◽  
Torsional Coupling

The vibration characteristics of a high-speed planetary gear transmission (PGT) are studied in this paper. A transverse-torsional coupling dynamic model is developed using lumped parameter method. In order to ensure the accuracy of the analysis model, the mesh stiffness, support stiffness and the influence of eccentric masses on system are also considered. By solving the eigenvalue of differential equations of motion, the natural frequencies and mode shapes are calculated. According to the vibration modes of this gear transmission, the characteristics of translational mode and torsional mode are described in details. The influence of mesh stiffness, support stiffness and modal modes are discussed by adopting strain and kinetic energy.

Split beam method for determining mode shapes of cantilever beams under multiple loading conditions

2021 ◽  
pp. 107754632110276
Author(s):  
Jun-Jie Li ◽  
Shuo-Feng Chiu ◽  
Sheng D Chao
Keyword(s):  
Operation Mode ◽  
Point Contact ◽  
Mode Shapes ◽  
Cantilever Beams ◽  
Loading Conditions ◽  
Mechanical Responses ◽  
Relative Contribution ◽  
Resonance Frequencies ◽  
Beam Method ◽  
Multiple Loading

We have developed a general method, dubbed the split beam method, to solve Euler–Bernoulli equations for cantilever beams under multiple loading conditions. This kind of problem is, in general, a difficult inhomogeneous eigenvalue problem. The new idea is to split the original beam into two (or more) effective beams, each of which corresponds to one specific load and bears its own Young’s modulus. The mode shape of the original beam can be obtained by linearly superposing those of the effective beams. We apply the split beam method to simulating mechanical responses of an atomic force microscope probe in the “dynamical” operation mode, under which there are a stabilizing force at the positioner and a point-contact force at the tip. Compared with traditional analytical or numerical methods, the split beam method uses only a few number of basis functions from each effective beam, so a very fast convergence rate is observed in solving both the resonance frequencies and the mode shapes at the same time. Moreover, by examining the superposition coefficients, the split beam method provides a physical insight into the relative contribution of an individual load on the beam.

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