scholarly journals Discussion of the Improved Methods for Analyzing a Cantilever Beam Carrying a Tip-Mass under Base Excitation

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Wang Hongjin ◽  
Meng Qingfeng ◽  
Feng Wuwei
Keyword(s):  
Cantilever Beam ◽  
Analytical Methods ◽  
Natural Frequencies ◽  
Separation Of Variables ◽  
Inertial Force ◽  
Mode Shapes ◽  
Base Excitation ◽  
Inertial Moment ◽  
Tip Mass ◽  
Improved Methods

Two improved analytical methods of calculations for natural frequencies and mode shapes of a uniform cantilever beam carrying a tip-mass under base excitation are presented based on forced vibration theory and the method of separation of variables, respectively. The cantilever model is simplified in detail by replacing the tip-mass with an equivalent inertial force and inertial moment acting at the free end of the cantilever based on D’Alembert’s principle. The concentrated equivalent inertial force and inertial moment are further represented as distributed loads using Dirac Delta Function. In this case, some typical natural frequencies and mode shapes of the cantilever model are calculated by the improved and unimproved analytical methods. The comparing results show that, after improvement, these two methods are in extremely good agreement with each other even the offset distance between the gravity center of the tip-mass and the attachment point is large. As further verification, the transient and steady displacement responses of the cantilever system under a sine base excitation are presented in which two improved methods are separately utilized. Finally, an experimental cantilever system is fabricated and the theoretical displacement responses are validated by the experimental measurements successfully.

scholarly journals Effects of Cable on the Dynamics of a Cantilever Beam with Tip Mass

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Yi-Xin Huang ◽  
Hao Tian ◽  
Yang Zhao
Keyword(s):  
Cantilever Beam ◽  
Spectral Element Method ◽  
Natural Frequencies ◽  
Spectral Element ◽  
Mode Shapes ◽  
Beam System ◽  
Double Beam ◽  
Tip Mass ◽  
Beam Mode ◽  
Element Method

The dynamic effects of cable attachment on a cantilever beam with tip mass are investigated by an improved Chebyshev spectral element method. The cabled beam is modeled as a double-beam system connected by springs at several discrete locations. By utilizing high order Chebyshev polynomials as basis functions and meshing the system at the locations of connections, precise numerical results of the natural frequencies and mode shapes can be obtained using only a few elements. The accuracy of this method is validated through comparing the results of finite element method and those of spectral element method in literature. The validated method is implemented to investigate the effects of parameters, including spring stiffness, number of connections, density, and Young’s modulus of cable. The results show that the mode shapes of the cabled beam system can be classified into two types: beam mode shapes and cable mode shapes, according to their main deformation. Their corresponding natural frequencies change in very different ways with the variation of system parameters. This work can be applied to optimize the dynamic characteristics of precise spacecraft structures with cable attachments.

Wave-Induced Vibrations of an Axial-Loaded Immersed Timoshenko Beam Carrying an Eccentric Tip Mass with Rotary Inertia

2010 ◽  
Vol 54 (01) ◽  
pp. 15-33
Author(s):  
Jong-Shyong Wu ◽  
Chin-Tzu Chen
Keyword(s):  
Closed Form ◽  
Timoshenko Beam ◽  
Forced Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Rotary Inertia ◽  
Mode Shapes ◽  
Mode Superposition Method ◽  
Tip Mass

Under the specified assumptions for the equation of motion, the closed-form solution for the natural frequencies and associated mode shapes of an immersed "Euler-Bernoulli" beam carrying an eccentric tip mass possessing rotary inertia has been reported in the existing literature. However, this is not true for the immersed "Timoshenko" beam, particularly for the case with effect of axial load considered. Furthermore, the information concerning the forced vibration analysis of the foregoing Timoshenko beam caused by wave excitations is also rare. Therefore, the first purpose of this paper is to present a technique to obtain the closed-form solution for the natural frequencies and associated mode shapes of an axial-loaded immersed "Timoshenko" beam carrying eccentric tip mass with rotary inertia by using the continuous-mass model. The second purpose is to determine the forced vibration responses of the latter resulting from excitations of regular waves by using the mode superposition method incorporated with the last closed-form solution for the natural frequencies and associated mode shapes of the beam. Because the determination of normal mode shapes of the axial-loaded immersed "Timoshenko" beam is one of the main tasks for achieving the second purpose and the existing literature concerned is scarce, the details about the derivation of orthogonality conditions are also presented. Good agreements between the results obtained from the presented technique and those obtained from the existing literature or conventional finite element method (FEM) confirm the reliability of the presented theories and the developed computer programs for this paper.

EFFECT OF ADDED TIP MASS ON THE NONLINEAR FLAPWISE AND CHORDWISE VIBRATION OF CANTILEVER COMPOSITE BEAM UNDER BASE EXCITATION

2012 ◽  
Vol 12 (02) ◽  
pp. 285-310 ◽  
Author(s):  
M. EFTEKHARI ◽  
M. MAHZOON ◽  
S. ZIAEI-RAD
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Laminated Composite ◽  
Base Excitation ◽  
System A ◽  
Autoparametric Resonance ◽  
The Stability ◽  
Tip Mass ◽  
Made In

In this paper, a comparative study is performed for a symmetrically laminated composite cantilever beam with and without a tip mass under harmonic base excitation. The base is subjected to both flapwise and chordwise excitations tuned to the primary resonances of the two directions and conditions of 2:1 autoparametric resonance. In the literature, the governing nonlinear equations of the same problem without tip mass have been derived using the extended Hamilton's principle. Extension is made in this study to include the effect of a tip mass on the response of the beam. The natural frequencies are obtained numerically using the diversity guided evolutionary algorithm (DGEA). Next, the multiple scales method is applied to determine the nonlinear response and stability of the system. A set of four first-order differential equations describing the modulation of the amplitudes and phases of interacting modes are derived for the perturbation analysis. For verification, the above equations are reduced to the special case of the cantilever beam without tip mass for comparison with existing results. Finally, the effect of the tip mass on the stability of the fixed points and on the amplitude of oscillation about the equilibrium points in both the frequency and force modulation responses is examined.

scholarly journals Nonlinear Dynamic Modeling and Analysis of an L-Shaped Multi-Beam Jointed Structure with Tip Mass

Materials ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 7279
Author(s):  
Jin Wei ◽  
Tao Yu ◽  
Dongping Jin ◽  
Mei Liu ◽  
Dengqing Cao ◽  
...  
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Natural Frequencies ◽  
Great Influence ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Orthogonality Relations ◽  
Tip Mass

A dynamic model of an L-shaped multi-beam joint structure is presented to investigate the nonlinear dynamic behavior of the system. Firstly, the nonlinear partial differential equations (PDEs) of motion for the beams, the governing equations of the tip mass, and their matching conditions and boundary conditions are obtained. The natural frequencies and the global mode shapes of the linearized model of the system are determined, and the orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to derive a set of nonlinear ordinary differential equations (ODEs) that govern the motion of the L-shaped multi-beam jointed structure. The accuracy of the model is verified by the comparison of the natural frequencies solved by the frequency equation and the ANSYS. Based on the nonlinear ODEs obtained in this model, the dynamic responses are worked out to investigate the effect of the tip mass and the joint on the nonlinear dynamic characteristic of the system. The results show that the inertia of the tip mass and the nonlinear stiffness of the joints have a great influence on the nonlinear response of the system.

Frequency-based damage detection in cantilever beam using vision-based monitoring system with motion magnification technique

2018 ◽  
Vol 29 (20) ◽  
pp. 3923-3936 ◽  
Author(s):  
Andrew Jaeyong Choi ◽  
Jae-Hung Han
Keyword(s):  
Genetic Algorithm ◽  
Damage Detection ◽  
Monitoring System ◽  
Cantilever Beam ◽  
Structural Damage ◽  
Natural Frequencies ◽  
Solution Space ◽  
Mode Shapes ◽  
High Resolution Data ◽  
Motion Magnification

This article proposes a method for damage detection using vision-based monitoring with motion magnification technique. The methods based on the vibration characteristics of structures such as natural frequency, mode shapes, and modal damping have been applied to structural damage detection. However, the conventional methods have limitations for practical applications. Vision-based monitoring system can be employed as a new structural monitoring system because of its simplicity, potentially low cost, and unique capability of collecting high-resolution data. A methodology called video motion magnification has been developed to amplify non-visible small motions in a video to reveal the dynamic response. The video motion magnification method can be applied to measure small displacements to calculate the natural frequencies and the operational deflection shapes of the structures. Unlike conventional optimization methods, a genetic algorithm explores the entire solution space and can obtain the global optimum. In this article, identification of the location and magnitude of damage in a cantilever beam is formulated as an optimization problem using a real-value genetic algorithm by minimizing the objective function, which directly compares the first three natural frequencies changes from the phase-based motion magnification measurement and from the analytical model of a damaged cantilever beam.

Closed-Form Solution of Natural Frequencies of a Cantilever Beam Under Longitudinal Rotation of the Support

2005 ◽  
Author(s):  
Mehdi Esmaeili ◽  
Mohammad Durali ◽  
Nader Jalili
Keyword(s):  
Closed Form ◽  
Cantilever Beam ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Longitudinal Direction ◽  
Form Solution ◽  
Governing Equations ◽  
Well Drilling ◽  
General Base ◽  
Tip Mass

This paper presents the modeling steps towards development of frequency equations for a cantilever beam with a tip mass under general base excitations. More specifically, the beam is considered to vibrate in all the three directions, while subjected to a base rotational motion around its longitudinal direction. This is a common configuration utilized in many vibrating beam gyroscopes and well drilling systems. The governing equations are derived using Extended Hamilton’s Principle with general 6-DOF base motion. The natural frequency equations are then extracted in closed-form for the case where the base undergoes longitudinal rotation. For validation purposes, the resulting natural frequencies are compared with two example case studies; one with a beam on a stationary base and the other one with a rotor having flexible shaft.

Exact Vibration Solution for Exponentially Tapered Cantilever With Tip Mass

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
C.Y. Wang ◽  
C. M. Wang
Keyword(s):  
Numerical Methods ◽  
Free Vibration ◽  
Cantilever Beam ◽  
Technical Note ◽  
Mode Shapes ◽  
Constant Thickness ◽  
Vibration Frequencies ◽  
Tapered Cantilever Beam ◽  
Tip Mass ◽  
First Time

This technical note is concerned with the free vibration problem of a cantilever beam with constant thickness and exponentially decaying width. Existing analytical results for such a vibration beam problem are found to be incomplete because lower frequencies could not be obtained. Presented herein is the exact characteristic equation for generating the complete vibration frequencies for the considered vibrating beam problem. Also the note treated for the first time such a tapered cantilever beam with a tip mass. The exact solutions (frequencies and mode shapes) are important to engineers designing such tapered beams and the results serve as benchmarks for assessing the validity, convergence and accuracy of numerical methods and solutions.

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