Closed-Form Exact Solutions for Viscously Damped Free and Forced Vibrations of Longitudinal and Torsional Bars

2017 ◽  
Vol 17 (08) ◽  
pp. 1750093 ◽  
Author(s):  
Jae-Hoon Kang
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Closed Form Solution ◽  
Forced Vibrations ◽  
Form Solution ◽  
Forced Response ◽  
Mode Shapes ◽  
Viscous Damping ◽  
Vibration Displacement ◽  
Free And Forced Vibrations

This paper studies the viscously damped free and forced vibrations of longitudinal and torsional bars. The method is exact and yields closed form solution for the vibration displacement in contrast with the well-known eigenfunction superposition (ES) method, which requires expression of the distributed forcing functions and displacement response functions as infinite series sums of free vibration eigenfunctions. The viscously damped natural frequency equation and the critical viscous damping equation are exactly derived for the bars. Then the viscously damped free vibration frequencies and corresponding damped mode shapes are calculated and plotted, aside from the undamped free vibration and corresponding mode shapes typically computed and used in vibration problems. The longitudinal or torsional amplitude versus forcing frequency curves showing the forced response to distributed loadings are plotted for various viscous damping parameters. It is found that the viscous damping affects the natural frequencies and the corresponding mode shapes of longitudinal and torsional bars, especially for the fundamental frequency.

scholarly journals A closed-form solution for free vibration of multiple cracked Timoshenko beam and application

2017 ◽  
Vol 39 (4) ◽  
pp. 315-328
Author(s):  
Nguyen Tien Khiem ◽  
Duong The Hung
Keyword(s):  
Free Vibration ◽  
Closed Form ◽  
Timoshenko Beam ◽  
Crack Depth ◽  
Closed Form Solution ◽  
Beam Theory ◽  
Frequency Equation ◽  
Form Solution ◽  
Mode Shapes ◽  
Timoshenko Beams

A closed-form solution for free vibration is constructed and used for obtaining explicit frequency equation and mode shapes of  Timoshenko beams with arbitrary number of cracks. The cracks are represented by the rotational springs of stiffness calculated from the crack depth.  Using the obtained frequency equation, the sensitivity of natural frequencies to crack of the beams is examined in comparison with the  Euler-Bernoulli beams. Numerical results demonstrate that the Timoshenko beam theory is efficiently applicable not only for short or fat beams but also for the long or slender ones. Nevertheless, both the theories are equivalent in sensitivity analysis of fundamental frequency to cracks and they get to be different for higher frequencies.

Complex Modal Analysis of Non-Self-Adjoint Hybrid Serpentine Belt Drive Systems

2000 ◽  
Vol 123 (2) ◽  
pp. 150-156 ◽  
Author(s):  
Lixin Zhang ◽  
Jean W. Zu ◽  
Zhichao Hou
Keyword(s):  
Modal Analysis ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mode Shapes ◽  
Belt Drive ◽  
Serpentine Belt ◽  
Complex Modal Analysis ◽  
Drive Systems ◽  
Complex Modal

A linear damped hybrid (continuous/discrete components) model is developed in this paper to characterize the dynamic behavior of serpentine belt drive systems. Both internal material damping and external tensioner arm damping are considered. The complex modal analysis method is developed to perform dynamic analysis of linear non-self-adjoint hybrid serpentine belt-drive systems. The adjoint eigenfunctions are acquired in terms of the mode shapes of an auxiliary hybrid system. The closed-form characteristic equation of eigenvalues and the exact closed-form solution for dynamic response of the non-self-adjoint hybrid model are obtained. Numerical simulations are performed to demonstrate the method of analysis. It is shown that there exists an optimum damping value for each vibration mode at which vibration decays the fastest.

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.

Free Vibration of a Multilayered One-Dimensional Quasi-Crystal Plate

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Natalie Waksmanski ◽  
Ernian Pan ◽  
Lian-Zhi Yang ◽  
Yang Gao
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Closed Form Solution ◽  
Form Solution ◽  
Crystal Plate ◽  
Mode Shapes ◽  
Sandwich Plates ◽  
One Dimensional ◽  
Propagator Matrix ◽  
Multilayered Plates

An exact closed-form solution of free vibration of a simply supported and multilayered one-dimensional (1D) quasi-crystal (QC) plate is derived using the pseudo-Stroh formulation and propagator matrix method. Natural frequencies and mode shapes are presented for a homogenous QC plate, a homogenous crystal plate, and two sandwich plates made of crystals and QCs. The natural frequencies and the corresponding mode shapes of the plates show the influence of stacking sequence on multilayered plates and the different roles phonon and phason modes play in dynamic analysis of QCs. This work could be employed to further expand the applications of QCs especially if used as composite materials.

Closed Form Solutions for the Motion of Electrically Excited Micro-Cantilever Beams

2006 ◽  
Author(s):  
Seyed Babak Ghaemi Oskouei ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Mode Shapes ◽  
Dynamical Response ◽  
Accurate Analysis ◽  
Nonlinear Part ◽  
Closed Form Solutions ◽  
Micro Cantilever ◽  
The Galerkin Method

The differential equation governing the motion of an electrically excited capacitive microcantilever beam is a nonlinear PDE [1]. Accurate analysis about its motion is of great importance in MEMS' dynamical response. In this paper first the nonlinear 4th order 2 point boundary value problem (ODE) governing the static deflection of the system is solved using three methods. 1. The nonlinear part is linearized and its exact solution is obtained. 2. For low applied DC voltages (not near pull-in) the solutin is found using the direct straight forward perturbation analysis. 3. Numerical computer solutions which are used for the previous solution's verifications. The next parts are devoted to the dynamic solution. The nonlinear time variant 4th order PDE governing the dynamic deflection of an electrically excited microbeam is scrutinized. First using the Galerkin Method the mode shapes and the first three mode temporal equations of the linearized equation are found. Considering no damping, using the perturbations method the temporal equations are solved in three states: far from resonance, near 1:1 resonance and near 1:2 resonance. Finally the damped equation is solved using the aforementioned method. In the literature no closed form solution for this problem is presented.

scholarly journals Buckling Behavior of Nanobeams Placed in Electromagnetic Field Using Shifted Chebyshev Polynomials-Based Rayleigh-Ritz Method

Nanomaterials ◽  
2019 ◽  
Vol 9 (9) ◽  
pp. 1326 ◽  
Author(s):  
Subrat Kumar Jena ◽  
Snehashish Chakraverty ◽  
Francesco Tornabene
Keyword(s):  
Boundary Condition ◽  
Closed Form ◽  
Chebyshev Polynomials ◽  
Ritz Method ◽  
Closed Form Solution ◽  
Form Solution ◽  
Buckling Load ◽  
Mode Shapes ◽  
Critical Buckling Load ◽  
Buckling Behavior

In the present investigation, the buckling behavior of Euler–Bernoulli nanobeam, which is placed in an electro-magnetic field, is investigated in the framework of Eringen’s nonlocal theory. Critical buckling load for all the classical boundary conditions such as “Pined–Pined (P-P), Clamped–Pined (C-P), Clamped–Clamped (C-C), and Clamped-Free (C-F)” are obtained using shifted Chebyshev polynomials-based Rayleigh-Ritz method. The main advantage of the shifted Chebyshev polynomials is that it does not make the system ill-conditioning with the higher number of terms in the approximation due to the orthogonality of the functions. Validation and convergence studies of the model have been carried out for different cases. Also, a closed-form solution has been obtained for the “Pined–Pined (P-P)” boundary condition using Navier’s technique, and the numerical results obtained for the “Pined–Pined (P-P)” boundary condition are validated with a closed-form solution. Further, the effects of various scaling parameters on the critical buckling load have been explored, and new results are presented as Figures and Tables. Finally, buckling mode shapes are also plotted to show the sensitiveness of the critical buckling load.

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