Nonmodal Solution of Spherical Shells With Cutouts Excited by High-Frequency Axisymmetric Forces

1970 ◽  
Vol 37 (4) ◽  
pp. 977-983 ◽  
Author(s):  
M. C. Junger
Keyword(s):  
Boundary Conditions ◽  
High Frequency ◽  
Near Field ◽  
Closed Form Solution ◽  
Radial Force ◽  
Form Solution ◽  
Mode Shapes ◽  
High Frequency Response ◽  
Circular Cutout ◽  
Line Loads

A closed-form solution is obtained for the high-frequency response of a thin spherical shell embodying a circular cutout and excited axisymmetrically by a concentrated radial force. The solution is constructed by combining the shell response to the radial exciting force with its response to radial, tangential, and moment line loads applied along the cutout boundary, these line loads being selected to match the boundary conditions. Concise expressions for the shell response are obtained by applying the Sommerfeld-Watson transformation to the slowly converging high-frequency modal series which is thereby reduced to only two terms, viz., an exponentially decaying near-field and a standing or propagating-wave field. These two terms are in the nature of the creeping waves commonly used to formulate electromagnetic or acoustic diffracted wave fields in the short-wavelength limit. The method is illustrated for the simple case of a circular cutout with a clamped boundary, but lends itself to more complicated boundary conditions, viz., intersecting shells or wave guides. The natural frequencies and mode shapes are found from a single, characteristic equation involving trigonometric functions.

Coupled Flexural and Torsional Vibration Analysis of Composite Beams

2019 ◽  
Author(s):  
Ehsan Sarfaraz ◽  
Jeremiah O. Afolabi ◽  
Hamid R. Hamidzadeh
Keyword(s):  
Boundary Conditions ◽  
Torsional Vibration ◽  
Analytical Method ◽  
Composite Beam ◽  
Equations Of Motion ◽  
Closed Form Solution ◽  
Composite Beams ◽  
Form Solution ◽  
Mode Shapes ◽  
Loss Factors

Abstract An analytical method is presented to determine the dynamic response of a coupled flexural and torsional vibration of composite beams. The general governing equations of motion are presented and a closed form solution for free vibration of the composite beam that demonstrates both geometric and material coupling is developed. The proposed solution is used to compute the natural frequencies, modal loss factors, and mode shapes of the composite beam for several modes of the coupled bending and torsional vibrations for any boundary conditions. In this analysis, hysteretic damping for the composite beam is considered and its effects on mode shapes and modal loss factors are determined. In addition, the variation of modal parameters such as bending displacement, bending slope, torsional rotation, shear force, bending moment, and torque along the beam for the first four mode shapes are presented for the clamped-free boundary conditions. Moreover, to verify the validity of the presented analytical method, results for the cases with no damping are compared with the previously established results and good agreement is achieved. The presented results can provide a guideline for selecting appropriate geometries and materials to design composite beams.

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.

High-Frequency Response of an Elastic Spherical Shell

1969 ◽  
Vol 36 (4) ◽  
pp. 859-864 ◽  
Author(s):  
David Feit ◽  
M. C. Junger
Keyword(s):  
Spherical Shell ◽  
Flat Plate ◽  
Wave Field ◽  
Classical Solution ◽  
High Frequency ◽  
Natural Frequencies ◽  
Near Field ◽  
Flexural Wave ◽  
High Frequency Response ◽  
Field Response

The classical solution of the point-excited spherical shell, in the form of a normal-mode series, converges poorly for large frequencies. By applying the Watson-Sommerfeld transformation to this series, the response is expressed as a sum of only two terms. These terms can be interpreted, respectively, as the near-field response and propagating flexural wave field of an infinite flat plate, the latter term being multiplied by a factor whose maxima coincide with the natural frequencies of the shell.

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.

scholarly journals The Chow-Lin method extended to dynamic models with autocorrelated residuals

2017 ◽  
Vol 10 (1) ◽  
Author(s):  
Aurélien Poissonnier
Keyword(s):  
Closed Form ◽  
Capital Stock ◽  
High Frequency ◽  
Dynamic Models ◽  
Closed Form Solution ◽  
Dynamic Structure ◽  
Form Solution ◽  
Static Models ◽  
Temporal Disaggregation ◽  
Communication Equipment

AbstractI provide a closed-form solution to temporal disaggregation or interpolation models which is both general in terms of dynamic structure of the model (lags of the high-frequency variable) and flexible in terms of autocorrelation of its residual. As for static models, I show that assuming autocorrelated residuals in dynamic models is practically convenient. To illustrate the potential of the solution proposed, I provide an example for quarterly non-financial corporations’ capital stock in computers and communication equipment.

Postbuckling and Free Vibrations of Composite Beams

2007 ◽  
Author(s):  
Samir A. Emam ◽  
Ali H. Nayfeh
Keyword(s):  
Natural Frequency ◽  
Axial Load ◽  
Closed Form Solution ◽  
Composite Beams ◽  
Free Vibrations ◽  
Form Solution ◽  
Mode Shapes ◽  
Axial Stiffness ◽  
Fundamental Natural Frequency ◽  
Transverse Vibrations

An exact solution for the postbuckling configurations of composite beams is presented. The equations governing the axial and transverse vibrations of a composite laminated beam accounting for the midplane stretching are presented. The inplane inertia and damping are neglected, and hence the two equations are reduced to a single equation governing the transverse vibrations. This equation is a nonlinear fourth-order partial-integral differential equation. We find that the governing equation for the postbuckling of a symmetric or antisymmetric composite beam has the same form as that of a metallic beam. A closed-form solution for the postbuckling configurations due to a given axial load beyond the critical buckling load is obtained. We followed Nayfeh, Anderson, and Kreider and exactly solved the linear vibration problem around the first buckled configuration to obtain the fundamental natural frequencies and their corresponding mode shapes using different fiber orientations. Characteristic curves showing variations of the maximum static deflection and the fundamental natural frequency of postbuckling vibrations with the applied axial load for a variety of fiber orientations are presented. We find out that the line-up orientation of the laminate strongly affects the static buckled configuration and the fundamental natural frequency. The ratio of the axial stiffness to the bending stiffness is a crucial parameter in the analysis. This parameter can be used to help design and optimize the composite beams behavior in the postbuckling domain.

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.

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