Large Amplitude Forced Vibrations of Simply Supported Thin Cylindrical Shells

1973 ◽  
Vol 40 (2) ◽  
pp. 471-477 ◽  
Author(s):  
J. H. Ginsberg
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Boundary ◽  
Circular Cylindrical Shell ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Inertia Effects ◽  
Modal Expansion ◽  
Wide Range ◽  
Nonlinear Equations Of Motion

The response of a thin circular cylindrical shell to resonant harmonic excitation is examined by a modal expansion approach. The nonlinear strain-displacement relations lead to a nonlinear boundary condition, as well as nonlinear equations of motion. The solution, which retains tangential inertia effects, is obtained by a perturbation technique that yields a consistent first approximation of the nonlinear response. The results are applicable for a wide range of parameters and to cases of excitation near any of the three lowest natural frequencies corresponding to given axial and circumferential wavelengths. For situations where shallow shell theory is valid, the results of previous studies, which were based upon such a theory, are in close agreement.

Periodic Response of Front End Accessory Drives With Dry Friction

1997 ◽  
Author(s):  
Michael J. Leamy ◽  
Noel C. Perkins
Keyword(s):  
Dry Friction ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Stick Slip ◽  
Automotive Engine ◽  
Belt Drives ◽  
Wide Range ◽  
Serpentine Belt ◽  
Incremental Harmonic Balance ◽  
Nonlinear Equations Of Motion

Abstract Belt drives have long been utilized in engine applications to power accessories such as alternators, pumps, compressors and fans. Drives employing a single, flat, ‘serpentine belt’ tensioned by an ‘automatic tensioner’ are now common in automotive engine applications. The automatic tensioner helps maintain constant belt tension and to dissipate unwanted belt drive vibration through dry friction. The objective of this study is to predict the periodic rotational response of the entire drive to harmonic excitation from the crankshaft. To this end, a multi-degree of freedom incremental harmonic balance (IHB) method is utilized to compute periodic solutions to the nonlinear equations of motion over a wide range of engine speeds. Computed results illustrate primary and secondary resonances of accessories and tensioner stick-slip motions.

Nonlinear Periodic Response of Engine Accessory Drives With Dry Friction Tensioners

1998 ◽  
Vol 120 (4) ◽  
pp. 909-916 ◽  
Author(s):  
M. J. Leamy ◽  
N. C. Perkins
Keyword(s):  
Dry Friction ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Harmonic Excitation ◽  
Incremental Harmonic Balance Method ◽  
Stick Slip ◽  
Automotive Engine ◽  
Wide Range ◽  
Serpentine Belt ◽  
Nonlinear Equations Of Motion

Belt drives have long been utilized in engine applications to power accessories such as alternators, pumps, compressors and fans. Drives employing a single, flat, “serpentine belt” tensioned by an “automatic tensioner” are now common in automotive engine applications. The automatic tensioner helps maintain constant belt tension and to dissipate unwanted belt drive vibration through dry friction. The objective of this study is to predict the periodic rotational response of the entire drive to harmonic excitation from the crankshaft. To this end, a multi-degree of freedom incremental harmonic balance method (IHB) is utilized to compute periodic solutions to the nonlinear equations of motion over a wide range of engine speeds. Computed results illustrate primary and secondary resonances of the accessory drive and tensioner stick-slip motions.

scholarly journals Comparison of Galerkin, POD and Nonlinear-Normal-Modes Models for Nonlinear Vibrations of Circular Cylindrical Shells

2006 ◽  
Author(s):  
M. Amabili ◽  
C. Touze´ ◽  
O. Thomas
Keyword(s):  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Harmonic Excitation ◽  
Nonlinear Responses ◽  
Nonlinear Normal ◽  
The Galerkin Method ◽  
Proper Orthogonal

The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

scholarly journals Nonlinear tuning of microresonators for dynamic range enhancement

2015 ◽  
Vol 471 (2179) ◽  
pp. 20140969 ◽  
Author(s):  
M. Saghafi ◽  
H. Dankowicz ◽  
W. Lacarbonara
Keyword(s):  
Nonlinear Response ◽  
Dynamic Range ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equations Of Motion ◽  
Path Following ◽  
Critical Amplitude ◽  
Inertia Effects ◽  
Response Characteristics ◽  
The Galerkin Method

This paper investigates the development of a novel framework and its implementation for the nonlinear tuning of nano/microresonators. Using geometrically exact mechanical formulations, a nonlinear model is obtained that governs the transverse and longitudinal dynamics of multilayer microbeams, and also takes into account rotary inertia effects. The partial differential equations of motion are discretized, according to the Galerkin method, after being reformulated into a mixed form. A zeroth-order shift as well as a hardening effect are observed in the frequency response of the beam. These results are confirmed by a higher order perturbation analysis using the method of multiple scales. An inverse problem is then proposed for the continuation of the critical amplitude at which the transition to nonlinear response characteristics occurs. Path-following techniques are employed to explore the dependence on the system parameters, as well as on the geometry of bilayer microbeams, of the magnitude of the dynamic range in nano/microresonators.

The Forced Oscillation of the Centrifugal Pendulum With Wide Angles

1953 ◽  
Vol 20 (1) ◽  
pp. 41-47
Author(s):  
F. R. E. Crossley
Keyword(s):  
Forced Oscillation ◽  
Torsional Oscillation ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Harmonic Order ◽  
Pendulum Motion ◽  
Type Curves ◽  
Effective Dynamic ◽  
Nonlinear Equations Of Motion ◽  
Effective Resonance

Abstract The simple pendulum mounted to a rotor suffering torsional oscillation is used as an effective dynamic damper which may be tuned to any one harmonic order of vibration. The nonlinear equations of motion are here developed to investigate whether wider angles of swing, or whether larger or smaller sizes of pendulum relative to its carrier, are more effective. Resonance-type curves are drawn by assuming a single harmonic excitation and hyperelliptic pendulum motion without damping; it is shown that theoretically oscillations up to 90 deg may be used which are predictable, and that in all cases the tuning must be higher than that indicated by small-angle theory.

Forced Wave Propagation in Elastic Cables With Small Curvature

1995 ◽  
Author(s):  
M. Behbahani-Nejad ◽  
N. C. Perkins
Keyword(s):  
Nonlinear Response ◽  
Asymptotic Form ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Forced Response ◽  
Transverse Waves ◽  
Propagating Waves ◽  
Distinct Frequency ◽  
Elastic Cables

Abstract This study presents an investigation of the coupled longitudinal-transverse waves that propagate along an elastic cable. The coupling considered derives from the equilibrium curvature (sag) of the cable. A mathematical model is presented that describes the three-dimensional nonlinear response of a long elastic cable. An asymptotic form of this model is derived for the linear response of cables having small equilibrium curvature. Linear in-plane response is described by coupled longitudinal-transverse partial differential equations of motion, which are comprehensively evaluated herein. The spectral relation governing propagating waves is derived using transform methods. In the spectral relation, three qualitatively distinct frequency regimes exist that are separated by two cut-off frequencies. This relation is employed in deriving a Green’s function which is then used to construct solutions for in-plane response under arbitrarily distributed harmonic excitation. Analysis of forced response reveals the existence of two types of periodic waves which propagate through the cable, one characterizing extension-compressive deformations (rod-type) and the other characterizing transverse deformations (string-type). These waves may propagate or attenuate depending on wave frequency. The propagation and attenuation of both wave types are highlighted through solutions for an infinite cable subjected to a concentrated harmonic excitation source.

Harmonically Forced Wave Propagation in Elastic Cables With Small Curvature

1997 ◽  
Vol 119 (3) ◽  
pp. 390-397 ◽  
Author(s):  
M. Behbahani-Nejad ◽  
N. C. Perkins
Keyword(s):  
Nonlinear Response ◽  
Asymptotic Form ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Forced Response ◽  
Periodic Waves ◽  
Transverse Waves ◽  
Propagating Waves ◽  
Elastic Cables

This study presents an investigation of coupled longitudinal-transverse waves that propagate along an elastic cable. The coupling considered derives from the equilibrium curvature (sag) of the cable. A mathematical model is presented that describes the three-dimensional nonlinear response of an extended elastic cable. An asymptotic form of this model is derived for the linear response of cables having small equilibrium curvature. Linear, in-plane response is described by coupled longitudinal-transverse partial differential equations of motion, which are comprehensively evaluated herein. The spectral relation governing propagating waves is derived using transform methods. In the spectral relation, three qualitatively distinct regimes exist that are separated by two cut-off frequencies which are strongly influenced by cable curvature. This relation is employed in deriving a Green’s function which is then used to construct solutions for in-plane response under arbitrarily distributed harmonic excitation. Analysis of forced response reveals the existence of two types of periodic waves which propagate through the cable, one characterizing extension-compressive deformations (rod-type) and the other characterizing transverse deformations (string-type). These waves may propagate or attenuate depending on wave frequency. The propagation and attenuation of both wave types are highlighted through solutions for an infinite cable subjected to a concentrated harmonic excitation source.

Nonlinear Response of a Cylindrical Shell to an Impulsive Pressure

1969 ◽  
Vol 36 (2) ◽  
pp. 277-284 ◽  
Author(s):  
E. G. Lovell ◽  
I. K. McIvor
Keyword(s):  
Cylindrical Shell ◽  
Nonlinear Response ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Fundamental Difference ◽  
Large Displacements ◽  
Flexural Response ◽  
Modal Coupling ◽  
Impulsive Pressure

When a circular cylindrical shell (plane strain) is subjected to a uniform radial impulse, the resulting circular mode may be unstable. In such a case flexural motion is excited, resulting in rather large displacements and stress. A previous nonlinear analysis [1]1 used a linear inextensionality constraint and displacement representation for the flexural response. A formulation employing a nonlinear inextensionality constraint is presented in this paper, and a comparison is made with the earlier work. The most significant result is a fundamental difference between the equations of motion; in this analysis the nonlinear modal coupling is primarily inertial. The condition for stability of the circular mode is unaffected, but substantial differences may occur in the long-term (nonlinear) response.

Vibration Control of the Flexible Robotic Arm through Modal Interaction

2014 ◽  
Vol 490-491 ◽  
pp. 1142-1145
Author(s):  
Zhi Hui Gao ◽  
Bing Dong Liu ◽  
Bo Shan
Keyword(s):  
Vibration Control ◽  
Control Method ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Rigid Motion ◽  
Robotic Arm ◽  
Vibration Absorber ◽  
Modal Interaction ◽  
Flexible Arm ◽  
Nonlinear Equations Of Motion

A vibration control method is proposed to suppress nonlinear large vibration of the flexible robotic arm undergoing rigid motion. The method takes advantage of modal interaction and is implemented based on internal resonance. To attenuate vibration of the flexible arm, another vibrating system, consisting of a rigid link, a flexible joint and a damper, is introduced as a vibration absorber. Perturbation technique is used to study the transient response of the nonlinear equations of motion. Numerical simulation results preliminarily verify that the proposed control strategy is able to effectively reduce vibration of the flexible robotic arm.

Non-Linear Vibrations of Fluid-Filled, Simply Supported Circular Cylindrical Shells: Theory and Experiments

2000 ◽  
Author(s):  
M. Amabili ◽  
F. Pellicano ◽  
M. P. Païdoussis
Keyword(s):  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Harmonic Excitation ◽  
Travelling Wave ◽  
Galerkin Projection ◽  
Inviscid Fluid ◽  
Simply Supported ◽  
Longitudinal Modes ◽  
Circular Cylindrical Shells

Abstract The large-amplitude response of thin, simply supported circular cylindrical shells to a harmonic excitation in the spectral neighbourhood of one of the lowest natural frequencies is investigated. Donnell’s nonlinear shallow-shell theory is used and the solution is obtained by Galerkin projection. A mode expansion including driven and companion modes, axisymmetric modes and additional asymmetric modes is used. In particular, asymmetric modes with twice the number of circumferential waves of driven and companion modes are included in the analysis. The boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satisfied. The effect of internal quiescent, incompressible and inviscid fluid is investigated. The equations of motion are studied by using a code based on the Collocation Method. Validation of the present model is obtained by comparison with other authoritative results and new experimental results. The effect of the number of axisymmetric modes used in the expansion on the response of the shell is investigated, clarifying questions open for a long time. The contribution of additional longitudinal modes is absolutely insignificant in both the driven and companion mode responses. The effect of modes with harmonics of the circumferential mode number n under consideration is limited so far as the trend of nonlinearity is concerned, but is significant in the response with companion mode participation for lightly damped shells (empty shells). Results show the occurrence of travelling wave response in the proximity of the resonance frequency, the fundamental role of the first and third axisymmetric modes in the expansion of the radial deflection with one longitudinal half-wave, and limit cycle responses. A liquid (water) contained in the shell generates a much stronger softening behaviour of the system. Experiments with a water-filled circular cylindrical shell made of steel are in very good agreement with the present theory.

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