Analysis of the Primary and Secondary Resonances of Viscoelastic Beams Made of Zener Material

2019 ◽  
Vol 14 (9) ◽  
Author(s):  
Przemysław Wielentejczyk ◽  
Roman Lewandowski
Keyword(s):  
Steady State ◽  
Dynamic Behavior ◽  
Harmonic Balance ◽  
Virtual Work ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Steady State Solution ◽  
Response Curves ◽  
Zener Model ◽  
Secondary Resonances

The problem of geometrically nonlinear, steady-state vibrations of beams made of viscoelastic (VE) materials is considered in this paper. The Euler–Bernoulli and the von Kármán theories are used to describe the dynamic behavior of beams. The VE material of the beams is modeled using the Zener model. Two harmonics are present in the assumed steady-state solution of the problem at hand, which enables an analysis of both the primary and secondary resonances. The virtual work equation and the harmonic balance method are used to derive the amplitude equations in the explicit form. The response curves are determined using the continuation method and treating the frequency of excitation as the main parameter. The results of several examples, which illustrate the dynamic behavior of the considered beams, are presented and discussed.

Effects of nonlinear interactions of flexural modes on the performance of a beam autoparametric vibration absorber

2019 ◽  
Vol 26 (7-8) ◽  
pp. 459-474
Author(s):  
Saeed Mahmoudkhani ◽  
Hodjat Soleymani Meymand
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Primary System ◽  
Lumped Mass ◽  
Response Curves ◽  
Internal Resonances ◽  
The One

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

Multiple Harmonic Balance Method for Aperiodic Vibration of a Piecewise-Linear System

1998 ◽  
Vol 120 (1) ◽  
pp. 181-187 ◽  
Author(s):  
Y. B. Kim
Keyword(s):  
Linear System ◽  
Harmonic Balance ◽  
Piecewise Linear ◽  
Harmonic Balance Method ◽  
Steady State Solution ◽  
Balance Method ◽  
Piecewise Linear System ◽  
Higher Dimensional ◽  
Computational Procedures ◽  
The Stability

A multiple harmonic balance method is presented in this paper for obtaining the aperiodic steady-state solution of a piecewise-linear system. As the method utilizes general and systematic computational procedures, it can be applied to analyze the multi-tone or combination-tone responses for the higher dimensional nonlinear systems such as rotors. Moreover, it is capable of informing the stability of the obtained solution using Floquet theory. To demonstrate the systematic approach of the new method, the almost periodic forced vibration of an articulated loading platform (ALP) with a piecewise-linear stiffness is computed as an example.

A study on nonlinear, parametric aeroelastic energy harvesters under oscillatory airflow

2020 ◽  
pp. 107754632097447
Author(s):  
Mohammad Mehdi Meshki ◽  
Ali Salehzadeh Nobari ◽  
Mohammad Homayoune Sadr
Keyword(s):  
Parametric Excitation ◽  
Indirect Effects ◽  
Harmonic Balance ◽  
Energy Harvester ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Direct And Indirect Effects ◽  
Flow Oscillation ◽  
Energy Harvesters ◽  
Wide Range

In this study, based on parametric excitation originating from airflow oscillation, a novel nonlinear aeroelastic energy harvester is proposed. In this respect, first, the governing equation of the system is derived and studied thoroughly to understand the direct and indirect effects of airflow oscillation on the local and global responses of the system. Then, by using a pseudo-arclength continuation method based on the harmonic balance method, the stable and unstable periodic and quasi-periodic responses of the system are tracked and analyzed. It is demonstrated that the proposed self-parametric (combination parametric and self-excitation) energy harvester can extract more power than the respective nonparametric system for a wide range of amplitudes and frequencies. The gained knowledge of parametric, aeroelastic systems is applicable for both aero-harvesters and other aeroelastic systems undergoing flow oscillation.

An Investigation of Steady-State Dynamic Response of a Sphere-Plane Contact Interface With Contact Loss

2006 ◽  
Vol 74 (2) ◽  
pp. 249-255 ◽  
Author(s):  
Q. L. Ma ◽  
A. Kahraman ◽  
J. Perret-Liaudet ◽  
E. Rigaud
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Numerical Solutions ◽  
Fourier Transforms ◽  
Damping Ratio ◽  
Harmonic Balance Method ◽  
Hertzian Contact ◽  
Contact Interface ◽  
Discrete Fourier Transforms ◽  
Plane Contact

In this study, the dynamic behavior of an elastic sphere-plane contact interface is studied analytically and experimentally. The analytical model includes both a continuous nonlinearity associated with the Hertzian contact and a clearance-type nonlinearity due to contact loss. The dimensionless governing equation is solved analytically by using multi-term harmonic balance method in conjunction with discrete Fourier transforms. The accuracy of the dynamic model and solution methods is demonstrated through comparisons with experimental data and numerical solutions for both harmonic amplitudes of the acceleration response and the phase difference between the response and the force excitation. A single-term harmonic balance approximation is used to derive a criterion for contact loss to occur. The influence of harmonic external excitation f(τ) and damping ratio ζ on the steady state response is also demonstrated.

scholarly journals Theoretical Design and Characteristics Analysis of a Quasi-Zero Stiffness Isolator Using a Disk Spring as Negative Stiffness Element

2015 ◽  
Vol 2015 ◽  
pp. 1-19 ◽  
Author(s):  
Lingshuai Meng ◽  
Jinggong Sun ◽  
Wenjuan Wu
Keyword(s):  
Damping Ratio ◽  
Low Frequency ◽  
Harmonic Balance Method ◽  
Steady State Solution ◽  
Excitation Amplitude ◽  
Response Curves ◽  
Disk Spring ◽  
Performance Effects ◽  
Characteristics Analysis ◽  
Isolation Performance

This paper presents a novel quasi-zero stiffness (QZS) isolator designed by combining a disk spring with a vertical linear spring. The static characteristics of the disk spring and the QZS isolator are investigated. The optimal combination of the configurative parameters is derived to achieve a wide displacement range around the equilibrium position in which the stiffness has a low value and changes slightly. By considering the overloaded or underloaded conditions, the dynamic equations are established for both force and displacement excitations. The frequency response curves (FRCs) are obtained by using the harmonic balance method (HBM) and confirmed by the numerical simulation. The stability of the steady-state solution is analyzed by applying Floquet theory. The force, absolute displacement, and acceleration transmissibility are defined to evaluate the isolation performance. Effects of the offset displacement, excitation amplitude, and damping ratio on the QZS isolator and the equivalent system (ELS) are studied. The results demonstrate that the QZS isolator for overloaded or underloaded can exhibit different stiffness characteristics with changing excitation amplitude. If loaded with an appropriate mass, excited by not too large amplitude, and owned a larger damper, the QZS isolator can possess better isolation performance than its ELS in low frequency range.

Simulation of Unsteady Turbomachinery Flows Using an Implicitly Coupled Nonlinear Harmonic Balance Method

2011 ◽  
Author(s):  
Jonathan M. Weiss ◽  
Venkataramanan Subramanian ◽  
Kenneth C. Hall
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Time Derivative ◽  
Computational Cost ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Computationally Efficient ◽  
State Equations ◽  
Approximate Factorization ◽  
Efficient Manner

A nonlinear harmonic balance method for the simulation of turbomachinery flows is presented. The method is based on representing an unsteady, time periodic flow by a Fourier series in time and then solving a set of mathematically steady-state equations to obtain the Fourier coefficients. The steady-state solutions are stored at discrete time levels distributed throughout one period of unsteadiness and are coupled via the physical time derivative and at periodic boundaries. Implicit coupling between time levels is achieved in a computationally efficient manner through approximate factorization of the linear system that results from the discretized equations. Unsteady, rotor-stator interactions are performed to validate the implementation. Results based on the harmonic balance method are compared against those obtained using a full unsteady, time-accurate calculation using moving meshes. The implicitly coupled nonlinear harmonic balance method is shown to produce a solution of reasonable accuracy compared to the full unsteady approach but with significantly less computational cost.

RESONANCE RESPONSE OF A SIMPLY SUPPORTED ROTOR-MAGNETIC BEARING SYSTEM BY HARMONIC BALANCE

2012 ◽  
Vol 22 (06) ◽  
pp. 1250136 ◽  
Author(s):  
A. Y. T. LEUNG ◽  
ZHONGJIN GUO
Keyword(s):  
Harmonic Balance ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Balance Method ◽  
Magnetic Bearing ◽  
Homotopy Continuation ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Resonance Response ◽  
Strongly Nonlinear System

Both the primary and superharmonic resonance responses of a rigid rotor supported by active magnetic bearings are investigated by means of the total harmonic balance method that does not linearize the nonlinear terms so that all solution branches can be studied. Two sets of second order ordinary differential equations governing the modulation of the amplitudes of vibration in the two orthogonal directions normal to the shaft axis are derived. Primary resonance is considered by six equations and superharmonic by eight equations. These equations are solved using the polynomial homotopy continuation technique to obtain all the steady state solutions whose stability is determined by the eigenvalues of the Jacobian matrix. It is found that different shapes of frequency-response and forcing amplitude-response curves can exist. Multiple-valued solutions, jump phenomenon, saddle-node, pitchfork and Hopf bifurcations are observed analytically and verified numerically. The new contributions include the foolproof multiple solutions of the strongly nonlinear system by means of the total harmonic balance. Some predicted frequency varying amplitudes could not be obtained by the multiple scales method.

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