Nonlinear Vibrations of a Beam Under Harmonic Excitation

1970 ◽  
Vol 37 (2) ◽  
pp. 292-297 ◽  
Author(s):  
W. Y. Tseng ◽  
J. Dugundji
Keyword(s):  
Harmonic Balance ◽  
Stability Problem ◽  
Nonlinear Vibrations ◽  
Harmonic Balance Method ◽  
Type Equation ◽  
Harmonic Excitation ◽  
Duffing Equation ◽  
Straight Beam ◽  
The Stability ◽  
Supporting Base

A straight beam with fixed ends, excited by the periodic motion of its supporting base in a direction normal to the beam span, was investigated analytically and experimentally. By using Galerkin’s method (one mode approximation) the governing partial differential equation reduces to the well-known Duffing equation. The harmonic balance method is applied to solve the Duffing equation. Besides the solution of simple harmonic motion (SHM), many other branch solutions, involving superharmonic motion (SPHM) and subharmonic motion (SBHM), are found experimentally and analytically. The stability problem is analyzed by solving a corresponding variational Hill-type equation. The results of the present analysis agree well with the experiments.

Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation

1971 ◽  
Vol 38 (2) ◽  
pp. 467-476 ◽  
Author(s):  
W.-Y. Tseng ◽  
J. Dugundji
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Type Equation ◽  
Harmonic Excitation ◽  
Harmonic Motion ◽  
Buckled Beam ◽  
Second Mode ◽  
The Stability ◽  
Supporting Base ◽  
Theoretical Results

A buckled beam with fixed ends, excited by the harmonic motion of its supporting base, was investigated analytically and experimentally. Using Galerkin’s method the governing partial differential equation reduced to a modified Duffing equation, which was solved by the harmonic balance method. Besides the solution of simple harmonic motion (SHM), other branch solutions involving superharmonic motion (SPHM) were found experimentally and analytically. The stability of the steady-state SHM and SPHM solutions were analyzed by solving a variational Hill-type equation. The importance of the second mode on these results was examined by a similar stability analysis. The Runge-Kutta numerical integration method was used to investigate the snap-through problem. Intermittent, as well as continuous, snap-through behavior was obtained. The theoretical results agreed well with the experiments.

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.

scholarly journals A Harmonic-Based Method for Computing the Stability of Periodic Oscillations of Non-Linear Structural Systems

2010 ◽  
Author(s):  
O. Thomas ◽  
A. Lazarus ◽  
C. Touze´
Keyword(s):  
Numerical Method ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Computation Time ◽  
Fourier Series Expansion ◽  
Simple Strategy ◽  
The Stability ◽  
Linear Periodic System ◽  
Continuation Technique

In this paper, we present a validation on a practical example of a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Based on Floquet theory and Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eigenvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing the Harmonic Balance Method and Asymptotic Numerical Method continuation technique with the developed Hill method, we obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions in a reduced computation time. This procedure is validated by considering an externally forced string and computing the complete bifurcation diagram with the stability of the periodic solutions. The particular coupled regimes are exhibited and found in excellent agreement with results of the literature, allowing a method validation.

Parametric Stability of a Nonlinear Rotating Blade

2013 ◽  
Author(s):  
Fengxia Wang ◽  
Albert C. J. Luo
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Parametric Stability ◽  
Rotating Blade ◽  
Geometric Term ◽  
Stability And Bifurcations ◽  
The Stability ◽  
The Galerkin Method ◽  
Gyroscopic Systems ◽  
Generalized Harmonic Balance Method

The stability of period-1 motions of a rotating blade with geometric nonlinearity is studied. The roles of cubic stiffening geometric term are considered in the study of nonlinear periodic motions and its stability and bifurcations of a rotating blade. The nonlinear model of a rotating blade is reduced to the ordinary differential equations through the Galerkin method, and the gyroscopic systems with parametric excitations are obtained. The generalized harmonic balance method is employed to determine the period-1 solutions and the corresponding stability and bifurcations.

scholarly journals Calculating periodic vibrations of nonlinear autonomous multi-degree-of-freedom systems by the incremental harmonic balance method

2004 ◽  
Vol 26 (3) ◽  
pp. 157-166
Author(s):  
Nguyen Van Khang ◽  
Thai Manh Cau
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Van Der Pol ◽  
Incremental Harmonic Balance ◽  
The Stability

In this paper the incremental harmonic balance method is used to calculate periodic vibrations of nonlinear autonomous multip-degree-of-freedom systems. According to Floquet theory, the stability of a periodic solution is checked by evaluating the eigenvalues of the monodromy matrix. Using the programme MAPLE, the authors have studied the periodic vibrations of the system multi-degree van der Pol form.

Effect of Fluid Mass on Non-Linear Natural Frequencies of a Rotating Beam

2003 ◽  
Author(s):  
Ahmad A. Al-Qaisia
Keyword(s):  
Harmonic Balance ◽  
Natural Frequencies ◽  
Harmonic Balance Method ◽  
Continuous System ◽  
Assumed Mode Method ◽  
Non Linear ◽  
Mode Method ◽  
The Stability ◽  
The Mathematical Model ◽  
Analytical Expressions

The non-linear natural frequencies of the first three modes of a beam clamped to a rigid rotating hub and carrying a distributed fluid along its span are investigated. The mathematical model is derived using the Lagrangian method and the continuous system is discretized using the assumed mode method. The resulted unimodal nonlinear equation of motion was solved using two methods; harmonic balance (HB) and time transformation (TT), to obtain approximate analytical expressions for the nonlinear natural frequencies. Results have shown that the two terms harmonic balance method (2THB) is more accurate than the time TT method. Results for the effect and type of distribution, i.e. uniform or linearly distributed, on the variation of the nonlinear natural frequency with the rotational speed of the system and how they affect the stability are studied and presented in non-dimensional form.

Nonlinear Vibrations of Piecewise-Linear Systems by Incremental Harmonic Balance Method

1992 ◽  
Vol 59 (1) ◽  
pp. 153-160 ◽  
Author(s):  
S. L. Lau ◽  
W.-S. Zhang
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Vibrations ◽  
Piecewise Linear ◽  
Practical Interest ◽  
Harmonic Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
Stiffness Characteristics ◽  
Ihb Method ◽  
Piecewise Linear Stiffness

The incremental harmonic balance (IHB) method is extended to analyze the periodic vibrations of systems with a general form of piecewise-linear stiffness characteristics. An explicit formulation has been worked out. This development is of significance as many structural and mechanical systems of practical interest possess a piecewise-linear stiffness. Typical examples show that the IHB method is very effective for analyzing this kind of systems under steady-state vibrations.

FEED FORWARD RESIDUE HARMONIC BALANCE METHOD FOR A QUADRATIC NONLINEAR OSCILLATOR

2011 ◽  
Vol 21 (06) ◽  
pp. 1783-1794 ◽  
Author(s):  
AYT LEUNG ◽  
ZHONGJIN GUO
Keyword(s):  
Fourier Series ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
High Order ◽  
Balance Method ◽  
Quadratic Nonlinearity ◽  
Analytical Prediction ◽  
Original Solution ◽  
Feed Forward ◽  
The Stability

The harmonic balance method truncates the Fourier series in a finite number of terms. In this paper we show that the truncated residues may be important to determine the stability of the approximated solution and that the truncated residues in the stability analysis can fully be considered without increasing the number of equations in the original solution. Therefore, the high order superharmonic and subharmonic responses and the cascade of bifurcations to irregular attractor can be accurately approximated by just the first few terms of the Fourier series so that analytical prediction is possible. A harmonically driven oscillator with quadratic nonlinearity is taken as examples. The explicitly analytical solutions are obtained for the steady state solutions and for the high order superharmonic approximation. The stabilities of the solutions are determined by the Floquet theory. It is shown that the predicted stability of the solution can be qualitatively different with and without the consideration of the feed forward residues. The second-, fourth- and eighth-order subharmonic analytical bifurcation solutions are calculated to obtain the cascades of bifurcations to irregular attractor. The improved analytical harmonic approximations are compared with other results and with numerical solutions. It is proved that a two superharmonic expansion with appropriate subharmonic is sufficient for determining the characteristics of the solutions of a harmonically driven oscillator with quadratic nonlinearity.

Nonlinear System Identification of a MEMS Dual-Backplate Capacitance Microphone by Harmonic Balance Method

2005 ◽  
Author(s):  
Jian Liu ◽  
David T. Martin ◽  
Karthik Kadirvel ◽  
Toshikazu Nishida ◽  
Louis N. Cattafesta ◽  
...  
Keyword(s):  
Steady State ◽  
Governing Equation ◽  
Harmonic Balance ◽  
Nonlinear System Identification ◽  
Harmonic Balance Method ◽  
Harmonic Excitation ◽  
Algebraic Equations ◽  
Nonlinear Identification ◽  
System Parameters ◽  
Mems Microphone

This paper presents the nonlinear identification of system parameters for a capacitive dual-backplate MEMS microphone. First, the microphone is modeled by a single-degree-of-freedom (SDOF) second order differential equation with electrostatic and cubic mechanical nonlinearities. A harmonic balance nonlinear identification approach is then applied to the governing equation to obtain a set of algebraic equations that relate the unknown system parameters to the steady-state response of the microphone under the harmonic excitation. The microphone is experimentally characterized and a nonlinear least-squares technique is implemented to identify the system parameters from experimental data. The experimentally extracted bandwidth of the microphone is over 218 kHz. Finally, numerical simulations of the governing equation are performed, using the identified system parameters, to validate the accuracy of the approximate solution. The differences between the properties of the integrated measured center velocity and simulated center displacement responses in the steady state are less than 1%.

Sign in / Sign up

Export Citation Format

Share Document