COMMENTS ON “A RATIONAL HARMONIC BALANCE APPROXIMATION FOR THE DUFFING EQUATION OF MIXED PARITY” (WITH AUTHORS' REPLY)

1998 ◽  
Vol 216 (1) ◽  
pp. 187-189 ◽  
Author(s):  
R.E. Mickens
Keyword(s):  
Harmonic Balance ◽  
Duffing Equation

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.

TRANSITION CURVES AND BIFURCATIONS OF A CLASS OF FRACTIONAL MATHIEU-TYPE EQUATIONS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250275 ◽  
Author(s):  
A. Y. T. LEUNG ◽  
ZHONGJIN GUO ◽  
H. X. YANG
Keyword(s):  
Fractional Order ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Duffing Equation ◽  
Response Amplitude ◽  
Transition Curve ◽  
Homotopy Continuation ◽  
Transition Curves ◽  
Fractional Orders

A general version of the fractional Mathieu equation and the corresponding fractional Mathieu–Duffing equation are established for the first time and investigated via the harmonic balance method. The approximate expressions for the transition curves separating the regions of stability are derived. It is shown that a change in the fractional derivative order remarkably affects the shape and location of the transition curves in the n = 1 tongue. However, the shape of the transition curve does not change very much for different fractional orders for the n = 0 tongue. The steady state approximate responses of the corresponding fractional Mathieu–Duffing equation are obtained by means of harmonic balance, polynomial homotopy continuation and technique of linearization. The curves with respect to fractional order versus response amplitude, driving amplitude versus response amplitude with different fractional orders are shown. It can be found that the bifurcation point and stability of branch solutions is different under different fractional orders of system. When the fractional order increases to some value, the symmetric breaking, saddle-node bifurcation as well as period-doubling bifurcation phenomena are found and exhibited analytically by taking the driving amplitude as the bifurcation parameter.

Non-linear two-frequency analysis of SIS mixers through harmonic balance

2002 ◽  
Vol 12 (3) ◽  
pp. 165-168
Author(s):  
S. Withington ◽  
P. Kittara ◽  
G. Yassin
Keyword(s):  
Frequency Analysis ◽  
Harmonic Balance ◽  
Non Linear

THE NEW ALGORITHM OF SOLVING HARMONIC BALANCE EQUATIONS

2019 ◽  
Author(s):  
Vladimir Lantsov ◽  
A. Papulina
Keyword(s):  
Harmonic Balance ◽  
Model Order Reduction ◽  
Order Reduction ◽  
Balance Equations ◽  
Model Order ◽  
Cad Systems ◽  
Computational Costs ◽  
Reduction Methods ◽  
Model Equations

The new algorithm of solving harmonic balance equations which used in electronic CAD systems is presented. The new algorithm is based on implementation to harmonic balance equations the ideas of model order reduction methods. This algorithm allows significantly reduce the size of memory for storing of model equations and reduce of computational costs.

A Scheduling Approach To Parallel Harmonic Balance Simulation

2009 ◽  
Author(s):  
David L. Rhodes ◽  
Apostolos Gerasoulis
Keyword(s):  
Harmonic Balance

Aerodynamic Asymmetry Analysis of Unsteady Flows in Turbomachinery

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
Kivanc Ekici ◽  
Robert E. Kielb ◽  
Kenneth C. Hall
Keyword(s):  
Harmonic Balance ◽  
Coupling Effect ◽  
Unsteady Flows ◽  
Unsteady Aerodynamic ◽  
Blade Row ◽  
Balance Technique ◽  
Turbomachinery Blades ◽  
Aerodynamic Asymmetry ◽  
Frequency Domain Approach ◽  
Harmonic Balance Technique

A nonlinear harmonic balance technique for the analysis of aerodynamic asymmetry of unsteady flows in turbomachinery is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of turbomachinery blades to self-excited vibrations. Traditionally, researchers have investigated the unsteady response of a blade row with the assumption that all the blades in the row are identical. With this assumption the entire wheel can be modeled using complex periodic boundary conditions and a computational grid spanning a single blade passage. In this study, the steady/unsteady aerodynamic asymmetry is modeled using multiple passages. Specifically, the method has been applied to aerodynamically asymmetric flutter problems for a rotor with a symmetry group of 2. The effect of geometric asymmetries on the unsteady aerodynamic response of a blade row is illustrated. For the cases investigated in this paper, the change in the diagonal terms (blade on itself) dominated the change in stability. Very little mode coupling effect caused by the off-diagonal terms was found.

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