HARMONIC BALANCE METHOD USED FOR CALCULATING THE STEADY STATE OSCILLATIONS OF A SIMPLE ONE-CYLINDER COLD ENGINE

2001 ◽  
Vol 241 (4) ◽  
pp. 541-565 ◽  
Author(s):  
F. ALBERTSON ◽  
J. GILBERT
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method

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.

Steady State Solutions to a Forced Nonlinear Oscillator: A Comparison of Results Using a Generalized Harmonic Balance Method and by Numerical Integrations

1993 ◽  
Author(s):  
Ben Noble ◽  
Julian J. Wu
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Initial Point ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Original Equation ◽  
Long Time ◽  
Steady State Solutions ◽  
Generalized Harmonic Balance Method

Abstract Steady state solutions for nonlinear dynamic problems are interesting because (1) the long time behaviors of many problems are of practical concern, and, (2) these behaviors are often difficult to predict. This paper first presents a brief description of a generalized harmonic balance method (GHB) for steady state solutions to nonlinear problems via a nonlinear oscillator problem with a quadratic nonlinearity. Using this approach, steady state solutions are obtained for problems with several parameters: damping, nonlinearity and frequency (subharmonic, superharmonic and primary resonance). These results, plotted in time evolution curves and phase diagrams are compared with those obtained by numerically integrating the original differential equations. The effect of initial conditions on long time solutions is discussed. This investigation indicates that (1) the GHB steady state is an excellent approximate solution to that of the original equation if such a solution is numerically stable, and (2) the GHB steady state simply indicates a region of instability when the numerical solution to the original equation, using a point in that region as the initial point, is unstable.

Computing Fluid Structure Interaction Coupling Time Spectral Method (TSM) and Harmonic Balance Method (HBM)

2017 ◽  
Author(s):  
Aude Cadel ◽  
Ghislaine Ngo Boum ◽  
Fabrice Thouverez ◽  
Alain Dugeai ◽  
Marie-Océane Parent
Keyword(s):  
Steady State ◽  
Flow Field ◽  
Spectral Method ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Forced Response ◽  
Balance Method ◽  
Integration Scheme ◽  
Fluid Structure ◽  
Time Spectral Method

This paper deals with fluid-structure interactions (FSI), involving a blade profile, submitted to different sources of excitations, as if it were included in a real engine. Two forces of excitation will be considered on the NACA 64A010 airfoil, described in : an external force, due to a forced rotation motion of the blade, and an aerodynamic force, induced by fluid flow around the structure. By using the Harmonic Balance Method, the airfoil’s motion equation becomes an algebraic problem. Then, this system is solved for each frequency of a chosen range. Therefore, the fluid effect on the translation motion of the profile is studied. To compute the time periodic aerodynamic field, the Time Spectral Method, implemented in the Onera’s elsA solver, is used for a fast and efficient resolution. This method relies on a time-integration scheme that turns the resolution of the turbulent Navier-Stokes problem into the resolution of several coupled steady state problems computed at different instants of the time period of the movement. The Theodorsen approach with several hypothesis exposed in allows an analytic estimation of the unsteady lift effort. The two approaches are compared for an imposed motion. In order to predict the dynamic behavior of the system, a fully coupled numerical methodology is developed. For each frequency and at each iteration, TSM supplies the flow field which is used by HBM as a nonlinear excitation on the structure to computate a periodic response and conversely, HBM supplies the new deformed mesh used by TSM to compute the flow field. This strategy has the advantage that all computations take place in the spectral domain, allowing thus to find the steady-state behavior of the fluid and the structure without computing any transient state. The analysis provides the Frequency Forced Response. Some frequencies in the range corresponding to a contribution change between structure and fluid damping are precisely highlighted.

scholarly journals The Elliptic Harmonic Balance Method for the Performance Analysis of a Two-Stage Vibration Isolation System with Geometric Nonlinearity

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Weilei Wu ◽  
Bin Tang
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Vibration Isolation ◽  
Harmonic Balance Method ◽  
Elliptic Functions ◽  
Balance Method ◽  
Damping Force ◽  
Two Stage ◽  
Isolation System ◽  
Vibration Isolation System

This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.

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