A New Spatial and Temporal Incremental Harmonic Balance Method for Obtaining Steady-State Responses of a One-Dimensional Continuous System

2016 ◽  
Author(s):  
X. F. Wang ◽  
W. D. Zhu
Keyword(s):  
Steady State ◽  
Computer Program ◽  
Harmonic Balance ◽  
Jacobian Matrix ◽  
Harmonic Balance Method ◽  
Continuous System ◽  
One Dimensional ◽  
Spatial Coordinate ◽  
Steady State Responses ◽  
State Responses

A new spatial and temporal incremental harmonic balanced (STIHB) method is developed for obtaining steady-state responses of a one-dimensional continuous system. In the STIHB method, Galerkin procedure for a governing partial differential equation (PDE) in the spatial coordinate to obtain a set of ordinary differential equations (ODEs) and the harmonic balance procedure for the set of ODEs in the temporal coordinate to obtain the harmonic balanced residual are combined to be Galerkin procedures for the PDE in the spatial and temporal coordinates to simultaneously obtain the spatial and temporal harmonic balanced residual, and integrations in Galerkin procedures are replaced by the fast discrete sine transform (DST) or fast discrete cosine transform (DCT) in the spatial coordinate and the fast Fouriour transform (FFT) in the temporal coordinate, which is referred to as a DST-FFT or DCT-FFT procedure. The harmonic balanced residual for an arbitrary second-order PDE can be automatically and efficiently obtained by a computer program when the expression of the PDE is given, where numbers of basis functions in the spatial and temporal coordinates can be arbitrarily selected and no more extra derivations are needed. There are two versions of the STIHB method. In the simple version, the DST-FFT or DCT-FFT procedure to calculate the harmonic balanced residual and Broyden’s method that is a quasi-Newton method are combined to find solutions that make the residual vanish, which can be used to construct steady-state solutions of the PDE. In the complex version, the exact Jacobian matrix is derived and used in Newton-Raphson method to achieve faster convergence. While its derivation is complex, the exact Jacobian matrix for the arbitrary PDE can be automatically and efficiently obtained by following a calculation routine when the linearized expression of the PDE is given, and it can be easily implemented by a computer program. The exact Jacobian matrix can also be used to study stability of steady-state responses, where no more extra derivations are needed. The STIHB method is demonstrated by studying the transverse vibration of a string with geometric nonlinearity; its frequency-response curves with weak and strong nonlinearities and different numbers of trial functions are calculated, and stability of solutions on the curves is studied.

A New Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Steady-State Responses of a One-Dimensional Second-Order Continuous System

2016 ◽  
Vol 84 (1) ◽  
Author(s):  
X. F. Wang ◽  
W. D. Zhu
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Term ◽  
Jacobian Matrix ◽  
Continuous System ◽  
Basis Functions ◽  
One Dimensional ◽  
Steady State Responses ◽  
Periodic Steady State ◽  
State Responses ◽  
Order Continuous

A new spatial and temporal harmonic balance (STHB) method is developed for obtaining periodic steady-state responses of a one-dimensional second-order continuous system. The spatial harmonic balance procedure with a series of sine and cosine basis functions can be efficiently conducted by the fast discrete sine and cosine transforms, respectively. The temporal harmonic balance procedure with basis functions of Fourier series can be efficiently conducted by the fast Fourier transform (FFT). In the STHB method, an associated set of ordinary differential equations (ODEs) of a governing partial differential equation (PDE), which is obtained by Galerkin method, does not need to be explicitly derived, and complicated calculation of a nonlinear term in the PDE can be avoided. The residual and the exact Jacobian matrix of an associated set of algebraic equations that are temporal harmonic balanced equations of the ODEs, which are used in Newton–Raphson method to iteratively search a final solution of the PDE, can be directly obtained by STHB procedures for the PDE even if the nonlinear term is included. The relationship of Jacobian matrix and Toeplitz form of the system matrix of the ODEs provides an efficient and convenient way to stability analysis for the STHB method; bifurcations can also be indicated. A complex boundary condition of a string with a spring at the boundary can be handled by the STHB method in combination with the spectral Tau method.

The Analysis of Linear Rotor-Bearing Systems: A General Transfer Matrix Method

1993 ◽  
Vol 115 (4) ◽  
pp. 490-497 ◽  
Author(s):  
An-Chen Lee ◽  
Yuan-Pin Shih ◽  
Yuan Kang
Keyword(s):  
Steady State ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Three Dimensional ◽  
Continuous System ◽  
State Variables ◽  
Rotor Bearing ◽  
Steady State Responses ◽  
State Responses

A general transfer matrix method (GTMM) is developed in the present work for analyzing the steady-state responses of rotor-bearing systems with an unbalancing shaft. Specifically, we derived the transfer matrix of shaft segments by considering the state variables of shaft in a continuous system sense to give the most general formulation. The shaft unbalance, axial force, and axial torque are all taken into consideration so that the completeness of transfer matrix method for steady-state analysis of linear rotor-bearing systems is reached. To demonstrate the effectiveness of this approach, a numerical example is presented to estimate the effect of three-dimensional distribution of shaft unbalance on the steady-state responses by GTMM and finite element method (FEM).

Steady-State Responses of a Beam on Idealized Strain-Hardening Foundations for a Moving Load

1973 ◽  
Vol 40 (4) ◽  
pp. 1040-1044 ◽  
Author(s):  
T. M. Mulcahy
Keyword(s):  
Steady State ◽  
Strain Hardening ◽  
Elastic Foundation ◽  
Moving Load ◽  
Elastic Beam ◽  
Point Load ◽  
One Dimensional ◽  
Wide Range ◽  
Steady State Responses ◽  
State Responses

The steady-state responses to a point load moving with constant velocity on an elastic beam which rests on two types of idealized strain-hardening foundations are considered. The one-dimensional elastic-rigid foundation problem is shown to be equivalent to an elastic foundation with two traveling point loads. The opposing loads produce deflections which remain bounded for all load velocities and less than the corresponding elastic foundation results. The deflections of a one-dimensional elastic-perfectly plastic foundation are shown to be bounded for all load velocities. However, deflections significantly larger than the corresponding elastic foundation results occur over a wide range of velocities which are less than the elastic foundation critical velocity.

Investigation on the Steady-State Responses of Asymmetric Rotors

1992 ◽  
Vol 114 (2) ◽  
pp. 194-208 ◽  
Author(s):  
Yuan Kang ◽  
Yuan-Pin Shih ◽  
An-Chen Lee
Keyword(s):  
Harmonic Balance ◽  
Mathematical Formulation ◽  
Harmonic Balance Method ◽  
Element Formulation ◽  
Internal Damping ◽  
Gyroscopic Moment ◽  
Steady State Responses ◽  
Angular Displacements ◽  
Stiffness And Damping ◽  
State Responses

This paper is to generalize the previous work by utilizing finite element formulation to accommodate the effects of both deviatoric inertia and stiffness due to asymmetry of flexible shaft and disk. A Timoshenko beam element is employed to simulate rotor-bearing systems by taking the gyroscopic moment, rotary inertia, shear deformation of shaft and, asymmetry of disk and shaft into account. Internal damping is not included but the extension is straightforward. Eulerian angles are used to describe the orientations of shaft element and disk, by which, in opposite to the vectorial approach, the mathematical formulation will be symmetric for angular displacements in two directions. The effects of the angle between the major axes of shaft and disk, deviatoric inertia of the asymmetric shaft, and characteristics of bearing on synchronous critical and subcritical speeds are estimated in conjunction with the harmonic balance method. Numerical examples show that the resonant speeds, at which peak responses occur, change due to various angles between major axes, asymmetry of shaft, stiffness, and damping of bearing.

Dynamic Analysis of an Automotive Belt-Drive System With a Noncircular Sprocket by a Modified Incremental Harmonic Balance Method

2016 ◽  
Vol 139 (1) ◽  
Author(s):  
X. F. Wang ◽  
W. D. Zhu
Keyword(s):  
Steady State ◽  
Harmonic Balance Method ◽  
Drive System ◽  
Belt Drive ◽  
Angular Variation ◽  
Incremental Harmonic Balance Method ◽  
Steady State Responses ◽  
Incremental Harmonic Balance ◽  
Drive Model ◽  
State Responses

A dynamic model of an automotive belt-drive system with a noncircular sprocket instead of a round sprocket is developed in this work to study the effect of reducing the angular variation of camshafts. There are two submodels in the belt-drive system, which are an engine model and a belt-drive model, and they are decoupled to simplify the analysis. When the belt-drive system operates at a steady-state, it is described by a nonlinear model with forced excitation, which can be approximated by a linear model with combined parametric and forced excitations. Steady-state responses of the engine and belt-drive models are calculated by a modified incremental harmonic balance method that incorporates fast Fourier transform and Broyden's method, which is efficient and accurate to obtain a periodic response of a multi-degree-of-freedom system. Steady-state responses of the angular variation of camshafts with different values of sprocket parameters are compared to investigate their optimal values to reduce the angular variation of camshafts. The optimal eccentricity and installation angle are larger and smaller than those from the kinematic model, respectively, which is consistent with published experimental results. This study first shows from a dynamic point of view why use of a noncircular sprocket can reduce the angular variation of camshafts in the operating speed of an engine. Simulation of a speed-up procedure for different sprocket parameters shows results that are consistent with steady-state responses. The belt-drive model developed in this work can be used to select sprocket parameters to minimize the angular variation of camshafts and numerically evaluate the dynamic performance of a belt-drive system with a given design of sprocket parameters.

Steady-State Dynamic Behavior of a Flexible Rotor With Auxiliary Support from a Clearance Bearing

1999 ◽  
Vol 121 (1) ◽  
pp. 78-83 ◽  
Author(s):  
H. Xie ◽  
G. T. Flowers ◽  
L. Feng ◽  
C. Lawrence
Keyword(s):  
Steady State ◽  
Dynamic Behavior ◽  
Harmonic Balance Method ◽  
Operating Conditions ◽  
State Behavior ◽  
Rotor Imbalance ◽  
Wide Range ◽  
Steady State Responses ◽  
Stiffness And Damping ◽  
State Responses

This paper investigates the steady-state responses of a rotor system supported by auxiliary bearings in which there is a clearance between the rotor and the inner race of the bearing. A simulation model based upon the rotor of a production jet engine is developed and its steady-state behavior is explored over a wide range of operating conditions for various parametric configurations. Specifically, the influence of rotor imbalance, clearance, support stiffness and damping is studied. Bifurcation diagrams are used as a tool to examine the dynamic behavior of this system as a function of the aforementioned parameters. The harmonic balance method is also employed for synchronous response cases. The observed dynamical responses is discussed and some insights into the behavior of such systems are presented.

Dynamic Analysis of an Automotive Belt-Drive System With a Noncircular Sprocket by a Modified Incremental Harmonic Balance Method

2016 ◽  
Author(s):  
X. F. Wang ◽  
W. D. Zhu
Keyword(s):  
Steady State ◽  
Dynamic Model ◽  
Kinematic Model ◽  
Harmonic Balance Method ◽  
Drive System ◽  
Belt Drive ◽  
Angular Variation ◽  
Steady State Responses ◽  
Optimal Values ◽  
State Responses

A dynamic model of an automotive belt-drive system with a noncircular sprocket instead of a round sprocket is developed in this work to study the effect of reducing the angular variation of camshafts. There are two sub-models in the belt-drive system, which are an engine model and a belt-drive model, and they are decoupled to simplify the analysis. When the belt-drive system operates at a steady state, it is described by a nonlinear model with forced excitation, which can be approximated by a linear model with combined parametric and forced excitations. Steady-state responses of the engine and belt-drive models are calculated by a modified incremental harmonic balanced method that incorporates fast Fourier transform and Broyden’s method, which is efficient and accurate to obtain a periodic response of a multi-degree-of-freedom system. Steady-state responses of the angular variation of camshafts with different values of sprocket parameters are compared to investigate their optimal values to reduce the angular variation of camshafts. Optimal values of sprocket parameters obtained from the dynamic model of the belt drive differ from those from its kinematic model in some previous studies, which is consistent with published experimental results. This study first shows from a dynamic point of view why use of a noncircular sprocket can reduce the angular variation of camshafts in the operating speed of an engine. Simulation of a speed-up procedure for different sprocket parameters shows results that are consistent with steady-state responses.

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