High Dimensional Harmonic Balance Research Articles

Supplemental-frequency harmonic balance: A new approach for modeling aperiodic aerodynamic response

Presented in this work is a novel and easy-to-implement supplemental-frequency harmonic balance (SF-HB) approach to efficiently compute dynamically aperiodic systems, which can be cumbersome to model using the nominal high-dimensional harmonic balance (HDHB) technique. The stability of the time-spectral operator and the SF-HB solver involving multiple excitation frequencies is ensured by introducing a group of supplemental frequencies based on the fact that aperiodic (or almost aperiodic) response would contain all possible frequencies. Together with the excitation (primary) frequencies, the frequency set forms a nearly-harmonic series leading to a small condition number for the Fourier transformation matrices directly impacting the stability of the HDHB solver. In contrast to the similar work reported in the literature where optimal unequally-spaced sub-time levels are determined through a rather complicated procedure, the current approach simply uses equally-spaced sub-time levels spanning a time period that depends on a predicted base frequency. At convergence, the original excitation frequency modes dominate the solution as desired. This new technique is verified for a forced pitching airfoil in the transonic inviscid flow regime subjected to two excitation frequencies. Results of both periodic and aperiodic cases show that with the help of Fourier interpolation, the entire time history of the dynamic response can be obtained through a single run of the SF-HB solver, for which the efficiency and robustness of the traditional HDHB method is preserved.

Prediction of Period-1 oscillations of the state-dependent delayed compound TCP model with PIE queue management policy via HDHBM

State-dependent time-delayed systems are more realistic delayed systems in nature and human society. Their dynamic problems are hot issues and difficult questions due to the fact that the classical RFDE theory is not suitable for them any more. And its corresponding rigorous theory is to be established. The study on the existence and stability of their periodic solutions is an important topic and a difficult question. Compound TCP is a widely used transport control protocol, and proportional integral enhanced (PIE) queue policy is an effective and simple queue policy in the Internet. Thus it is significant to study the dynamics and control of the compound TCP-PIE system. In this paper, Period-1 oscillations of the state-dependent delayed TCP models with PIE queue management policy are predicted by the high-dimensional harmonic balance method (HDHBM). Compared with the traditional harmonic balance method, HDHBM avoids the complex integration of the Fourier coefficients of the nonlinear right hand of the systems by solving the time domain variables of the Fourier coefficients at equally spaced sub-time over the period of one cycle of motion. Thus it transforms the problem into a relatively simple nonlinear algebraic equations, which greatly reduces the computation. By HDHBM, the approximate expressions of the Period-1 solutions for this state-dependent delayed compound TCP-PIE system are obtained. We compare their amplitudes and phase portraits with those of the DDE-BIFTOOL. They agree well with each other. It suggests the validity, effectiveness and simplicity of the employed method. It also turns out that many parameters of the state-dependent delayed TCP model with PIE queue management policy play a very important role in its dynamic behaviours. It follows that PIE parameter μ and update time T may play the most important roles in the Compound TCP-PIE congestion control model and packet loss probability p(t˜) may dominate the dynamics and performances of the network. As a result, we should control the parameters in a proper range to achieve our desired network performances.

Reducing parametric uncertainty in limit-cycle oscillation computational models

ABSTRACTThe assimilation of discrete data points with model predictions can be used to achieve a reduction in the uncertainty of the model input parameters, which generate accurate predictions. The problem investigated here involves the prediction of limit-cycle oscillations using a High-Dimensional Harmonic Balance (HDHB) method. The efficiency of the HDHB method is exploited to enable calibration of structural input parameters using a Bayesian inference technique. Markov-chain Monte Carlo is employed to sample the posterior distributions. Parameter estimation is carried out on a pitch/plunge aerofoil and two Goland wing configurations. In all cases, significant refinement was achieved in the distribution of possible structural parameters allowing better predictions of their true deterministic values. Additionally, a comparison of two approaches to extract the true values from the posterior distributions is presented.

Nonlinear dynamic analyses on a magnetopiezoelastic energy harvester with reversible hysteresis

This paper presents a systematic approach through a series of nonlinear analyses to predict and design the nonlinear resonance characteristics and energy-harvesting performance of a magnetopiezoelastic vibration energy harvester with reversible hysteresis. To this end, a mathematical model of the energy harvester system, composed of a bimorph cantilever beam along with three permanent magnets, is first derived. With this model, frequency response analyses are conducted using the methods of multiple scales and harmonic balance. In the case of weak excitation, the system’s stationary forced response is obtained through multiple-scale analysis (MSA). With this MSA solution, analytical design criteria in terms of the position parameters of the magnets are derived to determine the hysteresis type of the nonlinear resonance (stiffness hardening or stiffness softening). However, in the case of a relatively strong excitation, the high-dimensional harmonic balance analysis (HDHBA) shows that the stiffness-softening effect tends to strengthen for the oscillation amplitude in a specific condition, and this effect can even lead to hysteresis transition or potential well escape. Using the HDHBA, design criteria are set up and evaluated in terms of source vibration strength to detect the hysteresis transition or the condition required to determine the potential well escape. Finally, based on the present systematic analyses, the complicated resonant responses and energy-harvesting performance of the present system are examined with respect to several design parameters, and the results are discussed in detail.

Stabilization of High-Dimensional Harmonic Balance Solvers Using Time Spectral Viscosity

We investigate the application of a temporal spectral viscosity operator to eliminate aliasing errors associated with the high-dimensional harmonic balance technique, which is an efficient method for modeling nonlinear time-periodic problems. Previous studies have shown that aliasing errors resulting from the discrete Fourier transformation may slow down convergence, trigger a nonlinear instability, or lead to nonphysical solutions. A temporal spectral viscosity operator, similar to that used for pseudospectral methods, is introduced. The temporal spectral viscosity is added to the high-frequency modes of the solution to eliminate aliasing errors so as to ensure the convergence to the physical solution. The implementation of the technique is straightforward and can be incorporated into the high-dimensional harmonic balance solver as a matrix product operator. The accuracy and effectiveness of the modified method is demonstrated for different test cases including a Duffing oscillator and unsteady flow about an oscillating circular cylinder. Finally, the temporal spectral viscosity is applied to a turbomachinery aeroelasticity problem to investigate the effect of added dissipation on the accuracy of unsteady solutions.

A time domain collocation method for studying the aeroelasticity of a two dimensional airfoil with a structural nonlinearity

A time domain collocation method for the study of the motion of a two dimensional aeroelastic airfoil with a cubic structural nonlinearity is presented. This method first transforms the governing ordinary differential equations into a system of nonlinear algebraic equations (NAEs), which are then solved by a Jacobian-inverse-free NAE solver. Using the aeroelastic airfoil as a prototypical system, the time domain collocation method is shown here to be mathematically equivalent to the well known high dimensional harmonic balance method. Based on the fact that the high dimensional harmonic balance method is essentially a collocation method in disguise, we clearly explain the aliasing phenomenon of the high dimensional harmonic balance method. On the other hand, the conventional harmonic balance method is also applied. Previous studies show that the harmonic balance method does not produce aliasing in the framework of solving the Duffing equation. However, we demonstrate that a mathematical type of aliasing occurs in the harmonic balance method for the present self-excited nonlinear dynamical system. Besides, a parameter marching procedure is used to sufficiently eliminate the effects of aliasing pertaining to the time domain collocation method. Moreover, the accuracy of the time domain collocation method is compared with the harmonic balance method.

A time domain collocation method for obtaining the third superharmonic solutions to the Duffing oscillator

In this study, a simple time domain collocation method (TDC) is applied to investigate the third superharmonic solutions of the Duffing oscillator. Upon using the proposed scheme, the multivaluedness, jump phenomenon, and transitional region of the third superharmonic response are explored. The amplitude frequency response curves for various values of damping, nonlinearity, and external force are obtained and compared. In addition, instead of collocating at N points so that the resulting nonlinear algebraic system is well determined, we extend the time domain collocation method to a new version by collocating at M>N points. The resulting over determined system is solved by the least square method. The extended time domain collocation method can significantly relieve the nonphysical solution phenomenon, which may be severe in the time domain collocation method, and its equivalent high dimensional harmonic balance method. Finally, numerical examples confirm the simplicity, efficiency, and accuracy of the proposed scheme.

High-Dimensional Harmonic Balance Analysis for a Turning Process with State-Dependent Delay

The present study considers the state-dependent delay differential equations (SD-DDEs) for the turning process. In general, series expansion of the SD-DDEs turning system is essential in the nonlinear analysis such as the conventional methods of multiple scales and harmonic balance. Unfortunately, the mathematical theory of SD-DDEs, especially for those with an implicit function of delay, was just recently developed and any rigorous mathematical theory has not yet been proven. As one approach for the nonlinear analysis of the SD-DDEs, physically reasonable results could be obtained by extending the general theory of DDEs to the SD-DDEs through the use of the series expansion in conjunction with the implicit function, although there still remains an open issue of its mathematical rigorousness. The other approach may be treating the original SD-DDEs directly. To this end, the high-dimensional harmonic balance (HDHB) analysis is performed in this study in order to investigate the nonlinear behaviors of the turning system in the form of SD-DDEs without its series expansion. The results obtained by HDHB analysis are validated by comparing results with those of direct time integration. Using the resulting bifurcation diagrams, nonlinear chatter behaviors of the turning system are examined and discussed.

A Simple Collocation Scheme for Obtaining the Periodic Solutions of the Duffing Equation, and its Equivalence to the High Dimensional Harmonic Balance Method: Subharmonic Oscillations

A Simple Collocation Scheme for Obtaining the Periodic Solutions of the Duffing Equation, and its Equivalence to the High Dimensional Harmonic Balance Method: Subharmonic Oscillations

A harmonic balance approach for large-scale problems in nonlinear structural dynamics

Harmonic balance (HB) methods allow for rapid computation of time-periodic solutions for nonlinear dynamical systems. We present a filtered high dimensional harmonic balance (HDHB) approach, which operates in the time domain, and provide a framework for implementation into an existing finite element solver. To demonstrate its capabilities, the method is used to solve a set of nonlinear structural dynamics problems related to the field of flapping flight. For each example, the HDHB approach produces accurate steady-state solutions orders of magnitude faster than a traditional time-marching scheme.

High-dimensional Harmonic Balance Analysis for Second-order Delay-differential Equations

This paper demonstrates the utility of the high-dimensional harmonic balance (HDHB) method for locating limit cycles of second-order delay-differential equations (DDEs). A matrix version of the HDHB method for systems of DDEs is described in detail. The method has been successfully applied to capture the stable and/or unstable limit cycles in three different models: a machine tool vibration model, the sunflower equation and a circadian rhythm model. The results show excellent agreement with collocation and continuation-based solutions from DDE-BIFTOOL. The advantages of HDHB over the classical harmonic balance method are highlighted and discussed.

High dimensional harmonic balance dealiasing techniques for a Duffing oscillator

A dealiasing study of the high dimensional harmonic balance method is conducted using a Duffing oscillator as a prototypical nonlinear dynamical system. Previous study has shown that aliasing can occur when the high dimensional harmonic balance method is used to solve systems that contain nonlinearities. We demonstrate that frequency filtering techniques, such as the Fourier smoothing method, can sufficiently reduce or eliminate the effects of aliasing. The drawback is that Fourier filtering requires redundant coordinate transformations between the time and frequency domains, resulting in unnecessary computational expense. As an alternative, temporal filters are constructed based upon compact finite difference schemes with spectral-like resolution. It is shown that the temporal filtering schemes successfully replicate the performance of the Fourier smoothing method with improved computational economy.

A novel harmonic balance analysis for the Van Der Pol oscillator

This study focuses on a novel harmonic balance formulation, the high-dimensional harmonic balance method. To investigate a non-linearity in the damping term, the system chosen for study is the Van der Pol's oscillator. Both unforced and forced oscillators are analyzed. The results from the analysis are compared with those obtained from the classical harmonic balance and the time marching (Runge–Kutta) methods.

A high dimensional harmonic balance approach for an aeroelastic airfoil with cubic restoring forces

This is a study of a two-dimensional airfoil including a cubic spring stiffness placed in an incompressible flow. A new formulation of the harmonic balance method is employed for the aeroelastic airfoil to investigate the amplitude and frequency of the limit cycle oscillations. The results are compared with the results from the classical harmonic balance approach and from the conventional time marching integration method.

A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator

The present study focuses on a novel harmonic balance formulation, which is much easier to implement than the standard/classical harmonic balance method for complex nonlinear mathematical models and algorithms. Both harmonic balance approaches are applied to Duffing’s oscillator to demonstrate the advantages and disadvantages of the two approaches. A fundamental understanding of the difference between these two methods is achieved, and the properties of each method are analyzed in detail.