A Modified Incremental Harmonic Balance Method Combined with Tikhonov Regularization for Periodic Motion of Nonlinear System

In this paper, a modified incremental harmonic balance (IHB) method combined with Tikhonov regularization has been proposed to achieve the semi-analytical solution for the periodic nonlinear system. To the best of our knowledge, the convergence of the traditional IHB method is bound up with the iterative initial values of harmonic coefficients, especially near the bifurcation point. To this end, the Tikhonov regularization is introduced into the linear incremental equation to tackle the ill-posed situation in the iteration. To this end, the convergence performance of the traditional IHB method has been improved significantly. Moreover, convergence proof of the proposed method also has been given in this paper. Finally, a van der Pol–Duffing oscillator with external excitation and a cubic nonlinear airfoil system with the external store are adopted as numerical examples to illustrate the efficiency and the performance of the presented modified IHB method. The numerical examples show that the results achieved by the proposed method are in excellent agreement with the Runge–Kutta method, and the accuracy is not significantly reduced compared with the traditional IHB method. Especially, the modified IHB method also can converge to the exact solution from the initial values that the traditional IHB method cannot converge in both examples.

Nonlinear Vibration of a Nonlocal Functionally Graded Beam on Fractional Visco-Pasternak Foundation

This paper investigates the nonlinear dynamic behavior of a nonlocal functionally graded Euler–Bernoulli beam resting on a fractional visco-Pasternak foundation and subjected to harmonic loads. The proposed model captures both, nonlocal parameter considering the elastic stress gradient field and a material length scale parameter considering the strain gradient stress field. Additionally, the von Karman strain-displacement relation is used to describe the nonlinear geometrical beam behavior. The power-law model is utilized to represent the material variations across the thickness direction of the functionally graded beam. The following steps are conducted in this research study. At first, the governing equation of motion is derived using Hamilton's principle and then reduced to the nonlinear fractional order differential equation through the single-mode Galerkin approximation. The methodology to determine steady-state amplitude-frequency responses via incremental harmonic balance method and continuation technique is presented. The obtained periodic solutions are verified against the perturbation multiple scales method for the weakly nonlinear case and numerical integration Newmark method in the case of strong nonlinearity. It has been shown that the application of the incremental harmonic balance method in the analysis of nonlocal strain gradient theory-based structures, can lead to more reliable studies for strongly nonlinear systems. In the parametric study is shown that, on one hand, parameters of the visco-Pasternak foundation and power-law index remarkable affect the response amplitudes. On the contrary, the nonlocal and the length scale parameters are having a small influence on the amplitude-frequency response. Finally, the effects of the fractional derivative order on the system's damping are displayed at time response diagrams and subsequently discussed.

A Universal Nonlinear Analyzer for Rigid Multibody Systems Based on the Efficient Galerkin Averaging-Incremental Harmonic Balance Method

The bridge between the multibody dynamic modeling theory and nonlinear dynamic analysis theory is built for the first time in this work by introducing an efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method for steady-state nonlinear dynamic analysis of index-3 differential algebraic equations (DAEs) for general rigid multibody systems. The multibody dynamic modeling theory has made significant advances in generality and simplicity, and multibody systems are usually governed by DAEs. Since the fast Fourier transform and EGA are used, the EGA-IHB method has excellent robustness and computational efficiency. Since the Floquet theory cannot be directly used for stability analysis of periodic responses of DAEs, a new stability analysis procedure is developed, where perturbed, linearized DAEs are reduced to ordinary differential equations with use of independent generalized coordinates. A modified arc-length continuation method with a scaling strategy is used for calculating response curves and conducting parameter studies. Three examples are used to show the performance and capability of the current method. Periodic solutions of DAEs from the EGA-IHB method show excellent agreement with those from numerical integration methods. Amplitude-frequency and amplitude-parameter response curves are generated, and stability and period-doubling bifurcations are analyzed. The EGA-IHB method can be used as a universal solver and nonlinear analyzer for obtaining steady-state periodic responses of DAEs for general multibody systems.

A Modified Incremental Harmonic Balance Method for Periodic Forced Oscillations of a Dielectric Elastomer Membrane Undergoing In-Plane Deformation

Dielectric elastomers (DEs) are widely used in soft transducers with mechanical or electrical loads. DE devices are mainly used for applications under dynamic loads, such as, ocean wave generators, loudspeakers, oscillators, and artificial muscles. It is still a challenge to analytically solve the vibration equation of a DE transducer. For example, for a DE membrane undergoing stretching deformation that is studied in this paper, its vibration equation is highly nonlinear with high-order and fractional-order polynomials. Numerical integration (NI) methods or traditional harmonic balance (HB) methods were used in previous works, but the two methods have low efficiency for strong and complex nonlinearities, and it is difficult to improve the accuracy of the solution. In this work, a free-energy model is used to study the dynamic characteristics of a DE membrane undergoing in-plane deformation, which undergoes a combined load excited by mechanical compression and electric fields. To improve the calculation efficiency and accuracy, we employ a modified incremental harmonic balance (IHB) method based on the fast Fourier transform to solve the periodically-excited nonlinear dynamic equation of the DE membrane. Finally, results of the example verify that the modified IHB method is fast and accurate, and has a very good performance in solving a problem with high nonlinearities.

An Optimized Efficient Galerkin Averaging-Incremental Harmonic Balance Method for High-Dimensional Spatially Discretized Models of Continuous Systems Based on Parallel Computing

Modern computers are generally equipped with multi-core central processing units (CPUs). There has been great interest to introduce a parallelized harmonic balance method to make full use of computing resources and improve the efficiency in dealing with nonlinear dynamic analysis of high-dimensional spatially discretized models of continuous systems. In this work, an optimized efficient Galerkin averaging-incremental harmonic balance method for solving high-dimensional models of continuous systems based on parallel computing is introduced. Optimized parallel implementation based on tensor contraction is introduced in time-domain series calculations and the quasi-Newton method is used in the iteration procedure, which greatly accelerate computational speeds of both serial and parallel implementations. Especially, the parallel implementation achieves high parallel efficiency when multiple CPU cores are used. Due to its high computational efficiency and good robustness, the proposed method has the potential to be used as a powerful universal solver and analyzer for general types of continuous systems.

Semi-numerical analysis of a two-stage series composite planetary transmission considering incremental harmonic balance and multi-scale perturbation methods

This study conducts an analytical investigation of the dynamic response characteristics of a two-stage series composite system (TsSCS) with a planetary transmission consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, which solves the closed solution of incremental parameters passing through the singularity point of the analytical path, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method to verify the feasibility and effectiveness of the improved method. Following that, the multi-scale perturbation (MsP) method is applied to the TsSCS proposed in this subject to analyze the parameter excitation and gap nonlinear equations and then to obtain the analytical frequency response functions including the fundamental, subharmonic, and superharmonic resonance responses. The frequency response equations of the primary resonance, subharmonic resonance, and superharmonic resonance are solved to generate the frequency response characteristic curves of the planetary gear system (PGS) in this method. A comparison between the results obtained by the MsP method and the numerical integration method proves that the former is ideal and credible in most regions. Considering the parameters of TsSCS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The effects of the time-varying parameters and the nonlinear deenthing caused by the gear teeth clearance on the amplitude–frequency characteristics of TsSCS components are studied in this special topic.

Dynamic Characteristics of the Bouc–Wen Nonlinear Isolation System

The Bouc–Wen nonlinear hysteretic model has many control parameters, which has been widely used in the field of seismic isolation. The isolation layer is the most important part of the isolation system, which can be effectively simulated by the Bouc–Wen model, and the isolation system can reflect different dynamic characteristics under different control parameters. Therefore, this paper mainly studies and analyzes the nonlinear dynamic characteristics of the isolation system under different influence factors based on the incremental harmonic balance method, which can provide the basis for the dynamic design of the isolation system.