Design and Theoretical Analysis of High-Static-Low-Dynamic Stiffness Torsion Vibration Isolator Based on Oblique Springs

A torsion vibration isolator composed of oblique springs with high-static-low-dynamic stiffness (HSLDS) is proposed to attenuate the transmission of torsion vibration along the shipping shaft in this paper. It is good at in low frequency vibration isolation as it can significantly reduce the resonance frequency of the system with the same load capability. Firstly, the model of HSLDS torsion vibration isolator is introduced in this paper. Secondly, the non-dimensional torsion stiffness is formulated using mechanics theory, and the HSLDS characteristic of designed torsion vibration isolator is verified. Finally, the torque transmissibility is analyzed using the Increment Harmonic Balance (IHB) method, and the effects of the system parameters on it are analyzed. The results show that the resonant frequency increases accordingly as the stiffness ratio and the excitation torque are increased. However, the peak value of the torsion transmissibility is decreased as the damper ratio increasing.

Analysis of Dynamic Characteristics of a Fractional-Order Spur Gear Pair with Internal and External Excitations

The nonlinear frequency response characteristics of a spur gear pair with fractional-order derivative under combined internal and external excitations are investigated based on the incremental harmonic balance (IHB) method. First, a pure torsional vibration model is proposed that contains various complex factors, such as the time-varying mesh stiffness, transmission error, the fluctuation of input torque, backlash. Then, the IHB method is developed to calculate the higher-order approximate solution of the system and the correctness of the results is verified by comparing with numerical simulation results obtained by the Power Series Expansion (PSE) method. Furthermore, the types of various impact situations and their judgment conditions are discussed, and the different impact behaviors are analyzed in detail when w?[0,1.5] by using phase diagrams and amplitude-frequency response curves. The influence of important parameters on the dynamic characteristics of gear pair is analyzed at last. The results indicate that the analytical solution derived by IHB method is sufficiently precise. Significantly, the dynamic characteristics of the system could be effectively controlled by adjusting time-varying mesh stiffness coefficient, the order and coefficient of fractional-order term and the amplitudes of internal excitation or external excitation. As a part of the theory of fractional-order mechanical system, the impact performance of fractional-order gear pair is approached for the first time by analytical method.

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.

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 Efficient Galerkin Averaging-Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Tensor Contraction

An efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method is developed based on the fast Fourier transform (FFT) and tensor contraction to increase efficiency and robustness of the IHB method when calculating periodic responses of complex nonlinear systems with non-polynomial nonlinearities. As a semi-analytical method, derivation of formulae and programming are significantly simplified in the EGA-IHB method. The residual vector and Jacobian matrix corresponding to nonlinear terms in the EGA-IHB method are expressed using truncated Fourier series. After calculating Fourier coefficient vectors using the FFT, tensor contraction is used to calculate the Jacobian matrix, which can significantly improve numerical efficiency. Since inaccurate results may be obtained from discrete Fourier transform-based methods when aliasing occurs, the minimal non-aliasing sampling rate is determined for the EGA-IHB method. Performances of the EGA-IHB method are analyzed using several benchmark examples; its accuracy, efficiency, convergence, and robustness are analyzed and compared with several widely used semi-analytical methods. The EGA-IHB method has high efficiency and good robustness for both polynomial and nonpolynomial nonlinearities, and it has considerable advantages over the other methods.

Comparison Between the Incremental Harmonic Balance Method and Alternating Frequency/Time-Domain Method

Two widely used semi-analytical methods: the incremental harmonic balance (IHB) method and alternating frequency/time-domain (AFT) method are compared, and some long-standing discussions on frameworks of these two methods are cleared up. The IHB and AFT methods are proved for the first time to be theoretically equivalent when spectrum aliasing does not occur in the AFT method. Based on this equivalence, the minimal nonaliasing sampling rate for the AFT and fast Fourier transform (FFT)-based IHB methods can be obtained for a system with polynomial nonlinearities. While spectrum aliasing is theoretically inevitable for nonpolynomial nonlinearities, a sufficiently large sampling rate can be usually used with acceptable accuracy and efficiency for many systems. Convergence and efficiency of the IHB method, AFT method, and several FFT-based IHB methods are compared. Accuracy and convergence can be affected when the sampling rate is insufficient. This comparison can provide some insights to avoid misuse of these methods and choose which methods to use in engineering applications.

An Efficient Galerkin Averaging-Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Tensor Contraction

An efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method is developed based on the fast Fourier transform (FFT) and tensor contraction to increase efficiency and robustness of the IHB method when calculating periodic responses of complex nonlinear systems with non-polynomial nonlinearities. As a semi-analytical method, derivation of formulae and programming are significantly simplified in the EGA-IHB method. The residual vector and Jacobian matrix corresponding to nonlinear terms in the EGA-IHB method are expressed using truncated Fourier series. After calculating Fourier coefficient vectors using the FFT, tensor contraction is used to calculate the Jacobian matrix, which can significantly improve numerical efficiency. Since inaccurate results may be obtained from discrete Fourier transform-based methods when aliasing occurs, the minimal non-aliasing sampling rate is determined for the EGA-IHB method. Performances of the EGA-IHB method are analyzed using several benchmark examples; its accuracy, efficiency, convergence, and robustness are analyzed and compared with several widely used semi-analytical methods. The EGA-IHB method has high efficiency and good robustness for both polynomial and non-polynomial nonlinearities, and it has considerable advantages over the other methods.