Arc-length Continuation Method Research Articles

Design and optimization of quasi-zero-stiffness dual harvester-absorber system

The single quasi-zero-stiffness harvester-absorber system has shown its effectiveness in harvesting vibration energy and suppressing low-frequency vibrations simultaneously. Nevertheless, drawbacks of narrow frequency range of vibration suppression and low output power are still challenges. In this paper, a novel quasi-zero-stiffness dual harvester-absorber system (QZS DHAS) is proposed for broadband synergistic vibration suppression and energy harvesting at low frequency. The QZS feature is realized by mounting an equal-strength piezoelectric beam and a pair of magnet rings in parallel. Firstly, the restoring force of both the piezoelectric beam and the magnet rings are theoretically derived, respectively, which are verified by finite element simulations. Then, the governing equations of a host oscillator with QZS DHAS are derived based on Newton's law and Kirchhoff's law, which are solved by the harmonic balanced method combined with the pseudo arc-length continuation method. Moreover, the effects of mass ratios, frequency ratios and damping ratios on the performance of the QZS DHAS are investigated in details. Finally, the parameters of the QZS DHAS are optimized to improve the performance of vibration suppression and energy harvesting by using the multi-objectives genetic algorithm in combination with the technique of establishing order preference by similarity to the ideal solution. The results show that the peak response of the host oscillator is reduced by 33.48 % and the peak power is increased by 531.38 % after optimization, which indicates that the dual objectives of broadband vibration absorption and high-efficiency energy harvesting are fulfilled.

A computationally efficient method for dynamic response analyses of accessory drive systems using harmonic balance method together with alternating frequency/time domain technique

Serpentine belt accessory drive system (SBADS) is a typical coupled vibration system with rigid body vibrations and continuum vibrations. Vibration behaviors of a SBADS consist of rotational vibrations of pulleys and tensioner arm and transverse vibrations of belt spans. In modeling of a SBADS, transverse vibration of a belt span consisting of time and space coordinates is discretized into multiple degrees of freedom with only time variable. As a result, number of degrees of freedom of a SBADS is large, and computational efficiency of dynamic responses of a SBADS is low. To solve this issue, a calculation method for nonlinear dynamic characteristics of a SBADS is proposed based on the harmonic balance method with alternating frequency/time domain technique in the paper. An engine front end accessory drive system is taken as a study example, the coupling vibration model of the SBADS is established, and hysteretic behavior of the tensioner is considered in the model. In addition, a dimensionality reduction method for the dynamic matrix is proposed based on local nonlinearity of the model. A calculation method for nonlinear dynamic responses of the SBADS is subsequently developed by combining the harmonic balance method and alternating frequency/time domain technique. Nonlinear dynamic responses estimated by the proposed method are validated by comparing with numerical simulations. Finally, amplitude-frequency responses of the SBADS are calculated using arc-length continuation method, and influences of nonlinear tensioner on frequency domain responses and internal resonance phenomena of the SBADS are analyzed.

Nonlinear Vibration of Truncated Open Conical Nanoshells Under Harmonic Excitation

The nonlinear forced vibration behavior of truncated open conical nanoshells is investigated in this paper. Nonlocal elasticity theory has been employed to modify the size-dependent effect of micro/nanostructure. Von Karman nonlinear strains are used to consider the geometric nonlinearity of the nanoshells. Using Hamilton’s principle, the equations of motion and boundary conditions of truncated open conical nanoshells are derived. Galerkin method is used to solve the governing equations. Then, to determine the nonlinear forced vibration behavior of truncated open conical nanoshells, complex averaging and arc-length continuation methods are employed. Finally, the effect of nonlocal parameters, cone apex angle, the amplitude of harmonic excitation, structural damping coefficient, and geometrical parameters on the nonlinear frequency response of the truncated open conical nanoshells are discussed.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

COMPUTATION OF DISPERSION CURVES OF RAYLEIGH-LAMB WAVES BY NEWTON-RAPHSON METHOD

The paper treats the numerical construction of dispersion curves which belong to the propagation of Rayleigh-Lamb waves in isotropic elastic media. The Newton-Raphson method.is applied for computation of real, imaginary and complex wavenumbers in dependence on the frequency of the wave. Due to multivaluedness, unexpected jumps between modes of dispersion curves occur in branch tracing, which prevents successful direct application of the method. This shortcoming is eliminated in the article by introducing the arc-length continuation method. The efficiency of this supplement to the Newton-Raphson method is shown on the example of an aluminum plate with constructions of dispersion curves for real, imaginary and complex wave numbers.

Nonlinear vibrations of a dielectric elastomer cantilever combined-stepped-plate actuator

A dielectric elastomer (DE) cantilever combined-stepped-plate actuator is proposed and its nonlinear vibration enhancement behaviors are investigated. The irregular geometry of the actuator is transformed into a rectangular computational domain with the ideal of the iso-parametric element transformation; then the governing equations describing the nonlinear vibration of the actuator are formulated by the variational method. The nonlinear periodic responses are solved through the harmonic balance method and the arc-length continuation method. The results indicate that the single side (upper or lower) voltage excitation is more efficient in the actuating performance. The asymmetrical combined plate presents a better motion performance comparing with the symmetrical one. Not only can more drive modes be achieved (For instance, both symmetrical flapping mode and asymmetrical torsional mode can be excited by changing the voltage frequency), but also higher responses can be obtained. Additionally, isolated responses are found for some particular dimension parameters, which provides a potential design strategy to regulate the frequency domain of significant responses. This research concludes that, the asymmetrical combined plate subjected to an asymmetrical voltage loading would show a more efficient locomotion performance, which can provide an insight into the structural design and the excitation strategy for DE actuators.

Vibration Suppression on the Composite Laminated Plates Subjected to Aerodynamic and Harmonic Excitations Based on the Nonlinear Piezoelectric Shunt Damping

This paper proposes a nonlinear piezoelectric shunt damping circuit to mitigate the vibration of the plate structures subjected to the aerodynamic and harmonic excitations. The nonlinear piezoelectric shunt damping circuit consisting of the piezoelectric patch, inductor, resistor, negative capacitor and nonlinear capacitor is considered. Firstly, the governing equations of the electro-mechanical coupling nonlinear system are formulated based on the energy method and modified Ritz method. Furthermore, the complexification-averaging method combined with pseudo arc-length continuation method is adopted to analyze the steady-state response of the coupled system with aerodynamic and harmonic forces. A considerable number of numerical cases are performed to validate the proposed model and the time domain and frequency domain dynamic responses of the plate structures with and without control are further compared to validate the effectiveness of the proposed control strategy. Meanwhile, the energy transfer analysis between the plate structure and external piezoelectric shunt damping circuit is investigated. Finally, the effects of key parameters of the nonlinear piezoelectric shunt damping circuit including the negative capacitance and nonlinear capacitance on the vibration suppression performance of the piezoelectric shunt damping are discussed.

Nonlocal couple stress-based quasi-3D nonlinear dynamics of agglomerated CNT-reinforced micro/nano-plates before and after bifurcation phenomenon

In present research exploration, the nonlinear dynamic stability characteristics of axially compressed nanocomposite plates at micro/nano-scale reinforced with randomly oriented carbon nanotubes (CNTs) are investigated within the both prebuckling and postbuckling regimes. To accomplish this examination, the nonlocal couple stress (NCS) continuum elasticity is incorporated to a quasi-3D plate theory which separates the plate deformation to the bending and shear parts considering simultaneously the transverse shear and normal displacements. In addition, a two-parameter homogenization scheme is utilized to obtain the effective characters of the randomly oriented CNT-reinforced nanocomposites. The NCS-based nonlinear differential equations of motion are discretized using the Kronecker tensor product together with the shifted Chebyshev-Gauss-Lobatto gridding pattern. Thereafter, the Galerkin technique together with the pseudo arc-length continuation method are employed to achieve the NCS-based fRequency-load and nonlinear frequency ratio-deflection curves before and after of the bifurcation point. It is deduced that for a randomly oriented CNT-reinforced heterogeneous micro/nano-plate in which the most CNTs are located inside clusters, increasing the value of cluster volume fraction leads to increase a bit the significance of the softening and stiffing characters related to the nonlocal and couple stress tensors before the bifurcation phenomenon, but it causes to decrease them after the critical bifurcation point. Opposite patterns before and after the bifurcation phenomenon are predicted for the agglomeration in which the most CNTs are located outside clusters.

Modelling and nonlinear dynamics of dielectric elastomer sandwich cantilever-plate actuators

Inspired from the bimorph actuators for flapping and propelling swimming vehicles, a soft actuator made of two dielectric elastomer layers and a hyperelastic material core is presented. This actuator is also termed as a sandwich cantilever rectangular plate, which is stimulated by alternating voltages with a phase difference. Based on the classical plate theory and the variational method, the governing equation describing the nonlinear vibration of the sandwich cantilever plate is formulated. Employing the harmonic balance method and the arc-length continuation method, periodic solutions are obtained, then the convergence analyses of the numbers of discrete modes and harmonics are conducted. In order to illustrate the frequency features more clearly, a modified mode numbering scheme is developed. The numerical results manifest that, the combined DC-AC voltage excitation and the AC voltage excitation can excite abundant nonlinear behaviors of the soft actuator, including the local peak and the isolated bubble aroused by the nonlinear coupling. Additionally, the fundamental mode response keeps dominating the dynamic behaviors of the cantilever-plate actuator. The nonlinear dynamics broaden the resonant domain significantly, which can be potentially applied to the design and manufacture of high-performance soft actuators through the vibration enhancement strategy.

Size-dependent nonlinear vibration and instability of a damped microplate subjected to in-plane parametric excitation

The semi-analytical framework for nonlinear vibration and dynamic instability of a damped microplate under periodic parametric excitation is presented. The microplate is modeled using the higher-order shear deformation theory (HSDT) in conjugation with the modified strain gradient theory (MSGT). The governing partial differential equations of motion are obtained using Hamilton’s principle and further solved using Galerkin’s method. The ordinary differential equations without the nonlinear terms are solved using Bolotin’s method to obtain the dynamic instability region. A combination of the incremental harmonic balance (IHB) and the arc-length continuation methods is used to obtain the nonlinear forced vibration response. The effect of initial displacements on the steady-state response of the microplate is discussed. The Newmark-β method is used to obtain the time-history response plots. A comparison of results with those obtained from modified couple stress theory (MCST) and classical continuum theory (CCT) are examined. The effect of various parameters, such as the size of a plate, damping coefficient, static and dynamic loading factors, different boundary conditions, and different loading profiles, on the width of linear and nonlinear instability regions, are also studied.

A comparative study on multi- and variable-coefficient harmonic balance methods for quasi-periodic solutions

Quasi-periodic solutions widely exist in nonlinear dynamical systems, which are the one of the most important dynamical behaviors but much more difficult to obtain full- or semi-analytically than equilibria and periodic solutions. Recently, multi-harmonic balance method (MHBM) and variable-coefficient harmonic balance method (VCHBM) have been proposed to determine the quasi-periodic solutions semi-analytically. In this paper, after a brief review on the two methods, that is, MHBM uses a multi-dimensional Fourier series and VCHBM a variable-coefficient Fourier series, the relationship between them is unveiled particularly for the case of quasi-periodic solutions consisting of two frequency components. The transform matrices between the algebraic equations of the Fourier coefficients in the two methods are derived. Furthermore, a new formulation for alternating Frequency-Time method (AFT) and a phase condition for arc-length continuation method (ALC) in VCHBM are proposed to make the method more efficient and robust. Through the application of two examples, it is found that when the two methods choose equal harmonic order and thus have the same number of unknowns, MHBM demonstrates higher computational efficiency but VCHBM shows more robust and accurate, especially for the quasi-periodic solutions in the nonlinear dynamical systems with a single-frequency excitation. This paper provides insight views into the features of MHBM and VCHBM..

Hyperelastic Microcantilever AFM: Efficient Detection Mechanism Based on Principal Parametric Resonance

The impetus of writing this paper is to propose an efficient detection mechanism to scan the surface profile of a micro-sample using cantilever-based atomic force microscopy (AFM), operating in non-contact mode. In order to implement this scheme, the principal parametric resonance characteristics of the resonator are employed, benefiting from the bifurcation-based sensing mechanism. It is assumed that the microcantilever is made from a hyperelastic material, providing large deformation under small excitation amplitude. A nonlinear strain energy function is proposed to capture the elastic energy stored in the flexible component of the device. The tip–sample interaction is modeled based on the van der Waals non-contact force. The nonlinear equation governing the AFM’s dynamics is established using the extended Hamilton’s principle, obeying the Euler–Bernoulli beam theory. As a result, the vibration behavior of the system is introduced by a nonlinear equation having a time-dependent boundary condition. To capture the steady-state numerical response of the system, a developed Galerkin method is utilized to discretize the partial differential equation to a set of nonlinear ordinary differential equations (ODE) that are solved by the combination of shooting and arc-length continuation method. The output reveals that while the resonator is set to be operating near twice the fundamental natural frequency, the response amplitude undergoes a significant drop to the trivial stable branch as the sample’s profile experiences depression in the order of the picometer. According to the performed sensitivity analysis, the proposed working principle based on principal parametric resonance is recommended to design AFMs with ultra-high detection resolution for surface profile scanning.

Nonlinear free and forced vibrations of a fiber-reinforced dielectric elastomer-based microbeam

This paper investigates the nonlinear free and forced vibrations of a fiber-reinforced dielectric elastomer-based microbeam using the Euler–Bernoulli beam theory. The nonlinear material properties have been taken into account based on the original Gent hyperelastic approach. Also, the generalized von Kármán theory is utilized for obtaining nonlinear strains. We assumed that the elastomeric microbeam is incompressible. The system’s free energy fibrous component has been formulated using the Holzapfel-Gasser-Ogden model, which is appropriate to the two families of collagen fibers with the same mechanical properties. An electric part is connected to the system to aim to excite the microbeam considering two types of voltage: a DC (static) and a DC+AC (dynamic) voltage. Governing equations of motion have been derived using Hamilton’s principle. Subsequently, the reduced-order vibration model of the fiber-reinforced microbeam has been obtained by using the Galerkin method. We have calculated the nonlinear natural frequencies (free vibration) via the Hamiltonian scheme. Also, for the forced vibration, the shooting method combined with the arc-length continuation method has been applied to explore the resonant response of the system. The results show that fiber reinforcement affects the performance of the dielectric elastomers. Eventually, the present study can be considered to develop biological structures such as arterial walls and soft tissues because of their fibrous structures.

Study on nonlinear force transmissibility of flywheel rotor system considering periodic base motions

This paper reports on the study of the dynamic force transmissibility (DFT) of flywheel rotor system (FRS), considering periodic base motions and nonlinear support stiffness of angular contact ball bearings (ACBBs). The energy method and Lagrange equation are adopted to deduce the base induced additional excitations, including the internal (gyroscopic and stiffness) and external excitations. The influence of combined loads and contact angle variation is introduced into the Jones formula to accurately solve for the internal load distribution and nonlinear stiffness of the ACBBs. The lateral vibration model of FRS considering base motions and ACBB nonlinear support stiffness is then established. The DFT of the system is obtained by combining the harmonic balance method and alternate frequency/time domain method. The arc length continuation method is employed to track the multi-solution region owing to bifurcation. In addition, the stability of harmonic balance solutions is determined through eigenvalue analysis in the frequency domain. Both numerical integration and comparison with literature results are used to verify the accuracy of the proposed DFT model and solution method. On the basis of these results, the effects of base motion amplitude, ACBB axial preload, and rotor damping on the DFT of the system are discussed. With the increase of radial deformation, the support stiffness of ACBB under axial preload shows “soft” and “stiff” characteristics in turn. The additional excitation caused by the base motions is the key factor determining the nonlinear characteristics of FRS. These results support designing a reasonable vibration isolation system and reducing the transmission of FRS dynamic response to the spacecraft platform.

Rotational nonlinear double-beam energy harvesting

This paper presents an investigation of the performance of a coupled rotational double-beam energy harvester (DBEH) with magnetic nonlinearity. Two spring-connected cantilever beams are fixed on a rotating disc. Repelling magnets are attached to the frame and to the lower beam tip, and an equal-mass block is attached to the tip of the upper beam. To describe the dynamic response, a theoretical model related to the rotational motion of the coupled cantilever beam is derived from the Lagrange equations. In addition, the harmonic balance method, together with the arc-length continuation method, is applied to obtain the frequency response functions (FRFs). Parametric studies are then conducted to analyze the effect of varying the parameters on the energy harvesting performance, and numerical analysis is performed to validate the analytical solutions. Finally, the theoretical model is verified by forward- and reverse-frequency-sweeping experiments. The DBEH in rotational motion can perform effective energy harvesting over a wide range of rotational frequencies (10–35 rad s−1 ). The upper beam is found to exhibit better energy harvesting efficiency than the lower beam around the resonant frequency. This study effectively broadens the energy harvesting bandwidth and provides a theoretical model for the design of nonlinear magnet-coupled double-beam structure in rotational energy harvesting.

Stability and bifurcation analysis of super- and sub-harmonic responses in a torsional system with piecewise-type nonlinearities

The nonlinear dynamic behaviors induced by piecewise-type nonlinearities generally reflect super- and sub-harmonic responses. Various inferences can be drawn from the stability conditions observed in nonlinear dynamic behaviors, especially when they are projected in physical motions. This study aimed to investigate nonlinear dynamic characteristics with respect to variational stability conditions. To this end, the harmonic balance method was first implemented by employing Hill’s method, and the time histories under stable and unstable conditions were examined using a numerical simulation. Second, the super- and sub-harmonic responses were investigated according to frequency upsweeping based on the arc-length continuation method. While the stability conditions vary along the arc length, the bifurcation phenomena also show various characteristics depending on their stable or unstable status. Thus, the study findings indicate that, to determine the various stability conditions along the direction of the arc length, it is fairly reasonable to determine nonlinear dynamic behaviors such as period-doubling, period-doubling cascade, and quasi-periodic (or chaotic) responses. Overall, this study suggests analytical and numerical methods to understand the super- and sub-harmonic responses by comparing the arc length of solutions with the variational stability conditions.

Nonlinear dynamics of drill string in horizontal well under axial load and supercritical regime of flowing fluid

In this article, the nonlinear vibration of the spinning pipe conveying fluid in a borehole drilling is investigated under the axial compression load, nonlinear fluid forces, internal and external resonances. The spinning pipe is modeled as a simply supported rotor with imbalance mass and eccentricity. When the flowing fluid is in supercritical regime the pipe oscillates around an equilibrium position in the working frequency range of external excitation. Two to one internal resonance condition between forward and backward natural frequencies of the pipe is found at specific values of flow velocity and rotating speed. The multiple scales method is occupied to solve the governing ordinary differential equations and solvability condition is applied to get the modulation equations. The Runge–Kutta integration and the arc-length continuation method are used for directly solving ordinary differential equations. Bifurcation diagrams are plotted and time responses, phase portraits and Fast Fourier transformation of amplitudes are achieved. The Wolf algorithm is used to determine the Lyapunov Exponents. Periodic, multi-periodic, quasi-periodic, double jump and chaotic behaviors can be seen in simulation results.

Disconnected stationary solutions for 3D Kolmogorov flow problem: preliminary results

The extension of the classical A.N. Kolmogorov’s flow problem for the stationary 3D Navier-Stokes equations on a stretched torus for velocity vector function is considered. A spectral Fourier method with the Leray projection is used to solve the problem numerically. The resulting system of nonlinear equations is used to perform numerical bifurcation analysis. The problem is analyzed by constructing solution curves in the parameter-phase space using previously developed deflated pseudo arc-length continuation method. Disconnected solutions from the main solution branch are found. These results are preliminary and shall be generalized elsewhere.

Bifurcation scenario for a two-dimensional static airfoil exhibiting trailing edge stall

We numerically investigate stalling flow around a static airfoil at high Reynolds numbers using the Reynolds-averaged Navier–Stokes equations (RANS) closed with the Spalart–Allmaras turbulence model. An arclength continuation method allows to identify three branches of steady solutions, which form a characteristic inverted S-shaped curve as the angle of attack is varied. Global stability analyses of these steady solutions reveal the existence of two unstable modes: a low-frequency mode, which is unstable for angles of attack in the stall region, and a high-frequency vortex shedding mode, which is unstable at larger angles of attack. The low-frequency stall mode bifurcates several times along the three steady solutions: there are two Hopf bifurcations, two solutions with a two-fold degenerate eigenvalue and two saddle-node bifurcations. This low-frequency mode induces a cyclic flow separation and reattachment along the airfoil. Unsteady simulations of the RANS equations confirm the existence of large-amplitude low-frequency periodic solutions that oscillate around the three steady solutions in phase space. An analysis of the periodic solutions in the phase space shows that, when decreasing the angle of attack, the low-frequency periodic solution collides with the unstable steady middle-branch solution and thus disappears via a homoclinic bifurcation of periodic orbits. Finally, a one-equation nonlinear stall model is introduced to reveal that the disappearance of the limit cycle, when increasing the angle of attack, is due to a saddle-node bifurcation of periodic orbits.

Nonlinear vibration analysis of horizontal axis wind turbine blades using a modified pseudo arc-length continuation method

The horizontal axis wind turbine (HAWT) blades are exposed to parametric and direct excitations of harmonic nature due to the gravity and aerodynamic loads. Hence, the study of the periodic response of the HAWT blades under these operating loads is of great importance in structural design. In this paper, a new computationally efficient dynamic model for the blade is presented including the pre-twist effects. The Rayleigh-Ritz technique is applied to the derived dynamic model and a reduced order model (ROM) is devised in a systematic manner. An efficient method based on the pseudo arc-length continuation method is introduced to find out the edge-wise and flap-wise amplitude-frequency response (AFR) curves of the blade under harmonic wind inflow. Furthermore, the forced vibration of the blade under different operating loads including gravity, centrifugal and aerodynamic loads (due to the uniform and shear wind inflows) is studied separately. Applying the proposed method, the effects of structural damping, nonlinear terms, pre-twist ratio, and cyclic excitation due to the gravity force as well as the centrifugal force on the AFR curves are investigated. Eventually, the dynamic model and the presented method for the frequency analysis are validated with the available data.