Frequency Response Equations Research Articles

Integral resonant negative derivative feedback suppression control strategy for nonlinear dynamic vibration behavior model

One of the major problems in robotics research has been developing an actuator system for extremely dynamic-legged robots. High torque density and the capacity to control dynamic physical interactions are two design requirements for high-speed locomotion that are challenging for conventional actuators used in manufacturing applications to meet. To address this system and apply the desired control to reach the best stability position, the robot's foot was simulated with the Van der Pol equations, applied the required control, and studied that application. This work describes the actions of a new novel control mechanism known as the Integral Resonant Negative Derivative Feedback (IRNDF) controller, which reduces the vibration response of a double Van der Pol oscillator subjected to external excitations. This unique controller combines integral resonant control (IRC) and negative derivative feedback (NDF) controllers to provide a new controller effect for double Van der Pol oscillators. The multiple scale perturbation technique (MSPT) has been applied to solve the controlled system analytically. The MATLAB and MAPLE programs have been used to complete and clarify all of the numerical talks. The frequency response curves have been used to study the impact that altering the parameter values had on the amplitude. The controlled system vibration amplitude is governed by frequency-response equations (FREs), which have been constructed. In the vibration system, the IRC, NDF, and IRNDF controllers were compared to see which one was the best. Numerical results show that the unique IRNDF controller is the best at reducing oscillations and decreasing amplitude values. The effects of the effective parameters on the controlled system have been identified. The frequency-response equation that was derived has been used to plot the various response curves for the framework that show the stable and unstable zones when the controller is off and on. Lastly, excellent agreement between the derived numerical findings and the analytical ones was observed. Lastly, utilizing time histories and response curves to compare analytical and numerical solutions was fascinating and significant.

Stability and Control of Car Dynamics with a Quarter Model via a Novel Simple Harmonic Hump under External Force

Effects of the road, such as speed bumps, can significantly affect a car’s stability. This study focuses on how a quarter-car model is affected by a basic harmonic speed hump and how Cubic Negative Velocity Control (CNVC) is used to control the amplitude of disturbances. This study differs from earlier research in considering various control and force kinds that impact the system. The external forces in this context are a component of a non-linear dynamic system. Two-degree-of-freedom (2DOF) differential coupled equations describe the system’s equation. Numerous numerical experiments have been conducted, including proportional derivative (PD), negative derivative feedback (NDF), positive position feedback (PPF), linear negative velocity control (LNVC), and CNVC; the results show that when the hump is represented as a simple harmonic hump, CNVC has the best effect and can regulate vibrations more precisely than the other approaches on this system. Subsequently, the vibration value of the system was numerically analyzed both before and after the control was implemented. Using the frequency response equation and phase plane approaches in conjunction with the Runge–Kutta fourth order method (RK-4) in the context of resonance situation analysis, the stability of the numerical solution has been evaluated.

A novel optimal design method for tuned mass dampers with elastic motion‐limiting stoppers

AbstractTuned Mass Dampers (TMDs) are commonly used passive control devices in practical engineering applications. However, motion‐limiting stoppers are usually installed to control the excessive TMD displacement due to the building space limitation, resulting in piecewise nonlinearity and detuning of TMD. This paper studies the influence of elastic motion‐limiting stoppers on the optimal design of TMDs through a piecewise stiffness TMD (PSTMD) model. Performance of a PSTMD with classical design is first investigated and proven to be ineffective. To optimize the PSTMD parameters, the motion of PSTMD is decoupled from the controlled structure, and the frequency response equation of PSTMD is obtained analytically through the averaging method. Subsequently, the solution of the optimal design frequency for PSTMD is transformed into the solution of the jump frequency in the frequency response equation. With the optimal frequency of PSTMDs, the optimal damping and control performance of PSTMDs are discussed and analyzed compared with classical linear design, which fully showcases the effectiveness of the novel design method. Finally, the effectiveness of the novel design method is verified using a nine‐story benchmark frame structure, and the results demonstrate that the control performance of the optimal PSTMD can be improved by nearly under specific seismic excitation, compared to the PSTMD with classical linear method. In summary, the novel design method can effectively take into account the influence of piecewise nonlinearity caused by elastic motion‐limiting stoppers and improve the optimal control performance of TMD in a more realistic engineering environment.

Controlling wind turbine tower vibration under external force by applying control systems combination

The global focus has recently shifted away from fuel-based power sources, and one of the most important projects for energy production is wind energy. To maintain low costs, the current research examines the problem of vibrations affecting wind turbine towers’ performance (WTTs). In particular, the tower, resulting from excessive vibrations, can negatively affect a structure’s power output and service life, as it can cause fatigue. Therefore, we conducted numerical tests on various types of controlled systems. Our tests revealed that combining a new technique cubic negative velocity control (CNVC) and linear negative acceleration control (LNAC) was the most effective and cost-efficient option for vibration damping. This solution was derived by using an approximation method for the averaging technique. The external force is an important component of a nonlinear dynamic system and can be characterized by two-degree-of-freedom (2-DOF) differential coupled equations. After implementing the control measures, we conducted a numerical analysis of the vibration values before and after the operation. Stability is studied numerically. The numerical and approximate solutions were confirmed through the frequency response equation and time history with fourth-order Runge–Kutta (RK-4). Finally, we investigated the effect of parameters and compared our results with those of previously published studies.

Performance Improvement of Three-Spring Quasi-Zero-Stiffness Isolator by Shape Memory Alloy Damper

Quasi-zero stiffness vibration isolators (QZSI) can effectively resolve the contradiction between the load bearing and the vibration isolation for low-frequency vibration systems. The three-spring QZSI is a classical QZSI. However, constrained by its geometrical relationship, the working stroke of the QZSI is limited, and the isolation performance will deteriorate when the displacement response is significant. To solve this question, the shape memory alloy (SMA) damper, characterized by lower secondary stiffness, partially replaces the positive-stiffness spring of the three-spring QZS isolator to widen its effective working stroke, which forms a novel SMA-QZSI. Based on a piecewise linear constitutive model of SMA, a theoretical model of the force–displacement relationship of SMA-QZSI is established. Subsequently, the approximate analytic amplitude–frequency response equation of the non-smooth nonlinear SMA-QZSI system under harmonic excitation is obtained by the average method and verified numerically. In addition, the effects of mechanical parameters of SMA springs and external excitation parameters on mechanical behaviors and low-frequency isolation performance of the SMA-QZSI isolator are discussed. Finally, the static and dynamic experiments of SMA-QZSI are conducted to verify its force–displacement relationship and isolation performance. The results show that the proposed SMA-QZSI has remarkable low-frequency vibration isolation performance.

Powering piezoelectric frequency up-converter with rotary magnetic forces for torque-sensing application

A piezoelectric frequency up-energy harvester (PFU-EH) utilizing an SECE circuit (synchronized electric charge extraction) is developed for torque-sensing application in rotating machinery. Development of an explicit theoretical model for the system with the PFU-EH, considering phase angle shifts and magnetic impulsive forces is proposed to formulate the frequency response equations for voltage and power outputs using Fourier analysis. In this study, harvesting electrical energy can be achieved through the magnetic force interaction between the cantilevered piezoelectric beam, which carries two attached magnets, and the compound rotating disk structure, also carrying two attached magnets. The compound rotating disk structure, composed of a metal ring (outer disk) and a flywheel (inner disk) connected by a spring system, can create a phase angle caused by an applied load. As a result, mechanical torque is formed for inducing magnetic excitation to the PFU-EH and subsequently generating an AC voltage signal. The functionality of the AC voltage harvested by the PFU-EH can be further leveraged for two primary benefits. First, it can be converted into DC voltage output using the SECE circuit, enhancing the effectiveness of the electric signal without requiring load matching under different modal vibrations. Second, the SECE harvesting voltage can simultaneously function as a sensing signal for detecting mechanical torque itself. The sensing signal detection process is further analyzed by applying feature extraction from the spectrogram and utilizing a 2D-CNN for auxiliary segment identification. Finally, the experimental results demonstrate reasonable agreement with the predicted outcomes.

Energy harvesting for a vibration isolation system with negative stiffness under harmonic excitation

ABSTRACT In this paper, an energy harvesting system with a negative stiffness structure is established, whose main purpose is to achieve good vibration isolation, and at the same time, the system can also achieve the purpose of energy harvesting. In order to analyse the system model, the amplitude–frequency response equations for the displacement, induced voltage and electric power of the system when the system is in the main resonance state are obtained by using the multiscale method, and their amplitude–frequency response curves are plotted. In addition, we obtained the existence conditions for the system to satisfy Smale's horseshoe chaos using the Melnikov function. And the phase diagram and Poincaré mapping of the system were obtained using the fourth-order Runge–Kutta method. Numerical results show that the system has some energy harvesting performance along with vibration isolation. Variations in the parameters of the system affect the resonance frequency band and co-oscillation amplitude of the system as well as the nonlinear characteristics of the system itself.

Analytical Study of Nonlinear Flexural Vibration of a Beam with Geometric, Material and Combined Nonlinearities

This article explores the nonlinear vibration of beams with different types of nonlinearities. The beam vibration was modeled using Hamilton’s principle, and the equation of motion was solved using method of multiple time scales. Three models were developed assuming (a) geometric nonlinearity, (b) material nonlinearity and (c) combined geometric and material nonlinearity. The material nonlinearity also included both third and fourth nonlinear elasticity terms. The frequency response equation of these models were further evaluated quantitatively and qualitatively. The models capture the hardening effect, i.e., increase in resonant frequency as a function of forcing amplitude for geometric nonlinearity, and the softening effect, i.e., decrease in resonant frequency for material nonlinearity. The model is applied on the first three bending modes of the cantilever beam. The effect of the fourth-order material nonlinearity was smaller compared to the third-order term in the first mode, whereas it is significantly larger in second and third mode. The combined nonlinearity models shows a discontinuous frequency shift, which was resolved by utilizing a set of transition assumptions. This results in a smooth transition between the material and geometric zones in amplitude. These parametric models allow us to fine tune the nonlinear response of the system by changing the physical properties such as geometry, linear and nonlinear elastic properties.

Combination resonance of a moving ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field

The combination resonance of a moving rectangular ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field is investigated. Initially, the kinetic and potential energies of the system, considering geometrical nonlinearity, are obtained based on Kirchhoff's theory of thin plates. Additionally, the works of external forces are determined using the principle of virtual work. Subsequently, taking into account the nonlinear magnetization phenomenon of ferromagnetic materials, the magnetization force and Lorentz force acting on the moving ferromagnetic plate are calculated according to electromagnetic theory. Finally, the nonlinear magnetoelastic vibration equation is established by applying the Hamiltonian variational principle. The algebraic equation for static deflection and the differential equation for disturbed deflection are derived by applying the Galerkin method to the vibration equation. Then, an analytical solution for the frequency response equation under combination resonance is obtained using the multiple scale method. The proposed model and analytical research methods are validated by comparison with existing literature and numerical methods. Furthermore, the stability of steady-state solutions is determined, and static bifurcation is analyzed through singularity analysis. With the analytical results, numerical examples are plotted with varying frequency tuning values, magnetic field intensities, load amplitudes and load positions. The effects of these parameters on vibration characteristics are examined by comparing amplitude curves of systems with different physical parameters. The results demonstrate significant nonlinear vibration characteristics of the thin plate under the influence of a constant magnetic field and motion effects.

A vibration analysis of the permanent magnet synchronous motor under the effect of proportional derivative control

In industrial environments, excessive vibration can pose a safety risk since it can weaken the structural integrity of PMSMs and adjacent equipment. Vibration levels can be understood and controlled to help assure safe operation and avoid mishaps or equipment breakdowns. Friction and mechanical resistance are examples of vibration-induced losses that can lower PMSMs’ overall efficiency. Through the optimization of motor design and operation, engineers may minimize vibration and increase energy efficiency while lowering operating costs. Since an electric motor’s tendency to overheat after a brief period of use is linked to oscillating vibration, this study employs a proportional derivative control (PD control) to demonstrate the potential effectiveness of a permanent magnet synchronous motor (PMSM). The external force is a component of the non-linear dynamical system. The equation of the (PMSM) system is explained with two-degree-of-freedom (2dof) differential coupled equations, consisting of the major body (motor current) and cutting drive system. The present study uses the approximate method of multiple scales perturbation technique (MSPT) to get an approximate solution, showing the response equation before using (PD) and to achieve the highest effect on the control system so as to ensure the highest efficiency at the lowest possible cost. Different numerical tests have been conducted (NPDCVF, NVC, PPF, PD, delay velocity) and showed that (PD) has the best effect and is able to control the vibrations with more accuracy than others. Then, the vibration value of the system has been studied numerically before and after applying the control method of (MSPT). By combining the analysis of the resonance situation via both the phase plane techniques and the frequency response equation ω1≅ω2,ω≅ω2, within the utilization of Runge–Kutta of the fourth-order, the stability of the numerical solution has been investigated. Then, by employing the MATLAB tool, the impact of various parameters on model performance has been examined numerically. Lastly, a comparison is made with earlier research and other techniques used in prior work using other types of control mechanisms for other systems.

A Ball-Contacting Dynamic Vibration Absorber with Adjustable Stiffness and Nonlinear Characteristics

Structural vibration has always been a major concern in the engineering field. A dynamic vibration absorber in the form of contacts with adjustable stiffness (CDVA) offers effective vibration suppression and can improve conventional dynamic vibration absorbers with high sensitivity to frequency deviation and difficulty in adjusting the frequency. In this research, first, based on the theoretical model of the contact between a rubber ball and an inner cone, the feasibility of changing the axial contact state to change the structure’s natural frequency was verified using an ANSYS simulation. A theoretical model of the static contact stiffness between the ball and the inner cone was constructed using Hertzian contact theory and Hooke’s law, and a theoretical model of the cubic nonlinear elastic restoring force was used to characterize the stiffness properties of the rubber ball during compressive rebound. The steady-state frequency response equations of the main vibration structure were derived using the averaging method in conjunction with the two-degree-of-freedom dynamics model, and the stability of the solutions to the frequency response equations was obtained in conjunction with the stability determination criterion. Then, the impact of the CDVA’s design parameters on the nonlinear dynamic response of the primary vibration structure was simulated and analyzed. The resulting findings can serve as guidance for designing dynamic vibration absorber parameters. Based on the principles of ball-inner cone contact, a dynamic vibration absorber structure was proposed. A design test was conducted to verify the correctness of the contact stiffness model, and an experimental study was carried out to investigate the law of change in the dynamic stiffness and damping of the principle structure of CDVA under dynamic excitation conditions. Finally, the vibration test platform of the solidly supported beam structure was constructed, and vibration suppression tests of the CDVA in different compression states were conducted to investigate the tunability and feasibility of CDVA vibration suppression. The results showed that the dynamic vibration absorber had good vibration absorption characteristics and could be used for single-mode vibration suppression of multimodal main structures.

Nonlinear phenomena in vibrations of embedded carbon nanotubes conveying viscous fluid

Various nonlinear phenomena such as bifurcations and chaos in the responses of carbon nanotubes (CNTs) are recognized as being major contributors to the inaccuracy and instability of nanoscale mechanical systems. Therefore, the main purpose of this paper is to predict the nonlinear dynamic behavior of a CNT conveying viscous fluid and supported on a nonlinear elastic foundation. The proposed model is based on nonlocal Euler–Bernoulli beam theory. The Galerkin method and perturbation analysis are used to discretize the partial differential equation of motion and obtain the frequency-response equation, respectively. A detailed parametric study is reported into how the nonlocal parameter, foundation coefficients, fluid viscosity, and amplitude and frequency of the external force influence the nonlinear dynamics of the system. Subharmonic, quasi-periodic, and chaotic behaviors and hardening nonlinearity are revealed by means of the vibration time histories, frequency-response curves, bifurcation diagrams, phase portraits, power spectra, and Poincaré maps. Also, the results show that it is possible to eliminate irregular motion in the whole range of external force amplitude by selecting appropriate parameters.

Effect of nonlinear inertia and Duffing-type nonlinearity on the dynamic behaviour of nonlinear parametrically excited cantilever beams with tip mass

The dynamic response of parametrically excited cantilever beams with a tip mass (PECBM) is investigated in this paper. Employing the Method of Multiple Scales (MMS) and the Method of Varying Amplitudes (MVA), the frequency-response equations are obtained and the stability of the response is analysed. MVA results show that, for frequency ranges around the principal parametric resonance, the term containing a linear combination of nonlinear inertia and Duffing-type nonlinearity in the frequency response equation can tend to zero, resulting in an exponential growth of the vibrations. These results are supported by numerical simulations obtained by direct integration (DI) of the equation of motion, while the MMS fails to predict the critical frequency at which this phenomenon occurs. Furthermore, based on the MVA, a criterion for determining the hardening and softening characteristics of PECBM is developed, showing that both Duffing nonlinearity and nonlinear inertia terms contribute to determining this characteristic. To verify the results, experimental measurements of the response of a PECBM are presented, showing good agreement with analytical and numerical results. It is shown that the mass added at the cantilever tip can change the characteristics of the system's dynamics, enhancing the softening behaviour of the PECBM.

On controlling, Vibration Performance and Energy Transfer of an Offshore Wind Turbine Tower System via PPF Controller

In this paper, we focus on the control, energy transfer, and vibration performance under multi mixed excitations for the offshore wind turbine tower (OWTT) system. For reducing the controlled system oscillations, the positive position feedback (PPF) controller is applied. The energy transfers occur in the system of wind turbine by adding the PPF controller to the system equations. With the help of the phase plane approach, frequency response equations, and Poincare maps, the bifurcation and stability at worst resonance cases are sought and investigated. The vibration behaviors are studied numerically at different parameters values for the wind turbine system. Additionally, the response and numerical outcomes are examined, also, the approach of multiple scales is used to establish the approximate solutions of the wind turbine-controlled system. Besides that, MAPLE and MATLAB algorithms are used to implement the numerical results and compare analytical solutions with numerical behavior. The results also be compared to previous research that has been published.

Principal-Internal Joint Resonance of an Axially Moving Beam with Elastic Constraints and Excited by Current-Carrying Wires

Principal-internal joint resonance of an axially moving beam with elastic constraints is investigated, where the beam is located in magnetic field excited by current-carrying wires. Based on the magnetoelasticity theory and Hamilton principle, the nonlinear magneto-elastic vibration equations are derived. The displacement function is obtained by the elastic constraint boundary condition, and then the equation is discretized into 2-DOF ordinary differential equations using the Galerkin integral method. The method of multiple scales is employed for obtaining the amplitude–frequency response equations with coupling of the first two vibration modes. Through examples, amplitudes varying with frequency tuning parameter, axial velocity, current intensity, and external excitation force are exhibited. Results indicate that as current intensity increases, electromagnetic damping increases and the amplitude decreases; As external excitation force increases and axial velocity decreases respectively, the amplitude increases and the system changes from single periodic motion to quasi-periodic motion and then to single periodic motion; The lower-order modes are always dominant when the principle-internal resonance occurs.

Modelling of vortex-induced force and prediction of vortex-induced vibration of a bridge deck using method of multiple scales

With the increase of span length, long-span bridges become more flexible and susceptible to vortex-induced vibrations (VIV). Several models for vortex-induced force (VIF) and various methods for predicting VIV have been developed accordingly but without consensus. This paper uses the method of multiple scales, from a different angle of view, to analyze the general polynomial model for VIF and present a concise polynomial model for VIF and a simple method for predicting VIV of a bridge deck. The aeroelastic vibration of a deck section is simulated by the computational fluid dynamics (CFD) and the results are used to verify the concise polynomial model of VIF. The forced vibration of a deck section is also simulated by CFD to yield the simple method for predicting VIV together with the frequency response equation and the critical equation of VIV derived by the method of multiple scales. The results show that the even and cross terms of aerodynamic damping and stiffness in the general polynomial VIF model are small compared with the odd terms and can be thus omitted. The lock-in region and the amplitude of VIV of the deck section can be accurately predicted by the proposed simple method.

A time-delayed proportional-derivative controller for a dielectric elastomer circular membrane

• In this paper, an internal resonance of a circular DE membrane is investigated within mechanical and electrical excitations in order to find internal resonances for improving its performance for future applications. • This work proposes a time-delayed proportional-derivative (TDPD) controller for decreasing the oscillations of a DE system with quadratic and cubic nonlinearities. • Averaging method is used to solve the controlled system to obtain the first-order approximation solutions and response equations for the DE system. • The TDPD controller is the best control strategy that can reduce the framework vibrations regardless of the excitation frequency. Due to the substantial deformation generated by voltage excitation, the dielectric elastomer (DE) is a new type of functional polymer that can be employed as a smart actuator. In this paper, an internal resonance of a circular DE membrane is investigated within mechanical and electrical excitations in order to find internal resonances for improving its performance for future applications. All programs that contain DE will perform flawlessly if DE is managed. This work proposes a time-delayed proportional-derivative (TDPD) controller for decreasing the oscillations of a DE system with quadratic and cubic nonlinearities. The system is researched and investigated when using a Proportional–derivative (PD) controller to apply time-delayed control on displacement and velocity. Stability is improved, maximum peak overshoot is reduced, settling time is reduced, and the framework transient response is recovered using this control method. The first-order approximation solutions for the DE system are calculated using the averaging method. Simultaneous resonance is regarded as the worst type of resonance. To obtain the dynamic solution of the device with TDPD controllers, analytical and numerical approaches are used. The Routh-Huriwitz approach is used to review and analyze the consistency of the steady state solution in the near-resonance case. The effects of various factors on the steady-state solution are identified and discussed. The effects of time delay are investigated in order to determine the most stable range of time delays for the best performance. The MATLAB software package is used to generate simulation effects. When the results are compared to the numerical simulations, they show that the approximate solution and the control algorithm used in this paper are well validated. At the end, there is a comparison with previously published work. The focus of this paper is on using a TDPD controller to manage the nonlinear dynamic response and probable internal resonances of a circular DE membrane under mechanical and electrical excitations. The approximate solution of a nonlinear system of differential equations is investigated using the averaging technique in the cases of primary and 1:2 internal resonance, which are the worst-case scenarios that should be avoided. To display and compare controller effects at various system settings, several response curves are used. The TDPD controller is used to research nonlinear dynamics control of the DE system with harmonic force. The stability analysis and the effects of parameter behavior are tested using MATLAB program. To examine the controller effect, several response curves are used. The collected findings demonstrated that the TDPD control efficiency in reducing the system's nonlinear oscillations. The frequency response curves of the system were studied at various uncontrolled and regulated coefficients. Before and after control, response curve diagrams employing frequency response equations are created. Finally, validation curves are shown to measure the degree of similarity between analytical and numerical simulation predictions. At the end, there is a comparison with previously published works.

Piezoelectric plate generator from consecutive periodic rotary magnetic excitations

The piezoelectric device, connected to the circuit interface and subjected to the consecutive periodic rotary magnetic excitations, is presented for an energy harvesting application. The proposed system consists of a cantilevered piezoelectric plate structure with multiple attached magnets, along with a revolving disc carrying a single magnet. This study sets out to explore the primary development of explicit analytical and finite element formulations that consider magnetic coupling and time-delayed magnetic response. The two theoretical models are used to formulate the optimal power output frequency response equations using Fourier analysis. Due to the asymmetrical magnet locations and time-delayed magnetic response, the consecutive dynamic excitations periodically occur, resulting in dipole–dipole interactions in both spatial and temporal conditions. Consequently, the piezoelectric plate structure, designed to operate higher vibration modes, exhibits a predominant effect of frequency up-converting energy harvesters. The previous work is initially validated by using a single magnetic excitation and examining both cases with and without time-delayed magnetic responses. The analyses of broadband optimal power generations are based on Fourier’s harmonic numbers and frequency domains, including time-tracking magnetic excitations. Finally, various case studies are presented to examine the effects of different physical components, such as magnet numbers, magnet locations and masses, piezoelectric materials and thicknesses, and locations of electrode segments.

The Magneto–Thermo–Elastic Primary Resonance of Rotating Ferromagnetic Functional Gradient Cylindrical Shell in a Transverse Magnetic Field

The primary resonance behavior of a rotating ferromagnetic functional gradient cylindrical shell in a magnetic field, temperature field, and excitation force is investigated. Based on the physical neutral surface deformation theory and the Donnell theory, considering the effect of geometric nonlinearity, expressions of strain energy and kinetic energy of the shell and the work of forces are given, respectively. Applying the Hamilton principle, the magneto–thermo–elastic equation of a functional gradient cylindrical shell is derived by considering the magnetization effect of ferromagnetic metal. The question is discretized by Galerkin method and solved by the multi-scale method to obtain the amplitude–frequency response equation. The stability of the solution is discriminated by using the Lyapunov theory. Through numerical examples, the response curves of the system under different parameters are plotted, and the parameter ranges corresponding to multi-valued solution regions and single-valued solution regions are determined. The effects of parameter changes on the dynamic response and stability of system are analyzed. The results show that a coupling mechanism between temperature field, magnetic field, and excitation force affects the response and stability of the system, and the change of parameters have a significant effect on the vibration characteristics and stability. The dynamics model established in this paper is a theoretical reference for investigation on the multi-physics field coupling dynamic behaviors of structures.

Parametric resonance of multi-frequency excited MEMS based on homotopy analysis method

The nonlinear dynamics and control of micro electromechanical systems (MEMS) is an important topic at present and homotopy analysis method (HAM) is an effective semi-numerical and semi-analytical method for solving strongly nonlinear problems. In this paper, the parametric resonance of MEMS with multi-frequency excitation is studied by HAM. Firstly, the differential equation of electrostatically driven microbeam is processed by Taylor expansion, and it is transformed into the parametric motion model with multi-frequency excitation. Then, the approximate solution and amplitude–frequency response equation of the system are obtained by HAM, and compared with the numerical solution. Finally, the influence of direct current (DC) and alternating current (AC) on principal parametric resonance and superharmonic resonance is discussed. The results show that HAM is an effective method to analyze the parametric vibration of multi-frequency excitation system, and the amplitude–frequency response curve of microbeam about parametric motion depends on the time scale in high DC and high AC state. This study effectively extends the application of HAM in parametric resonance, which is of great significance to the study of nonlinear vibration of MEMS.