Amplitude-frequency Response Equation Research Articles

Nonlinear combination resonance analysis of parametric-forced excitation for an axially moving piezoelectric rectangular thin plate in thermal- electromechanical coupling field

In this paper, the combination resonance of parametric-forced excitation characteristics for an axially moving rectangular piezoelectric plate under a thermal-electromechanical field is studied. Based on Kirchhoff-Love plate theory and Von Karman theory, the transverse vibration governing equations are derived from Hamilton's principle. The equations are discretized to ordinary differential equations by the Galerkin method. Then, the multiple scales method is applied to solve the system combination resonance equation, two different resonance states and corresponding amplitude-frequency response equations are obtained by eliminating the secular term, respectively. Additionally, the stability of the steady-state responses is analyzed by Lyapunov stability. Based on the numerical analysis, the influence of axial velocity, external voltage, central temperature difference, structural damping, and other parameters on nonlinear combination resonance response are investigated. The effect of parameter variation on period-doubling bifurcation and the chaotic motion of the system are also discussed by the system global bifurcation diagram.

Simultaneous vibration isolation and energy harvesting via a new negative stiffness system with inertial amplifier

The negative stiffness system provides significant vibration isolation. The conversion of mechanical energy from structural vibrations into electrical energy is an effective method of energy harvesting. In this paper, a new negative stiffness system (NSS) with inertial amplifier for vibration isolation and energy harvesting is established, whose main purpose is to achieve good vibration isolation effect while the system can achieve the purpose of energy harvesting. Inertial amplifiers are used to improve the dynamic effective mass of the system, thereby reducing the natural frequency of the system. Using the multi-scale method, the amplitude-frequency response equation of the system is derived and the amplitude-frequency characteristic curve is plotted. The Lyapunov exponent, the root mean square of the voltage of the piezoelectric system and the basin of attraction are introduced to analyze the energy harvesting performance of the system. Numerical results show that the inertial amplifier improves the vibration isolation capacity of the negative stiffness structure in the low-frequency band. There is a correlation between the basin of attraction, the Lyapunov exponent, and the root mean square of voltage.

Dual-mode coupling resonance and dynamic stability of axially moving ferromagnetic thin plate strips in alternating magnetic field

This paper investigates the dual-mode coupling resonance and dynamic stability of axially moving ferromagnetic thin plate strips in alternating magnetic field. According to the basic electromagnetic theory and the magnetoelastic mechanics model, Lorentz electromagnetic force and torque generated by eddy current effect and the magnetizing force generated by magnetization are determined for the ferromagnetic plate strip. Introducing Von Karman geometric nonlinearity, the nonlinear governing equation of axially moving ferromagnetic plate strips under the combined action of mechanical load and magnetizing force is established by Hamilton's principle. For the simply supported plate strip, Galerkin method and the multi-scale method are employed to solve the modal coupling resonance dominated by electromagnetic excitation, obtaining the amplitude-frequency response equations. The stability of resonance responses is analyzed by the Lyapunov stability theory. Through parametric research, the amplitude-frequency characteristic curves, bifurcation diagrams and Lyapunov exponent spectra of each mode are plotted; the effect of different parameters on vibration characteristics and dynamic stability under the modal coupling is analyzed for the system. The results show that as one of the modal amplitudes enters the resonance region, the other modal amplitude decays into the "collapse" region due to energy exchange between modes; each modal amplitudes has more complex multi-valued and jumping phenomena for the nonlinear coupling effect. The increase in velocity will affect the stability of system; the enhancement of magnetic field will make the system change into chaos at high velocity.

The harmonic resonance and singularity analysis of bifurcation for the magnetized elastic plate with action of time-varying magnetic potential

This paper deals with the superharmonic resonance of a ferromagnetic thin rectangular plate in the air-gap magnetic field excited by armature magnetic potential. Electromagnetic forces applied on the plate by the air-gap magnetic field cause the plate to transversal vibrate, which affects the air-gap magnetic field in turn, resulting the vibration and magnetic field coupling. According to the basic electromagnetic field theory and considering the magnetoelastic coupling effect, the air-gap magnetic field intensity is obtained by solving the Laplace's equation satisfied the air-gap magnetic boundary conditions. The electromagnetic force model of soft ferromagnetic plates is determined based on theories of electromagnetic and elasticity. According to the large deflection theory of plates, basic energy relationships and variational equations of the elastic plate are given. Eventually, the nonlinear magnetoelastic vibration equation of ferromagnetic thin plates is derived using Hamilton's principle and Galerkin method. The multi-scale method is used to solve the superharmonic resonance to obtain the amplitude-frequency response equation and the stability discriminant of solutions. The topological analysis of amplitude-frequency equation is carried out using singularity theory, and the bifurcation characteristics of systems on physical parameter planes in different regions are obtained according to the transition set. The correctness of analytical solutions is verified by comparison with numerical solutions. Through numerical calculations, curves of the static deflection and equivalent magnetic force of plate with parameters are given, and the amplitude curves, dynamic phase plane trajectories and time history diagrams of system response with changes in electromagnetic and structural parameters are plotted. Results show that both the decrease of armature magnetic potential amplitude and the increases of plate thickness and initial air-gap thickness reduce the static deflection. The increase of armature magnetic potential amplitude increases the equivalent magnetic force. As the decrease of initial air-gap thickness and the increases of armature magnetic potential amplitude and excitation force amplitude, the amplitudes of the upper branch and lower branch curves representing stable solutions decrease and increase, respectively, and the single-value solution region increases.

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.

Analysis on Thermo-Electrical Principal Parametric Resonance of an Axially Moving Piezoelectric Thin Plate

In this paper, principal parametric resonance of an axially moving piezoelectric rectangular thin plate under thermal and electric field is investigated. Based on Kirchhoff–Love plate theory and Von Karman theory, the transverse vibration differential equation of a piezoelectric rectangular thin plate under thermal and electric field is derived by using Hamilton’s principle. The dimensionless vibration equation of piezoelectric rectangular thin plate with parametric excitations is discretized by Galerkin’s method. Then, the multiple scales method is applied to derive amplitude-frequency response equation and the stability conditions of the steady-state solution are obtained by Lyapunov stability theory. Numerical method is used to find the influences of specific parameters on the vibration performance and stability of the system. Based on the global bifurcation diagram and corresponding response diagram, the influences of bifurcation control parameters on the nonlinear dynamic characteristics of the system are discussed. Numerical results illustrate that the system amplitude frequency characteristic curve presents soft spring characteristics. There are periodic and chaotic motions with the increase of velocity and the central temperature difference, and the decrease of plate thickness and velocity will result in the decrease of chaotic threshold. The results also show that increase the velocity perturbation amplitude can prolong the chaotic motion.

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.

Internal Resonance of the Coupling Electromechanical Systems Based on Josephson Junction Effects.

The internal resonances of the coupling vibration among electro-dynamic modes of an NEMS are studied for the coupling resonators connected on a Josephson junction. The methodology adopted involves coupling a resonator connected on a Josephson junction. The mathematical model of the coupled system is then obtained by considering the regulatory nonlinear effect of the phase difference of that Josephson junction. The resulting dynamic differential equation is deduced by considering the nonlinear terms of the Josephson junction and the nanobeam. The multi-scale method is then used to obtain the 1:1:1 resonant amplitude-frequency response equation of the coupled electromechanical system. The influence of the phase difference of the Josephson junction, magnetic field, external excitation and other factors are analyzed based on the internal resonant amplitude of the coupled system. The simulation results illustrate that the changes in the values of the magnetic field, excitation amplitude and divided resistances can lead to a remarkable change in the values of the nanobeam frequency and amplitude. The internal resonance principle is used to generate a mutual conversion and amplification among electrical signals and mechanical signals. This research provides a theoretical framework and a numerical approach for improving the sensitivity of magnetic quality detection.

Subharmonic Resonance of Duffing Oscillator With Dry Friction Under Foundation Excitation

Abstract The 1/3 subharmonic resonance response of Duffing oscillator with Coulomb dry friction under foundation excitation is investigated, and the approximate analytical solution of the subharmonic resonance of the system is obtained by using the incremental averaging method. Based on the approximate analytical solution of the primary resonance obtained by the averaging method, the approximate analytical solution of subharmonic resonance is solved by the averaging method according to the incremental equation, and the amplitude–frequency response equation of subharmonic resonance is obtained. It is found that the Coulomb friction affects the amplitude–frequency response of both the primary resonance and subharmonic resonance of the nonlinear dry friction system in the form of equivalent damping. The comparison between the approximate analytical solution and the numerical solution shows that the approximate analytical solutions of the primary resonance and subharmonic resonance are both in very good agreement with the numerical solution. The existence condition of the 1/3 subharmonic resonance for the nonlinear dry friction system is presented, and the stability of the steady-state solution of subharmonic resonance is also judged. Based on the approximate analytical solution, the effects of the nonlinear stiffness and the Coulomb friction on the amplitude–frequency response of resonance and critical frequency of 1/3 subharmonic resonance of the nonlinear dry friction system are analyzed in detail. The analysis results show that the incremental averaging method can effectively obtain the approximate analytical solution in unified form for the subharmonic resonance of nonlinear system with Coulomb friction.

Localizing Bifurcations in Non-Linear Dynamical Systems via Analytical and Numerical Methods

In this paper, we study the bifurcations of non-linear dynamical systems. We continue to develop the analytical approach, permitting the prediction of the bifurcation of dynamics. Our approach is based on implicit (approximate) amplitude-frequency response equations of the form FΩ,A;c̲ =0, where c̲ denotes the parameters. We demonstrate that tools of differential geometry make possible the discovery of the change of differential properties of solutions of the equation FΩ,A;c̲ =0. Such qualitative changes of the solutions of the amplitude-frequency response equation, referred to as metamorphoses, lead to qualitative changes of dynamics (bifurcations). We show that the analytical prediction of metamorphoses is of great help in numerical simulation.

Study on a Class of Piecewise Nonlinear Systems with Fractional Delay

In this paper, a dynamic model of piecewise nonlinear system with fractional-order time delay is simplified. The amplitude frequency response equation of the dynamic model of piecewise nonlinear system with fractional-order time delay under periodic excitation is obtained by using the average method. It is found that the amplitude of the system changes when the external excitation frequency changes. At the same time, the amplitude frequency response characteristics of the system under different time delay parameters, different fractional-order parameters, and coefficient are studied. By analyzing the amplitude frequency response characteristics, the influence of time delay and fractional-order parameters on the stability of the system is analyzed in this paper, and the bifurcation equations of the system are studied by using the theory of continuity. The transition sets under different piecewise states and the constrained bifurcation behaviors under the corresponding unfolding parameters are obtained. The variation of the bifurcation topology of the system with the change of system parameters is given.

Bifurcation and stability of resonance of electric vehicle powertrain: Focus on the influence of electromagnetic parameters

The influence of the electromagnetic parameters on the torsional dynamics of the electric vehicle powertrain is studied by considering the electromechanical coupling effect. By adding the electromagnetic torque on the drive side, the powertrain is simplified as nonlinear drive-shaft model. The number, stability, and bifurcation conditions of the equilibrium points of the nonlinear drive-shaft model are deduced. Based on the averaged equations and the amplitude-frequency response equation, the stability and bifurcation conditions, such as fold bifurcation and Hopf bifurcation, of the resonance curve are discussed. The influence of electromagnetic parameters on the torsional dynamics is studied by simulation. It is shown that with the change of the parameters, the number as well as the stability of the equilibrium points may be changed which is affected by fold bifurcation. It is also shown that the resonance curve may lose its stability when fold bifurcation happens. By limiting the parameters in the region without fold bifurcation, the unstable dynamics of the resonance curve can be controlled.

COMBINED CONTROL OF VEHICLE SUSPENSION WITH FRACTIONAL-ORDER MAGNETORHEOLOGICAL FLUID DAMPER MODEL

The fractional-order Bingham model of magnetorheological fluid damper has simple structure and can better describe the hysteretic characteristics of the system. The vibration control of a nonlinear vehicle suspension system with magnetorheological fluid damper under harmonic excitation is studied, where the single-degree-of-freedom 1/4 vehicle suspension system with fractional-order Bingham model of magnetorheological fluid damper is considered. The primary resonance response of suspension system with fractional-order Bingham model under sky-hook damping semi-active control is analyzed, and the approximate analytical solution is obtained by means of averaging method. The amplitude-frequency response equation of the steady-state solution of the suspension system is obtained, and the stability condition of the suspension system is also obtained according to Lyapunov’s stability theory. By comparing the amplitude-frequency response curves of the numerical solution and approximate analytical solution, the accuracy of the approximate analytical solution has been verified. The influence of semi-active control on the ride comfort of the vehicle is illustrated by the root mean square values of acceleration of the sprung mass in the vertical direction, it is found that the semi-active control strategy of sky-hook damping can not improve the ride comfort of vehicle in low frequency excitation region of road. Therefore, a combined control strategy of passive control and semi-active control is proposed, and the influence of semi-active control parameter on the vibration control effect is analyzed. The results show that the combined control strategy can not only improve the ride comfort of the vehicle, but also effectively suppress the primary resonance vibration amplitude of suspension system.

Nonlinear Vibration Control for Nanobeam with Time Delay by Field Effect Transistor Sensing

A vibration control signal sensing method is proposed utilizing the grid electrode displacement sensitive effect of field-effect transistors. The signal sensing method is applied to nonlinear resonance delay control of a nanobeam that is used as the core component of nano–microdevices. The nonlinear vibration dynamical model of the nanobeam based on the field-effect tube sensing is established and the differential equation of motion with time delay control is presented. The amplitude frequency response equation and phase frequency response equation of the nanobeam are obtained by analyzing the first-order approximate solution of the primary and superharmonic vibration of the nonlinear equation with multi-scale method. The vibration feedback gain and time delay can affect the vibration amplitude and nonlinear behavior of the nanobeam. The nonlinear vibration of the nanobeam can be adjusted and effectively controlled by selecting appropriate feedback gains and time delays.

Vibration Characteristics of Rolling Mill System under Constraints of the Nonlinear Spring Force and Friction Force from Hydraulic Cylinder

Considering the two kinds of nonlinear constraints of rolling mill hydraulic cylinder, spring force and friction force, the vibration model of rolling mill system is established. The amplitude frequency response equations are obtained by using the average method. Comparing the time history curves of vertical vibration displacement of rolling mill system under the nonlinear spring force and friction force, the amplitude frequency characteristic curves are simulated. The external excitation amplitude is viewed as the bifurcation parameter, and the system bifurcation response changing with the external excitation amplitude is analyzed. The influence of the external excitation amplitude on the system stability is studied. The results indicate that the increase of the nonlinear spring force makes the rolling mill system’s unstable area to become wider, and the influence on the rolling mill system of nonlinear friction force behaves as the damping characteristics; the vibration of rolling mill system is alternating between the periodic, period-doubling, and the chaotic motion. The research results provide a theoretical support for restraining the vibration of the rolling mill system in the actual production process.

Vertical–Horizontal Coupling Vibration of Hot Rolling Mill Rolls under Multi-Piecewise Nonlinear Constraints

This study establishes a vertical–horizontal coupling vibration model of hot rolling mill rolls under multi-piecewise nonlinear constraints considering the piecewise nonlinear spring force and piecewise nonlinear friction force constraints of the hydraulic cylinder in the vertical direction of the rolls, the piecewise stiffness constraints in the horizontal direction, and the influence of the nonlinear dynamic rolling force in the rolling process. Using the average method to solve the amplitude–frequency response equation of the coupled vibration system and taking the actual parameters of a 1780 mm hot rolling mill (Chengde Steel Co., Ltd., Chengde, China) as an example, we study the amplitude–frequency characteristics of the mill rolls under different parameter settings. The results show that the amplitude and resonance region can be reduced by appropriately reducing the external disturbance force and the nonlinear spring force of the hydraulic cylinder, appropriately increasing the nonlinear friction force, and eliminating the gap between the bearing seat and the mill housing, to avoid the amplitude jump phenomenon due to piecewise variation. Furthermore, using the singularity theory to study the static bifurcation characteristics of the coupled vibration system, we establish a relationship between the vibration parameters and the topological bifurcation solution of the coupled system. The transition sets and their corresponding bifurcation topological structure in three cases are given, and the steady and unsteady process parameter regions of the rolls are obtained. The dynamic behavior of the coupled vibration system can be controlled by varying the bifurcation parameter. This study provides a theoretical basis for restraining the vibration of hot rolling mill rolls and optimizing the process parameters.

Subharmonic resonance bifurcation and chaos of simple pendulum system with vertical excitation and horizontal constraint

In order to improve the working performance and optimize the working parameters of the typical engineering pendulum of a typical system that it is abstracted as a physical simple pendulum model with vertical excitation and horizontal constraint. The dynamical equation of the system with vertical excitation and horizontal constraint is established by using Lagrange equation. The multiple-scale method is used to analyze the subharmonic response characteristics of the system. The amplitude-frequency response equation and the phase-frequency response equation are obtained through calculation. The effects of the system parameters on the amplitude resonance bandwidth and variability are clarified. According to the singularity theory and the universal unfolding theory, the bifurcation topology structure of the subharmonic resonance of the system is obtained. The Melnikov function is applied to the study of the critical conditions for the chaotic motion of the system. The parameter equation of homoclinic orbit motion is obtained through calculation. The threshold conditions of chaos in the sense of Smale are analyzed by solving the Melnikov function of the homoclinic motion orbit. The dynamic characteristics of the system, including single-parameter bifurcation, maximum Lyapunov exponent, bi-parameter bifurcation, and manifold transition in the attraction basin, are analyzed numerically. The results show that the main path of the system entering into the chaos is an almost period doubling bifurcation. Complex dynamical behaviors such as periodic motion, period doubling bifurcation and chaos are found. The bi-parameter matching areas of the subharmonic resonance bifurcation and chaos of the system are clarified. The results reveal the global characteristics of the system with vertical excitation and horizontal constraint, such as subharmonic resonance bifurcation, periodic attractor multiplication, and the coexistence of periodic and chaotic attractors. The results further clarify the mechanism of the influence of system parameters change on the movement form transformation, energy distribution and evolution law of the system. The mechanism of the influence of relevant parameters on the performance of the engineering system with vertical excitation and horizontal constraint is also obtained. The results of this research provide theoretical bases for adjusting the parameters of working performances of this typical physical system in engineering domain and the vibration reduction and suppression of the system in actual working conditions.

Experimental Modelling and Amplitude-Frequency Response Analysis of a Piecewise Linear Vibration System

The amplitude-frequency response of a nonlinear vibration system with the coexistence of stiffness and viscous damping piecewise linearities are analysed by means of analytical, numerical and experimental investigations. First, a mechanical model of the piecewise linear system under simple harmonic base excitation is established, and the amplitude-frequency response equation is obtained by the averaging method. Second, an experimental device is built to realize the piecewise linear system. The stiffness and damping coefficients are identified by the least square method. Third, case studies are conducted to illustrate the effect of the clearance and base excitation amplitude on the amplitude-frequency response. The experimental results show that the introduction of the piecewise linear stiffness and damping significantly decreases the response amplitude at the primary resonance. The piecewise linear stiffness, damping coefficients, primary resonance frequency and frequency range of the bi-stable state depend on the clearance and excitation amplitude. The experimental results are consistent with the theoretical predictions and numerical simulation results of the method of backward differentiation formulas. This research provides instructive ideas to the design of the nonlinear isolator in practical engineering.

DYNAMICAL ANALYSIS ON A KIND OF SEMI-ACTIVE VIBRATION ISOLATION SYSTEMS WITH DAMPING CONTROL 1)

A series of single-degree-of-freedom semi-active nonlinear vibration isolation system with cubic stiffness based on relative velocity feedback on-off control is studied. By using the averaging method, the approximate analytical solutions of the primary resonance of the system are obtained, which is controlled by the acceleration and relative velocity based acceleration drive damping control, the velocity and relative velocity based sky-hook damping control, and the displacement and relative velocity based ground-hook damping control, respectively. A further comparison between the numerical solutions and the approximate analytical solutions under different control strategies is fulfilled. And such comparison results suggest that the approximate analytical results are quite consistent with the numerical solutions, which verifies the effectiveness of the approximate analytical solutions. Moreover, the stability conditions of the steady-state solution of the system under different control strategies are analyzed by Lyapunov theory, and the influences of system parameters on the control effect are discussed. In order to compare the control performances in the three different control approaches, the amplitude-frequency response equation is conducted. The results show that the similar expressions can be found in the switching conditions in the process of the analytical analysis of the three relative velocity feedback control strategies. In the aspect of suppressing resonance response amplitude, the sky-hook damping control strategy based on velocity and relative velocity feedback has the best damping effect in low frequency band, while the acceleration drive damping control strategy based on acceleration and relative velocity feedback has the best damping effect in high frequency band. The sky-hook damping control strategy also shows good performance in reducing the amplitude response of transient response. This analytical research method can also be applied to the system with other semi-active on-off control strategies, and it provides an effective way for the control strategy research of semi-active vibration isolation system.