Amplitude-frequency Response Equation Research Articles

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.

Analytical Analysis for the Nonlinear Phenomena of a Dual-Rotor System at the Case of Primary Resonances

This paper proposes an analytical method to study the nonlinear phenomena of a simple dual-rotor system in the case of primary resonances. According to the assumed mode method (AMM), the dynamic equations of the system with four degrees of freedom (4DOF) are established by considering the nonlinearities of the inter-shaft bearing as a nonlinear spring. The 4DOF dynamic equations in real coordinate are transferred into 2DOF dynamic equations in complex coordinate because of the symmetry of the system in two directions. Then the amplitude-frequency response equations for both primary resonances are obtained by the multiple scales method and the numerical verification shows that the simplified method of the dynamic equation is correct. Moreover, the primary resonances affected by the typical parameters such as the linear stiffness and the nonlinear stiffness of the inter-shaft bearing, and the rotation speed ratio are discussed in detail afterwards. The results show that the system will show the jump phenomenon and resonance hysteresis phenomenon when the linear stiffness is rather small and the nonlinear stiffness is rather large. The linear stiffness will suppress these nonlinear phenomena while the nonlinear stiffness will promote them. The results obtained in this paper will contribute to the mechanism analysis of the nonlinear phenomena in a dual-rotor system, which is beyond the reach of the numerical method.

Research on a nonlinear quasi-zero stiffness vibration isolator with a vibration absorber.

A negative stiffness mechanism consisting of a spring and cylinder is proposed, and a grounded dynamic vibration absorber is designed based on a quasi-zero stiffness vibration isolator to constitute the vibration isolator with a vibration absorber system. The range of parameters for attaining zero stiffness is derived from static analysis. The dynamic analysis of the vibration isolator with a vibration absorber system is carried out by a multiscale method, and the amplitude-frequency response equation of the system is obtained. The influence of different system parameters on the amplitude-frequency response is analyzed. The amplitude-frequency response of the quasi-zero stiffness vibration isolator is compared with that of the vibration isolator with a vibration absorber, and the linear and nonlinear analytical solutions of the vibration isolator with a vibration absorber system are also compared. The results show that the designed vibration isolator with a vibration absorber is an ideal choice for low-frequency vibration isolation, with no large resonance peak throughout the system and significantly improved reliability of the system.

Nonlinear superharmonic and subharmonic resonances of piezoelectric elliptic thin plates under thermoelastic coupling effect

This paper aims at investigating the nonlinear vibration characteristics of the piezoelectric elliptic thin plate under the combined action of external harmonic excitation and temperature field. Based on the Von-Karman plate theory of large deflection, the nonlinear governing equation is derived by using the Bubnov-Galerkin method. Then, the amplitude-frequency response equations are further obtained using the multi-scale method, and the stability conditions of the steady solution are determined according to the Routh-Hurwitz criterion. The effects of the temperature difference at the plate center, the effective damping coefficient, the external excitation amplitude, and the ratio of the semi-major axis to the semi-minor axis on superharmonic and subharmonic resonances behaviors of the piezoelectric elliptic thin plate are analyzed. It is found that the coexistence region of multiple solutions and the resonant amplitude decrease for superharmonic resonance, as the temperature difference at the plate center increases. For subharmonic resonance, with the increase in the temperature difference at the plate center, the response amplitude increases, while the distance between two branches of the amplitude-frequency response curve has no obvious change. The softened/hardened nonlinear characteristics of the system are determined by the ratio of the semi-major axis to the semi-minor axis of the piezoelectric elliptic thin plate.

Coupled Vibration Behavior of Hot Rolling Mill Rolls under Multinonlinear Effects

A coupled vibration model of hot rolling mill rolls under multiple nonlinear effects is established by considering the nonlinear spring force produced by the hydraulic cylinder, the nonlinear friction between the work rolls, the dynamic variation of rolling force, and the effect of external excitation as well as according to the structural constraints of a four-high hot rolling mill in the vertical and horizontal directions. The amplitude-frequency response equation of rolling mill rolls is determined by using a multiple-scale approximation method. Furthermore, use of actual data for simulation indicates that the internal resonance is the main cause of coupling vibration of the rolling mill rolls. In addition, changes in the movement displacement of the hydraulic cylinder and the coupling parameters strongly affect the coupling system of the rolling mill rolls. Finally, the study of the dynamic bifurcation characteristics of the rolling mill rolls indicates that, with varying external excitation amplitude, the vibration of rolls alternates between periodic motion, period-doubling motion, and chaotic motion in both vertical and horizontal directions. This is one of the reasons for the appearance of periodic light and dark stripes on the strip surface. Furthermore, the range of the external excitation amplitude (F0) at which the rolling mill roll system vibrates violently, that is, 5.68e5 N < F0 < 5.84e5 N and F0 > 6.12e5 N, must be avoided. The research results can provide a theoretical reference for further exploration of the coupling vibration mechanism of hot rolling mills.

Stability analysis of floating raft system under multiexcitation condition

A floating raft system is subjected to multiple excitation sources with multiple frequencies for each excitation source. Considering the two characteristics of excitation source, the stability of floating raft system was analyzed. A vibration equation for the floating raft system under multiexcitation condition was established. A multiscale method was then used to solve the equation. The amplitude–frequency response equation and unstable region of solution are discussed. The results show that the vibration of raft frame fits the pattern of soft-spring vibration. This indicates that the excitation of main raft unit with a rigid connection compromises the stability of the system, whereas the excitation of unit with elastic connection increases stability.

Nonlinear vibration control of nano-beam based on capacitive acoustic sensors

The nonlinear vibration control of the Euler–Bernoulli beam is studied based on capacitive micro-mechanical acoustic sensors The graphene film has the characteristics of high sensitivity and high accuracy which can be applied to sense the vibration signal. Capacitive micro-mechanical acoustic sensors can be used to detect the acoustic signal of the vibration of the nano-beam. The nonlinear vibration control equation of nano-beam can be established with the displacement and velocity voltage feedback controller based on capacitive micro-mechanical acoustic sensors. The amplitude-frequency response equation of the primary resonance of nano-beam can be gotten by using the multiple scales method. The relationship between the nonlinear vibration of nano-beam and system parameters is investigated. The influencing factors of how to weak system nonlinearity and enhance system stability are analyzed. The static bifurcation behavior of the system is discussed. The numerical results show that the nonlinearity of vibration can be reduced and the stability of the system can be improved by selecting the appropriate control gains and appropriately reducing the amplitude of DC and AC excitation voltages.

Periodic response characteristics on a piecewise hysteresis nonlinear system

A piecewise hysteresis nonlinear dynamic model with nonlinear stiffness and damping is established for a system with alternating elastic constraints and hysteresis nonlinearity. First, the amplitude–frequency response equation of the system is solved by using the averaging method under periodic excitation. Two frequency regions with multiple solutions are then identified in the system, and the amplitude–frequency response characteristics under different system parameters are obtained. Thus, the influence law of different piecewise nonlinear factors on system stability is explored. Second, the bifurcation behavior of the system under different external disturbances is analyzed. Given the change of perturbation parameters, the system is found to be a complex dynamic system with alternating periodic, double periodic, and chaotic motions, and many other forms of movement. Moreover, lower buffer damping leads to the complex and unpredictable chaotic state of the system’s dynamic behavior.

Chaos detection of Duffing system with fractional-order derivative by Melnikov method.

The chaos detection of the Duffing system with the fractional-order derivative subjected to external harmonic excitation is investigated by the Melnikov method. In order to apply the Melnikov method to detect the chaos of the Duffing system with the fractional-order derivative, it is transformed into the first-order approximate equivalent integer-order system via the harmonic balance method, which has the same steady-state amplitude-frequency response equation with the original system. Also, the amplitude-frequency response of the Duffing system with the fractional-order derivative and its first-order approximate equivalent integer-order system are compared by the numerical solutions, and they are in good agreement. Then, the analytical chaos criterion of the Duffing system with the fractional-order derivative is obtained by the Melnikov function. The bifurcation and chaos of the Duffing system with the fractional-order derivative and an integer-order derivative are analyzed in detail, and the chaos criterion obtained by the Melnikov function is verified by using bifurcation analysis and phase portraits. The analysis results show that the Melnikov method is effective to detect the chaos in the Duffing system with the fractional-order derivative by transforming it into an equivalent integer-order system.

Resonances and chaos of electrostatically actuated arch micro/nanoresonators with time delay velocity feedback

Abstract Time-delay velocity feedback is introduced into micro- and nano-electro-mechanical systems (M/NEMS). First, the multi-scale method is used to obtain the amplitude-frequency response equation of the system under the influence of delay parameters. The numerical simulation method is used to verify the conclusion. Second, the Melnikov function method is extended to the time-delay system and is used to obtain the chaotic amplitude threshold of the system. Through the Hopf bifurcation condition of the time-delay feedback system, the range of control parameters suitable for the Melnikov function method is obtained, and then, the critical conditions for the chaotic motion of the system are given. Finally, the fourth-order Runge–Kutta method, point-mapping method and spectrum diagram are used to numerically simulate the evolution of the dynamic behaviour of the time-delay controlled system with time delay parameters. The results show that under the condition of a positive gain coefficient and a small amount of delay, the feedback of time delay can effectively restrain the amplitude and chaos of the system. However, under negative feedback, the delayed feedback induces chaotic motion of the system instead. This result shows that time-delay velocity feedback is a good method for the control system. Therefore, reasonable selection of control system parameters can improve the efficiency of vibration control, thus laying the foundation for further study on the vibration of the more complex micro-nano system.

Dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative

In this paper, the dynamic response of a piecewise linear single-degree-of-freedom oscillator with fractional-order derivative is studied. First, a mathematical model of the single-degree-of-freedom system is established, and the approximate steady-state solution associated with the amplitude–frequency equation is obtained based on the averaging method. Then, the amplitude–frequency response equations are used for stability analysis, and the stability condition is founded. To validate the correctness and precision, the approximate solutions determined by the analytical method are compared with the solutions based on the numerical integration method. It is found that the approximate solutions and the numerical solutions are in excellent agreement. Finally, the effects of system parameters, such as fractional-order coefficient, order, clearance, and piecewise stiffness, on the complex dynamical behaviors of the piecewise linear single-degree-of-freedom oscillator are studied. The results show that the system parameters not only influence resonance amplitude and resonance frequency but also affect the size of the unstable region.

Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field

The nonlinear combined resonance problem of a ferromagnetic circular plate in a transverse alternating magnetic field is investigated. On the basis of the deformation potential energy, the strain potential energy, and the kinetic energy of the circular plate, the Hamilton principle is used to induce the magnetoelastic coupling transverse vibration dynamical equation of the ferromagnetic circular plate. Based on the basic electromagnetic theory, the expressions of the magnet force and the Lorenz force of the circular plate are presented. A displacement function satisfying clamped-edge combined with the Galerkin method is used to derive the Duffing vibration differential equation of the circular plate. The amplitude-frequency response equations of the system under various combined resonance forms are obtained by means of the multi-scale method, and the stability of the steady-state solutions is analyzed according to the Lyapunov theory. Through examples, the amplitude-frequency characteristic curves with different parameters, the amplitude of resonance varying with magnetic field intensity and excitation force, and the time-course response diagram, phase diagram, Poincare diagram of the system vibration are plotted, respectively. The effects of different parameters on the amplitude and stability of the system are discussed. The results show that the electromagnetic parameters have a significant effect on the multi-valued attribute and stability of the resonance solutions, and the system may exhibit complex nonlinear dynamical behavior including multi-period and quasi-periodic motion.

Traverse Vibration of Axially Moving Laminated SMA Beam considering Random Perturbation

The nonlinear random vibration of axially moving shape memory alloy (SMA) laminated beam under transverse loads is investigated. Considering the effects of axial movement and random perturbation, the dynamic equation of the SMA laminated beam is established by means of physical equation, force balance conditions, deformation compatibility equation, and the constitutive relation by polynomial function. The transverse vibration differential equation of axially moving SMA laminated beam with two simply‐supported edges is obtained through the Galerkin method. The amplitude‐frequency response equation under primary resonance and the probability density function under random perturbation are derived by using the averaging method and stochastic averaging method, respectively, and the theoretical results are numerically validated. In addition, under the combination of the harmonic excitation and random perturbation, the effects of axial movement velocity and random perturbation intensity on system steady‐state response are analyzed.

一类含五次非线性恢复力的Duffing系统共振与分岔特性分析

Some Duffing systems with external excitation and quintic nonlinear restoring forces were considered, the amplitude-frequency response equation for the system was obtained with the multi-scale method, and the amplitude-frequency characteristic curves and their changing rules under different parameter changes were given. At the same time, the singularity theory was applied to get the transition sets and the corresponding topological structures of the system in 3 cases. Second, the fixed point of the system was determined, and the Hamiltonian function was used to get the heteroclinic orbit of the system, so the threshold of chaos in the Smale horseshoe sense was obtained with the Melnikov method. Then, the dynamic bifurcation and chaotic behavior of the system under external excitation and quintic nonlinear coefficients were given through numerical simulation. It is found that there are nonlinear phenomena such as periodic motion, period doubling motion, quasi periodic motion and chaos. The correctness of the theory was verified with nonlinear methods such as the Lyapunov exponent, the phase diagram and the Poincaré sections. The work provides a theoretical reference for further understanding of the nonlinear characteristics of Duffing systems and their evolution laws.

Vortex-Induced Vibration Response Bifurcation Analysis of Top-Tensioned Riser Based on the Model of Variable Lift Coefficient

Top-tensioned riser (TTR) is one of the most frequently used pieces of equipment in offshore petroleum engineering. At present, the research of its vortex-induced vibration (VIV) is mainly focused on numerical simulation with few analytical approaches. In this paper, the nonlinear dynamic equation of VIV about TTR is developed by introducing the third-order variable lift coefficient hydrodynamic model. The amplitude-frequency response equation under 1:1 primary resonance excitation is deduced based on the single mode discrete results. The nonlinear dynamic responses characteristics of the TTR vibration under the excitation of vortex-induced resonance are discussed. Bifurcation theory is used to study the singularity of the analytical solution of the riser system. The hysteresis and solitary solutions from the results of singularity analysis are found. Such results can provide guidance for the design and optimization of riser structural parameters.

Primary resonance of fractional-order Duffing–van der Pol oscillator by harmonic balance method** Project supported by the National Natural Science Foundation of China (Grant Nos. 11872254 and 11672191).

The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.