Nonlinear Stiffness Coefficients Research Articles

Bifurcation Characteristic Research on the Load Vertical Vibration of a Hydraulic Automatic Gauge Control System

As the core control system of a rolling mill, the hydraulic automatic gauge control (HAGC) system is key to ensuring a rolling process with high speed, high precision and high reliability. However, a HAGC system is typically a mechanical-electric-hydraulic coupling system with nonlinear characteristics. The vertical vibration of the load easily occurs during the working process, which seriously affects the stability of the system and the causes are difficult to determine. In this work, the theory and method of nonlinear dynamics were employed. The load vertical vibration model of the HAGC system was established. Then, the multi-scale method was utilized to solve the obtained model, and the singularity theory was further applied to derive the transition set. Moreover, the research object of this article focused on some nonlinear factors such as excitation force, elastic force and damping force. The effects of the above feature parameters on bifurcation behavior were emphatically explored. The bifurcation characteristic of the load vertical vibration of the HAGC system was revealed. The research results indicate that the bifurcation curves in each sub-region, divided by the transition set, possess their own topological structure. The changes of the feature parameters, such as the nonlinear stiffness coefficient, liquid column height, nonlinear damping coefficient, and external excitation have an influence on the vibration amplitude of the HAGC system. By reasonably adjusting the nonlinear stiffness coefficient to effectively avoid the resonance region, the stability of the system will be facilitated. Furthermore, this is conducive to the system’s stability as it properly controls the size of the liquid column height of the hydraulic cylinder. The appropriate nonlinear damping coefficient can decrease the unstable area, which is beneficial to the stability of the system. However, large external excitation is not conducive to the stability of the system.

Superharmonic Resonance of Fractional-Order Mathieu–Duffing Oscillator

The superharmonic resonance of fractional-order Mathieu–Duffing oscillator subjected to external harmonic excitation is investigated. Based on the Krylov–Bogolubov–Mitropolsky (KBM) asymptotic method, the approximate analytical solution for the third superharmonic resonance under parametric-forced joint resonance is obtained, where the unified expressions of the fractional-order term with fractional order from 0 to 2 are gained. The amplitude–frequency equation for steady-state solution and corresponding stability condition are also presented. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional-order term, excitation amplitudes, and nonlinear stiffness coefficient on the superharmonic resonance response of the system are analyzed in detail. The results show that the KBM method is effective to analyze dynamic response in a fractional-order Mathieu–Duffing system.

On vibration behavior and motion bifurcation of a nonlinear asymmetric rotating shaft

This article investigates the nonlinear vibrations of asymmetric vertically supported Jeffcott rotor system. Asymmetry in both linear and nonlinear stiffness coefficients of the rotating shaft is considered. The disk eccentricity and its orientation angle are included in the system model. Asymptotic analysis is sought to obtain an analytical approximate solution for the considered system model in the primary resonance case. Bifurcation diagrams for the different system parameters are obtained to explore the system steady-state lateral vibrations. The main acquired results revealed that (1) the symmetric system can oscillate by one of three stable forward whirling amplitudes at the same rotational speed depending on the initial position of the rotating disk. (2) Asymmetry in the linear stiffness coefficient does not affect the symmetry of the whirling motion, but it may change the system natural frequency. (3) Asymmetry in the nonlinear stiffness coefficient is responsible for both asymmetrical and backward whirling motions. All obtained analytical results have been verified via solving the system original equations numerically, where the analytical and numerical results are in excellent agreement.

Parameter-splitting perturbation method for the improved solutions to strongly nonlinear systems

A procedure named parameter-splitting perturbation method for improving the perturbation solutions to the forced vibrations of strongly nonlinear oscillators is proposed. The idea of the proposed procedure is presented in general first. After that, it is applied to optimize the solutions obtained by the multiple-scales method which is one of well-known perturbation methods. The harmonically forced Duffing oscillator, the harmonically forced oscillator with both nonlinear restoring force and nonlinear inertial force, the harmonically forced purely nonlinear oscillator and harmonically forced two-degree-of-freedom system with cubic nonlinearity are analyzed in various cases to show the advantages of the proposed method. The ratio of nonlinear stiffness coefficient to linear stiffness coefficient is chosen to be larger than one to highlight the validity of the propose method when dealing with strongly nonlinear oscillators. The validity of the proposed procedure is examined by comparing the frequency–response curves obtained by the proposed method, conventional multiple-scales method and numerical continuation method. Moreover, the errors corresponding to the results obtained by multiple-scales method are compared with those obtained by the proposed method to examine the performance of the proposed method. The results show that the proposed method can give much improved solutions in comparison with those obtained by multiple-scales method.

Mechanics of Fluid-Conveying Microtubes: Coupled Buckling and Post-Buckling

This paper investigates the coupled mechanics of a fluid-conveying microtube embedded inside an elastic medium and subject to a pretension. The fluid-structure interaction model of the microsystem is developed based on Lagrange’s equations for the open system of a clamped-clamped microtube. A continuation model is used to examine the nonlinear mechanics of this microsystem prior to and beyond losing stability; the growth and the response in the supercritical regime is analysed. It is shown that the microtube stays stable prior to losing stability at the so-called critical flow velocity; beyond that point, the amplitude of the buckled microsystem grows with the velocity of the flowing fluid. The effects of different system parameters such as the linear and nonlinear stiffness coefficients of the elastic medium as well as the length-scale parameter and the slenderness ratio of the microtube on the critical speeds and the post-buckling behaviour are analysed.

Stochastic bifurcations in a nonlinear tri-stable energy harvester under colored noise

In this paper, the stochastic bifurcations and the performance analysis of a strongly nonlinear tri-stable energy harvesting system with colored noise are investigated. Using the stochastic averaging method, the averaged Fokker–Plank–Kolmogorov equation and the stationary probability density (SPD) of the amplitude are obtained, respectively. Meanwhile, the Monte Carlo simulations are performed to verify the effectiveness of the theoretical results. D-bifurcation is studied through the largest Lyapunov exponent calculations, which implies the system undergoes D-bifurcation twice with increasing the nonlinear stiffness coefficients. The effects of the nonlinear stiffness coefficients, noise intensity and correlation time on P-bifurcation are discussed by the qualitative changes of the SPD. Moreover, the relationship between D- and P-bifurcation is explored. If the strength of stochastic jump has obvious gap with respect to the two statuses before and after the occurrence of P-bifurcation, the D-bifurcation will happen, too. Finally, the performance and the capability of harvesting energy from ambient random excitation are analyzed.

Contact Model of Revolute Joint with Clearance Based on Fractal Theory

The contact stiffness of a mechanical bonding surface is an important parameter in determining the normal and radial contact force. To improve the calculation accuracy of the contact force model, the surface roughness of the bonding surface and the energy loss that necessarily occurs during the impact process should be considered comprehensively. To study the normal contact force of a revolute joint with clearance more accurately in the case of dry friction, a nonlinear stiffness coefficient model considering the surface roughness was established based on fractal theory, which considers the elastic, elastic-plastic, and plastic deformations of the asperities of the contact surface during the contact process. On this basis, a modified nonlinear spring damping model was established based on the Lankarani–Nikravesh contact force model. The laws influencing the surface roughness, recovery coefficient, initial velocity, and clearance size on the impact force were revealed, and were compared with the Lankarani–Nikravesh model and a hybrid model using MATLAB. The maximum impact force was obtained using a modified contact force model under different initial velocities, different clearances, and different degrees of surface roughness, and the calculated results were then compared with the experiment results. This study indicates that the modified model can be used more widely than other models, and is suitable for both large and small clearances. In particular, the modified model is more accurate when calculating the contact force of a revolute joint with a small clearance.

Enhancement of tristable energy harvesting using stochastic resonance

In this paper, stochastic resonance (SR) is exploited within a tristable electromechanical coupled system to enhance energy harvesting. Assuming that the effective capacitance of the piezoelectric element can be neglected, the coupled electromechanical equations are converted to a second-order stochastic differential equation. Using the adiabatic approximation and the linear response theory, the spectral amplification of the equivalent tristable mechanical system is obtained to characterize the SR. It is observed that the spectral amplification exhibits a maximum as a function of the nonlinear stiffness coefficients and noise intensity. That is, both parameter-optimized SR and traditional SR are found in this system, which can be used to amplify the amplitude of the system response and enhance the performance of nonlinear vibration energy harvesting. Moreover, the efficiency in power conversion and the RMS voltage obtained under the SR effect are much higher than those harvested under ambient random vibration alone. The theoretical results are well verified through the numerical simulations of the original electromechanical coupled system.

Stochastic resonance in an underdamped triple-well potential system

In this paper, stochastic resonance (SR) in an underdamped triple-well potential system driven by Gaussian white noise and a parametric harmonic excitation is investigated. The analytical expressions of the output signal-to-noise ratio (SNR), together with the mean first-passage times (MFPTs), are derived for the triple-well potential system involving damping in adiabatic limit. The effects of noise intensity, damping coefficient and triple-well potential on MFPTs and SNR are analyzed. The results suggest the existence of two critical damping values, giving rise to the onset and the disappearance of SR, respectively. Since the system is unstable in weak damping regime, emergence of SR is prohibited. Under the weak noise level, SNR exhibits a prominent resonance-like behavior at the optimal value of damping coefficient. Moreover, the two nonlinear stiffness coefficients of restoring force play an opposite role in the enhancement of SR. Thus, the SR effect significantly depends on the change of the two-side potential wells. Particularly, the appropriate choice of triple-well potential function and damping coefficient can improve the response of the system to an external periodic excitation according to the damping-induced resonance effect. Finally, the numerical results confirm the effectiveness of the theoretical analyses.

Nonlinear magnetic vibration absorber for passive control of a multi–storey structure

A nonlinear magnetic vibration absorber is presented and used to control vibration of a three–storey structure. A distinctive feature of the absorber concerns its versatility for tuning the linear and nonlinear stiffness coefficients, depending on simple geometric design parameters such as the distance between fixed magnets and the moving one. In particular, the absorber can be set either as a nonlinear tuned vibration absorber, a nonlinear energy sink, or a bistable tuned vibration absorber, according to whether the linear stiffness term is positive, vanishing, or negative. The response of the primary structure and the vibration mitigation are investigated in the cases of impulsive shock, free vibration with imposed initial displacement, and single frequency excitation. Significant reductions of the primary structure vibrations are obtained for the three cases investigated, showing the ability of using a vibration absorber only relying on magnetic forces for passive control. The detailed comparisons of the absorbers performance show that, in this case study, no general guidelines can be easily deduced for selecting one of the three tunings for a nonlinear absorber. Depending on the excitation, the vibratory levels, and the frequency content of the excitation, the three configurations show advantages and drawbacks that are discussed.

Analysis of dynamic characteristics of nonlinear vehicle suspension system with five degrees of freedom

With the rapid development of the automobile industry, the performance requirements of the automobile has been becoming higher and higher which shows the great significance of the study on vehicle suspension characteristics. Based on the structural characteristics of the vehicle, a five degree of freedom vehicle model is established. The dynamic equation of the vehicle model is listed by using the vibration law of the multi degree of freedom damping system. The simulation model of five degree of freedom linear and nonlinear system of automobile is established by using MATLAB/SIMULINK. The simulation analysis of ride comfort is carried out based on simulated nonlinear stiffness: Compared with the results of linear and nonlinear parameters, the nonlinear stiffness coefficient alpha is monotonically decreasing in -0.4∼0, which can improve the ride comfort of the vehicle. The nonlinear stiffness mainly affects the seat acceleration, the centroid acceleration and the pitching acceleration. The nonlinear system is optimized by the obtained law concluded from simulation. The results show that the performance of the suspension system can be greatly improved, the mean square root value of the centroid acceleration is reduced by 38.67 % and the mean square root value of the pitching acceleration is reduced by 44.54 %, the mean square root value of the seat acceleration is reduced by 11.31 %, and the improvement of vehicle ride comfort is very obvious.

Independent nonlinearity tuning of planar spring via geometrical design for wideband vibration energy harvesting

This paper reports a novel nonlinearity tuning method through sophisticated geometrical design for wideband energy harvesting application. By carefully controlling the thickness to length ratio of the planar spring beams, the nonlinear stiffness coefficient can be intentionally and precisely adjusted, while the linear stiffness coefficient can be kept unchanged. Thanks to the different effects on the linear and nonlinear parts, both the resonant frequency of the linear part and the bandwidth range of nonlinear vibration can be designed separately. As a proof of concept, MEMS-based electromagnetic vibrational energy harvesters employing the independent nonlinearity tuning method have been designed and fabricated. Both simulation and experiment were conducted to validate our method. At the acceleration of 0.5g, the micro-fabricated device is able to tune the operating bandwidth of the output voltage from 37Hz to as wide as 75Hz. This design can easily tune the length and thickness of the spring beams without changing the volume of the whole energy harvester device, which is compatible with MEMS batch fabrication process.

Nonlinear vibrations of large structures with uncertain parameters

The effects of uncertainties on the nonlinear dynamics of complex structures remain poorly mastered and most methods deal with the linear case. This article deals with a model of a large and complex structure with uncertain parameters for the nonlinear dynamic case, and the reduction in the model discretized by the finite element method is obtained by reducing the degrees of freedom in the numerical model. This is achieved by the development of the unknown displacement vector on the basis of the eigenmodes; a particular attention is paid to the calculation of the nonlinear stiffness coefficients of the model. The method combines the stochastic finite element methods with a modal reduction class based on sub-structuring the component mode synthesis method. The reference method is the Monte Carlo simulation which consists in making several simulations for different values of the uncertain parameters. The simulation of complex and nonlinear structures is costly in terms of memory and computation time. To solve this problem, the perturbation method combined with the component mode synthesis reduction method significantly reduces the computational cost by preserving the physical content of the original structure. The numerical integration by the Newmark schema is used; the first statistical moments (mean and variance) of the nonlinear dynamic response are computed. Numerical simulations illustrate the accuracy and effectiveness of the proposed methodology.

Study on the nonlinear transient response for the non-contact mechanical face seal

Considering the tilt of the seal ring, the transient vibration response analysis model of the non-contact mechanical seal is presented. The model is consisted of the transient Reynolds equation, the equation of motion and the equation for solving the high order nonlinear dynamic coefficients of seal. The relative error of the high order nonlinear film force to the linear film force is also obtained. With Euler method, the characteristic parameters of the transient vibration response are obtained, which include the axial vibration displacements and the angle-swing of the static ring. The 14 nonlinear force and 14 nonlinear overturning moment dynamic coefficients for the non-contact mechanical seal are calculated. The results show that the influence of the damping effects of the sealed fluid between the seal gap on the axial vibration displacements and the angle-swing is linear. The film thickness distribution changes with the axial vibration of seal, which will lead to static ring swing, and the swing also can cause the axial vibration of the seal. With the increase of the nonlinear order, the relative error of the nonlinear film force decreases. All of the nonlinear film forces, the non-linear stiffness coefficient and damping coefficient decrease with the seal film thickness increases.

Analysis of dynamic characteristics and stability prediction of gas bearings

PurposeThe prediction model to estimate the stability of a rotor-bearing system is established, which can predict the stability of gas bearings by applying Routh–Hurwitz stability criterion. This paper aims to provide the theoretical foundation for controlling actively the bearing running stiffness and damping and stemming the instability of a gas film.Design/methodology/approachThe nonlinear dynamic lubrication analysis mathematical model of spherical hybrid gas bearings is established. Perturbation control equation is derived by the partial derivative method. The finite difference method is used to discrete the perturbation control equation in generalized coordinate system, and the difference expression of perturbation pressure is derived. The relational expression which involves the relationship between the dynamic characteristic coefficients of HSGHGB systems and perturbation pressure is deduced. So, the transient perturbation pressure distribution of a three-dimensional micro gas film, nonlinear gas film force, dynamic stiffness and dynamic damping coefficients of bearings are numerically computed using VC++6.0 programs.FindingsThe results show that the influence of supply pressure, speed and eccentricity on the dynamic characteristics of bearings is significant.Originality/valueThe influence law of supply pressure, speed and eccentricity ratio on the dynamic stiffness and damping coefficients of HSGHGB systems is researched. The prediction model to estimate the stability of rotor-bearing system is established, which can predict the stability of gas bearings by applying the Routh–Hurwitz stability criterion.

Nonlinear dynamic analysis of a parametrically excited vehicle–bridge interaction system

In this paper, a nonlinear supported Euler–Bernoulli beam under harmonic excitation coupled to a 2 degree of freedom vehicle model with cubic nonlinear stiffness and damping is investigated. The equations of motion are derived by Newton’s law and discretized into a set of coupled second-order nonlinear differential equations via Galerkin’s method with cubic nonlinear terms. Based on the created model, numerical simulations have been conducted using the Runge–Kutta integration method to perform a parametric study on influences of the nonlinear support stiffness coefficient, mass ratio, excitation amplitude and position relation for the vehicle–bridge interaction (VBI) system by using bifurcation diagram and 3-D frequency spectrum. The results indicate that depending on different parameters, a diverse range of periodic motion, quasi-periodic response, chaotic behavior and jump discontinuous phenomenon are observed. And the chaotic regions are scattered between a number of periodic/quasi-periodic motions. The study may contribute to a further understanding of the dynamic characteristics and present useful information to dynamic design and vibration control for the VBI system.

Optimization under uncertainty of parallel nonlinear energy sinks

Nonlinear Energy Sinks (NESs) are a promising technique for passively reducing the amplitude of vibrations. Through nonlinear stiffness properties, a NES is able to passively and irreversibly absorb energy. Unlike the traditional Tuned Mass Damper (TMD), NESs do not require a specific tuning and absorb energy over a wider range of frequencies. Nevertheless, they are still only efficient over a limited range of excitations. In order to mitigate this limitation and maximize the efficiency range, this work investigates the optimization of multiple NESs configured in parallel. It is well known that the efficiency of a NES is extremely sensitive to small perturbations in loading conditions or design parameters. In fact, the efficiency of a NES has been shown to be nearly discontinuous in the neighborhood of its activation threshold. For this reason, uncertainties must be taken into account in the design optimization of NESs. In addition, the discontinuities require a specific treatment during the optimization process. In this work, the objective of the optimization is to maximize the expected value of the efficiency of NESs in parallel. The optimization algorithm is able to tackle design variables with uncertainty (e.g., nonlinear stiffness coefficients) as well as aleatory variables such as the initial velocity of the main system. The optimal design of several parallel NES configurations for maximum mean efficiency is investigated. Specifically, NES nonlinear stiffness properties, considered random design variables, are optimized for cases with 1, 2, 3, 4, 5, and 10 NESs in parallel. The distributions of efficiency for the optimal parallel configurations are compared to distributions of efficiencies of non-optimized NESs. It is observed that the optimization enables a sharp increase in the mean value of efficiency while reducing the corresponding variance, thus leading to more robust NES designs.

On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force

This paper deals with the vibration analysis of a horizontally supported Jeffcott rotor system. Both nonlinear restoring force and the rotor weight are considered in the system modeling. The model shows a small difference between the natural frequencies of the vertical and horizontal mode. The multiple scales perturbation technique is utilized to obtain a second-order approximate solution at the simultaneous resonance case. The bifurcation analyses are conducted. The stability of the obtained solution is investigated by applying Lyapunov first method. The influences of all the parameters on the system behavior are explored. The Effect of both the negative and positive values of the nonlinear stiffness coefficient is studied. At the large rotor eccentricity, the analysis revealed the following: (1) the existence of three different stable solutions in an interval of the rotational speed. (2) The disk exposed to two consecutive jumps if its speed crossed the resonant speed. (3) For a soft spring, localized and nonlocalized oscillation in both the horizontal and vertical mode occurs. (4) For a hard spring, nonlocalized oscillation occurs in the two directions in addition to the localized motion in the vertical direction only (5) The system is very sensitive to initial conditions. Then, numerical simulations are performed to confirm the accuracy of the approximate results. It is found that the predictions from the analytical solutions are in a good agreement with the numerical simulations. Finally, a comparison with previously published work is included.

Dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation

In this paper, the computation schemes for periodic solutions of the forced fractional-order Mathieu-Duffing equation are derived based on incremental harmonic balance (IHB) method. The general forms of periodic solutions are founded by the IHB method, which could be useful to obtain the periodic solutions with higher precision. The comparisons of the approximate analytical solutions by the IHB method and numerical integration are fulfilled, and the results certify the correctness and higher precision of the solutions by the IHB method. The dynamical analysis of strongly nonlinear fractional-order Mathieu-Duffing equation is investigated by the IHB method. Then, the effects of the excitation frequency, fractional order, fractional coefficient, and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed. At last, the detailed results are summarized and the conclusions are made, which present some useful information to analyze and/or control the dynamical response of this kind of system.

Dynamical analysis of Duffing oscillator with fractional-order feedback with time delay

With increasingly strict requirements for control speed and system performance, the unavoidable time delay becomes a serious problem. Fractional-order feedback is constantly adopted in control engineering due to its advantages, such as robustness, strong de-noising ability and better control performance. In this paper, the dynamical characteristics of an autonomous Duffing oscillator under fractional-order feedback coupling with time delay are investigated. At first, the first-order approximate analytical solution is obtained by the averaging method. The equivalent stiffness and equivalent damping coefficients are defined by the feedback coefficient, fractional order and time delay. It is found that the fractional-order feedback coupling with time delay has the functions of both delayed velocity feedback and delayed displacement feedback simultaneously. Then, the comparison between the analytical solution and the numerical one verifies the correctness and satisfactory precision of the approximately analytical solution under three parameter conditions respectively. The effects of the feedback coefficient, fractional order and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed, including the locations of bifurcation points, the stabilities of the periodic solutions, the existence ranges of the periodic solutions, the stability of zero solution and the stability switch times. It is found that the increase of fractional order could make the delay-amplitude curves of periodic solutions shift rightwards, but the stabilities of the periodic solutions and the stability switch times of zero solution cannot be changed. The decrease of the feedback coefficient makes the amplitudes and ranges of the periodic solutions become larger, and induces the stability switch times of zero solution to decrease, but the stabilities of the periodic solutions keep unchanged. The sign of the nonlinear stiffness coefficient determines the stabilities and the bending directions of delay-amplitude curves of periodic solutions, but the bifurcation points, the stability of zero solution and the stability switch times are not changed. It could be concluded that the primary system parameters have important influences on the dynamical behavior of Duffing oscillator, and the results are very helpful to design, analyze or control this kind of system. The analysis procedure and conclusions could provide a reference for the study on the similar fractional-order dynamic systems with time delays.