Effect Of Nonlinear Stiffness Research Articles

Forced vibration analysis of a beam-plate system coupled through multiple nonlinear single-freedom-degree systems

A beam-plate coupling system is widely used in the construction of different frameworks across industries according to engineering application requirements. In this paper, a nonlinear single-degree-of-freedom structure is introduced as a coupling element between the beam and the plate. Therefore, a coupling system connected through a plurality of the nonlinear single-degree-of-freedom structures is proposed with exposure of its operational mechanism and the dynamic behavior analysis as required. The nonlinear coupled governing equations of the beam-plate system with series nonlinear energy sinks are established according to the Hamilton principle and solvable by the Galerkin truncation method. Meanwhile, the nonlinear characteristics of transverse vibration of the coupling system are studied for discussing the effect of nonlinear stiffness on a low-frequency response of the beam-plate system. Further, the transverse vibration of the system may vary with changes in the nonlinear stiffness, the external viscous damping, and the motion mass of the nonlinear single-degree-of-freedom structures. Accordingly, the variations along with these factors are investigated. The results indicate that a reasonable nonlinear single-degree-of-freedom structure can have a significant influence on the transverse vibration characteristics of the beam-plate system.

Effect of nonlinear stiffness on single link flexible joint robotic manipulator and its control

This study aims to investigate the significant dynamics of a Single Link Flexible Joint (SLFJ) Robotic Manipulator and implement chaotic control using the Adaptive Sliding Mode Control (ASMC) algorithm. The mathematical model of the system incorporates stiffness nonlinearity by introducing the function (x-x3) Various dynamical tools such as stability analysis of equilibrium points, sensitivity to initial conditions, orbit diagrams for parameter variations, Lyapunov spectrum, and frequency spectrum analysis are employed. The ASMC algorithm is utilized to suppress the chaotic behavior of the system. The equilibrium points are determined for critical parameter values, with the Jacobian matrix and corresponding eigenvalues exhibiting conjugate complex with negative real parts, indicating stability resembling a spiral saddle. Phase portraits illustrate the location and range of the chaotic attractor. Investigation of parameter ′a′ using orbit diagrams and Lyapunov spectrum reveals the transition from periodic to chaotic oscillations. The frequency spectrum confirms the chaotic nature of the system. Parameter estimation, state trajectories, and sliding surface analysis demonstrate that the ASMC scheme can effectively control the system in less than 0.3 s. Nonlinear stiffness has not been previously considered in SLFJ Robotic Manipulators, highlighting a novel aspect of this study. The results unveil new chaotic ranges for parameter variation previously undocumented. Frequency spectrum analysis is employed to distinguish between noise and chaotic oscillations, addressing a significant challenge in real-time systems. The ASMC algorithm is specifically designed to control chaotic oscillations, demonstrating its effectiveness in this context.

Study on vortex-induced multimodal coupled vibration of arch bridge suspenders

The objective of this paper was to investigate the 2-DOF suspender response under the condition of vortex-induced force and reveal the multimodal coupling mechanism. Firstly, an elastically mounted segmental model of suspender was established. Wind tunnel experiments were conducted to obtain the time history displacement signals in cross-flow and in-line directions, the empirical parameters were obtained by identify those signals. The nonlinear partial differential motion equation of suspender with two degrees of freedom was established based on Hamilton's principle, which considered the dynamic strain, bending stiffness and vortex-induced force of suspender. The Galerkin method was used to discretize the nonlinear partial differential equations into nonlinear motion governing equations, which considered the first three order modes. A numerical method was used to solve the motion governing equation, the time history displacement curve, phase diagram, amplitude spectrum and Poincare map of the multi-modal responses in the cross-flow and in-line directions were obtained. With the weak coupling effect of nonlinear stiffness and vortex-induced force, the cross-flow and in-line responses demonstrate period doubling bifurcation, quasi-periodic vibration, period doubling vibration, crises and transient chaotic phenomena. However, with the strong coupling effect, the crises and transient chaos in response amplitude disappear. More importantly, planar/non-planar multimodal internal resonance were found in this work. With the weak multimodal coupling effect in suspender response, the low-order modal response is dominant; as the multimodal coupling effect increases, the high-order modal response becomes predominant. In addition, the system energy not only transfers from high-order modal response to low-order modal response, but also the system energy transfer from low-order modal response to high-order modal response. These conclusions are useful for the design and practical engineering of suspenders, which have high theoretical research value in understanding the mechanism of multimodal coupling effects on suspender response.

Coupling Effect of Nonlinear Stiffness of Tape Spring Hinges and Flexible Deformation of Panels during Orbit Maneuvers

Many solar panels for spacecrafts are deployed by Tape Spring Hinges (TSHs) which have changeable stiffness. The stiffness of TSH is small when panels are folded, and it becomes large quickly in its deployed status. Since the solar panel is a thin sheet, flexible deformation is easily generated by orbit maneuvers. The coupling effect between the nonlinear TSHs and the flexible panels generates obvious vibration which affects the operational stability of the satellite. To investigate this coupling effect, non-deformable, linear deformable and nonlinear deformable panels were modelled by rigid body, modal order reduction method (MORM) and finite element method (FEM), respectively. The driving torque of TSH was described as a function of the rotation angle and angular velocity. The nonlinear properties of the TSH were reflected by one angle-stiffness spline multiplied by one stiffness coefficient. Dynamic responses of a satellite in deployment and orbit steering were analyzed by numerical simulations. Analysis results indicate the local deformation of panels keeps the stiffness of the TSH within a large range which accelerates the orbit maneuvers. However, much vibration is generated by the coupling effect if the luck-up status is broken up. The coupling effect affects the sequence of deployment, overshoot phenomenon and acceleration magnitude of the panels. Although the MORM is more efficient than FEM in computation, we propose FEM is better suited in the design of TSH and in studying the precise control of spacecraft with flexible solar panels and TSHs.

Nonlinear Dynamic Analysis of Rotor-Bearing System with Cubic Nonlinearity

Nonlinear dynamic characteristics of a rotor-bearing system with cubic nonlinearity are investigated. The comprehensive effects of the unbalanced excitation, the internal clearance, the nonlinear Hertzian contact force, the varying compliance vibration, and the nonlinear stiffness of support material are considered. The expression with the linear and the cubic nonlinear terms is adopted to characterize the synthetical nonlinearity of the rotor-bearing system. The effects of nonlinear stiffness, rotating speed, and mass eccentricity on the dynamic behaviors of the system are studied using the rotor trajectory diagrams, bifurcation diagrams, and Poincaré map. The complicated dynamic behaviors and types of routes to chaos are found, including the periodic doubling bifurcation, sudden transition, and quasiperiodic from periodic motion to chaos. The research results show that the system has complex nonlinear dynamic behaviors such as multiple period, paroxysmal bifurcation, inverse bifurcation, jumping phenomena, and chaos; the nonlinear characteristics of the system are significantly enhanced with the increase of the nonlinear stiffness, and the material with lower nonlinear stiffness is more conducive to the stable operation of the system. The research will contribute to a comprehensive understanding of the nonlinear dynamics of the rotor-bearing system.

Reduced order modelling of the non-linear stiffness in MEMS resonators

We propose a simplified numerical technique to quantify the effects of non-linear stiffness in MEMS resonators and we validate our approach on a clamped–clamped beam, a softening Disk Ring Gyroscope (DRG) and a shallow arch showing internal resonance. We generate a Reduced Order Model (ROM) which is integrated with either a direct integration approach or a continuation technique with arc length control. Finally we compare and validate the results with a full FEM model.

The Influence of Nonlinear Stiffness of Novel Flexible Road Wheel on Ride Comfort of Tracked Vehicle Traversing Random Uneven Road

The linear wheel model is the most widely used in ride comfort analysis of tracked vehicles. However, as the speed of the vehicle increases and the application of new materials, and the road wheel becomes lighter and softer, its nonlinear characteristics become more and more obvious and can't be ignored in vehicle dynamics simulation. This paper aims to study the effect of nonlinear stiffness of a novel flexible road wheel (FRW) with unique suspension bearing structure on the ride comfort of a tracked vehicle traversing random uneven road. The linear and nonlinear models of FRW were established by fitting the load-deflection data obtained from the static loading test of the physical prototype. The established linear and nonlinear models of FRW were added to a half-vehicle model of a tracked vehicle which has been proved by published test results. The ride comfort of the half-vehicle model of the tracked vehicle with linear and nonlinear models of FRW on random uneven road surface was studied in detail. The study results show that, compared with the linear wheel model, the nonlinear model has a tremendous influence on the dynamic response of the tracked vehicle, effectively suppressing the vibration, especially for the high frequency excitation. In addition, the nonlinear factor has a greater impact on the dynamic performance of the wheel than on the suspension and the body. The research results enrich the study of nonlinear dynamics of tracked vehicles, and provide reference for nonlinear modeling of other pneumatic tires and non-inflatable wheels.

Dynamics of Nonlinear Primary Oscillator with Nonlinear Energy Sink under Harmonic Excitation: Effects of Nonlinear Stiffness

The dynamics of a system consisting of a nonlinear primary oscillator, subjected to a harmonic external force, and a nonlinear energy sink (NES) are investigated. The analytical solutions for the steady-state responses are obtained by the complexification-averaging method and the analytical model is confirmed by numerical simulations. The results indicate that the introduction of the NES can effectively suppress the vibrations of the primary oscillator. However, as the excitation amplitude increased, the NES may lose its efficiency within certain frequency range due to the appearance of the high response branches. Following the results analysis, it is concluded that this failure can be eliminated by reducing the nonlinear stiffness of the NES properly. The effects of nonlinear stiffness of the primary oscillator on the corresponding responses are also studied. The increase in this nonlinear stiffness can reduce the response amplitude and alter the frequency band where the high branches exist.

Nonlinear stability analysis of whirl flutter in a rotor-nacelle system

Whirl flutter is an aeroelastic instability that affects propellers/rotors and the surrounding airframe structure on which they are mounted. Whirl flutter analysis gets progressively more complicated with the addition of nonlinear effects. This paper investigates the impact of nonlinear pylon stiffness on the whirl flutter stability of a basic rotor-nacelle model, compared to a baseline linear stiffness version. The use of suitable nonlinear analysis techniques to address such a nonlinear model is also demonstrated. Three types of nonlinearity were investigated in this paper: cubic softening, cubic hardening and a combined cubic softening—quintic hardening case. The investigation was conducted through a combination of eigenvalue and bifurcation analyses, supplemented by time simulations, in order to fully capture the effects of nonlinear stiffness on the dynamic behaviour of the rotor-nacelle system. The results illustrate the coexistence of stable and unstable limit cycles and equilibria for a range of parameter values in the nonlinear cases, which are not found in the linear baseline model. These branches are connected by a number of different bifurcation types: fold, pitchfork, Hopf, homoclinic and heteroclinic. The results also demonstrate the importance of nonlinear whirl flutter models and analysis methods. Of particular interest are cases where the dynamics of the nacelle are unstable despite linear analysis predicting stable behaviour. A more complete stability envelope for the combined model was generated to take account of this phenomenon.

Use of the dynamic stiffness method to interpret experimental data from a nonlinear system

The interpretation of experimental data from nonlinear structures is challenging, primarily because of dependency on types and levels of excitation, and coupling issues with test equipment. In this paper, the use of the dynamic stiffness method, which is commonly used in the analysis of linear systems, is used to interpret the data from a vibration test of a controllable compressed beam structure coupled to a test shaker. For a single mode of the system, this method facilitates the separation of mass, stiffness and damping effects, including nonlinear stiffness effects. It also allows the separation of the dynamics of the shaker from the structure under test. The approach needs to be used with care, and is only suitable if the nonlinear system has a response that is predominantly at the excitation frequency. For the structure under test, the raw experimental data revealed little about the underlying causes of the dynamic behaviour. However, the dynamic stiffness approach allowed the effects due to the nonlinear stiffness to be easily determined.

Natural frequencies of a super-critical transporting Timoshenko beam

Abstract Super-critical transporting continua are often discussed in the context of an Euler beam model. Here Timoshenko beam theory is applied to study free vibration of high-speed transporting continua. Therefore, effects of rotary inertia and shear deformation on transverse vibration of super-critical transporting beams are discovered for the first time. The Galerkin method is applied to solve natural frequencies with the simply supported boundary conditions. Meanwhile, the weighted residual method (WRM) is employed to deal with the fixed ends. The natural frequencies of super-critical continua are verified by using the discrete Fourier transform (DFT). In the super-critical regime, the straight configuration of the beam loses its stability. The buckling configurations, due to the nonlinear stiffness, are deduced from the nonlinear static equilibrium equation. Furthermore, the governing equation for the transverse vibration of the super-critical transporting Timoshenko beam is derived based on the buckling shape. Time histories are calculated by using the finite difference method (FDM). Furthermore, natural frequencies of nonlinear free vibration are determined by using the DFT. In addition, the effect of nonlinear stiffness on the natural frequencies is discovered. On the other hand, a partially linearized equation is deduced by regarding buckling configurations and its spatial derivatives as constant coefficients. Then, natural frequencies are extracted by using the Galerkin method. The two different approaches are compared. Numerical results show that the rotary inertia and the shear deformation significantly affect vibration characteristics of the super-critical transporting beam for some configurations. Moreover, comparisons with an Euler-Bernoulli beam model reveal that the fundamental frequency of the Timoshenko beam is higher when the speed is slightly greater than the critical value.

Analytical study of electromechanical buckling of a micro spherical elastic film on a compliant substrate part II: Postbuckling analysis

This work presents an analytical study of the electromechanical postbuckling of a spherical elastic thin film electrode that is bonded to a spherical dielectric compliant substrate. The compliant substrate is bonded to an inner rigid, fixed and electrically grounded spherical electrode. As will be shown, when the applied voltage reaches a critical value, the film buckles into a well ordered periodic surface patterns. The electromechanical postbuckling problem will analytically be solved for the one-dimensional, square checkerboard, hexagonal and herringbone periodic patterns. This work studies the effect of different geometry and material parameters on the selection and wavelength of the preferred/stable surface pattern of the buckled film. As will be deduced, the preferred buckling pattern can be determined to be either hexagonal or herringbone pattern depending on the geometry and material parameters of the film/substrate system. Furthermore, we study the effect of nonlinear stiffness of the compliant substrate on the electromechanical behavior of the film. The simplicity of generating and removing elastic surface wrinklings by On/Off voltage switching sufficiently increases the potential of the electromechanical buckling response to be employed in different MEMS application, such as, micro sensors, micro optical switches and deformable micro mirrors.

A robust nonlinear output feedback control method for limit cycle oscillation suppression using synthetic jet actuators

A synthetic jet actuator-based output feedback control method is presented, which achieves asymptotic limit cycle oscillation regulation in small unmanned aerial vehicle wings, where the dynamic model contains uncertainty and unmodeled external disturbances. In addition, the proposed control method compensates for the parametric uncertainty and nonlinearity inherent in the synthetic jet actuator dynamics. Motivated by the limitations characteristic of small unmanned aerial vehicles, the control method is designed to be computationally inexpensive, eliminating the need for time-varying parameter update laws, function approximators, or other computationally heavy techniques. To this end, a computationally minimal robust-inverse control method is utilized, which is proven to compensate for the uncertainties in both the aerial vehicle dynamics and the synthetic jet actuator dynamics. By endowing the robust-inverse control law with a bank of dynamic filters, asymptotic limit cycle oscillation regulation is achieved using only pitching and plunging displacement measurements in the feedback loop. The result is an asymptotic synthetic jet actuator-based limit cycle oscillation regulation control method, which does not require velocity measurements, adaptive laws, or function approximators in the feedback loop. To achieve the result, a detailed mathematical model of the limit cycle oscillation dynamics is utilized, which includes nonlinear stiffness effects, unmodeled external disturbances, and dynamic model uncertainty, in addition to the parametric uncertainty in the synthetic jet actuator dynamic model. A rigorous Lyapunov-based stability analysis is utilized to prove asymptotic regulation of limit cycle oscillations, and numerical simulation results are provided to demonstrate the performance of the proposed control law.

Prediction of the vibration characteristics for wheeled tractor with suspended driver seat including air spring and MR damper

It’s essential to establish a dynamic model of a wheeled tractor with suspended driver seat including air spring and Magnetorheological (MR) damper to analyze its vibration characteristics and performance. The equivalent linearized model of the wheeled tractor always neglects the effect of nonlinear stiffness for scissors linkage driver seat and air spring with auxiliary chamber as well as dynamic characteristics of real tractor tyre and considers the driver seat as linear spring and damper elements, which causes the analysis to have a lower precision. We considered the effect of nonlinear stiffness for scissors linkage seat and air spring with auxiliary chamber as well as MR damper and dynamic characteristics of real tyre and developed a complete nonlinear dynamic model of wheeled tractor with suspended driver seat including air spring and MR damper. The experimental results demonstrate that the improved nonlinear dynamic model is more consistent with the actual state than the equivalent linearized model, and the validity of the proposed model is verified. Furthermore, the effect of forward speed, ratio of volume for air spring with respect to volume of auxiliary chamber, and area of orifice on the acceleration of wheeled tractor was also investigated, which can provide a theoretical basis for control algorithm design of semiactive vibration isolation system.

Store-Induced Limit-Cycle Oscillations Due to Nonlinear Wing-Store Attachment

Several high-performance fighter aircraft exhibit store-induced limit-cycle oscillations, leading to pilot discomfort, potential structural fatigue, and flight envelope restrictions. The roles of various aerodynamic and structural factors causing the limit-cycle oscillation are not sufficiently understood, and their numerical exploration via time marching is computationally expensive. In this paper, the effects of nonlinear stiffness and damping in the wing-store attachments of the F-16 aircraft are examined, in the presence of steady flow aerodynamic nonlinearity, using the computationally efficient harmonic balance method. Structural mechanisms including cubic restoring force of both softening and hardening types, freeplay, and Coulomb friction are systematically evaluated, and the most likely among these are identified by comparing the computed limit-cycle oscillation results to flight data. An extension of the harmonic balance method to handle nonlinear unsteady aerodynamics along with structural nonlinearity is also proposed to enable rapid and accurate limit-cycle oscillation assessment of candidate store configurations.

Analysis of 1/2 sub-harmonic resonance in a maneuvering rotor system

This paper focuses on the 1/2 sub-harmonic resonance of an aircraft’s rotor system under hovering flight that can be modeled as a maneuver load G in the equations of motion. The effect on the rotor system is analyzed by using theoretical methods. It is shown that the sub-harmonic resonance may occur due to maneuvering flight conditions. The larger the eccentricity E and the maneuver load G, the greater the sub-harmonic resonance. The effects of nonlinear stiffness, damping of the system, maneuver load, and eccentricity on the sub-harmonic resonance region in parameter planes are also investigated. Bifurcation diagrams of the analytical solutions are in good agreement with that of the numerical simulation solutions. These results will contribute to the understanding of the nonlinear dynamic behaviors of maneuvering rotor systems.

Effect of nonlinear stiffness on the total harmonic distortion and sound pressure level of a circular miniature loudspeaker-experiments and simulations

This paper reports our study on the nonlinear stiffness of a few diaphragm designs used for miniature loudspeakers. For this study, the sample diaphragms were experimented for their stiffness curves using the laser vibrometry. Additionally, the stiffness curves were also obtained using finite element analysis (FEA). For calculating total harmonic distortion (THD), the electroacoustic system is modeled using the equivalentcircuit analogy. As a result of solving the coupled system of ordinary differential equations by the Runge-Kutta method, THD of three sample designs, characterized by different track patterns, were calculated and plotted as a function of frequency. The specimens were tested in a standard anechoic chamber for recording THD. Additionally, the sound pressure level (SPL) curves of these specimens were also recorded for comparison. It turns out that, despite no significant SPL difference, diaphragms with different track patterns shall, as a result, lead to substantial variations of THD. The consistence of our simulations with the measured THD has promised the feasibility of simulating THD by FEA for optimizing the design of track patterns for diaphragm.

Vibration of two beams connected by nonlinear isolators: analytical and experimental study

To study the coupling vibration of nonlinear isolators and flexible bodies, test rigs of two flexible beams connected by wire mesh isolators are constructed and investigated both experimentally and analytically. A five-parameter polynomial model of wire mesh isolators is derived by identifying parameters in the frequency domain with the sine-sweep test. For obtaining the parameters that are valid in a wide range of frequency, a numerically assisted identification method is developed. With this model, the vibration of two flexible beams connected by wire mesh isolators is studied. The frequency response is obtained analytically by employing the Green’s function method and harmonic balance method. Sine-sweep test results with three test rigs show good coherence with the corresponding numerical results. With obtained experimental results and numerical results, effect of connection parameters is studied in detail. It is found that traditional design rules for isolators are no longer effective and the coupling vibration must be investigated in the design phase. Another phenomenon is that the damping has a function of weakening the effect of nonlinear stiffness. Nonlinear stiffness and nonlinear damping can decrease the transmissibility along with the increase of the excitation level.

A Reduced-Order Dynamic Model of Nonlinear Oscillating Devices

A reduced-order dynamic model is presented for nonlinear devices subjected to in-plane oscillatory motion. Comparisons between numerical and finite element results demonstrate that the nonlinear behavior of a planar resonator can be predicted accurately by the derived dynamic model with significantly less computation. Simulation results illustrate the effects of nonlinear stiffness, damping ratio, electrostatic driving force, and device dimensions on the nonlinear dynamic behavior. The analysis yields two possible stable responses, depending on the initial rotation angle and rotation rate. The present dynamic model can be easily modified to analyze the nonlinear response of various planar resonators.

Nonlinear Stiffness of a Magneto-Rheological Damper

The last decade has witnessed an important role of magneto-rheological dampers in the semi-active vibration control on the basis of empirical models. Those models established by fitting experimental data, however, do not offer any explicit expressions for the stiffness and the damping of magneto-rheological dampers. Hence, it is not easy for engineers to get any intuitive information about the effects of stiffness and damping of a magneto-rheological damper on the dynamic performance of a controlled system. To manifest the nonlinear properties of a magneto-rheological damper, this paper presents the hysteretic phenomena and the additional nonlinear stiffness of a typical magneto-rheological damper in terms of equivalent linear stiffness and equivalent linear damping. Then, it gives a brief discussion about the effect of nonlinear stiffness on the vibration control through the numerical simulations and an experiment for the semi-active suspension of a quarter car model with a magneto-rheological damper installed. Both numerical simulations and experimental results show that the additional nonlinear stiffness in the magneto-rheological damper is remarkable, and should be taken into consideration in the design of vibration control.