Time-varying Stiffness Research Articles

Dynamic analysis of gear system under fractional-order PID control with the feedback of meshing error change rate

Based on the fractional-order proportional-integral-derivative (PID) control of the feedback of change rate of gear meshing error, the nonlinear dynamics problem of a spur gear pair is investigated. A model is established including backlash, time-varying stiffness, internal and external excitations and other factors, where the fractional-order PID control terms are also considered. The periodic solution is obtained based on the incremental harmonic balance method, and it is verified by the numerical solution. The effects of the coefficients and the orders of the fractional-order PID controller are analyzed in detail. The control performances of fractional-order PID controller and integer-order PID controller are compared. The results show that the fractional-order PID controller has good control performance for the gear meshing system. By changing the parameters of the controller, the resonance amplitude of the gear meshing system can be suppressed, and the resonance frequency of the gear meshing system can also be adjusted.

Fuzzy adaptive hybrid impedance control for mirror milling system

In this paper, a fuzzy adaptive hybrid impedance control scheme designed for the supporting side of mirror milling system is proposed, which is capable of solving the force supporting problem with time-varying environment stiffness. In the process of milling large thin-walled sides, maintaining a constant supporting force is crucial both for suppressing vibration and for reducing deformation. Moreover, a superior impedance control scheme is beneficial in terms of enhancing the system stability, which is characterized by easy-implementation in the industrial robots. Hence, the force error triggered by the time-varying stiffness environment can be compensated by using our fuzzy adaptive algorithm. Simulations are tested on a parallel robot, i.e., Tricept, in order to verify the force control accuracy as well as its robustness in terms of a time-varying stiffness environment. Both the simulation and the experiment show that our proposed support scheme has a better performance on maintaining the desired contact force than hybrid impedance (HI) control and adaptive hybrid impedance (AHI) control.

Effect of vehicle vibration environment of high-speed train on dynamic performance of axle box bearing

ABSTRACTIn this study, we developed a comprehensive three-dimensional vehicle–track coupled dynamics model considering the traction drive system and axle box bearing. In this model, dynamic interactions between the axle box bearing and other components, such as the wheelset and bogie frame, are considered based on a detailed analysis of the structural properties and working mechanism of the axle box bearing. A few complicated dynamic excitations, such as the time-varying mesh stiffness of gears, time-varying stiffness of bearing, bearing gaps and track irregularities, are considered. Then, the dynamic responses of the vehicle–track system are demonstrated via numerical simulations based on the established dynamics model. The results indicate that the traction drive system and track irregularities can significantly influence the dynamic interactions of the axle box bearing. The necessity of considering the excitation caused by gear meshing and track irregularities when assessing the dynamic performance of the axle box bearing is demonstrated.

Time-varying stiffness model of spur gear considering the effect of surface morphology characteristics

The nonuniform cantilever beam and Hertzian contact model have been widely used to derive the mesh stiffness of spur gear assuming that the contact surface is absolutely frictionless. However, studies have confirmed that machined surfaces are rough in microscale and can be simulated by the Weierstrass–Mandelbort function. In order to get a reasonable and precise mesh stiffness model, the M-B contact model and finite element method are combined to express the local contact stiffness Kh. Through the simulation and comparison, the analytical finite element method is proved to be consistent with the traditional models and introduces the roughness parameters of machined tooth surface into the meshing process. Furthermore, the results also show that it is advantageous to improve the total mesh stiffness by increasing the fractal dimension D and input torque T as well as decreasing the roughness parameter G. In this paper, a relationship is built between the total mesh stiffness of gear sets with tooth surface characters and input torque, which can be a guidance in the design of the tooth surface parameters and the choice of the processing method in the future.

Parametric resonance analysis of rectangular plates subjected to moving inertial loads via IHB method

This paper deals with the induced instability due to parametric resonance of rectangular plates traversed by inertial loads and lying on elastic foundations. The extended Hamilton's principle is employed to derive the partial differential equation associated with the transverse motion of the plate. Subsequently, this equation is transformed into a set of ordinary differential equations by the Galerkin method. Including vertical, centripetal and Coriolis acceleration terms related to the moving mass transition in the analysis leads to governing equations with time-varying mass, damping and stiffness coefficients. Particularly, the intermittent passage of masses along rectilinear paths, or the motion of an individual mass along an orbiting path, permits to subcategorize the problem as a parametrically excited system with periodic coefficients. By applying the incremental harmonic balance (IHB) method, the stability of the induced plate vibrations is investigated, revealing an emersion of instability tongues in the parameters plane. Semi-analytical results are provided for various boundary conditions of the plate which got verified through direct numerical simulations and other results reported in the literature.

Nonlinear dynamics of heave motion of the sandglass-type floating body with piecewise-nonlinear, time-varying stiffness

In this paper, the dynamic behaviors and stability of the nonlinear heave response of the sandglass-type Floating Production, Storage, and Offloading unit (FPSO) under harmonic wave excitation force are studied. With consideration of the special shape characteristics, the heave restoring stiffness is modeled as a piecewise-nonlinear, time-varying (PNTV) function. The incremental harmonic balance (IHB) method in conjunction with the incremental arc-length method are employed to perform an elaborate investigation on its dynamic behaviors. Floquet theory is applied to determine the stability of the periodic solutions. The accuracy of the approach is verified through a comparison with the results of direct numerical integration using 4th-order Runge-Kutta method. Then, hardening-type nonlinearity of the heave motion is proved by parametric studies concerning the effects of damping ratio, wave excitation force, wave elevation and its phase difference with the wave excitation force on the nonlinear response. Based on this, a more reasonable approach considering frequency-dependent characteristic and memory effect of the hydrodynamic forces are established by the modification of traditional IHB method. The validity of the improvement is demonstrated by the widely used hybrid time-domain numerical simulation method in engineering application. Finally, by comparing the motion responses of two floating bodies with the corresponding linear heave RAO, the results show that even though the hardening-type nonlinearity can lead to the occurrence of unexpected severe heave motion, the wave-free characteristic of the heave motion for this type of floating body can suppress the development of the hardening-type nonlinearity and thus the established wave-free guideline is reasonable. Nevertheless, the frequency of the maximum RAO can exceed the wave-free frequency in the case of strong nonlinearity. Consequently, the nonlinearity of the heave motion should be fully considered and introduced in the existing design criteria.

A new analytical model for viscous wall dampers and its experimental validation

This paper presents a new analytical model for viscous wall dampers (VWDs) and their effectiveness in reducing seismic structural distress. A series of performance tests on VWDs is conducted to measure the response of the dampers under various loading conditions. Based on the non-linear responses elicited in these mechanical tests, a generalised Maxwell model for VWDs with time-varying stiffness and damping is proposed to assess their amplitude dependencies. Shaking table tests are conducted on a 3-storey steel space frame at a 1:4 scale with and without VWDs, using three types of earthquake excitations, to study the effectiveness of these devices. The additional stiffness of VWDs can significantly increase the natural frequencies of the steel frame, and the effect of vibration control on the displacements of the structural model is remarkable. The added damping contributed by VWDs can significantly reduce the acceleration responses and the inter-storey shear forces of the steel frame. The proposed VWD model is implemented using the OpenSees software, and is subsequently used for the dynamic simulation of the shaking table tests. The effectiveness and good adaptability of the proposed model are verified based on the agreement between the numerical simulation and shaking table test results.

Simulation of meshing characteristics of harmonic reducer and experimental verification

This article studies the causes of the time-varying stiffness of harmonic drive gear and the number of meshing teeth and the meshing length under different torques. Multi-pair tooth meshing charact...

Normal contact stiffness identification-based force compensation for a hardware-in-the-loop docking simulator

Docking is a very important and challenging space operation. To ensure the safety and reliability of the docking, the thorough ground verification is necessary. Hardware-in-the-loop (HIL) simulation is a good choice, especially when the geometric structure of the docking device is complex. However, the HIL simulation suffers from energy increase and instability since time delay in the system is unavoidable. In this study, a six degree-of-freedom (DOF) parallel robot is used as the motion simulator to physically perform the contact in the simulation. To achieve system stability and high fidelity, a normal contact stiffness identification-based force compensation method is proposed. The contact force is compensated through identifying the time-varying stiffness along the normal direction of the contact surface. The 6-DOF force-moment compensation algorithm for the HIL parallel robot system is presented in detail. Simulations and experiments are both carried out, and the results show that the proposed compensation method is effective.

Nonlinear dynamic response analysis of two-stage spur gear space driving mechanism with large inertia load

Large inertia load has been widely used in space driving mechanisms, but the research concerning theory is still an unexplored scientific field. Towards the problem of nonlinear disturbance in space driving mechanism with large inertia load, a 14-DOF (Degree of Freedom) nonlinear, time-varying, dynamic model of two-stage spur gear system was established, taking into consideration time-varying stiffness, backlash and transmission error. The dynamic response of the load, under large or small inertia was investigated, basing on the dynamic model. The results indicate that at starting, normal operation and braking, large inertia load system has obvious hysteresis, compared to small inertia. The factors that improve dynamic response speed under large inertia load were studied. The results indicate that improving the stiffness and damping of the output shaft and changing the material of second gear pair to titanium alloy are helpful in improving the dynamic response speed of the system. Some results enrich the research of two-stage spur gear nonlinear model and large inertia load, since they provide important reference for the actual design of the gear system.

Volterra approach – a new method to accurately calculate the non-linear and time-varying roll restoring arm of ships in irregular longitudinal seas

ABSTRACTWith several incidents of large amplitude roll motion of ships observed at sea over the past two decades, the dynamic stability of modern ship types has assumed significant importance. The move by the International Maritime Organization to develop a second generation level-2 criteria has resulted in considerable attention to the development of simplified analytical models for roll motion. Especially for the problem of parametric roll, there is still no analytical model which accurately captures the non-linear roll restoring stiffness taking into account the combined effect of incident waves and dynamic heave and pitch motions of the vessel. This paper extends our previous work to accurately capture both the linear and higher-order time-varying roll stiffness coefficients through an analytical model while taking into account incident waves and dynamic motion of the vessel. This approach called the Volterra method is compared against the existing simplified models and a non-linear time-domain simulation program to demonstrate better agreement of the proposed approach.

Adaptive Hybrid System Framework for Unified Impedance and Admittance Control

Impedance and Admittance Control are two well-known controllers to accomplish the same goal: the regulation of the mechanical impedance of manipulators interacting dynamically with the environment. However, they both are affected by a strong limitation deriving from their fixed causality, which causes their inability to provide good performance over a large spectrum of environment stiffnesses. In this paper an adaptive hybrid system framework is proposed to unify Impedance and Admittance formulations and consequently overcome this limit. Indeed, the hybrid framework interpolates the opposite performance and stability characteristics of the above-mentioned impedance-based control strategies leading to a family of controllers with intermediate properties, and thus suitable for several conditions. Moreover, the adaptivity allows the hybrid system to operate properly in an environment characterized by unknown and even time-varying stiffness. Especially, the work focuses on the development of this latter aspect and an adaptive solution based on a feedforward Neural Network is presented. The effectiveness of the novel control strategy is demonstrated by means of numerical simulations.

A new model for calculating time-varying gearmesh stiffness

Time-varying gearmesh stiffness (TVGS) is the main cause of gear vibration, and its accuracy affects the responses of dynamic models. An exponential curve model based on the Saint Venant’s Principle is proposed to calculate the gearmesh stiffness of cracked spur gears in this paper. With the proposed model, the TVGS under the circumstances of healthy condition and four crack cases are computed, whose results have a good agreement with those of finite element method (FEM). Therefore, the exponential curve model can be used to estimate the TVGS and an alternative to FEM in gearmesh stiffness calculation is provided.

Stiffness variation method for milling chatter suppression via piezoelectric stack actuators

Abstract In cutting process, chatter is an inevitable phenomenon that greatly affects workpiece surface quality, tool life and machining efficiency. The stiffness variation (SV) method has been proposed and applied in chatter suppression for a long time. However, the early studies focused on boring and turning with one degree of freedom. For milling process with two degrees of freedom, there is no related research about SV. In this paper, SV method is employed through modulating the stiffness around a nominal value in order to suppress milling chatter. The milling dynamic model of SV with two degrees of freedom is constructed. In this model, the classical delay differential equation (DDE), which governs the milling process, is replaced with a DDE with a time-varying stiffness term. The stability analysis with different SV is completed using the semi-discretization method (SDM) and results show that the stable region with SV is larger than that under most of the traditional conditions. The results of stability analysis are verified by time domain simulation. In addition, the influences on stability lobe diagram (SLD), which is caused by different waveforms, amplitudes and frequencies of SV, are also analyzed specifically. The analysis results can provide the optimal parameters combination for milling. Cutting experiments are implemented on a three-axis milling machine to validate the effectiveness of SV. In the experiment, piezoelectric stack actuators are used to modulate the stiffness with time-varying preload. The milling forces signals are acquired by data collecting instrument, whose root-mean-square (RMS) is used as the metric of cutting vibrations. Experiment results are in good agreement with theoretical prediction.

Nonlinear dynamic model of a gear-rotor-bearing system considering the flash temperature

The instantaneous flash temperature is an important factor for gears in service. To investigate the effect of the flash temperature of a tooth surface on the dynamics of the spur gear system, a modified nonlinear dynamic model of a gear-rotor-bearing system is established. The factors such as the contact temperature of the tooth surface, time-varying stiffness, tooth surface friction, backlash, the comprehensive transmission error and so on are considered. The flash temperature of a tooth surface of pinion and gear is formulated according to Blok's flash temperature theory. The mathematical expression of the contact temperature of the tooth surface varied with time is derived and the tooth profile deformation caused by the change of the flash temperature of the tooth surface is calculated. The expression of the mesh stiffness varied with the flash temperature of the tooth surface is derived based on Hertz contact theory. The temperature stiffness is proposed and added to the nonlinear dynamic model of the system. The influence of load on the flash temperature of the tooth surface is analyzed in the parameters plane. The variation of the flash temperature of the tooth surface is studied. The numerical results indicate that the calculated method of the flash temperature of the gear tooth surface is effective and it can reflect the rules for the change of gear meshing temperature and sliding of the gear tooth surface. The effects of frequency, backlash, bearing clearance, comprehensive transmission error and time-varying stiffness on the nonlinear dynamics of the system are analyzed according to the bifurcation diagrams, Top Lyapunov Exponent (TLE) spectrums, phase portraits and Poincaré maps. Some nonlinear phenomena such as periodic bifurcation, grazing bifurcation, quasi-periodic bifurcation, chaos and its routes to chaos are investigated and the critical parameters are identified. The results provide an understanding of the system and serve as a useful reference in designing such systems.

A comparative study of surface waviness models for predicting vibrations of a ball bearing

Surface waviness models utilized to describe the excitation events in the ball bearings (BABs) play an important role in predicting the vibrations of the ball bearing systems. This work proposes a comparative study on the most relevant existing surface waviness models. One model is named as time-varying displacement (TVD) model, which can only describe the time-varying displacement excitation (TDIE). Another model is named as TVDS model, which can describe the TDIE and time-varying contact stiffness coefficient excitation (TCSE). The influences of the wave number, maximum amplitude, nonuniform distribution on the contact stiffness coefficients are studied, as well as vibrations of the BAB. The comparative results demonstrate that the time-varying displacement and stiffness (TVDS) model can provide a more accurate impulses caused by the uniform and nonuniform surface waviness on the races of the BABs. It also seems that the surface waviness distribution has a significantly influence on vibrations of the BABs.

Research on thermally induced vibration of large hoop truss antenna moving through earth’s shadow

The coupled thermal-structural analysis of a spacecraft including a large hoop truss antenna in the on-orbital environment is studied. The structural strain is considered as the superposition of the thermal strain and the elastic strain. The thermo-elastic coupling equation of the cable-truss composited structure for the hoop truss antenna is established by introducing the thermo-elastic linear time-varying stiffness. The time-varying thermal field is modeled by considering the change of the attitude of the spacecraft and the influence of the spacecraft occlusion on the temperature changing of components. The obtained thermal field is used as the external load, and the influence of temperature changing when the spacecraft structure moves through earth’s shadow on the thermally induced vibration of the structure is analyzed. The results show that the temperature environment on-orbit has little influence on the structure. The data of on-orbit of a spacecraft is analyzed, and the efficiency of the numerical analysis is verified.

Helical gear wear monitoring: Modelling and experimental validation

Gear tooth surface wear is a common failure mode. It occurs over relatively long periods of service nonetheless, it degrades operating efficiency and leads to other major failures such as excessive tooth removal and catastrophic breakage. To develop accurate wear detection and diagnosis approaches at the early phase of the wear, this paper examines the gear dynamic responses from both experimental and numerical studies with increasing extents of wear on tooth contact surfaces. An experimental test facility comprising of a back-to-back two-stage helical gearbox arrangement was used in a run-to-failure test, in which variable sinusoidal and step increment loads along with variable speeds were applied and gear wear was allowed to progress naturally. A comprehensive dynamic model was also developed to study the influence of surface wear on gear dynamic response, with the inclusion of time-varying stiffness and tooth friction based on elasto-hydrodynamic lubrication (EHL) principles. The model consists of an 18 degree of freedom (DOF) vibration system, which includes the effects of the supporting bearings, driving motor and loading system. It also couples the transverse and torsional motions resulting from time-varying friction forces, time varying mesh stiffness and the excitation of different wear severities. Vibration signatures due to tooth wear severity and frictional excitations were acquired for the parameter determination and the validation of the model with the experimental results. The experimental test and numerical model results show clearly correlated behaviour, over different gear sizes and geometries. The spectral peaks at the meshing frequency components along with their sidebands were used to examine the response patterns due to wear. The paper concludes that the mesh vibration amplitudes of the second and third harmonics as well as the sideband components increase considerably with the extent of wear and hence these can be used as effective features for fault detection and diagnosis.

An improved analytical model for a lubricated roller bearing including a localized defect with different edge shapes

Vibrations of a roller bearing (RB) with a localized defect (LOD) are determined by LOD edge shapes, which can be used to detect and diagnose the LODs. Therefore, it is very helpful to analyze the relationships between impulses and LOD edge shapes for detection and diagnosis of the early LODs. In this study, an improved analytical model for a lubricated RB with a LOD considering different edge shapes is proposed. The LOD edge propagation is determined by the size of small cylindrical surface at its edge. A time-varying impact force (TVIF) model for the LOD with different edge shapes is also presented depended on Hertzian contact theory. The time-varying displacement excitation (TVDE) and time-varying contact stiffness coefficient (TVSC) between the roller and LOD edges can be formulated by the presented model, which cannot be formulated by the previous models considering sharp edges in the literatures. Influences of LOD edge shapes on vibrations of the unlubricated and lubricated RB are investigated. The numerical results show that the amplitude and impulse waveform of the accelerations of the RB will be affected by the LOD edge shape and lubricated oil; however, the peak frequencies in the spectrum are slightly influenced by the LOD edge shape and lubricated oil. It seems that the presented numerical results can give some guidance for the incipient LOD detection and diagnosis for RBs.

Estimation of Time-Varying, Intrinsic and Reflex Dynamic Joint Stiffness during Movement. Application to the Ankle Joint.

Dynamic joint stiffness determines the relation between joint position and torque, and plays a vital role in the control of posture and movement. Dynamic joint stiffness can be quantified during quasi-stationary conditions using disturbance experiments, where small position perturbations are applied to the joint and the torque response is recorded. Dynamic joint stiffness is composed of intrinsic and reflex mechanisms that act and change together, so that nonlinear, mathematical models and specialized system identification techniques are necessary to estimate their relative contributions to overall joint stiffness. Quasi-stationary experiments have demonstrated that dynamic joint stiffness is heavily modulated by joint position and voluntary torque. Consequently, during movement, when joint position and torque change rapidly, dynamic joint stiffness will be Time-Varying (TV). This paper introduces a new method to quantify the TV intrinsic and reflex components of dynamic joint stiffness during movement. The algorithm combines ensemble and deterministic approaches for estimation of TV systems; and uses a TV, parallel-cascade, nonlinear system identification technique to separate overall dynamic joint stiffness into intrinsic and reflex components from position and torque records. Simulation studies of a stiffness model, whose parameters varied with time as is expected during walking, demonstrated that the new algorithm accurately tracked the changes in dynamic joint stiffness using as little as 40 gait cycles. The method was also used to estimate the intrinsic and reflex dynamic ankle stiffness from an experiment with a healthy subject during which ankle movements were imposed while the subject maintained a constant muscle contraction. The method identified TV stiffness model parameters that predicted the measured torque very well, accounting for more than 95% of its variance. Moreover, both intrinsic and reflex dynamic stiffness were heavily modulated through the movement in a manner that could not be predicted from quasi-stationary experiments. The new method provides the tool needed to explore the role of dynamic stiffness in the control of movement.