Time-varying Stiffness Research Articles

Time-varying stiffness characteristics of roller bearing influenced by thermal behavior due to surface frictions and different lubricant oil temperatures

In this paper, the time-varying stiffness of the cylindrical roller bearing due to the thermal behavior, i.e. warm-up process and different lubricant oil temperatures are experimentally investigated. Firstly, a special test rig that can measure the bearing stiffness is built. Then, a bearing stiffness identification method is proposed, which can avoid the influence of the clearance and deformation on measurement results. Finally, a series of measurements are carried out at different operational conditions. Results reveal that the bearing stiffness shows strong nonlinearity and time-varying characteristics due to the variation of structure and oil performance. In startup (quasi-isothermal condition) and thermal balance states (combined with the frictional heat generation), the influence of the rotational speed on stiffness is opposite.

Time-varying linear quadratic Gaussian optimal control for three-degree-of-freedom wave energy converters

This paper presents a time-varying linear quadratic control for a three-degree-of-freedom (3-DoF) wave energy converter (WEC). The dynamic model for a heave-surge-pitch WEC has a linear coupling between pitch and surge modes, as well as a nonlinear coupling between the heave and pitch modes that affect the pitch motion. It is observed, however, that the heave motion is decoupled from the other two modes, and that the heave effect on the pitch motion can be recognized as a time-varying stiffness. Hence it is proposed in this paper to design a time-varying linear quadratic control for the coupled pitch-surge modes. The goal is to maximize the energy harvested over a receding time horizon. Prediction of the excitation forces is needed and is obtained based an estimate of the excitation force at the current time. The latter is obtained using an extended Kalman filter. The WEC motion is assumed constrained in this study. The results show that the energy harvested using this 3-DoF WEC is about three-fold higher than the energy extracted in heave mode, depending on the sea state. The paper also shows, with proper design of the control, it is possible to exploit this coupling to harvest more energy.

Efficient response determination of a M-D-O-F gear model subject to combined periodic and stochastic excitations

The nonlinear response of a multi-degree-of-freedom (M-D-O-F) gear model subject to combined periodic and stochastic excitations is investigated by an efficient linearization scheme, accounting for the time-variant stiffness, and the backlash feature involved in the model. The steady-state response of the model is expressed as the sum of a periodic (deterministic) part, and of a zero-mean stochastic part. Further, the piecewise linear term accounting for the teeth backlash in the gear model is approximated by a 5-th order polynomial expression. Harmonic balance and statistical linearization schemes are used to derive a set of coupled nonlinear algebraic equations involving the amplitudes of the periodic part, and the statistical moments of the stochastic part of the solution. These equations can be solved iteratively by any of the apropos standard numerical algorithms. Examples of a two-stage gear system under combined excitations with wide-band spectrum stochastic excitation, and narrow-band spectrum stochastic excitation are considered. The derived numerical results demonstrate the reliability and efficiency of the proposed scheme by juxtaposition with data from pertinent Monte Carlo simulations.

Control of Period-Doubling and Chaos in Varying Compliance Resonances for a Ball Bearing

AbstractVarying compliance (VC) is an inevitable parametrical excitation to rolling bearing systems due to time-varying stiffness from rolling element revolution. Period-doubling instability in the VC primary resonances of ball bearing is presented in many studies. Recently, this instability was demonstrated to be a probable indicator of occurrence of strong one to two internal resonances and chaotic motions, which has potential effects on the stability and safety of the bearing-rotor system. However, few studies have directly attempted to suppress this bifurcation instability. Here, a dynamic stiffness evaluating method is presented for assessing the threshold of the period-doubling and complex motions in VC primary resonances of ball bearings, where the elaborate evolution of the bifurcating process is obtained by harmonic balance and alternating frequency/time domain (HB-AFT) method and using Floquet theory. Our analysis indicates that by introducing certain additional stiffness, the period-doubling and corresponding subharmonic internal resonances can be suppressed. Besides, the evolution and mechanism of type I intermittency chaos in ball bearings will be clarified in depth. It is also shown that extensive chaotic motions for large bearing clearances (e.g., 40 μm) can vanish perfectly by action of additional stiffness.

Time-Varying Stiffness Calculation of Spiral Bevel Gears Based On SIMPACK

Aiming at the difficulty in calculating the time-varying stiffness of spiral bevel gears, the time-varying stiffness acquisition and calculation of spiral bevel gears under the actual coincidence degree based on SIMPACK is proposed. Taking a pair of spiral bevel gears as an example, the process of modeling and simulation with SIMPACK is described in detail. The numerical results are compared with the ISO standard calculation method, which verifies its correctness and rationality. Through the analysis of the relationship between meshing stiffness and load and damping under different working conditions, it is shown that meshing stiffness increases nonlinearly with the increase of load, and load damping makes the change of stiffness curve gentler.

Nonharmonic Oscillations of Solar Pores at the Dissipation Stage

Long-term variations of the magnetic field of solitary solar pores with periods in the range of 4–12 h are detected with the Helioseismic and Magnetic Imager (HMI) of the Solar Dynamics Observatory (SDO). These long-term oscillations of the pores obviously reflect the movements of these structures as whole magnetic formations around their equilibrium position. In this article, long-period oscillations of the solar pore are studied based on the SDO data for the first time. The results are interpreted with the solution obtained earlier for eigen oscillations of the system with time-varying stiffness.

Dynamic effects on spur gear pairs power loss lubricated with axle gear oils

The dynamic model of 12-degrees-of-freedom for spur gears pair accounting for nonlinear time-varying stiffness, damping, and coefficient of friction along the path of contact obtained from experimental tests is investigated. The Newmark's integration method is used to solve the equations and obtain the dynamic responses. Elementary mass, stiffness, and damping matrices with torsional and translational coupled effects were detailed. The lens of this work is to start from a nonlinear dynamic model to evaluate the influence of dynamic effects and lubrication on meshing gears power loss for different spur gear geometries within various operating conditions. The results reveal some useful references to vibration control, dynamic design, and efficiency improvement.

Bifurcation Analysis and Chaotic Behaviors of a Spur Microsegment Gear Pair With Time-Varying Pressure Angle

Abstract The study performs a dynamic analysis of a microsegment gear system with respect to the time-varying stiffness, gear backlash, time-varying pressure angle, and comprehensive error. The period expansion method is proposed to construct a single-degree-of-freedom nonlinear dynamic model of a spur microsegment gear pair. To improve the accuracy of simulation results, the time-varying mesh stiffness and the time-varying pressure angle are expressed in terms of piecewise Fourier functions, respectively. The effect of mesh frequency, backlash, mesh damping, and input power on bifurcation, and chaos properties of the system are analyzed. The numerical results indicate that the system performs a chaotic motion after several frequency jumps as the mesh frequency increases. When the backlash increases, the frequency region of double-side impact and chaos tends to be decreased. The effect of mesh damping and input power on bifurcation characteristics is also investigated. The increase in mesh damping and input power can stabilize the system and avoid double-side impact. The results potentially present a useful source of reference for technicians and engineers for the dynamic design and vibration control of the aforementioned system with a nonconstant pressure angle.

Real-Time Tracking of Structural Stiffness Reduction with Unknown Inputs, Using Self-Adaptive Recursive Least-Square and Curvature-Change Techniques

In this paper, a new computationally efficient algorithm is developed for online and real-time identification of time, location, and severity of abrupt changes in structural stiffness as well as the unknown inputs such as earthquake signal. The proposed algorithm consists of three stages and is based on self-adaptive recursive least-square (RLS) and curvature-change approaches. In stage 1 (intact structure), a simple compact RLS is hired to estimate the unknown parameters and input of the structure such as stiffness and earthquake. Once the damage has occurred, its time and location are identified in stage 2, using two robust damage indices which are based on the structural jerk response and the error between measured and estimated responses of structure from RLS. Finally, the damage severity as well as the unknown excitations are identified in the third stage (damaged structure), using a self-adaptive multiple-forgetting-factor RLS. The method is validated through numerical and experimental case studies including linear and nonlinear buildings, a truss structure, and a three-story steel frame with different excitations and damage scenarios. Results show that the proposed algorithm can effectively identify the time-varying structural stiffness as well as unknown excitations with high computational efficiency, even when the measured data is contaminated with different levels of noise. In addition, as no optimization method is used here, it can be applied to real-time applications with computational efficiency.

Dynamic modeling and simulation of inchworm movement towards bio-inspired soft robot design

Inchworms have been one of the most widely used bionic templates for designing soft robotic devices. Bioresearch has shown that muscles of inchworms exhibit nonlinear hysteresis and their body structures are with hydrostatic skeleton. But effects of these properties on their dynamic movements have not been studied yet. In this work, a dynamic model based on the principle of virtual power of an inchworm is established to examine the problem. A spring-damper model with time-varying stiffness and damping coefficients is used to model controllable nonlinear properties of the inchworm muscles. The hydrostatic skeleton is applied to the model as a constant volume constraint for each segment. Based on this, simulations of three typical movements including omega-shaped arching motion, cantilevered probing motion and surprising fast looping motion are presented. The effects of the nonlinear properties including variable stiffness and damping properties of muscles on these dynamic behaviors of inchworms are illustrated. Some inspiration for designing bio-inspired crawling robots and soft slender robotic devices is obtained. And we think this work will hopefully provide better understanding and guidance for design and control of these robotic devices.

Modelling and Dynamic Analysis of the Spiral Bevel Gear-Shaft-Bearing-Gearbox Coupling System

To accurately study the dynamic characteristics of the spiral bevel gear transmission system in a helicopter tail transmission system, the finite element model of the gear shaft was established by a Timoshenko beam element, and the mechanical model of the spiral bevel gear was created by the lumped mass method. The substructure method is employed to extract the dynamic parameters from the gearbox’s finite element model, and the dynamic model of the spiral bevel gear-shaft-bearing-gearbox coupling system was built according to the interface coordination conditions. In the model, the influences of time-varying stiffness, a time-varying transmission error, gearbox flexibility, unbalance excitation, and a flexible shaft and bearing support on the system vibration were taken into account simultaneously. On this basis, the dynamic differential equations of the full coupling system of the spiral bevel gear were derived, and the effects of the gearbox flexibility, the shaft angle, and the unbalance on the dynamic properties of the system were analysed. The results show that the gearbox flexibility can reduce the gear meshing force and bearing force, in which there is a more significant impact on the bearing force. The shaft angle affects the position, size, and direction of the system’s axis trajectory. Meanwhile, the meshing force and the bearing force of the system are also varied because of the various pitch angles of the driving and driven gears under different shaft angles. The unbalance of the gear shaft has an effect on the vibration of the spiral bevel gear transmission system in all directions, wherein the influence on the torsional vibration is the most significant, and the influence increases as the unbalance rises. The unbalance of the gear shaft also affects the meshing force and bearing force, which increases as the rotational speed rises. This research provides a theoretical basis to optimize dynamic performance and reduce the vibration and noise of a spiral bevel gear full coupling system.

Analysis of load distribution and contact stiffness of the single-nut ball screw based on the whole rolling elements model

This paper aims to investigate the load distribution and contact stiffness characteristics of the single-nut ball screw pair (SNBSP). First, the transformed relationship of coordinate systems is established. Then, the whole rolling elements load distribution model of the SNBSP is presented. Based on this, the whole rolling elements contact stiffness model is obtained. Applying the Newton–Raphson iterative method to solve the model, the normal force of rolling elements and the contact angles between balls and raceway surface are determined. The calculation results are reasonably consistent with those of the half pitch model. Then, the local contact stiffness and global contact stiffness are obtained. Furthermore, the effects of axial load and structural parameters of the SNBSP on the normal contact force, contact angle, and local and global contact stiffness are discussed using numeric analysis. Finally, a dynamic model of the z-axis feed system with time-varying axial stiffness is established, and the accuracy of the model is verified by experiments.

CONTINUOUS MOTION AND TIME-VARYING STIFFNESS ESTIMATION OF THE HUMAN ELBOW JOINT BASED ON SEMG

The flexibility of body joints plays an important role in daily life, particularly when performing high-precision rapid pose switching. Importantly, understanding the characteristics of human joint movement is necessary for constructing robotic joints with the softness of humanoid joints. A novel method for estimating continuous motion and time-varying stiffness of the human elbow joint was proposed in the current study, which was based on surface electromyography (sEMG). We used the Hill-based muscle model (HMM) to establish a continuous motion estimation model (CMEM) of the elbow joint, and the genetic algorithm (GA) was used to optimize unknown parameters. Muscle short-range stiffness (SRS) was then used to characterize muscle stiffness, and a joint kinetic equation was used to express the relationship between skeletal muscle stiffness and elbow joint stiffness. Finally, we established a time-varying stiffness estimation model (TVSEM) of the elbow joint based on the CMEM. In addition, five subjects were tested to verify the performance of the CMEM and TVSEM. The total average root-mean-square errors (RMSEs) of the CMEM with the optimal trials were 0.19[Formula: see text]rad and 0.21[Formula: see text]rad and the repeated trials were 0.24[Formula: see text]rad and 0.25[Formula: see text]rad, with 1.25-kg and 2.5[Formula: see text]kg-loads, respectively. The values of elbow joint stiffness ranged from 0–40[Formula: see text]Nm/rad for different muscle activities, which were estimated by the TVSEM.

Dynamic analysis of a ball screw feed system with time-varying and piecewise-nonlinear stiffness

In this study, a single-degree-of-freedom model is established to investigate the dynamic characteristics of a single-nut double-cycle ball screw feed system by considering the contact states of the nonlinear kinematic joints. Based on fully considering the parameters of the ball screw feed system, the axial deformations and forces of the key components are calculated to construct a set of piecewise-nonlinear restoring force functions of the system displacement and worktable position. The variations of the contact stiffnesses of the kinematic joints and transmission stiffness of the system with different boundary conditions are analyzed and the results indicate that they all have abrupt changes when the system displacement reaches a critical value. The changing law of the system transmission stiffness in the whole stoke is discussed. Additionally, the effects of excitation force, worktable position and system mass on the dynamic characteristics of the system and its correlative components are analyzed.

Dynamical Analysis and Stabilization of Wind Turbine Drivetrain via Adaptive Fixed-Time Terminal Sliding Mode Controller

In order to study the stability of the wind turbine drivetrain in further depth, we present a nonlinear relative rotation mathematical model considering the nonlinear time-varying stiffness and the nonlinear damping force. Meanwhile, the nonlinear dynamics of the model under combined harmonic excitation are studied in detail. And some interesting dynamic phenomena are observed visually. Furthermore, to suppress chaotic oscillation within bounded time independent of initial conditions, a novel adaptive fixed-time terminal sliding mode controller is proposed. The stability of the final closed loop system is guaranteed according to Lyapunov stability theory. Rigorous mathematical analyses are used to prove the validity of the presented approach. Finally, compared with the existing finite-time stability method, simulation results are given to highlight the effectiveness and superiority of the proposed method and verify the theoretical analyses.

Reduced-Order Modeling for Stability and Steady-State Response Analysis of Asymmetric Rotor Using Three-Dimensional Finite Element Model

A generalized and efficient technique of reduced-order model (ROM) is proposed in this paper for stability and steady-state response analysis of an asymmetric rotor based on three-dimensional (3D) finite element model. The equations of motion of the asymmetric rotor-bearing system are established in the rotating frame. Therefore, the periodic time-variant coefficients only exist at a tiny minority of degrees-of-freedom (DOFs) of bearings. During the model reduction process, the asymmetric rotor-bearing system is divided into rotor and bearings. Only the rotor was reduced. And the physical coordinates of bearings are kept in the reduced model during reduction. Then, the relationship between the rotor and bearings is established by inserting periodic time-variant stiffness and damping matrix of bearings into the reduced model of rotor. There is no reduction to the matrices of bearings, which guarantees the accuracy of the calculation. This technique combined with fixed-interface component mode synthesis (CMS) and free-interface CMS is compared with other existing modal reduction method on an off-center asymmetric rotor and shows good performance.

Parametrically Excited Nonlinear Two-Degree-of-Freedom Electromechanical Systems

This paper presents a nonlinear parametrically excited cantilever beam with electromagnets. A parametrically excited two-degree-of-freedom (2-DOF) system with linear time-varying stiffness, nonlinear cubic stiffness, nonlinear cubic parametric stiffness and nonlinear damping is considered. In previous studies the stability and bifurcation of the nonlinear parametrically excited 2-DOF were investigated through analytical, semi-analytical and numerical methods. Unlike previous studies, in this contribution the system’s response amplitude and phase at parametric resonance and parametric combination resonance are demonstrated experimentally and some novel results are discussed. Experimental and analytical amplitude-frequency plots are presented to show the stable solutions. Solutions for the system response are presented for specific values of parametric excitation frequency and the energy transfer between modes of vibrations is observed. The results presented in this paper prove that the bifurcation point and hence the bandwidth of the parametric resonance can be predicted correctly with the proposed analytical method. The proposed nonlinear parametrically excited 2-DOF can be used to design Micro ElectroMechanical Systems (MEMS) actuators and sensors. Validating the experimental results with the theory can improve the efficiency of these electrical systems.

Time-varying system identification by enhanced Empirical Wavelet Transform based on Synchroextracting Transform

In this paper, an enhanced Empirical Wavelet Transform (EWT) approach based on Synchroextracting Transform (SET) is proposed for time-varying system identification. When a structure of time-varying physical properties, i.e. mass, stiffness or damping, is under external excitations, structural dynamic responses are usually non-stationary because the system has time-varying dynamic vibration characteristics. Under this circumstance, it would be difficult to determine the number of Intrinsic Mode Functions (IMFs) included in structural dynamic responses by using Fourier spectrum. Considering that the filtering boundaries of traditional EWT method are defined based on the segmental Fourier Spectrum of a processed signal, directly using it for non-stationary signal decomposition may not be effective and accurate. To apply the EWT method for time-varying system identification, in this study, time-frequency analysis based on SET is first performed to determine the frequency components of a non-stationary vibration signal instead of using Fourier spectrum. The filtering boundaries for EWT analysis are determined based on the time-frequency representation. Then, the IMFs are extracted from the non-stationary vibration signals by using EWT with the above defined filtering boundaries. When the IMFs are accurately obtained, the instantaneous frequencies of IMFs are identified by using Hilbert Transform (HT). In numerical simulations, a simulated signal with a high level noise is analyzed to verify the feasibility of using SET to define the filtering boundaries. Then the proposed approach is used to identify the instantaneous frequencies of a time-varying two-storey shear type building under earthquake and Gaussian white noise excitations, respectively. Experimental investigations on a time-varying bridge-vehicle system are conducted to verify the effectiveness of the proposed approach. The results in both numerical simulations and experimental validations demonstrate that the enhanced EWT approach can effectively and reliably identify the instantaneous frequencies of time-varying systems.

Significance of non-stationary characteristics of ground-motion on structural damage: shaking table study

This paper reports the results of a set of benchmark medium-scale shaking table tests to investigate the significance of the non-stationary characteristics of ground-motion on nonlinear dynamic responses and the structural damage of reinforced concrete (RC) columns. To examine the influence of ground-motion characteristics, four RC columns are tested under (1) near-field without pulse, (2) near-field pulse-like, and (3) far-field ground-motions. These ground-motion records were spectrally matched by the reweighted Volterra series algorithm without changing non-ergodic characteristics. To explore the confinement effects, two sets of column specimens are designed to represent the modern well-confined and older lightly-confined RC columns. Each column is tested in slight, extensive and complete damage limit states. Then aftershock excitations are conducted to investigate the performance of severely damaged RC columns. Low amplitude white-noise tests are conducted on pristine columns and after each damage limit state experiment to detect natural frequency variant of damaged columns using transfer function estimate. Furthermore, using time–frequency analysis, the real-time variant frequency of test specimens is estimated. The significant duration of ground-motions accounting for the effect of non-stationary characteristics of ground-motion is also estimated by time-cumulative damage analysis of the test results. Finally, the time-variant stiffness degradation of RC columns is estimated.

Investigation on Nonlinear Dynamic Characteristics of a New Rigid-Flexible Gear Transmission With Wear

Abstract At present, the mean value of the meshing stiffness and the gear backlash is a fixed value in the nonlinear dynamic model. In this study, wear is considered in the model of the gear backlash and time-varying stiffness. With the increase of the operating time, the meshing stiffness decreases and the gear backlash increases. A six degrees-of-freedom nonlinear dynamic model of a new rigid-flexible gear pair is established with time-varying stiffness and time-varying gear backlash. The dynamic behaviors of the gear transmission system are studied through bifurcation diagrams with the operating time as control parameters. Then, the dynamic characteristics of the gear transmission system are analyzed using excitation frequency as control parameters at four operating time points. The bifurcation diagrams, Poincaré maps, fast Fourier transform (FFT) spectra, phase diagrams, and time series are used to investigate the state of motion. The results can provide a reference for the gear transmission system with wear.