Time-varying Stiffness Research Articles

Force-induced deformation prediction and flexible error compensation strategy in flank milling of thin-walled parts

• A force-induced deformation prediction model of thin-walled parts based on the substructure method is proposed. • A flexible error compensation strategy based on the continuity of the compensated tool locations is proposed. • The proposed deformation prediction model can greatly improve the prediction efficiency. • The flexible error compensation strategy converges faster than the published methods. • The correctness of the proposed method is verified via flank milling of a 3 mm thin-walled part. Due to the weak rigidity of flexible thin-walled part and tool, deformation inevitably occurs under the action of a cutting force in the flank milling of thin-walled parts, resulting in reductions of the machining accuracy and efficiency. To solve the above problem, a force-induced deformation prediction model based on the static substructure method and a flexible error compensation strategy for the deformation of the thin-walled part and tool in flank milling are proposed. The tool is considered to be a cantilevered beam to calculate its deformation. The thin-walled part is divided into two substructures, namely, the substructure with the material to be cut (STBC) and the unmachined substructure, which remains unchanged. Deformation models of the two substructures are constructed based on the element stiffness matrix of the thin-walled part. The time-varying stiffness and the removed material from the thin-walled part are only updated in the deformation model of the STBC. The deformation model of the entire thin-walled part is constructed by coupling the deformation models of the two substructures. In this way, the degrees of freedom of the deformation model of the entire thin-walled part are significantly reduced. To reduce the deformation error, a flexible error compensation strategy is proposed. The tool location at the initial feed position is directly compensated by the mirror principle. For the other tool locations, the tool location at the next feed position is adjusted with the compensated tool location at the previous feed position as the initial iteration value. This flexible error compensation strategy is based on the continuity of the compensated toolpath, which is close to the actual cutting process and adjusts the tool locations with high convergence. Finally, the proposed methods are proven via flank milling experiments. The deformation prediction results agree with the experimental results, and the deformation errors are reduced greatly after error compensation by using the proposed strategy.

Study on the Dynamic Modeling for Free Beam Structure Based on Parameter Identification

To improve the tank aiming accuracy, this paper studies on the nonlinear vibration characteristics caused by the clearance between the barrel and lining. Based on the parameter identification, a free beam modeling method with clearance is proposed. Firstly, based on the Hamilton principle and hypothesis mode method, the free beam dynamic model framework with clearance is established. Secondly, the clearance time-varying stiffness and damping are obtained by the wavelet identification method from experimental data. Hence, the clearance dynamic model is established to describe the nonlinear dynamic behavior accurately. Finally, based on the similarity theory, the scaled model is used to verify the modeling method effectiveness by evaluating the frequency response function (FRF) error between the proposed model and test model. The results show that the FRF relative error is less than 2% between the measured and simulated value based on time-varying parameters, the coincidence index is more than 85%, under the 95% confidence index. It is proved that the dynamic modeling method could improve the barrel-lining dynamic model accuracy effectively with the clearance factor.

Effect of gear topology discontinuities on the nonlinear dynamic response of a multi-degree-of-freedom gear train

Weight reduction is a recurring concern in the design of modern mechanical systems. This search may lead engineers to resort to using gears with holes in order to meet their requirements. This paper presents a methodology to carry out nonlinear dynamic analyses of a gear transmission with holes in the gear blanks subjected to a multiharmonic internal excitation. This work investigates the influence of these holes on the vibration levels, occurrence of contact loss and possible bifurcations. The numerical model features two flexible shafts coupled by a spur gear pair with holes. The gear model consists in two lumped masses and inertias and includes the gear backlash as well as the internal excitation sources that are the time-varying stiffness and the static transmission error (STE). The resulting mechanical system is solved in the frequency domain by the Harmonic Balance Method (HBM) coupled with an arc-length continuation algorithm and its stability is evaluated with Hill’s method. Results show that adding holes not only impacts the STE but also the mesh stiffness. The interactions of these two quantities has a substantial influence on the bifurcation structure along the main solution branch and leads to a decrease of the span of vibro-impact regions. As the applied static torque is increased, it is found that using holed gear blanks can effectively prevent contact loss and lead to a linear response.

A recognition method for crack position on the outer ring of full ceramic bearing based on the synchronous root mean square difference

In this paper, a method for recognizing the crack location on the outer ring of the full ceramic bearing is proposed, which considers the transmission of the vibration signal on the outer ring when a crack occurs. A two-degree-of-freedom nonlinear dynamic model of the full ceramic bearing with a crack is established, and the impact of the crack is evaluated through the change of the strain energy. Then, considering the attenuation of the vibration signal along the outer ring, the dynamic model is improved with a time-varying stiffness related with the crack position. By analyzing the ratio of the synchronous root mean square difference of the vibration signal, the position of the outer ring crack is recognized. The simulation results are compared with the experimental results to verify the accuracy and feasibility of the position recognition method. The research provides a new recognition method for bearing fault location, and helps with the fault diagnosis and status monitoring of bearing systems.

A single‐mode recursive validation method for modal identification of linear time‐varying structures based on prior knowledge

Discerning the spurious modes for time-invariant or extremely slow time-varying structures with time-consuming tools (stabilization diagram and clustering), which is the main task of the automated modal identification, has been developed in last years. However, there is still a challenge the recursive identification of the linear time-varying structures. This study presents a single-mode recursive validation method for the recursive identification of the linear time-varying structures. Since one major issue is that the properties of the physical modes are time-varying, the proposed method extracts the information about the current state of the time-varying structures from the nonstationary vibration responses via deep learning method for the separation between physical and spurious modes. For the sake of effective elimination to spurious modes and control of computation efforts, the proposed method utilizes the application-dependent prior knowledge rather than iterations or high-dimensional optimizations, with robustness to hyperparameters. It can be combined with any parametric system identification method. A time-varying stiffness numerical example and a time-varying mass distribution experimental example illustrate the performance of the proposed modal validation method under various time-varying processes.

Dynamic modeling and vibration analysis of a cracked 3K-II planetary gear set for fault detection

Abstract. The 3K planetary gear system is a basic planetary transmission structure with many advantages over the 2K-H planetary gear system. However, the vibration characteristics will be more complicated due to the increase of central gears meshing with each planet gear simultaneously. In this paper, a lumped-parameter model for a 3K-II planetary gear set was developed to simulate the dynamic response. The time-varying stiffness of each meshing pair for different gear tooth root crack faults is calculated via the finite element method. By considering the effect of time-varying transmission paths, the transverse synthetic vibrations are obtained. Subsequently, the feasibilities of transverse synthetic vibration signals and output torsional vibration signals as reference for fault diagnosis are analyzed by studying the time-domain and frequency-domain characteristics of these two vibration signals. The results indicate that both the transverse synthetic vibration signals and output torsional vibration signals can be used for fault identification and localization of the 3K-II planetary gear train, and yet they both have their limitations. Some results of this paper are available as references for the fault diagnosis of 3K planetary gear trains.

Dynamic Modeling and Nonlinear Characteristics of a Cracked Bolt-Disk Combined Rotor System

Abstract Different from the crack on the rotating shaft, the crack on the bolt, which is a connecting part of the bolt-disk combined rotor, is a kind of local defect. The local crack on the bolt under high pretension is always in open state, and it increases the overall vibration of the combined rotor significantly in practice. This paper studies the modeling of the crack on the bolt and nonlinear dynamic behaviors of the cracked bolt-disk rotor system. The circumferential bolts with a transverse open crack are treated as several bar elements under the assumption that each bolt has the same original tensile extension length. The cracked correction coefficient is introduced to describe the decreasing amount of bolt's tension due to crack. After this coefficient is obtained according to finite element method, the stiffness matrix of circumferential bolts with crack is built based on total potential energy. The dynamic model consists of a time-independent stiffness matrix for perfect bolts, a time-variant reductive stiffness, and an additional moment. As a result, the crack in bolt reduces rotor's nonlinear stability and leads to greater vibration and fluctuation. In addition, crack depth has much larger influence than crack location on the dynamic behaviors.

Dynamics Analysis of Rolling Bearings for Motor of a Mobile Power Station

The electric motor is an important part of a mobile power station. Under the complex operating conditions, the stiffness of the rolling bearing exhibits strong time-varying characteristics and non-linear characteristics, which is the one of the main sources for the system non-linearity. Taking the angular contact bearing as the research object, the coupled dynamic model of the bearing-rotor system is established to calculate the time-varying stiffness. The bifurcation law exhibited by the system can be attributed to the excitation of the non-linear bearing force. The numerical calculation method is used to study the motion state bifurcation law and dynamic frequency response characteristics under the rated working condition of the motor. Considering bearing damping, clearance and radial load, the bifurcation and time domain diagrams of different parameters, displacements and rotation speeds in different directions are studied respectively. The results show that when the parameters change, the system will experience chaos, quasi-period and period-doubling motion state; Damping suppresses vibration very clearly, while changes in bearing clearance and radial load significantly cause the system’s vibration amplitude to increase, and radial load changes accelerate system vibration. A reasonable bearing is of great significance for motor the design of a mobile power station.

Numerical and experimental analysis of the effect of eccentric phase difference in a rotor-bearing system with bolted-disk joint

Bolted joints are widely used in industrial rotating machines to fasten the adjacent disks together and affect system dynamic properties. Therefore, there is a strong need to study their influence on the response of such systems. This paper investigated the effect of the eccentric phase difference of the disk and bolted-disk joint on rotor dynamics, which takes into account the time-varying bending stiffness of the bolted joint. The bolted-disk joint is modeled as a two-node element based upon the energy theorem and Lagrange’s principle, where the relative lateral displacement stiffness, relative bending stiffness, and coupling stiffness between the adjacent disks are considered. And then the dynamic model of the rotor-bearing system is derived based on the proposed bolted joint element and lumped mass modeling method. Combining with the Newmark-β integration scheme, the established model allows the dynamic response characteristics of the rotor-bearing system with the bolted joint to be predicted, and the impact of eccentric phase difference in the rotor system on the response to be investigated. The validity of the simulation results was confirmed by experiment. Through the modeling method proposed in this paper and obtained results, the bifurcation characteristics of the bolted joint rotor system can be predicted.

Dynamic analysis of traction motor in a locomotive considering surface waviness on races of a motor bearing

The traction motor is the power source of the locomotive. If the surface waviness occurs on the races of the motor bearing, it will cause abnormal vibration and noise, accelerate fatigue and wear, and seriously affect the stability and safety of the traction power transmission. In this paper, an excitation model coupling the time-varying displacement and contact stiffness excitations is adopted to investigate the effect of the surface waviness of the motor bearing on the traction motor under the excitation from the locomotive-track coupled system. The detailed mechanical power transmission path and the internal/external excitations (e.g., wheel–rail interaction, gear mesh, and internal interactions of the rolling bearing) of the locomotive are comprehensively considered to provide accurate dynamic loads for the traction motor. Effects of the wavenumber and amplitude of the surface waviness on the traction motor and its neighbor components of the locomotive are investigated. The results indicate that controlling the amplitude of the waviness and avoiding the wavenumber being an integer multiple of the number of the rollers are helpful for reducing the abnormal vibration and noise of the traction motor.

Nonlinear characteristic investigation of tribo-crack-dynamic rotor system based on full spectrum analysis

The tribo-crack-dynamic model and experimental method to research dynamic mechanism are proposed based on the full spectrum in this paper. In view of the nonlinear characteristics from rub-impact forces, oil-film forces and time-varying stiffness of crack shaft, the tribo-crack-dynamic rotor model is established to serve the analysis of dynamic mechanism and characteristics. The numerical simulation is developed to obtain the precession signal and bifurcation diagram for the tribo-crack-dynamic rotor system. The results show that the negative frequency components are more abundant and their amplitudes increase in the full spectrum, showing obvious reverse precession characteristics when cracks and rub-impact faults occur at the same time. Experiment investigation on nonlinear dynamics is designed, which verifies the tribo-crack-dynamic model and the positive and negative precession characteristics in multi-fault rotor system.

Effects of transmission shaft flexibility on rolling element bearing tribodynamics in a high-performance transmission

In a high-performance transmission, rolling bearings play a crucial role in supporting the loaded shafts and gear mesh whilst minimising friction. Prediction of critical speeds and dynamic responses is important to avoid prolonged periods of elevated vibration amplitudes, which can contribute to premature bearing and gearbox failure. Nonlinearities such as backlash and time-varying stiffnesses can lead to complex coupled dynamics between gears and bearings. In the current study, an 8-speed transmission is discretized and modelled by the finite element method. The eigenproblem is solved using damped modal analysis considering gyroscopic effects. Time-varying stiffness data and backlashes are introduced as nonlinearities. The system response is compared for cases of linear and nonlinear bearings, and rigid and flexible shafts to ascertain the significance of elasticity. The flexible model improves the accuracy of bearing load oscillation by altering the transfer path between gears and bearings. This provides an advancement upon existing works, where shaft dynamics are often simplified. Furthermore, it is shown that the combination of local nonlinear effects and flexible components significantly changes the response of the system and the bearing life prediction for a system-level analysis.

Adaptive bidirectional dynamic response control of offshore wind turbines with time‐varying structural properties

Wind, wave and seismic loading induced excessive vibration of offshore wind turbines (OWTs) impairs the wind turbine energy output and escalates its fatigue damage. The present study proposes a three-dimensional adaptive pendulum tuned mass damper (3D-APTMD) to control the bidirectional dynamic responses of monopile OWTs exposed to combined wind, wave, seismic loading, and time-varying structural properties. In terms of the proposed adaptive tuning algorithm, the natural frequency and damping property of the 3D-APTMD are tuned in real time to match the wind turbine time-varying dominant frequency caused by environmental and/or structural property variations. Performance of the developed adaptive damper is examined on a monopile 5-MW (megawatt) baseline wind turbine subjected to representative wind, wave, and seismic loading. Time-varying stiffness of the tower and foundation is introduced to simulate damage evolution during service life. A three-dimensional pendulum tuned mass damper (3D-PTMD) and dual linear TMDs (2TMD) are used for comparison. Research results show that when damage occurs, 3D-PTMD and 2TMD become partially off-tuned and less effective. In comparison, the 3D-APTMD which is tuned in real time maintains its effectiveness. It can improve the reduction effect by around 25% and 30% when compared with the 3D-PTMD and 2TMD, respectively. Therefore, the proposed 3D-APTMD is promising for real application to protect OWTs, high-rise buildings, chimneys, and other types of slender structures suffering from excessive bidirectional vibration, and environmental, operational, and/or structural property variations.

A novel geometric model of breathing crack and its influence on rotor dynamics

The present research focuses on performing the dynamic study of a cracked, internally damped, composite rotating shaft system with journal bearing end supports. A novel mathematical formulation is proposed to introduce a time-varying stiffness matrix for simulating the breathing behaviour of the crack. The random search algorithm is used as one of the metaheuristic processes to carry out the optimization and generate time-dependent geometric parameters of the cracked surface for one full rotation. The derived stiffness matrix is used in the composite shaft’s higher order motion equation obtained through equivalent modulus theory, whose internal damping properties are incorporated using operator-based viscoelastic model. Eigenanalysis is carried out to perform a thorough study of dynamic characteristics of cracked shaft considering the crack depth and crack position as two important parameters. The effect of crack on stacking sequence as well as mode shapes of the heterogeneous laminated shaft is also studied.

A method for calculating radial time-varying stiffness of flexible cylindrical roller bearings with localized defects

In this paper, the flexibility of bearing parts and localized defects are considered, and a method for calculating the radial time-varying stiffness of flexible cylindrical roller bearings (CRBs) with localized defects is proposed. The proposed method uses the finite element (FE) method to calculate the global flexibility of bearing parts and adopts the analytical contact theory to calculate the localized contact flexibility of bearing parts. Furthermore, based on the displacement excitation assumption, localized defect models of the outer race, inner race, and roller are established. The proposed method is verified by comparing with the methods in the references and the finite element method. The effect of different angles of defect width on time-varying stiffness is discussed. The results show that as the angle of defect width increases, the width of the stiffness saltation also gradually increases. Besides, the calculation accuracy of the proposed method is higher than that of the classical analytical method, and the computational efficiency is greatly improved compared with the finite element method. This study can provide a theoretical basis for defect diagnosis of CRBs.

Parameters research on time-varying stiffness of the ball bearing system without race control hypothesis

When the ball bearing serving under the combined loading conditions, the ball will roll in and out of the loaded zone periodically. Therefore the bearing stiffness will vary with the position of the ball, which will cause vibration. In order to reveal the vibration mechanism, the quasi static model without raceway control hypothesis is modeled. A two-layer nested iterative algorithm based on Newton–Raphson (N-R) method with dynamic declined factors is presented. The effect of the dispersion of bearing parameters and the installation errors on the time-varying carrying characteristics of the ball-raceway contact and the bearing stiffness are investigated. Numerical simulation illustrates that besides the load and the rotating speed, the dispersion of bearing parameters and the installation errors have noticeable effect on the ball-raceway contact load, ball-inner raceway contact state and bearing stiffness, which should be given full consideration during the process of design and fault diagnosis for the rotor-bearing system.

Determination of Mesh Stiffness of Gear—Analytical Approach vs. FEM Analysis

This paper focuses on modeling the time-varying stiffness of spur gearings, which in dynamic models of transmission systems acts as an important element of the internal excitation of the dynamic system. Here are introduced ways to approach the modeling of gear stiffness using analytical calculations, which allow to model the course of mesh stiffness depending on its rotation. For verification of used analytical model were created five different gearings, and based on their geometry, the respective stiffness curves were analytically determined. Subsequently, a finite element simulation was performed in the Abaqus CAE software. Due to this software, it was possible to identify and objectively compare the stiffness curves and further determine the suitability of using the analytical model to determine the mesh stiffness of gearing.

The vibration analysis of the CNC vertical milling machine spindle system considering nonlinear and nonsmooth bearing restoring force

The performance improvement of the computer numerically controlled (CNC) vertical milling machine is constrained by limited understanding of the dynamic behaviors of the spindle system. In order to study the dynamic characteristics of the CNC vertical milling machine spindle system under unbalanced force, the lumped mass method is applied to propose a novel dynamic model. In this comprehensive model, Hertz contact, radial clearance and time-varying stiffness of rolling bearings are considered. According to the contact behaviors of rolling bearings supporting the spindle, restoring force is a nonsmooth and nonlinear function of displacements and bearing radial clearance. Then, governing differential equations of the eighteen-degree-of-freedom system are derived using the Lagrange equation. In addition, the effectiveness of this dynamic model is verified by experiments. The influences of spindle speed and the bearing parameters on the vibration behaviors and stability of the spindle are analyzed by amplitude-frequency curve, bifurcation diagram, time domain waveform, spectrum diagram, trajectory and Poincaré. Moreover, we discussed how the dynamics of the spindle are affected by the eccentricity of the belt pulley, the extended length of the milling tool, and the position of the intermediate bearing set. The research results can benefit the structural design of the spindle, optimize the bearings selection, as well as improve the machining accuracy.

Vibration and Stability Analysis of a Bearing–Rotor System with Transverse Breathing Crack and Initial Bending

This paper focuses on the stability and nonlinear response of a bearing-rotor system affected by a transverse crack and initial bending which was thought to be part of an unbalance or had been neglected before. The differences of breathing functions for the transverse breathing crack caused by initial bending is presented here, and the calculation of time-varying finite elements stiffness matrix of the cracked shaft is improved by replacing traditional the approximate crack segment with an exact area. After establishing the dynamic model of the cracked rotor with initial bending, vibrational characteristics such as amplitude-speed diagram, frequency spectrogram and bifurcations are investigated in detail. The eigenvalues of the transition matrix are calculated and analyzed as an indicator of dynamic stability with the growths of crack depth and initial bending. Many differences are found between the two cases of dynamic response of rotor system by numerical simulation. The frequency change with the growth of initial bending is opposite to the change with the growth of crack depth, and the shapes of amplitude-speed also having great different features. Stable regions are reduced and extended laterally by initial bending. All these results obtained in this paper will contribute to identify the bending fault and assess the stability of the bearing-rotor systems.

Nonlinear dynamical behaviors of spiral bevel gears in transient mixed lubrication

Herein, a new and actual nonlinear dynamic model of spiral bevel gears (SBG) in transient mixed lubrication was firstly developed, which considered the coupling action of direct contact of asperity and the transient squeezing effect of lubricant along the entire contact trajectory in the form of time-variant contact stiffness and damping. Besides, the nonlinear backlash, actual time-variant rotating contact radii and static transmission error obtained by loaded tooth contact analysis (LTCA) method were all involved in this model. Based on the proposed model, notable deviations were found compared with traditional Hertz contact or full-film lubrication assumptions, which highlight the important role of film lubrication and asperity contact. After that, a series of investigations on nonlinear dynamic of SBG were implemented to reveal the influence of lubrication, roughness, applied torque, especially, the influence of contact trajectory which has not been reported. The results demonstrate that the influence of roughness on dynamic response are complicated, and the excessive roughness tends to arise bilateral impact of gear tooth. The effect of transmission torque is more significant than that of contact stiffness and damping. As for different contact trajectories, the vibration response of heel contact is the strongest, while toe contact leads to the weakest vibration.