Quasi-periodic Motions Research Articles

Revised method of multiple scales for 1:2 internal resonance piezoelectric vibration energy harvester considering the coupled frequency

The piezoelectric vibration energy harvester based on the 1:2 internal resonance possesses the advantages of wide-band and high-efficiency energy harvesting. The method of multiple scales (MMS) has been applied to derive the approximate analytical solution of the output response. However, the coupled frequency caused by internal resonance has strong effect on the output voltage response, which cannot be determined by traditional MMS. To overcome this shortage, the method of equivalent forced load is proposed to decouple the voltage frequency, and then obtain the revised approximate analytical solution (MMS-New) of the output responses. The responses of the electromechanical-coupled governing equations of the system are verified by experiments. Compared with the approximate solutions of the traditional MMS, the accuracy of the revised approximate solutions are significantly improved. In the internal resonance region, the maximum accuracy of output voltage response can be improved by 40.64%, and the maximum accuracy can be increased to 10 times outside the internal resonance region. In the whole excitation frequency range, the amplitude–frequency response curve exhibits nonlinear hardening and softening when the forward and reverse sweep frequency analysis are respectively performed. The Jacobi matrix and Lyapunov stability theory of the modulation equations are derived to determine the stability of the approximate solutions. The dynamic behavior in unstable region are investigated using means of time history curve, spectrum, phase orbit and Poincare section, the quasi periodic and chaotic motions are observed.

Nonlinear vibration response of a helicopter supercritical tail rotor drive shaft equipped with a dry friction damper

PurposeThe supercritical design of tail rotor drive shaft has attracted more attention in helicopter design due to its high power–weight ratio and low maintenance cost. However, there exists excessive vibration when the shaft passes through the critical frequency. Dry friction damper is the equipment applied to the drive shaft to suppress the excessive vibration. In order to figure out the damping mechanism of the dry friction damper and improve the damping efficiency, the dynamic model of the shaft/damper system is established based on the Jeffcott rotor model.Design/methodology/approachThe typical frequency response of the system is studied through bifurcation diagrams, amplitude-frequency characteristic curves and waterfall frequency response spectrum. The typical transient responses under frequency sweeps are also obtained.FindingsThe results show that the response of the system changes from periodic no-rub motion to quasi-periodic rub-impact motion, and then to synchronous full annular rub-impact, and finally, back to periodic no-rub motion. The slip of the rub-impact ring improves the stability of the system. Besides, the effects of the system parameters including critical dry friction force, rub-impact friction coefficient, initial clearance on the stability and the vibration damping capacity are studied. It is observed that the stability changes significantly varying the three parameters respectively. The vibration damping capacity is mainly affected by the critical dry friction force and the initial clearance.Originality/valuePresented results provide guidance for the design of the dry friction damper.

Simulation of Random Deformable Motion in Soft-Tissue Cone-Beam CT with Learned Models.

Cone-beam CT (CBCT) is widely used for guidance in interventional radiology but it is susceptible to motion artifacts. Motion in interventional CBCT features a complex combination of diverse sources including quasi-periodic, consistent motion patterns such as respiratory motion, and aperiodic, quasi-random, motion such as peristalsis. Recent developments in image-based motion compensation methods include approaches that combine autofocus techniques with deep learning models for extraction of image features pertinent to CBCT motion. Training of such deep autofocus models requires the generation of large amounts of realistic, motion-corrupted CBCT. Previous works on motion simulation were mostly focused on quasi-periodic motion patterns, and reliable simulation of complex combined motion with quasi-random components remains an unaddressed challenge. This work presents a framework aimed at synthesis of realistic motion trajectories for simulation of deformable motion in soft-tissue CBCT. The approach leveraged the capability of conditional generative adversarial network (GAN) models to learn the complex underlying motion present in unlabeled, motion-corrupted, CBCT volumes. The approach is designed for training with unpaired clinical CBCT in an unsupervised fashion. This work presents a first feasibility study, in which the model was trained with simulated data featuring known motion, providing a controlled scenario for validation of the proposed approach prior to extension to clinical data. Our proof-of-concept study illustrated the potential of the model to generate realistic, variable simulation of CBCT deformable motion fields, consistent with three trends underlying the designed training data: i) the synthetic motion induced only diffeomorphic deformations - with Jacobian Determinant larger than zero; ii) the synthetic motion showed median displacement of 0. 5 mm in regions predominantly static in the training (e.g., the posterior aspect of the patient laying supine), compared to a median displacement of 3.8 mm in regions more prone to motion in the training; and iii) the synthetic motion exhibited predominant directionality consistent with the training set, resulting in larger motion in the superior-inferior direction (median and maximum amplitude of 4.58 mm and 20 mm, > 2x larger than the two remaining direction). Together, the proposed framework shows the feasibility for realistic motion simulation and synthesis of variable CBCT data.

Three-dimensional dynamics of a cantilevered pipe conveying pulsating fluid

This paper investigates the stability and three-dimensional (3-D) nonlinear dynamics of a cantilevered pipe with an internal fluid having a harmonic component of flow velocity superposed on a constant mean value. The nonlinear equations of motion for an inextensible cantilevered pipe with the consideration of internal pulsating flow are presented. The nonlinear inertial terms in the governing equations are replaced by equivalent displacement and velocity terms by using a perturbation method. The partial differential equations are then transformed into a set of ordinary differential equations (ODEs) by using the Galerkin method. The instability regions of the subcritical and supercritical resonances of a linear system are determined via the Floquet theory. The effects of mean flow velocity and mass ratio are investigated. The resulting coupled nonlinear differential equations are numerically solved using a fourth-order Runge-Kutta integration scheme for the subcritical and supercritical flow velocities. The nonlinear dynamical responses are presented in the form of bifurcation diagrams, time histories, phase portraits, power spectral densities (PSDs) and Poincaré maps. Some interesting and sometimes unexpected results have been observed with different flow velocities. The analytical model is found to exhibit rich and variegated dynamical behaviors which include 2-D or 3-D periodic, quasiperiodic and chaotic motions. The convergence analysis of the number of truncating modes in the Galerkin approach is also conducted.

Dynamic analysis of an axially moving underwater pipe conveying pulsating fluid

In this paper, both linear and non-linear dynamics of a slender and uniform pipe conveying pulsating fluid, which is axially moving in an incompressible fluid, are comprehensively studied. The vibration equations of the system are established by considering various factors, including a coordinate conversion system, an “axial added mass coefficient” describing the additional inertia forces caused by the external fluid, the Kelvin–Voigt viscoelastic damping, a kind of non-linear additional axial tension, and the pulsating internal fluid. The vibration equations are discretized by the Galerkin procedure and solved by the Runge–Kutta approach, and the validity of the solution procedure is carefully checked. After that, the linear and non-linear responses of the system are studied when the internal flow velocity and the axially moving speed of the pipe are small. For linear responses, the Kelvin–Voigt viscoelastic damping has great influences on the second and third modes of the system. For the non-linear dynamic, the results are rich and changeful, including the first and second principal parametric resonances, the secondary resonance, the combination resonance, period-1 motion, quasi-periodic motion, and chaotic motion. Finally, the influence of several key system parameters on the non-linear responses is analyzed.

Effect of the unbalanced magnetic pull force and oil-film force on nonlinearity of a dual-rotor bearing system

Turbogenerator sets, the indispensible rotating machinery in thermal power plants, can be simplified into a dual-rotor bearing system. In this case, its nonlinear phenomenon largely results from the unbalanced magnetic pull (UMP) and the nonlinear oil film forces, which, however, are not well understood. In this study, a mathematical model for a dual-rotor bearing system is established, the effects of the UMP and the oil-film forces on its nonlinear phenomenon are investigated creatively. Of which, the UMP force is initially calculated by the integration of the air-magnetic energy, and the short journal bearing assumption is employed to derive the oil-film force. The fitness of the developed model is validated through the stability experiments of a 660 MW turbogenerator set. Results indicate that in the idling condition, nonlinear phenomenon of the system can be described as the period-1 and quasi-periodic motions as the rotational speed rises. At low speeds, the amplitude of oil whirl appears. With the rising rotational speed, the oil whirl turns into oil whip in the second quasi-periodic. Interestingly, the electromagnetic force can greatly alleviate the system amplitude and enhance its stability considering the UMP force. The nonlinear effect of air-gap distance of the system under load conditions is also investigated. The system is still in the quasi-periodic state with the increasing air-gap distance at low/high speed. But the system is still unstable owing to oil whirl/whip and UMP force, causing the system experiences multiple states and becomes more stable at the intermediate speed.

An Investigation of the Bifurcation Behavior of an F-18 Aircraft Model

Bifurcations of equilibria and limit cycles of an F-18 aircraft model have been investigated in this paper based on one or two continuation parameters. First, it is shown that the aircraft can be in a stable straight-and-level flight state by coordinating the elevator deflection and the engine thrust and fixing the other parameters. Second, the aircraft may be disturbed by perturbations out of the longitudinal plane and get into the lateral-directional motion mode by a branch point bifurcation. Third, when the straight flight state loses its stability, body axis angular rates will present periodic or quasi-periodic motion pattern by the Hopf or Neimark–Sacker bifurcation. Finally, bifurcation structure of limit cycles has been exhibited, including the generalized Hopf bifurcation, the Neimark–Sacker bifurcation, the Hopf-Hopf bifurcation and the fold-Neimark–Sacker bifurcation. Meanwhile corresponding bifurcation curves in two-parameter plane have been depicted with the help of numerical continuation techniques.

Nonlinear dynamic analysis of parked large wind turbine blade considering harmonic inertial excitation using continuum mathematical model

In this paper, an investigation on the nonlinear dynamics of parked large wind turbine blades considering harmonic inertial excitation is presented. The investigation is based on the continuum mathematical model which is derived using the extended Hamilton principle. Employing the Euler–Bernoulli beam model, the equations of motion are derived by considering the flap-wise and edge-wise deformations as well as the cubic nonlinearity. The proposed model is validated through comparisons with both published results and numerical simulations. The nonlinear dynamics of the NREL 5-MW wind turbine blade is investigated numerically using a bifurcation toolbox MATCONT. The steady state primary resonances of the first three modes are compared and analyzed with varying excitation frequency and excitation direction. Results show that for parked blades the aerodynamic damping increases with the increase of inflow wind speed, and the response under low wind speed becomes more unfavorable than high wind speed. The blade response can suddenly jump up or down when the excitation direction varies slightly because of the saddle node bifurcation. The 3:1 internal resonance between the 1st and 3rd modes is also observed in this study, suggesting the possible interaction of the two modes during large amplitude vibration. Quasi-periodic motions can also be induced because of the Hopf bifurcation in the case of negative aerodynamic damping.

The Nonlinear Dynamic Behavior of a Particle on a Vibrating Screen Based on the Elastoplastic Contact Model

The dynamic response of particles is closely related to screening efficiency. To study the dynamic response of particles, the dynamic equations of a particle on a screening surface are established based on the elastoplastic contact model of spherical particles and are solved for the coal particle. Then, the trajectories of the particles are given with different falling heights and particle radii. The completely different trajectories with slight changes in the falling height and particle radius indicate strong nonlinearity. Second, the nonlinear dynamic behavior under different amplitudes and frequencies is discussed, and the route of transition from quasiperiodic motion to chaotic motion is revealed. Finally, we discuss the average speed along the screening surface considering the frequency, amplitude, friction coefficient, inclination angle, and vibration direction angle. In addition, the convergence conditions of particle motion are proposed, and they are only affected by the inclination angle and friction angle. The results show that in the normal direction of the vibrating screen, the particle motion is quasiperiodic at low frequencies. With increasing frequency, the motion of the particle becomes chaotic, and its Poincaré map becomes petal-shaped. In addition, the number of petals increases at the mutation of the bifurcation diagram. The increase in frequency, amplitude and inclination angle and the decrease in friction coefficient lead to an increase in particle speed along the screen surface. In addition, the particle speed reaches a maximum when the vibration direction angle is 65°. This work provides a theoretical basis for controlling the thickness of granular material flow on a vibrating screen and selecting screening process parameters.

Chordwise flexible aft-tail suppresses jet-switching by reinstating wake periodicity in a flapping foil

The effect of a chordwise flexible aft-tail of a rigid heaving aerofoil on the dynamical transitions of the trailing-wake is studied here. The two-way coupled fluid–solid dynamics is simulated using an in-house fluid–structure interaction (FSI) platform, comprising a discrete forcing immersed boundary method based incompressible Navier–Stokes solver, weakly coupled with a finite difference method based structural solver. The FSI dynamics is studied in comparison to the corresponding rigid tail configuration. For the latter, mild jet-switching due to quasi-periodic movement of the wake vortices gives way to vigorous jet-switching as the dynamics transitions to a state of intermittency, where the quasi-periodic behaviour gets interspersed with chaotic windows. Introduction of a moderately flexible tail regularises this intermittent dynamics, eliminating jet-switching. The wake exhibits a deflected reverse Kármán pattern with fluctuating angles, governed by quasi-periodicity. With a highly flexible tail (very low rigidity), the wake shows almost a symmetric reverse Kármán street as periodicity is restored. Flexibility of the aft-tail is next controlled by changing its length, and flow is regularised and periodicity retained for moderate rigidity for increased length. Different dynamical states are established through robust nonlinear dynamical tools. The underlying flow-field behaviour, instrumental in suppressing the jet-switching phenomenon, is identified through a detailed investigation of the near-field vortex interactions dictated by the dynamics. A suite of measures has also been derived from the unsteady flow field to quantify the interactions of the key near-field vortices with a view to understanding the mechanism of switching and its subsequent suppression through flexibility.

Experimental Verification and Nonlinear Dynamic Response Analysis of a Rolling Element Bearing with Localized Defects

In this paper, the dynamic behavior of rolling element bearings with localized faults on the inner and outer rings is investigated. A nonlinear mathematical model is developed with five degrees of freedom considering rotor unbalance. In this bearing model, the nonlinearity is caused by the Hertzian contact forces and the radial internal clearance. The fourth-order Runge–Kutta technique is used to solve the coupled nonlinear equations of motion numerically. Nonlinear vibration response of the rotor and bearing housing can be obtained in both time and frequency domains. An experimental verification of the numerical model is presented where experimental measurements for defective ball bearings are compared with the numerical results. Envelope spectra of the numerical results show similar behavior to that of the measured experimental signals. A parametric analysis is conducted to investigate the effect of system parameters on the nonlinear dynamic response using time waveforms, orbit plots, frequency spectra and bifurcation diagrams. The presented results demonstrate that the dynamic response shows periodic, quasi-periodic and chaotic motions because of varying rotational speeds and defect width. The proposed model contributes toward improved design and better health monitoring of bearings in practice.

ECG Patient Simulator Based on Mathematical Models.

In this work, we propose a versatile, low-cost, and tunable electronic device to generate realistic electrocardiogram (ECG) waveforms, capable of simulating ECG of patients within a wide range of possibilities. A visual analysis of the clinical ECG register provides the cardiologist with vital physiological information to determine the patient’s heart condition. Because of its clinical significance, there is a strong interest in algorithms and medical ECG measuring devices that acquire, preserve, and process ECG recordings with high fidelity. Bearing this in mind, the proposed electronic device is based on four different mathematical models describing macroscopic heartbeat dynamics with ordinary differential equations. Firstly, we produce full 12-lead ECG profiles by implementing a model comprising a network of heterogeneous oscillators. Then, we implement a discretized reaction–diffusion model in our electronic device to reproduce ECG waveforms from various rhythm disorders. Finally, in order to show the versatility and capabilities of our system, we include two additional models, a ring of three coupled oscillators and a model based on a quasiperiodic motion, which can reproduce a wide range of pathological conditions. With this, the proposed device can reproduce around thirty-two cardiac rhythms with the possibility of exploring different parameter values to simulate new arrhythmias with the same hardware. Our system, which is a hybrid analog–digital circuit, generates realistic ECG signals through digital-to-analog converters whose amplitudes and waveforms are controlled through an interactive and friendly graphic interface. Our ECG patient simulator arises as a promising platform for assessing the performance of electrocardiograph equipment and ECG signal processing software in clinical trials. Additionally the produced 12-lead profiles can be tested in patient monitoring systems.

Dynamic modeling and analysis of the internal gear transmission with tooth crack subjected to the heavy torque

This work addresses the free and nonlinear vibrations of the internal gear transmission. A rigid-elastic coupled model is established using Hamilton's principle and the nonlinear in-extensional assumption, where the cracked time-varying mesh stiffnesses regarding torque-dependent and geometric nonlinearity caused by the heavy torque on the elastic ring gear are considered. Based on the coupled model, the natural frequencies and vibration mode shapes are given, and the distinctive modal properties associated with the torque are investigated by comparing with a rigid model. The differences of the nonlinear behaviors between the two models are presented through bifurcation diagrams, the largest Lyapunov exponent (LLE) diagrams, phase diagrams, Poincaré maps, time histories, and FFT spectra. The effects of parameters such as excitation frequency, torque, and crack depth on the dynamic responses are examined. The results show that the nonlinear behaviors are rich, and quasi-periodic motion replaces periodic-1 motion when the elasticity of the ring gear is considered. The dynamic responses indicate that the larger crack depth becomes, the more obvious the sidebands can be found in the frequency domain. As for lower excitation frequency, the main frequencies also include the mesh, elastic vibration, modulation, and natural frequencies.

Non-Linear Interactions of Jeffcott-Rotor System Controlled by a Radial PD-Control Algorithm and Eight-Pole Magnetic Bearings Actuator

Within this work, the radial Proportional Derivative (PD-) controller along with the eight-poles electro-magnetic actuator are introduced as a novel control strategy to suppress the lateral oscillations of a non-linear Jeffcott-rotor system. The proposed control strategy has been designed such that each pole of the magnetic actuator generates an attractive magnetic force proportional to the radial displacement and radial velocity of the rotating shaft in the direction of that pole. According to the proposed control mechanism, the mathematical model that governs the non-linear interactions between the Jeffcott system and the magnetic actuator has been established. Then, an analytical solution for the obtained non-linear dynamic model has been derived using perturbation analysis. Based on the extracted analytical solution, the motion bifurcation of the Jeffcott system has been investigated before and after control via plotting the different response curves. The obtained results illustrate that the uncontrolled Jeffcott-rotor behaves like a hard-spring duffing oscillator and responds with bi-stable periodic oscillation when the rotor angular speed is higher than the system’s natural frequency. It is alsomfound that the system, before control, can exhibit stable symmetric motion with high vibration amplitudes in both the horizontal and vertical directions, regardless of the eccentricity magnitude. In addition, the acquired results demonstrate that the introduced control technique can eliminate catastrophic bifurcation behaviors and undesired vibration of the system when the control parameters are designed properly. However, it is reported that the improper design of the controller gains may destabilize the Jeffcott system and force it to perform either chaotic or quasi-periodic motions depending on the magnitudes of both the shaft eccentricity and the control parameters. Finally, to validate the accuracy of the obtained results, numerical simulations for all response curves have been introduced which have been in excellent agreement with the analytical investigations.

Автоколивання тришарової усіченої конічної оболонки із стільниковим заповнювачем, виготовленим адитивними технологіями

This paper presents a nonlinear mathematical model of self-vibrations of conical sandwich shells with a honeycomb core made by additive technologies. The vibrations of the structure are described by fifteen unknowns. Each layer of the structure is described by five unknowns: three projections of the displacements of the layer middle surface and two rotation angles of the middle surface normal. Displacement continuity conditions at the layer interfaces are used. The higher-order shear theory is used to describe the stress-strain state of the structure. The case of conical sandwich shell ? supersonic gas flow interaction is considered. Due to this interaction, self-vibrations of the shell structure are set up. In their analysis, the geometrical nonlinearity of the structure is accounted for. Motion equations of the structure are derived using the assumed-mode method, which uses the kinetic and the potential energy of the structure. The self-vibrations are represented as eigenmode expansions, which contain a set of generalized coordinates. A system of nonlinear autonomous ordinary differential equations in the generalized coordinates is derived. The self-vibrations are studied using a combination of the shooting technique and the parameter continuation method. Multipliers are calculated to analyze the stability of periodic vibrations and their bifurcations. The dynamic instability of the structure’s trivial equilibrium is studied by numerical simulation. For clamped-clamped and cantilever shells, the properties of their periodic, quasiperiodic, and chaotic motions are analyzed in detail.

Quasi-periodic motions in a two-class economy with technology choice: an extreme case

Quasi-periodic motions in a two-class economy with technology choice: an extreme case

A collocation algorithm based on septic B-splines and bifurcation of traveling waves for Sawada–Kotera equation

In present work, a mathematical model standing for the numerical solutions of the nonlinear Sawada–Kotera (S–K) equation has been examined. For this purpose, we have used collocation finite element method for the model problem. The solution for each unknown is written as a linear combination of time parameters and combination of the septic B-splines. Next, applying Von-Neumann theory, we demonstrate that the scheme is marginally stable. Error norms L2 and L∞ are computed for single soliton solutions to display the practicality and robustness the suggested scheme. The obtained numerical results have been illustrated with tables. Furthermore, to show the numerical behavior of the single soliton, some of the simulations are depicted in 2D and 3D. Present results show that the method procures high accurate solutions so present scheme will be useful for the other nonlinear scientific problems. After that, bifurcation behavior of traveling waves for the S-K equation has been examined depending on the velocity (v) of the traveling wave. It has been seen that velocity (v) of the traveling wave plays a key role for the existence of different kind of quasiperiodic motions of the nonlinear S-K equation. Coexisting quasiperiodic motions for traveling wave solutions of the equation have been presented.

Dynamics simulation of rolling bearing system with pedestal looseness in speed-up train

The paper mainly studies the dynamics characteristics of the roller bearing system with pedestal looseness in speed-up train. Considering eccentricity, damping coefficient, stiffness of shaft, and support forces of roller bearing, a centralized mass model of roller bearing system of speed-up train is presented. Moreover, numerical analysis method is adopted to solve the dynamics responses. Based on the trajectory cures, phase plane portraits, Poincaré maps, and amplitude spectrum cures, the dynamics characteristics are analysed when the motor rotating speed and gap of the looseness are changed. The simulation results reveal some complex dynamic behaviour such as periodic motion, quasi-periodic motion, multi-periodic motion and chaotic motion. These results will help us understand and harness these dynamics behaviour of the proposed bearing system of speed-up train.

Dynamic Modeling and Stability Analysis for a Spur Gear System Considering Gear Backlash and Bearing Clearance

In practice, gear backlash and bearing clearance usually exist together in a gear system. They may induce complicated dynamic responses and degrade transmission performance. Up to now, although each of them has been researched, little attention has been paid to the coupling dynamic characteristics of gear backlash and bearing clearance. In a limited number of relevant studies, since the linear collision models they adopted are difficult to realistically depict actual collision behaviors caused by bearing clearance, these studies cannot accurately reveal the coupling dynamic characteristics of gear backlash and bearing clearance. Furthermore, system stability of the gear system considering gear backlash and bearing clearance has not been thoroughly investigated. In view of this, this paper contributes to the research on dynamic modeling and stability analysis for the spur gear system considering gear backlash and bearing clearance. A nonlinear collision model with time-varying contact stiffness/damping is suggested for describing the bearing collision behaviors. Based on the geometrical relationship of dynamic center distance, dynamic working pressure angle, and dynamic backlash, the coupling motion model of gear backlash and bearing clearance is developed. On this basis, the dynamic model of the spur gear system considering gear backlash and bearing clearance is established and verified by numerical simulations, virtual prototyping simulations and experiments. Afterwards, to thoroughly explore the complicated dynamic characteristics of the gear system considering gear backlash and bearing clearance, several important parameters, i.e., rotational speed, gear backlash and bearing clearance, are chosen as bifurcation parameters to study their influences on system stability via bifurcation diagrams, time-domain waveforms, FFT spectra, Poincaré maps, and phase diagrams. Various complicated nonlinear behaviors, such as hopping, multiple periodic motion, quasi-periodic motion, and chaotic motion, are revealed. This study can provide useful reference for the multi-clearance coupling research of the gear system in complicated working environments.

Nonlinear analysis of axial-torsional vibration of drill string based on a 3 DOF model

Based on a model with three-degree-of-freedom, the nonlinear coupled axial-torsional vibration of drill string is numerically studied considering a modified contact model of bit-rock interaction. The effects of driving angular velocities, nominal drilling pressures, and formation stiffness on response are discussed. The simulation results show that response can be periodic, quasi-periodic or chaotic motion with the variation of system parameters. Increasing driving angular velocity is helpful to reduce the duration of sticking and even make the system stick-slip free. However, bit-bounce may occur if driving angular velocity is too large. Increasing nominal drilling pressure can cause the system to be more prone to stick-slip, but it has dampening effect on bit-bounce. The larger the formation stiffness is, the smaller the amplitude of the axial vibration response of the system is, and the driving angular velocity corresponding to the maximum amplitude of the axial vibration response is also larger, whereas the variation of the amplitude of torsional vibration is smaller. Compared with the results from two-degree-of-freedom model, the amplitude of torsional vibration of the present model is remarkable greater. Meanwhile, the present model is more prone to stick-slip than the one with two-degree-of-freedom, and it is not easy to bit bounce.