Nonlinear Torsional Vibration Research Articles

Research on Nonlinear Vibration of Dual Mass Flywheel Considering Piecewise Linear Stiffness and Damping

Nonlinear torsional vibration differential equation of the nested arc-shaped short spring dual mass flywheel (DMF) is established, considering the piecewise linear stiffness and damping of the spring. The first-order approximate analytical solution under sinusoidal excitation and the amplitude–frequency characteristic function are obtained by means of the average method which verified by the Runge–Kutta (R–K) method. The effects of the parameters of input excitation, inertia, and piecewise linear stiffness and damping of DMF on the resonant amplitude, resonant frequency band, and equivalent linear natural frequency of the system are analyzed. The results show that the amplitude–frequency characteristic curve bending and jumping with the changes of excitation frequency and the peak of resonant amplitude can be obviously reduced by increasing the inertia of the primary flywheel and decreasing the inertia of the secondary flywheel. The complex nonlinear dynamic phenomena such as Period 1, quasi-periodic, and chaos are obtained by analyzing the forced vibration response under the different excitation frequencies.

Nonlinear torsional vibration and dynamic post-buckling responses of spiral stiffened functionally graded porous cylindrical shells

This study employs a semi-analytical approach to investigate the nonlinear torsional vibration and dynamic torsional post-buckling (DTPB) responses of spiral stiffened functionally graded (FG) porous (SSFGP) cylindrical shells. These shells are resting on a generalized nonlinear viscoelastic foundation (GNVEF). This foundation consists of a dual-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model. The model includes nonlinear cubic stiffness and takes damping effects into consideration. Within the scope of this research, two variations of SSFGP cylindrical shells are examined: those characterized by non-uniform and uniform porosity distributions. Employing the Donnell shell theory, von-Kármán nonlinear geometric assumptions, and Galerkin’s method, a discretized nonlinear governing equation is derived to analyze the behaviors of the shells. Consequently, explicit formulations for dynamic torsional load are meticulously obtained. The findings of the present study are validated by comparing them with the outcomes documented in existing literature, as well as through alignment with the P-T method. This research delves into the system’s nonlinear behaviors, with scrutinization of the effects of diverse factors such as material and geometrical parameters. The researchers and engineers in this field may use the findings of this research as benchmarks for their design and research of SSFGP cylindrical shells.

Nonlinear behavior of quasi-zero stiffness nonlinear torsional vibration isolator

Nonlinear behavior of quasi-zero stiffness nonlinear torsional vibration isolator

Three-dimensional magnetic field and thermal environment, and parameter uncertainty effects on nonlinear torsional vibration of an embedded rod composed of two dissimilar rods welded by friction welding

Three-dimensional magnetic field and thermal environment, and parameter uncertainty effects on nonlinear torsional vibration of an embedded rod composed of two dissimilar rods welded by friction welding

Modeling of nonlinear torsional vibrations of a truncated conical rod

Разработана нелинейная математическая модель нестационарных крутильных колебаний усеченного конического стержня из упругого материала с учетом нелинейной связи между напряжениями и деформациями. Выведено нелинейное уравнение для крутильных колебаний усеченного конического стержня относительно главной части крутильного перемещения оси симметрии стержня. Показано, что полученное уравнение нелинейных крутильных колебаний усеченного конического упругого стрежня в частных случаях совпадает с известными уравнениями, полученными другими авторами. С помощью полученного уравнения можно однозначно определить напряженно-деформированное состояние произвольного сечения конического стержня по пространственной координате и времени. На основе построенной модели численно решена задача о нестационарных крутильных колебаниях усеченного конического стержня при действии торцевой и поверхностной динамических нагрузок в условиях, когда широкий конец стержня жестко заделан, а узкий является свободным.

Nonlinear torsional vibration analysis of shearer semi-direct drive cutting transmission system subjected to multi-frequency load excitation

Nonlinear torsional vibration analysis of shearer semi-direct drive cutting transmission system subjected to multi-frequency load excitation

Analysis of the influence of piston–cylinder friction on the torsional vibration characteristics of compressor crankshaft system

With the development of compressors toward high speed and multiple columns, the torsional vibration phenomenon has become a major factor affecting the service life and reliability of the shaft system. Therefore, this paper considers the influence of friction between piston and cylinder on the instantaneous inertia of the crank connecting rod mechanism, establishes a nonlinear torsional vibration mechanics model of shale gas compressor shaft system, solves the natural frequency of the shaft system under undamped and damped conditions using the eigenvector method, and investigates the influence of the friction coefficient between piston and cylinder and the operating speed on the torsional vibration response of the shaft system under the self-excitation of the shaft system and the action of and excitation moment by the Runge–Kutta methods. The results show that after considering the friction between the piston and the cylinder, the second-order natural frequency of the shaft system shows a "high-low–high" fluctuation pattern. As the friction coefficient increases, the amplitude of the shaft system and the peak vibration speed show a rising trend. Meanwhile, when the speed increases, the vibration of the shaft system changes from chaotic \ period to the proposed periodic state, but the amplitude shows a decreasing trend. The research in this paper aims to improve the theory of nonlinear dynamics of compressor shaft systems, and the determined nonlinear parameters can be used to guide the operation and maintenance of compressors in engineering.

An indirect method to determine nonlinearly elastic shear stress-strain constitutive relationships for nonlinear torsional vibration of nonlinearly elastic shafts

PurposeIn this paper, the authors aim to propose an effective method to indirectly determine nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials, and then study the nonlinear free torsional vibration of Al–1%Si shaft.Design/methodology/approachIn this study the authors use BoxLucas1 model to fit the determined-experimentally nonlinear elastic normal stress–strain constitutive relationship curve of Al–1%Si, a typical case of isotropic nonlinear elasticity materials, and then derive its nonlinear shear stress-strain constitutive relationships based on the fitting constitutive relationships and general equations of plane-stress and plane-strain transformation. Hamilton’s principle is utilized to gain nonlinear governing equation and boundary conditions for free torsional vibration of Al–1%Si shaft. Differential quadrature method and an iterative algorithm are employed to numerically solve the gained equations of motion.Findings The effect of four variables, namely dimensionless fundamental vibration amplitude , radius and length , and nonlinear-elasticity intensity factor , on frequencies and mode shapes of the shafts is obtained. Numerical results are in good agreement with reference solutions, and show that compared with linearly elastic shear stress-strain constitutive relationships of the shafts made of the nonlinear elasticity materials, its actual nonlinearly elastic shear stress-strain constitutive relationships have smaller torsion frequencies. In addition, but having opposite hardening effect, the rest of the four variables have softening effect on nonlinearly elastic torsion frequencies. Eventually, taking into account nonlinearly elastic shear stress-strain constitutive relationships, changes of the four factors, i.e. , , and , cause inflation and deflation behaviors of mode shapes in nonlinear free torsional vibration.Originality/valueThe study could provide a reference for indirectly determining nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials and for structure design of torsional shaft made of nonlinear elasticity materials.

Internal resonances of nanorods in presence of surface energy effect: Nonlinear torsional vibration

In this paper, effects of the surface energy on the nonlinear torsional vibrations and internal resonances of nanorods are investigated. To this end, the second-order term with respect to angle of rotation is considered for the displacement field. In addition, the geometrically nonlinear strain–displacement relations are obtained based on the von-Kármán theory. Then, Hamilton’s principle is implemented to derive the governing equation of motion for torsional vibration of nanorod. In the governing equation of motion, the surface energy parameters are included by the surface elasticity theory. The multiple-scale method is employed to solve the governing equation of motion for fixed-free and fixed-fixed end conditions. The nanorod considered is made of aluminum and silicon because of different values of their surface parameters. The effect of surface energy parameters on the torsional frequencies of nanorods is investigated for different values of length, radius, frequency number, and amplitude of the nonlinear vibrations. In addition the cases of occurring, the internal resonances are reported. The results obtained in this research may be useful for better design of nanoelectromechanical devices such as nanobearings and rotary servomotors.

Three-Dimensional Thermal Stress Effects on Nonlinear Torsional Vibration of Carbon Nanotubes Embedded in an Elastic Medium

ABSTRACT Nonlinear torsional vibration of nanorods embedded in an elastic medium under three-dimensional thermal stresses is investigated in this study. The scale effect is introduced to the equation of motion using the nonlocal theory. The nanorods are under the effect of a three-dimensional thermal environment. The elastic medium is modeled by infinite rotational springs around the nanorod. Galerkin’s and He’s variational methods are used to solve the differential equation of motion and obtain torsional frequencies. An uncertainty analysis is done to show the effect of the uncertain parameters on the frequencies. The frequency sensitivities are obtained to demonstrate the frequency sensitivities to uncertain parameters. Effect of temperature changes, elastic medium stiffness, vibration amplitude, nonlocal scale coefficient, and nanorod length and diameter on the nonlinear torsional frequencies are investigated. The effect of temperature on the frequencies is dependent on the values of elastic medium stiffness, vibration amplitude, and nanorod length and diameter.

A nonlinear torsional vibration model of harmonic gear reducer and the effect of various factors on torsional vibration during start and stop

The torsional vibration problems often occur to industrial robots during start and stop due to the use of flexible wheel in harmonic gear reducer as a transmission pair. Because of the influence of nonlinear stiffness, transmission error, and other factors, the nonlinear torsional vibration of harmonic gear reducer will affect the stability and reliability of transmission system. In this article, based on the structural characteristics of the harmonic gear reducer, considering the nonlinear torsional stiffness, transmission error, meshing damping, and other factors, the nonlinear torsional dynamic model of the harmonic gear reducer was established with the theoretical analysis and the experimental results. Based on this model, the influence of various factors such as rotational speed, moment of inertia, torsional stiffness, and transmission error on the nonlinear torsional vibration of harmonic gear reducer was discussed. The results show that the vibration amplitude of harmonic gear reducer increases with the increase of speed, transmission error, and motor inertia, and decreases with the increase of load inertia and damping. The dynamic model and analysis method established in the study can provide the theoretical guidance for the optimal design and the vibration reduction of harmonic gear reducer.

An analysis of the nonlinear torsional vibrations in the powertrains caused by the idle parts in the automatic transmissions

An analysis of the nonlinear torsional vibrations in the powertrains caused by the idle parts in the automatic transmissions

Nonlinear torsional vibrations of electromechanical coupling of diesel engine gear system and electric generator

New approach for analysis and detuning of nonlinear torsional vibrations of diesel electric generator gear system is suggested. Nonlinear magnetic field of the generator is calculated by finite element method to obtain the disturbing moments, which act on the gear system. Moreover, others disturbing moments, which arise due to operation in diesel engine cylinders, are accounted. Essential nonlinear model of torsional vibrations with sixteen degrees-of- freedom is derived. This model accounts the nonlinear moments of two elastic clutches and the backlashes of gear pairs. The comparison of the numerical simulations data with experimental results is performed to verify the obtained model. The approach for essential nonlinear free torsional vibrations calculations is suggested. The harmonic balanced method is the basis of this approach. Sensitivity theory is used to detune the nonlinear forced torsional vibrations from resonances. The results of nonlinear torsional vibrations analysis are discussed.

Non-linear torsional vibration of lengthy thin-walled simply supported beam of open section resting on Winkler foundation

The aim of this paper is to study the non-linear torsional vibrations of lengthy thin-walled simply supported beam of open section resting on Winkler foundation. The governing differential equation of motion is derived. An effort has been made to obtain and solve the governing differential equation of large amplitude torsional vibrations of doubly symmetric thin-walled beams of open section with simply supported boundary. Approximate solutions are obtained for simply supported beams using the well-known Galerkin’s method. Plots specifying the effect of large amplitudes on nonlinear frequency of torsional vibrations for various non-dimensional beam constants are furnished. From the present study, we observe, that for lower values of Winkler stiffness parameter λ≤0.1, the non-linear frequency increases drastically as β increases. Also, it is noticed that as the value of Winkler stiffness parameter λ increases, the influence of β decreases. For higher values of λ≥30, the effect of β in the present range 0–0.1 on non-linear frequency seems to be negligible.

Nonlinear torsional vibration analysis of motor rotor system in shearer semi-direct drive cutting unit under electromagnetic and load excitation

Under the influence of electromagnetic excitation and terminal loads disturbance, the motor rotor shaft of the semi-direct cutting section of a shearer may produce electromechanical coupling resonance. Against this background, this work investigates the vibration properties of a motor rotor and the effect of electromechanical couplings. According to the electric machine theory and the Maxwell equation, the electromechanically coupled nonlinear torsional vibration model for the rotor system is first established under the influence of electromagnetic excitation and terminal load disturbance. Besides, its approximate solution is derived through the multi-scale method in the case of main parametric resonance, and the steady-state solution is determined through the numerical method. Moreover, the electromagnetic parameters including internal power factor angle, permanent magnet coefficient, number of pole pairs, and magnetic saturation coefficient and the mechanical parameters such as torsional stiffness of rotor and terminal load, which are responsible for the torsional vibration of rotor system, are studied, respectively. The results show that the stiffness failure and load mutation could cause chaotic motions, and a reasonable selection of electromagnetic and mechanical parameters for the permanent magnet synchronous motor (PMSM) could prevent the jumping and bifurcation from happening. This work is expected to benefit, both theoretically and practically, the design, optimization, and diagnosis of high-performance shearers that are operated in a complex working environment.

Nonlinear Torsional Vibration Analysis and Control of Semidirect Electromechanical Coupling Transmission System in Shearer

The nonlinear torsional vibration and instability oscillation caused by nonlinear damping in the shearer electromechanical coupling cutting transmission system in shearer driven by the permanent magnet synchronous motor (PMSM) is investigated in this paper. The electromechanical coupling transmission system in the shearer is equivalent to a concentrated mass model for the purpose of establishing the system dynamic model by the Lagrange–Maxwell equation. Then, the Routh–Hurwitz criterion is used to determine the torsional vibration critical point and stability region for the Hopf bifurcation for the cutting transmission system. According to the Routh–Hurwitz stability criterion, the Hopf bifurcation type and the effect of the supercritical Hopf bifurcation in the torsional vibration of the cutting transmission system are analyzed. Furthermore, based on the washout filter, the Hopf bifurcation controller is designed for suppressing the transmission system’s large vibration amplitude and unstable oscillation. In addition, the influences of the linear gain and nonlinear gain on the bifurcation point and the limit cycle amplitude are discussed. Finally, the numerical simulation results indicate the effectiveness of the designed controller. The research achievements can provide a theoretical basis for design or optimize the cutting transmission system of high‐reliability shearer driven by PMSM.

Nonlinear Torsional Vibration Analysis and Nonlinear Feedback Control of Complex Permanent Magnet Semidirect Drive Cutting System in Coal Cutters

The semidirect drive cutting transmission system of coal cutters is prone to unstable torsional vibration when the resistance values of its driving permanent magnet synchronous motor (PMSM) are affected by changes in temperatures and tough conditions. Besides, the system has the properties of complex electromechanically coupling such as the coupling between electrical parameters and mechanical parameters. Therefore, in this study, the nonlinear torsional vibration equation was established on the basis of the Lagrange‐Maxwell theory. Moreover, in light of the nonlinear dynamic bifurcation theory, the system stability was analyzed by taking the resistance value of power motor as the bifurcation parameter. In addition, the influence of subcritical bifurcation on the torsional vibration was studied by investigating the necessary and sufficient conditions for dynamic Hopf bifurcation and classifying the bifurcation types. At last, in order to suppress destabilizing oscillation induced by Hopf bifurcation, the nonlinear feedback controller was constructed, with the introduction of feedback from the motor velocity as well as the selection of voltage value on the q shaft as the controlled variable. Meanwhile, the three‐order normal form and controlling parameters of the system were obtained with the aid of the multiple scales method and the harmonic balance method. In this way, the Hopf bifurcation point was transferred to control the stability of Hopf bifurcation and the amplitude of limit cycle, thus guaranteeing reliable and safe operation of the system. The numerical simulation results indicate that the designed controller boosts an ideal controlling effect.

Investigation of electromechanical coupling torsional vibration and stability in a high-speed permanent magnet synchronous motor driven system

A permanent magnet synchronous motor (PMSM) driven system is a typical electromechanically coupled system. In this paper, to improve the operational performance and stability of the PMSM system, torsional vibrations due to the electromechanical coupling effects are studied. An electromagnetic excitation model was first established to consider the effect of torsional angle on magnetomotive force. Then, nonlinear torsional vibration equations of the driven system were obtained. In addition, an electromechanically-coupled torsional vibration mechanism based on natural frequency modulation of the system was revealed. Finally, the torsional vibration equations and coupled mechanism were validated against experimental results describing the actual torsional vibration and a numerical method was used to verify our theoretical analysis. The results provide a theoretical basis for the parameter design of electromechanical transmission systems.

Hysteretic bit/rock interaction model to analyze the torsional dynamics of a drill string

Abstract The present paper proposes a novel hysteretic (non-reversible) bit/rock interaction model for the torsional dynamics of a drill string. Non-reversible means that the torque-on-bit depends not only on the bit speed, but also on the bit acceleration, producing a type of hysteretic cycle. The continuous drill string system is discretized by means of the finite element method and a reduced-order model is constructed using the normal modes of the associated conservative system. The parameters of the proposed hysteretic bit/rock interaction model is fitted with field data. The non-linear torsional vibration and the stability map of the drill string system are analyzed employing the proposed bit/rock interaction model and also a commonly used reversible model (without hysteresis). It turns out that the hysteretic model affects the stability region of the system.

Analysis of Nonlinear Torsional Vibration Characteristics of EV Powertrain Considering Rotor Eccentricity of PMSM

Analysis of Nonlinear Torsional Vibration Characteristics of EV Powertrain Considering Rotor Eccentricity of PMSM