Nonlinear Torsional Vibration Research Articles

Non-linear torsional vibration characteristics of an internal combustion engine crankshaft assembly

Crankshaft assembly failure is one of the main factors that affects the reliability and service life of engines. The linear lumped mass method, which has been universally applied to the dynamic modeling of engine crankshaft assembly, reveals obvious simulation errors. The nonlinear dynamic characteristics of a crankshaft assembly are instructionally significant to the improvement of modeling correctness. In this paper, a general expression for the non-constant inertia of a crankshaft assembly is derived based on the instantaneous kinetic energy equivalence method. The nonlinear dynamic equations of a multi-cylinder crankshaft assembly are established using the Lagrange rule considering nonlinear factors such as the non-constant inertia of reciprocating components and the structural damping of shaft segments. The natural frequency and mode shapes of a crankshaft assembly are investigated employing the eigenvector method. The forced vibration response of a diesel engine crankshaft assembly taking into account the non-constant inertia is studied using the numerical integral method. The simulation results are compared with a lumped mass model and a detailed model using the system matrix method. Results of non-linear torsional vibration analysis indicate that the additional excitation torque created by non-constant inertia activates the 2nd order rolling vibration, and the additional damping torque resulting from the non-constant inertia is the main nonlinear factor. The increased torsional angular displacement evoked by the high order excitation torque relates to the non-constant inertia. This research project is aimed at improving nonlinear dynamics theory, and the confirmed nonlinear parameters can be used for the structure design of a crankshaft assembly.

Nonlinear free vibrations of thin-walled beams in torsion

Nonlinear torsional vibrations of thin-walled beams exhibiting primary and secondary warpings are investigated. The coupled nonlinear torsional–axial equations of motion are considered. Ignoring the axial inertia term leads to a differential equation of motion in terms of angle of twist. Two sets of torsional boundary conditions, that is, clamped–clamped and clamped-free boundary conditions are considered. The governing partial differential equation of motion is discretized and transformed into a set of ordinary differential equations of motion using Galerkin’s method. Then, the method of multiple scales is used to solve the time domain equations and derive the equations governing the modulation of the amplitudes and phases of the vibration modes. The obtained results are compared with the available results in the literature that are obtained from boundary element and finite element methods, which reveals an excellent agreement between different solution methodologies. Finally, the internal resonance and the stability of coupled and uncoupled nonlinear modes are investigated. This study can be a preliminary step in the understanding of complex dynamics of such systems in internal resonance excited by external resonant excitations.

Dynamics Behaviors of Rolling Mill's Nonlinear Torsional Vibration of Multi-degree-of-freedom Main Drive System with Clearance

The dynamic model of multi-degree-of-freedom main drive system with clearance is established.The equations are decoupled by transforming them into principal coordinates.The method of modified Lindstedt-Poincare(MLP) is employed to obtain the analytical solutions of the nonlinear system under loaded friction of rolling and out disturbance excitation.The bifurcation equation of the system is deduced by the modified Lindstedt-Poincare combined with the multiple scales.The characteristics of bifurcation of non-autonomy system are analyzed by means of singularity theory,respectively,and different topological structure of solution is obtained under different parameters and obtaining the variation area of parameters under which the rolling mill could vibrate stably.The numerical simulation bases on some steel mill.Rolling mill's main drive system which simplifies as a four-degree-of-freedom is given.It exhibits many changes of the relationship between amplitude and frequency under the variation of parameters like nonlinear stiffness,clearance,torque disturbance and rolling force and obtaining clearance and torque disturbance are the main factors which influent the vibration.To reduce the amplitude of torsional vibration,the value of clearance and torque disturbance are regulated.The research results provide theory basis and reference for analyzing torsional vibration of rolling mill's transmission system caused by clearance.

Dynamics Behaviors of Rolling Mill's Nonlinear Torsional Vibration of Multi-degree-of-freedom Main Drive System with Clearance

The dynamic model of multi-degree-of-freedom main drive system with clearance is established.The equations are decoupled by transforming them into principal coordinates.The method of modified Lindstedt-Poincare(MLP) is employed to obtain the analytical solutions of the nonlinear system under loaded friction of rolling and out disturbance excitation.The bifurcation equation of the system is deduced by the modified Lindstedt-Poincare combined with the multiple scales.The characteristics of bifurcation of non-autonomy system are analyzed by means of singularity theory,respectively,and different topological structure of solution is obtained under different parameters and obtaining the variation area of parameters under which the rolling mill could vibrate stably.The numerical simulation bases on some steel mill.Rolling mill's main drive system which simplifies as a four-degree-of-freedom is given.It exhibits many changes of the relationship between amplitude and frequency under the variation of parameters like nonlinear stiffness,clearance,torque disturbance and rolling force and obtaining clearance and torque disturbance are the main factors which influent the vibration.To reduce the amplitude of torsional vibration,the value of clearance and torque disturbance are regulated.The research results provide theory basis and reference for analyzing torsional vibration of rolling mill's transmission system caused by clearance.

Study on Dynamic Load Sharing Behavior of Two-Stage Planetary Gear Train Based on a Nonlinear Vibration Model

The nonlinear torsional vibration model of a two-stage planetary gear system is established taking errors of transmission, time varying meshing stiffness and multiple gear backlashes into account. The solution of the equations is determined by using ODE45. The influences of transmission errors on the load sharing behavior are assessed and some useful theoretical guidelines for the design of planetary gear systems are provided at last.

Strongly nonlinear shear perturbations in discrete and continuous cubic nonlinear systems

Systems with constraints, the masses in which move only along guides, can execute strongly nonlinear vibrations. This means that nonlinear phenomena manifest themselves at arbitrary small deviations from equilibrium. The form of vibrations of a single mass is described by elliptical Jacobi functions. The spectrum of these vibrations is found. With an increase in amplitude, the period of vibrations decreases. We deduce equations of strongly nonlinear vibrations of a chain of connected masses. In the continuum limit, we obtain a new nonlinear equation in partial derivatives. We devise transformation of variables leading to linearization of this equation. We implemented a factorization procedure that decreases the order of the equation in partial derivatives from second to first. Exact solutions to the first-order equation describe the slow evolution of the displacement profile in a distributed system. In the absence of preliminary tension of elastic elements in the continued model, traveling waves cannot be achieved; however, time-oscillating solutions like standing waves are possible. We obtain an equation for a field of strongly nonlinear deformations. Its exact solution describes periodic movement in time and space. As well, the period of time oscillations decreases with an increase in amplitude, and the spatial period, in contrast, increases. The product of the vibration frequency multiplied by the spatial period is a constant that depends on the deformation energy. We propose a scheme of the mechanical system producing strongly nonlinear torsional vibrations. We experimentally measured the period of torsional vibrations of a single disc. We show that with an increase in amplitude in the process of vibration attenuation, an increase in the period occurs, which agrees with calculations. We measure the shapes of nonlinear vibrations of a chain of connected discs. A strongly nonlinear behavior of the chain is observed.

Nonlinear Torsional Vibration Modeling and Bifurcation Characteristic Study of a Planetary Gear Train

A nonlinear torsional vibration model of a planetary gear train with errors of transmission,time varying meshing stiffness and gear backlashes is established and dimensionless equations of the system are derived.The solution of the equations is carried out by using the method of numerical integration.By comparing with Poincare maps and bifurcation diagrams,the bifurcation properties of the system are studied.The influences of some bifurcation parameters such as rotational speed,damping coefficient and gear backlashes on the bifurcation properties of the system are assessed.The study results reveal that the system's motion state will change into chaos in the way of crisis as speed increase and period doubling bifurcation is the way to chaos as backlashes increase.A smaller damping coefficient will make the system's periodic motion state change into complex state.Gear backlashes have a strong impact on the system's bifurcation characteristic when dimensionless the backlashes is bigger than 3.5.

Super-Harmonic Resonance Analysis on Torsional Vibration of Misaligned Rotor Driven by Universal Joint

The super-harmonic resonance of the nonlinear torsional vibration of misaligned rotor system driven by universal joint was studied considering both natural structure misalignment and actual error misalignment. Utilizing multi-scale method, the periodic solution of weakly nonlinear torsional vibration equation was obtained corresponding to super-harmonic resonance, including amplitude-frequency and phase-frequency characteristic expressions of steady periodic solution. The stability of equilibrium point was investigated using Lyapunov first approximate stability theory, then the stable region and unstable region on the amplitude of the super-harmonic resonance periodic solution, which varied with the detuning parameter. At last, the driving shaft’s steady periodic motion of the first approximation and its calculation simulation were carried out according to the kinetic relation about driven shaft and driving shaft. It is found that jumping phenomenon and dynamic bifurcation occur when the rotating angular velocity of driving shaft is half of the natural frequency of the deriving system. The results above indicate the fundamental characteristic of the nonlinear dynamic on the misaligned rotor, also applying the foundation for advanced bifurcation and singularities analysis.

Global Dynamic Characteristic of Nonlinear Torsional Vibration System under Harmonically Excitation

Torsional vibration generally causes serious instability and damage problems in many rotating machinery parts.The global dynamic characteristic of nonlinear torsional vibration system with nonlinear rigidity and nonlinear friction force is investigated.On the basis of the generalized dissipation Lagrange's equation, the dynamics equation of nonlinear torsional vibration system is deduced.The bifurcation and chaotic motion in the system subjected to an external harmonic excitation is studied by theoretical analysis and numerical simulation.The stability of unperturbed system is analyzed by using the stability theory of equilibrium positions of Hamiltonian systems.The criterion of existence of chaos phenomena under a periodic perturbation is given by means of Melnikov's method.It is shown that the existence of homoclinic and heteroclinic orbits in the unperturbed system implies chaos arising from breaking of homoclinic or heteroclinic orbits under perturbation.The validity of the result is checked numerically.Periodic doubling bifurcation route to chaos, quasi-periodic route to chaos, intermittency route to chaos are found to occur due to the amplitude varying in some range.The evolution of system dynamic responses is demonstrated in detail by Poincare maps and bifurcation diagrams when the system undergoes a sequence of periodic doubling or quasi-periodic bifurcations to chaos.The conclusion can provide reference for deeply researching the dynamic behavior of mechanical drive systems.

A nonlinear theory for electroelastic shells with relatively large in-plane shear deformation and its implications in nonlinear mode coupling

A set of nonlinear two-dimensional equations for thin electroelastic shells in vibrations with moderately large shear deformation in the tangent plane are obtained from the three-dimensional equations of nonlinear electroelasticity. As an example for application, the equations are used to study nonlinear torsional vibration of a circular cylindrical piezoelectric shell. It is shown that torsion is nonlinearly coupled to axial extension and circumferential extension. The results of this paper emphasize the need for further study of mode coupling induced by nonlinearity.

Frequency response calculation of non-linear torsional vibration in gear systems

The current paper focuses on the non-linear torsional vibration of a one-stage transmission gear system. Four different methods have been applied for solution of the equation of motion; the discretization method, the perturbation method, the Ritz method, and the step-wise time integration of the equation of motion. The time and frequency results from these analyses have been compared with each other, as well as those reported in literatures. Although all of these methods are accurate and computationally effective for finding the main spectral contribution, however, only the discretization method and the step-wise time-integration model are able to identify the other frequency components.

A non-linear dynamic model for planetary gear sets

A discrete non-linear torsional vibration model of a single-stage planetary set is proposed in the current study. The model includes all possible power flow configurations, any number of planets in any spacing arrangement and any planet mesh phasing configurations. It also includes time variation of gear mesh stiffnesses as well as clearance (backlash) non-linearities. The non-linear equations of motion are solved semi-analytically using multi-term harmonic balance method (HBM) in conjunction with inverse discrete Fourier transform and Newton-Rapson method. The HBM solutions are compared with numerical simulation results to demonstrate the accuracy of the HBM formulation. Another comparison with predictions of a deformable-body dynamic model is also provided to assess the accuracy of the discrete model. Limited parametric studies are presented at the end, to show the influence of key gear design parameters on dynamic response.

Nonlinear Torsional Vibration of a Circular Cylindrical Piezoelectric Rod with Relatively Large Shear Deformation

We show that, in a circular cylindrical rod, torsional modes are coupling to extension when the shear deformation associated with the torsional modes is no longer infinitesimal. A set of a couple equations is derived with which the effect of extension on the torsional frequency is examined. The results are useful to the understanding and design of devices operating with torsional modes.

Nonlinear torsional vibrations of a stretched rod

A thin elastic stretched rod is considered. Nonlinear torsional vibrations of the rod are investigated. The Reissner variational principle is used to determine the vibration amplitude. It is shown that, in contrast to the linear case, the amplitude is a limited quantity.

Investigation of Damping of Vibrations in a System With Two‐Disc Inseparable Clutch

AbstractThis paper presents the results of tests on free and forced harmonic torsional vibrations in a system with a two‐disc inseparable clutch, with structural friction taken into account. Nonlinear histeresis loop describing the frictional‐elastic properties of the system was introduced into the model. The mathematical model of the vibrating system containing two disks inseparable clutch was built. During free vibrations of the system, its damping characteristics were tested by a digital simulation method. The vibration damping decrement as a function of amplitude torsional displacement was determined. When vibrations were harmonically forced, the amplitude ‐ frequency characteristics of the system were determined numerically. The system was used as a nonlinear torsional vibration damper in a linear system with a harmonic force. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

LINEAR AND NONLINEAR AERODYNAMIC THEORY OF INTERACTION BETWEEN FLEXIBLE LONG STRUCTURE AND WIND

In light of the characteristics of the interactions between flexible structure and wind in three directions, and based on the rational mechanical section-model of structure, a new aerodynamic force model is accepted, i. e. the coefficients of three component forces are the functions of the instantaneous attack angle and rotational speed Ci = Ci(β(t), Θ), (i = D, L, M). So, a new method to formulate the linear and nonlinear aerodynamic items of wind and structure interacting has been put forward in accordance with “strip theory” and modified “quasi-static theory”, and then the linear and nonlinear coupled theory of super-slender structure for civil engineering analyzing are converged in one model. For the linear aerodynamic-force parts, the semi-analytical expressions of the items so-called “flutter derivatives” corresponding to the one in the classic equations have been given here, and so have the nonlinear parts. The study of the stability of nonlinear aerodynamic-coupled torsional vibration of the old Tacoma bridge shows that the form and results of the nonlinear control equation in rotational direction are in agreement with that of V. F. Bohm's.

Nonlinear Torsional Vibrations of a Shaft System with the Hysteretic and Two-Staged Torsional Stiffness.

This paper proposes torsional vibrations of the clearance problem in coupling, joint and clutch for a rotating shaft system. To make researches into the vibrational phenomena of the system, bilinear spring stiffness models with hysteretic torque in clearance are developed. The dynamic behavior of this system is investigated by using a numerical simulation. It is demonstrated that the steady-state amplitude shows nonlinear characteristics and changes remarkably with the hysteretic torque of the system. The behavior of resonance curves is examined in detail when the equilibrium point of the torsional stiffness changes. The experiments are performed to confirm the results of numerical simulations, and the experimental results agree well qualitatively with those from the simulations.

Simple resonance in nonlinear torsional vibration systems of three variable generalized masses

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Modelling and Control of a Flexible Spherical Wrist

SUMMARYIn this paper, a very inexpensive, lightweight and simple wrist mechanism is introduced. This wrist displays nonlinear torsional vibrations. This differs from conventional wrists in that structural flexibility in the mechanism is allowed to occur by design. In this paper, the dynamic equations of this wrist are derived. System identification techniques are then employed to obtain a linearized model. Various control strategies are studied. It is shown that the input-output feedback linearization technique is not feasible for these nonlinear dynamic equations. It is also shown that the use of conventional rigid body PID controllers on this proto-type is inadequate. A tracking controller which compensates for the flexible dynamics of the wrist is implemented with encouraging results. This controller allows the end-effector to be placed at an arbitrary orientation with little vibration. The effect of the controller is to make the wrist appear to have a much higher structural stiffness. The compliant nature of this wrist allows simple force control strategies to be implemented. It is the combination of the wrist with the control algorithm which makes this design viable.

On a model of the nonlinear torsional vibration with variable generalized masses

In this paper the establishment of the parametric vibration differential equations for a model of the nonlinear torsional vibration with variable generalized masses considered.