A DAE formulation for geared rotor dynamics including frictional contact between the teeth

A DAE approach is presented for geared rotor dynamics simulations with rigid helical evolvent gears. It includes the normal contact force between the teeth as well as tangential components. Given the evolvent tooth flank geometry of gear 1 and gear 2 [1], the contact line and the velocity difference in the contact are found. The requirement of no penetration of the teeth yields a non-holonomic constraint and the contact normal force. The friction caused force and moment are obtained by applying Coulomb's friction model. This approach is used to investigate the dynamics of two ideal rotors with translational DoFs, which are connected by gears to one another. The driving rotor has a given angular speed, while the driven rotates unrestrainedly and is connected to a rotational damper. Because of the periodic friction terms, the solution is periodic. A direct time integration or a harmonic approach can be used for the numerical computation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

弹流润滑条件下表面形貌对摩擦噪声的影响

研究了弹流润滑状态下表面形貌对摩擦噪声的影响.通过激光微加工方法在金属圆盘试件表面上制造了两种沟槽型织构表面形貌,在双盘摩擦磨损试验机上对不同表面形貌进行了摩擦噪声和摩擦特性试验,分析了线接触弹流润滑条件下不同工况和表面形貌影响摩擦噪声的机理,并结合有限元分析对结论加以验证.结果表明:载荷和转速的变化对线接触摩擦噪声有明显影响,由于线接触副工作在部分膜弹流润滑状态下,所以摩擦噪声的特性与干摩擦时类似,摩擦因数较大的表面会辐射出更强的摩擦噪声;特定结构的表面形貌能改善表面润滑特性,有效降低摩擦噪声声压级,沟槽型织构的存在可以打断接触区域应力分布,减轻接触面微凸体碰撞作用,从而降低了表面自激振动,同时合适的表面形貌结构也有利于润滑油膜的形成,减小了系统的摩擦能量,达到降低摩擦噪声的效果.     

Determination of wear coefficient in mixed lubrication using FEM

In a line or point contact with an elastohydrodynamic lubricant oil film, solid-to-solid contacts are common and wear will occur at these places. Given that there is only a portion of the load is supported by the direct interaction of roughness asperities, the wear coefficient should be less than that for dry contacts and account for the effect of surface roughness and oil film. Since it is difficult to obtain the wear coefficient value at different oil film thickness by experiments, this paper presents the methodology of determination of wear coefficient in mixed lubrication using finite element method (FEM). In this method, the roughness of contact surfaces is characterized as fractal surfaces by the Weierstrass–Mandelbrot (W–M) function, the sliding wear in mixed lubrication is simulated by the Coupled-Eulerian–Lagrangian (CEL) method and the wear volume is calculated according to the solid–solid contact load. Then the wear coefficient can be determined and the simulation example shows that the wear coefficient decreases nonlinearly with the increasing of oil film thickness and dynamic viscosity in mixed lubrication.     

Modeling of jointed structures using zero thickness interface elements

An important contribution to global damping of mechanical devices is structural damping due tomicroslip effects with friction in joint interfaces. In order to investigate the mechanical behaviour in these contact interfaces numerically, a contact element in the context of Finite Element Method (FEM) is presented. The suggested element is an isoparametric zero thickness element which is well suited for the present problem because the contact area is known and only small relative displacements occur. Arbitrary linear or nonlinear constitutive contact models for normal and tangential contact behaviour can be implemented. Using a proper parametrisation of the contact area, it is possible to apply the element in contact interfaces lying arbitrarily in space and in interfaces discretized with distorted elements. This method is described before a numerical example is compared with experimental results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

用平面弹性理论的复变函数解法精确确定直齿轮轮齿的挠度

本论文吸取并发展了目前受到国内外普遍注意的由会田俊夫、寺内喜男和永村和照所开创的用平面弹性理论的复变函数解法求直齿轮轮齿的应力和变形的方法,把它们的应力函数的近似解发展为精确解;提高了映射齿形的精度,提出了用五项分式项表达的、映射误差小于1％m的映射函数;提出了求直接影响齿轮啮合质量的啮合点相对于相邻轮齿中线位移的原则;编制了计算程序,并对计算结果进行了分析.     

On modelling and analysis of gear drives with nonlinear couplings

The paper deals with modelling of vibration of shaft systems with gears and rolling-element bearings using the modal synthesis method with DOF number reduction. The influence of the nonlinear bearing and gearing contact forces with the possibility of the contact interruption is respected. The gear drive nonlinear vibrations caused by internal excitation generated in gear meshing, accompanied by impact and chaotic motions are studied. The theory is applied to a simple test-gearbox. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Load share and finite element stress analysis for double circular-arc helical gears

The authors analyze the tooth surface contact and stresses for double circular-arc helical gear drives. The geometry of such gear drives has been represented by the authors in their previous paper [1]. The proposed approach is based on application of (i) computerized simulation of meshing and contact of loaded gear drives, and (ii) the finite element method. Load share between the neighboring pairs of teeth is based on the analysis of position errors caused by surface mismatch and elastic deformation of teeth. The authors have investigated the conditions of load share under a load and determined the real contact ratio for aligned and misaligned gear drives, respectively. Elastic deformation of teeth and the stress analysis of the double circular-arc helical gears are accomplished by using the finite element method. The finite element models for the pinion and gear are constructed, respectively. Contact pressure is spread over elliptical area. The stress analysis for aligned and misaligned gear drives, respectively, has been performed. The numerical results have been compared with those obtained by other approaches.     

Analysis of effect of random perturbation on dynamic response of gear transmission system

In order to investigate the effects of random perturbation of a low-frequency excitation caused by torque fluctuations, gear damping ratio, gear backlash, meshing frequency and meshing stiffness, the random dynamic model of a single pair of three-degree-of-freedom spur gear transmission system is established. With gear meshing frequency changing, the dynamic characteristics of the gear transmission system were analyzed by bifurcation diagram, phase diagram, time course diagram and Poincaré map of the system. The effects of random perturbation caused by a low-frequency excitation caused by torque fluctuations, gear damping ratio, gear backlash, meshing frequency and meshing stiffness were comparative analyzed. Numerical simulation shows that the gear transmission system with nonlinear clearance exists rich period-doubling bifurcation phenomenon. With the increasing of the gear meshing frequency, gear transmission system will be from the chaotic motion to periodic motion by inverse period-doubling bifurcation. The effect of the meshing frequency random perturbation on the gear transmission system movement is largest. On the contrary, the effect of the meshing stiffness random perturbation on the system is minimum.     

Multifield coupled analysis for a harmonic movable tooth drive system integrated with shape memory alloys

In this article, a harmonic movable tooth drive system integrated with shape memory alloys is proposed. The key of the system is to integrate the shape memory alloy drive principle with the harmonic movable tooth drive principle by which a small size and a large output torque can be achieved simultaneously. Here the structure of the drive system is determined, and its operating principle is illustrated. The output torque equation for the drive is deduced and coupled dynamics equations for the system are given. With use of these equations, the forces applied to the wave generator and the output torque of the drive system under pulsed current are investigated. The results show that with reasonable pulsed current parameters, a large and steady output torque can be obtained.     

Development of generalized Iwan model to simulate frictional contacts with variable normal loads

Most friction models are originally proposed to predict restoring forces in mechanical contacts with constant normal load. In practice the contact interface kinematics may involve normal motion in addition to the tangential displacements, leading to variation of the contact normal load. This phenomenon is observed most strongly in contacts with high lateral vibration amplitudes and is known as slap. The current study establishes a general friction model to account for variation in the normal load and enables one to predict the behavior of a contact more precisely. Iwan model (1966) [5] is a suitable candidate for contact interface modeling and is able to represent the stick-micro/macro slip behavior involved in a friction contact. This physical based model is employed in the current work and its physical parameters are generalized to include the normal load variation effects. The model is characterized by a slippage distribution density function and a linear stiffness at stick state. Both these parameters, defined in presence of constant normal load in the original model, are derived considering normal load variation leading to generalization of the contact model. Conventional models with constant normal loads produce symmetric contact interface hysteresis loops, but the developed generalized Iwan model is capable of generating asymmetric hysteresis loops similar to those frequently seen in experiments. The generalized contact model is employed to simulate the measured behavior of a beam with frictional support observed in an experimental test set-up. The contact slippage distribution function is first identified in a constant normal load condition. Next in low levels of contact preloads where variation of the normal load is significant, the identified distribution function in generalized form is employed to predict the experimental observations.     

Dynamic analysis and passive control of viscoelastically damped nonlinear dynamic contacts

This paper presents an analytical and finite element study on nonlinear contact dynamics and controls. Nonlinear dynamic contacts between eccentrically supported masses and simply supported beams are studied. Passive control of the dynamic contacts using viscoelastic dampers is also proposed and evaluated. A nonlinear contact finite element is modeled by a set of nonlinear stiffness and damping polynomial functions; and a nonlinear viscoelastic finite element is modeled by a Standard Linear Model with frequency-dependent nonlinear stiffness and damping functions. Analyses show that the dynamic contact force increases as the initial gap increases. Application of viscoelastic dampers can effectively reduce contact loads and prevent dynamic contacts. A simple design equation is also proposed.     

A micromechanics-based Cosserat-type model for dense particulate solids

A two-dimensional continuum theory is presented for cohesionless granular media consisting of identical rigid disks. While the normal deformation of contacting particles is constrained, the tangential frictional contact is modelled by a line spring with a constant stiffness. To describe the static frictional system transmitting couples at contacts, a Cosserat-type continuum including rotational degrees of freedom is appropriate. Contrary to the classical elastic medium, movement of particles within a granular system in response to applied loads can give rise to localisations of force chains and large voids. In addition to relative displacement and rotation, a director governing the direction of interparticle forces and a phase field delineating density variation, are therefore introduced. Total work done involving these two order parameters for a particle is attained on an orientation average. Based on the formulation of free energy, a concentration- and anisotropy-dependent formulation for static quantities (stress and couple stress) in the rate form is derived in light of the principles of thermodynamics. It is consistent with the requirement of observer independence and material symmetry. The governing equations for two order parameters are derived, in which void concentration and stress anisotropy are related to relative displacement and rotation. As an example, the proposed model is applied to the hardening regime of deformation of a dense particle assembly with initial perfect lattice under simple shear. It is demonstrated that in the presence of dilatancy and director variation, there exists a linear relation between the shear stress and strain, in coincidence with experimental observations. Received: February 24, 2005     

A micromechanics-based Cosserat-type model for dense particulate solids

A two-dimensional continuum theory is presented for cohesionless granular media consisting of identical rigid disks. While the normal deformation of contacting particles is constrained, the tangential frictional contact is modelled by a line spring with a constant stiffness. To describe the static frictional system transmitting couples at contacts, a Cosserat-type continuum including rotational degrees of freedom is appropriate. Contrary to the classical elastic medium, movement of particles within a granular system in response to applied loads can give rise to localisations of force chains and large voids. In addition to relative displacement and rotation, a director governing the direction of interparticle forces and a phase field delineating density variation, are therefore introduced. Total work done involving these two order parameters for a particle is attained on an orientation average. Based on the formulation of free energy, a concentration- and anisotropy-dependent formulation for static quantities (stress and couple stress) in the rate form is derived in light of the principles of thermodynamics. It is consistent with the requirement of observer independence and material symmetry. The governing equations for two order parameters are derived, in which void concentration and stress anisotropy are related to relative displacement and rotation. As an example, the proposed model is applied to the hardening regime of deformation of a dense particle assembly with initial perfect lattice under simple shear. It is demonstrated that in the presence of dilatancy and director variation, there exists a linear relation between the shear stress and strain, in coincidence with experimental observations.     

Development of a single-layer finite element and a simplified finite element modeling approach for constrained layer damped structures

A new finite element (FE) is formulated based on an extension of previous FE models for studying constrained layer damping (CLD) in beams. Most existing CLD FE models are based on the assumption that the shear deformation in the core layer is the only source of damping in the structure. However, previous research has shown that other types of deformation in the core layer, such as deformations from longitudinal extension, and transverse compression, can also be important. In the finite element formulated here, shear, extension, and compression deformations are all included. As presented, there are 14 degrees of freedom in this element. However, this new element can be extended to cases in which the CLD structure has more than three layers. The numerical study shows that this finite element can be used to predict the dynamic characteristics accurately. However, there is a limitation when the core layer has a high stiffness, as the new element tends to predict loss factors and natural frequencies that are too high. As a result, this element can be accepted as a general computation model to study the CLD mechanism when the core layer is soft. Because the element includes all three types of damping, the computational cost can be very high for large scale models. Based on this consideration, a simplified finite modeling approach is presented. This approach is based on an existing experimental approach for extracting equivalent properties for a CLD structure. Numerical examples show that the use of these extracted properties with commercially available FE models can lead to sufficiently accurate results with a lower computational expense.     

Bending of a two-layer beam with non-rigid contact between the layers

The bending, under plane stress state conditions, of a two-layer beam-strip with identical isotropic linearly elastic layers with non-rigid contact between them is considered. The effect of the contact interaction between the layers, simulated by an elastic or elastoplastic gasket of negligibly small thickness with a finite shear stiffness, on the deflection of the beam is studied. Absolute slippage and rigid contact between the layers are the two limiting values of the shear stiffness. The values of the flexural stiffness of the beam differ by a factor of four in these limiting situations. The problem is reduced to a one- dimensional problem in the case of harmonic external load and an asymptotic solution is constructed for it. In the case of a load of general form, the Kirchhoff - Love hypotheses are used to construct an approximate solution and the problem is reduced to a one-dimensional problem. The difficulties which arise in simulating of the interaction forces between the layers using Coulombic dry friction forces are discussed.     

Numerical simulation of three-phase flow in an external gear pump using immersed boundary approach

This paper presents a three-phase fully compressible model applied along with an immersed boundary model for predicting cavitation occurring in a two dimensional gear pump in the presence of non-condensable gas (NCG). Combination of these models is capable of overcoming numerical challenges such as modelling the contact between the gears and simulating the effect of NCG in cavitation. The model accounting for the effect of NCG also has broader applicability, since gas dissolved in liquids can come out of the solution when exposed to low pressures; this plays a significant role in the pump performance and cavitation erosion. Here the simulation results are presented for the gear pump at different operating conditions including the contact between gear, gear RPM and % of NCG; their effects on performance and cavitation is demonstrated. The results suggest that modelling the contact between the gears play a role in the cavitation prediction inside the gear pump. An increase in cavitation is observed when the contact is modelled even for the small pressure difference considered between the inlet and outlet. An increase in the RPM of the gears also results in increased cavitation within the pump, whereas an increase in the percentage of NCG content by a small amount can reduce the cavitation to a greater extent. This reduction is due to the expansion of the gas at a lower pressure which recovers the pressure and prevents or delays the phase-change process of the working fluid. The fluctuations in the outflow rate is also found to increase when the gears are in contact and also with increasing gas content.     

Dissipation induced instabilities in continuous non-conservative systems

In this contribution we analyze the stabilizing and destabilizing effect of small damping for rather general class of continuous non-conservative systems, described by the non-self-adjoint boundary eigenvalue problems. Explicit asymptotic expressions obtained for the stability domain allow us to predict when a given combination of the damping parameters leads to increase or to decrease in the critical non-conservative load. The results obtained explain why different types of internal and external damping so surprisingly influence on the stability of non-conservative systems. As a mechanical example the stability of a viscoelastic rod with small internal and external damping, loaded by tangential follower force, is studied in detail. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On some regularities in dynamic response of cyclic periodic structures

The paper deals with cyclic periodic structures modelling bladed disk assemblies of blades with friction elements for vibration damping. These elements placed between adjacent blades reduce the vibration amplitudes by means of dry friction resulting from centrifugal forces acting on the elements and relative displacements of the blades. However, the application of these friction elements results in an additional dynamical coupling which together with mistuning of some system parameters (e.g., blade eigenfrequency or contact parameters) may cause localization of vibration. In the present paper a linear approximation of such a system is investigated. The structure composed of cyclic periodic cells modelled each as a clamped-free beam interacting with each other by means of viscoelastic elements of complex stiffness is applied for dynamic system analysis. In case of free vibrations as well as in case of steady-state dynamic response to a harmonic pressure field, a perfect periodic structure and the structures with periodically mistuned parameters (blade eigenfrequencies and contact parameters) are studied. Some regularities in the dynamic response of the systems with mistuning have been noticed. Despite the fact that only a linear approximation has been used, the results and conclusions can be applied for models which describe the blade interaction in a nonlinear way.     

An analysis of contact deformation of auxetic composites

The adaptive mode of frictional interaction has been studied as a self-locking effect upon contact deformation of isotropic and anisotropic auxetic materials with a negative Poisson ratio. This effect manifests itself in the fact that the bearing capacity of the joint rises with increasing shear load. In particular, the parameters of stress state (contact load, tangential stresses, slippage, etc.) were determined for a double-lap joint under conditions of compression with or with out shear. The contact interaction was analyzed by the finite-element method for three profiles of symmetrically located contact elements (plane, cylindrical, and wedge-shaped). The Poisson ratio was varied within the range theoretically admissible for isotropic elastic media. Analogous calculations were also performed for a joint with a deformed element made of an anisotropic auxetic composite, whose reinforcement angle was varied. The maximum loads, tangential stresses, and slippage are obtained as nonlinear functions of Poisson ratio (in the isotropic case) and reinforcement angle of the composite material. The stress concentration and the increased ultimate shear forces are also estimated. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 5, pp. 681–692, September–October, 2006.     

Closed-form solution considering the tangential effect under harmonic line load for an infinite Euler–Bernoulli beam on elastic foundation

The dynamic response of an infinite Euler–Bernoulli beam resting on an elastic foundation, which considers the tangential interaction between the beam and foundation under harmonic line loads, is developed in this study in the form of a closed-form solution. Previous studies have focused on elastic Winkler foundations, wherein the tangential interaction between the bottom of the beam and the foundation is not considered. In this study, a series of separate horizontal springs is diverted to the contact surface between the foundation and beam to simulate the horizontal tangential effect. The horizontal spring reaction is assumed proportional to the relative tangential displacement. As the geometric equation and linear-elastic constitutive equation of beam under the condition of small deformation have been presented based on the basic principle of elasticity mechanics, the analysis model is built and the governing differential equations about normal and tangential deflections of beam are deduced. Double Fourier transformation and the residue theorem are used to derive the closed-form solution to this problem. The proposed solution is then validated by comparing the degraded solution with the known results and comparing the numerical solution with the analytical solution. We also discuss the case in which the load direction is not vertical to the beam. Results can be used as a reference for engineering design.