Torsional Stiffness Coefficients Research Articles

Research on dynamic transmission error of harmonic drive with uncertain parameters by an interval method

Transmission error is one of the most important performance indicators for evaluating harmonic drives, and can have crippling effects on positioning accuracy and stability of industrial robots. However, most of the existing error analysis methods focus on a single factor, and do not consider the uncertainty of dynamic parameters, leading to evident limitations. In the present study, static transmission error (caused by manufacturing and assembly error) and dynamic transmission error (generated by static transmission error and dynamic parameters) of a harmonic drive are modeled. An interval method is developed and used to numerically express uncertain dynamic parameters of the system. Chebyshev polynomials are used to approximate the dynamic differential equations of the harmonic drive, and then the distribution of dynamic transmission error and its relationship with uncertain parameters are discussed in detail. In addition, a global sensitivity analysis is carried out to intuitively demonstrate how much impact each parameter has on dynamic transmission error. Our results suggest that the moment of inertia Jin and the torsional stiffness coefficient k1 have a large influence on dynamic transmission error. Finally, the proposed method is validated by experiment. The method can be adopted to determine the upper and lower bounds of dynamic transmission error of dynamic systems under the influence of uncertain parameters and provides a theoretical basis for transmission error optimization and compensation.

Simulation analysis and experimental evaluation of the transient behaviour of a reluctance magnetic coupling

The aim of this study is to present the dynamic behaviour of two co-axial rotating shafts connected by a reluctance magnetic coupling. The analysis is based on non-linear differential equations of motion. The reluctance magnetic coupling is modelled by an equivalent torsional stiffness coefficient and an additional viscous damping coefficient to account for the effects of induced currents in the salient-pole during the transient. These coefficients are obtained from three-dimensional finite element electromagnetic models. In order to evaluate the accuracy of the proposed model, several transient responses for the shaft line are considered. The simulation results are compared with those obtained from measurements.

Tuning coupled wave dispersion in a granular chain on a V-shaped rail

This paper introduces a model for the propagation of multiple types of waves in a compressed granular chain placed on an elastic V-shaped rail. The system possesses translational, rotational, and torsional degrees of freedom, and the different waves interact due to the presence of Hertzian contacts between adjacent spheres and between the spheres and the rail. Additional tangential force and torque between the spheres and the rail governed by the Hertz–Deresiewicz law are accounted for. This allows an exclusive tuning of the shear and torsional stiffness coefficients and leads to a complex coupled wave dynamics. Tunable dispersion relations including zero group velocity mode, accidental degeneracy and mode hybridisation are investigated in the linear regime. It is shown that the tangential force-dependent shear stiffness between the spheres and the rail strongly influences the dispersion relations, while the influence of the torque-dependent torsional stiffness is much weaker. The inclination angle of the rail is examined as another tunable degree of freedom, which modifies the wave dispersion not only by changing the shear coupling but also by influencing the coupling among different types of waves. The tunability of the wave dynamics in the granular chain evidenced in this work offers a flexible way of designing acoustic systems for selective vibration filtering and wave manipulation.

Finite element analysis for engine crankshaft torsional stiffness

Accurate crankshaft torsional stiffness is important to build the lumped parameter model (LPM) for crankshaft torsional vibration analysis. For the crankshaft system of a four-cylinder in-line gasoline engine, a procedure to build the finite element model (FEM) is presented. And the applicability of two ways to set the equivalent mass or equivalent moment of inertia for piston-rod in FEM is declared. When building a lumped parameter model, the concept of rigid-rotation of the crank pin central section around the central line of the crankshaft is proposed, and a method to obtain the torsional angle of the central section is declared. What is more, three post-processing methods are presented to obtain the torsional stiffness coefficients. According to three load cases of no-load, half load and full load, the tests are carried out. The simulation results of FEM and LPM, and the test results have good agreement with each other.

Finite element analysis for engine crankshaft torsional stiffness

Accurate crankshaft torsional stiffness is important to build the lumped parameter model (LPM) for crankshaft torsional vibration analysis. For the crankshaft system of a four-cylinder in-line gasoline engine, a procedure to build the finite element model (FEM) is presented. And the applicability of two ways to set the equivalent mass or equivalent moment of inertia for piston-rod in FEM is declared. When building a lumped parameter model, the concept of rigid-rotation of the crank pin central section around the central line of the crankshaft is proposed, and a method to obtain the torsional angle of the central section is declared. What is more, three post-processing methods are presented to obtain the torsional stiffness coefficients. According to three load cases of no-load, half load and full load, the tests are carried out. The simulation results of FEM and LPM, and the test results have good agreement with each other.

Finite Element Contact Analysis for Sprocket of an Armoured Face Conveyor

Sprocket has an important role in improving the transport performance and life of the armoured face conveyor, which is the key component in the transmission system of armoured face conveyor (AFC). In this paper, true stress-true strain curves of materials of the sprocket and the ring chain are obtained by computing the data of static tensile test. A simplified symmetric finite element contact model is established based on the nonlinear finite element software ABAQUS. Stress field and torsional stiffness coefficient of the sprocket are calculated according to loads and boundary conditions. Simulation results are in good accordance with the test results. The finite element model and the simulation results provide useful guidance for design and test of the sprocket. Meanwhile, accurate torsional stiffness coefficient of the sprocket is obtained for simulation of the transmission system of AFC.

Elastically restrained Bernoulli-Euler beams applied to rotary machinery modelling

Facing the lateral vibration problem of a machine rotor as a beam on elastic supports in bending, the authors deal with the free vibration of elastically restrained Bernoulli-Euler beams carrying a finite number of concentrated elements along their length. Based on Rayleigh’s quotient, an iterative strategy is developed to find the approximated torsional stiffness coefficients, which allows the reconciliation between the theoretical model results and the experimental ones, obtained through impact tests. The mentioned algorithm treats the vibration of continuous beams under a determined set of boundary and continuity conditions, including different torsional stiffness coefficients and the effect of attached concentrated masses and rotational inertias, not only in the energetic terms of the Rayleigh’s quotient but also on the mode shapes, considering the shape functions defined in branches. Several loading cases are examined and examples are given to illustrate the validity of the model and accuracy of the obtained natural frequencies.

Contribution to the definition of the torsional stiffness of the crankshaft of a diesel engine used in heavy-duty vehicles

A new approach for determining the torsional stiffness coefficient of an equivalent mechanical system's crankshaft section, which can be used to study torsional vibrations, is presented. In order to describe mathematically the torsional vibration of a reciprocating engine, it is necessary to replace the actual system consisting of the crankshaft, connection rods, and reciprocating elements by an equivalent system, usually modelled as a lumped-parameter system. The present authors, in accordance with their personal experiences, have found the determination of the equivalent system's parameters to be a very important and demanding task, especially in regard to torsional stiffness and damping coefficients. This paper analyses all aspects of determining a crankshaft section's torsional stiffness coefficient, which is the most complex element in a torsional dynamic model of a crankshaft. The method analysing the crank torsional stiffness coefficient determination has been given in the literature. The real boundary conditions of the crankshaft are emphasized as a problem, which makes determination of the torsional stiffness coefficient of the crankshaft section even more difficult. An indirect method for this determination is explained in this paper. By using this method, the real boundary conditions of the crankshaft of the diesel engine used in heavy-duty vehicles are considered, and the results obtained are reliable and confirmed by experiment.

Torsion of Open and Closed Thin-Walled Laminated Composite Sections

Vlasov’s theory is expanded to perform the linear analysis of open and closed sections made of general laminated composites. Following the guidelines of Mechanics of Laminated Beams, the transverse shear deformations are included in the formulation. To overcome the overestimation of stiffness by the often assumption of negligible tangential stress, the laminate resultant force and moment are instead set to zero in this paper. The present model accounts for all possible elastic couplings in composite sections, such as extension- and bending-torsion. Applications of the formulation to flanged and box sections are presented and guidelines are provided on how to evaluate the torsional stiffness coefficients of the beam. A general solution for a fixed composite beam under tip torsional loads is derived, and the steps can be used to derive solutions for any other boundary conditions. The effect of warping-torsion on the torsional stiffness of the beam is investigated. Composite beams were tested in torsion and the angle of twist and shear strains were measured for verification of the analytical model predictions, which agree closely with the experimental results and finite element results.