Pascal Ziegler

Investigation of Gears Using an Elastic Multibody Model with Contact

The classical approach to simulate contacts between gears is to use rigid body models coupled with a parallel spring damper combination. However, these models had been developed for properly meshing gears with smooth contacts and cannot cover wave propagation caused by hard contacts or impacts. Moreover, as they are based on the assumption of rigidness, often light weight designs, resulting in very compliable gear bodies, cannot be considered appropriately. To evaluate how appropriate these rigid body models are to simulate impact forces, a very detailed finite element model is used to simulate several impacts and the results are compared to simulations with a rigid body model. The results reveal that for compliable gear bodies, there exist dynamic effects that considerably affect contact forces and motion and that these effects cannot be covered by rigid body models at all. Hence, a flexible model is imperative to precisely simulate impact forces. To reduce integration time, we present a modally reduced elastic multibody model including contact that allows very precise simulations in reasonable time. For the contact calculations a node-to-segment penalty formulation is introduced and is integrated using central differences. Even though the elastic model is a reduced model, it is still of huge size, as any node on any flank is a potential contact node. Also, the transformation data between modal and nodal coordinates must be accessible during integration. To reduce the required amount of memory a coarse collision detection is introduced that allows to dynamically reload only the transformation required in the current integration step. This approach allows very precise simulations of contacts between gears with integration times about 400 times faster than for associated finite element simulations. At the same time the model is robust and fast enough to allow the simulation of many contacts and many revolutions. To validate this approach basic experimental investigations with simple impact bodies have been carried out. The results from these experiments and related simulations agree very well.

Review on contact simulation of beveloid and cycloid gears and application of a modern approach to treat deformations

Gear drives are key components for all kinds of machines as well as of industrial equipment. Therein, beveloid gears and cycloid gears are increasingly used in industry. Gaining a more comprehensive understanding of those types of gears is essential. However, the measurement of the dynamic response of these gears is not an option due to the high cost of the required experiments. Along with the development of computer technology, several numerical tools and methods to study gears with standard and non-standard flank profiles have been introduced. Various works related to standard gears or beveloid and cycloid gears have been published. In this study, a contemporary review about the modelling and contact simulation of beveloid and cycloid gear drives will be given. Some studies will also be introduced to present an efficient approach to simulate contact forces and contact characteristics of gear wheels with standard and non-standard tooth profiles considering deformations too.

Impact Studies of Gears in Combustion Engines

Gears are commonly used design elements, typically used to convert torque. However, gears are also used in mechanisms or gear drives to transmit motion. Typical applications of gear drives are gear trains, used to drive the camshaft by the crankshaft in large-scale diesel engines. There, normally the transmitted rated torque is relatively small compared to the dynamical loads. They often introduce vibrations of the entire drive train, caused by gas forces or auxiliary devices. These dynamic loads cause the flanks of teeth to lift-off. The re-establishment of contact is mostly in the form of impacts and may occur on both sides of the teeth. Because of the noise induced by these impacts, this phenomenon is called gear hammering. In gear trains, the gears are often designed with thin gear bodies to reduce inertia effects and the total weight of the engine. As a result not only noise, but also endurance problems may arise due to the high peaks in the contact forces. Thus, the precise knowledge of the contact forces is necessary for the design process. However, contact forces between rotating gears are extremely difficult and expensive to measure. Therefore, the simulation of contact forces inheres great importance. Nowadays, the contact simulations are mainly done with commercial multibody packages, assuming rigid gears connected by elastic elements. These elastic elements somehow describe the contact stiffness as well as the elasticity of the gear bodies.

Dynamic stress recovery in gear train simulations using elastic multibody systems

A reduced elastic multibody model is employed for the transient simulation of spur gear, a pinion and a large gear. The elastic multibody model enables short computation times with high accuracy even for several full rotations. The numerical time integration is carried out in reduced space, whereas contact calculation is based on nodal contact calculation. Contact stresses and dedendum stresses are calculated using a matrix of stress ansatz functions, allowing the detection of critical stresses in the design process. The stress ansatz functions are obtained using reduction matrices with different kinds and numbers of modes.

Generalized Component Mode Synthesis for the Spatial Motion of Flexible Bodies With Large Rotations About One Axis

By far the most common approach to describe flexible multi-body systems in industrial practice is the floating frame of reference formulation (FFRF) very often combined with the component mode synthesis (CMS) in order to reduce the number of flexible degrees of freedom. As a result of the relative formulation of the flexible deformation with respect to the reference frame, the mass matrix and the quadratic velocity vector are state-dependent, i.e. non-constant. This requires an evaluation of both the mass matrix and the quadratic velocity vector in every integration step, representing a considerable numerical cost. One way to avoid the state-dependency is to use an absolute formulation as proposed in [2], which was extended in [4] for the use of the same shape functions as used in the classical CMS approach. In this approach, referred to as generalized component mode synthesis (GCMS), the total absolute displacements are approximated directly. Consequently, the mass matrix is constant, there is no quadratic velocity vector and the stiffness matrix is a co-rotated constant matrix. However, it was shown that when using the same shape functions as in the classical CMS approach, nine times the number of degrees of freedom are necessary to describe the same deformation shapes as in the CMS. Even though the integration times of the CMS and GCMS are of the same order, as presented in [5], in technical systems the majority of components are constrained to motions with only one single large rotation. Therefore, in this work the GCMS is formulated for large rotations around a fixed spatial axis. This allows to reduce the number of necessary flexible shape functions to three times the number of CMS shape functions and, consequently, further increases numerical efficiency compared to the GCMS for arbitrary large rotations. A piston engine composed of three flexible bodies, two of which rotate, is used as a test example for the planar formulation. It is compared to the GCMS and a classical FFRF with CMS. The results agree very well, while the GCMS for planar rotations is about three times faster than the other formulations.Copyright

PRELIMINARY STUDY TO INVESTIGATE THE EFFECT OF PISTON-LIKE AND ROCKING MOTIONS OF THE STAPES FOOTPLATE ON THE BASILAR MEMBRANE VIBRATION

The inner ear or cochlea is a bone structure of spiral shape and is composed of mainly two conical chambers which are filled with fluid and separated by a soft membrane, the basilar membrane. In case of a healthy ear, the closed hydraulic system is excited through the vibration of the stapes. According to present hearing theory, this leads to pressure waves in the cochlear fluid which in turn results in the characteristic vibration behavior of the basilar membrane. Related to the sound frequency, hair cells in certain areas of the basilar membrane are stimulated and cause hearing nerve stimulation. As reported in literature, the stapedial motion is mainly piston-like for low frequencies, whereas for higher frequencies rocking motions increasingly occur. Since purely rocking motions of the stapes footplate generate no net fluid displacement, several researches doubt that these motion components can lead to basilar membrane vibration and thus to hearing impression. Therefore, in this study a Finite Element model with simplified geometry of the human cochlea is developed using Eulerian-based acoustic elements to model the inner ear fluid. First the vibrations of the basilar membrane are calculated for a purely piston-like excitation mode. Then, these results are compared with the basilar membrane vibration pattern evoked by purely rocking motion around the short axis of the stapes footplate.

Simulation of Gear Hammering With a Fully Elastic Model

In large Diesel engines often geartrains are used to drive the camshafts. As the average transmitted load is small, dynamic loads, e.g. gas forces, are typically dominant. This can result in a rattling motion of teeth within the backlash, called gear hammering. We will show that for these impact-like contacts rigid body models are not sufficient, instead, elastic models are necessary to precisely simulate contact forces. The model proposed here is a modally reduced elastic multibody model including a contact algorithm. Due to the size of the modal transformation matrices, additional ways to reduce the computational effort are necessary, e.g. a dynamic reloading scheme. The model is robust and fast enough to allow simulations of many revolutions and many contacts. Furthermore, basic experiments have been carried out to validate the model. Experimental results are presented and agree very well with the simulations.