# Dynamics of Systems with Multiple Friction Contacts

## Quenching friction-induced oscillations by the use of high-frequency excitation

Dry friction is present in many technical systems and is the reason for a variety of undesired phenomena. The property of non-smoothness and a negative slope at low relative velocities of the friction force may cause friction induced oscillations or stick-slip motion. One attempt to quench these oscillations are superposed high-frequency vibrations. Thus, the effective friction characteristic is smoothed and undesired friction induced oscillations can be quenched. For systems with one degree of freedom, this method is investigated by simulations and experiments, which show good accordance. Systems with multiple degrees of freedom and multiple friction contacts however show a largely richer dynamic behavior. Also, the influence of high-frequency excitation is not investigated yet. To improve the comprehension of such systems, numerical and analytical approaches are inquired. Applied procedures are the excitation of single masses and investigating the effect on the whole system. Furthermore, the influence of the direction of the excitation and the importance of which masses are excited is examined.

## Superposed oscillations in revolutional joints with dry friction

Along with roller bearings, plain bearings are the type of bearings that are used most in technical systems. By the use of different lubricants and pairing materials, the bearings can be adapted to many different requirements. When lubricants are not deployable because of environmental restrictions, dry bearings have to be used. In such systems, undesired effects like stick-slip-motion or break-away can occur due to dry friction. Especially in positioning facilities, where small tolerances and high precision are required, this can be problematic. There are several approaches to compensate effects caused by dry friction and research still goes on. Especially in control technology, many sophisticated concepts are developed to obtain the desired system dynamics.

Another approach to influence the dynamical behavior of the system is the superposition of high-frequency oscillations. These vibrations can be in plane of the contact surface or normal to it. In-plane oscillations can be divided into transversal and longitudinal motion. The superposed oscillations have a smoothing effect on the system dynamics, which can yield a damping effect that can even suppress friction-induced oscillations. This topic has been subject of research in recent years and many papers were published, where theoretical and experimental results are discussed.

To contribute to the understanding and application of this effect, a model of a novel revolution joint with dry friction and axial excitation is built and investigated. A plain bearing rotates around a bolt, which cannot rotate and is excited to axial oscillations. Due to the rotational motion of the bearing, there is permanent sliding in the contact area between plain bearing and the bolt and no such effects like stick-slip-motion or break-away occur. Furthermore, this can lead to a reduction of the torque that is required to realize the rotation of the bearing.  However, the friction power leads to a heat input into the bolt which results in thermal expansion and a changed frictional behavior in the contact. To investigate the system dynamics and to analyze the influence of the  heat generation, the equation for heat conduction, the equation for the displacement field in the bolt and the equation of motion of the plain bearing have to be evaluated and solved. Numerical simulations show, that a stationary state is reached, where the bolt has a constant, elevated temperature and oscillates at a constant amplitude. The thermal expansion of the bolt results in an increased friction force, which limits the oscillation amplitude. This also results in a greater torque to realize the rotation of the plain bearing. Under certain parameters, this thermal property can even quench the friction reducing effect of superposed vibrations.

To validate this investigation, a test rig is constructed, where an electric motor realizes the rotation and piezo actuators are used for the axial excitation. The drive torque is measured by a sensor and the temperature at the contact area is measured by a pyrometer. The goal is to get insights in the relation between high-frequency excitation, heat generation and the friction torque.

Contact: Prof. A. Fidlin, S. Keller

Constantly increasing energy costs and restrictive legal regulations make it necessary to consistently increase the efficiency in all types of machines and systems. As a result, damping influences are systematically reduced. In addition, mechanical structures are increasingly being designed in terms of lightweight construction, which further increases their sensitivity to vibration excitation. That is why it is urgently necessary to reduce vibrations of mechanical structures effectively and in a focused manner without significantly influencing the efficiency of the machine as a whole. The nonlinearities of the damping forces, which are often neglected in the design, open up great potential for realizing situation-dependent behavior without having to resort to active control and external energy supply.

In particular, dry friction with the inherent stick-slide transitions enables the realization of mechanical switching elements that can serve as basic building blocks for adaptable dissipative devices. Above all, the aim is to investigate how novel devices based on dry friction can be used for the targeted reduction of externally excited, parametrically excited, and self-excited vibrations. In addition to the analysis of the dissipative device itself, methods for determining an optimal configuration and spatial placement of these devices are also proposed. In order to achieve these goals, various devices based on four basic elements (elasticity, play, friction with possibly modulated normal force and spring with distributed friction) and their combinations are compared in terms of their effectiveness and their self-adaptive properties. In addition, analytical methods are being developed in order to be able to reliably evaluate the efficiency of these devices. Prototypes for the most promising concepts are manufactured and experimentally tested.

Contact: Prof. A. Fidlin, J. Aramendiz

# Design of bipedal robots with optimized energy efficiency in varying environments

In the development of biped robots, energy efficient locomotion via walking or running is a major research priority. Due to the limited energy storage (battery), energy efficiency significantly determines the walking distance that can be covered. Energy efficiency depends not only on the controller used for stabilizing the motion, but also on the structural design and its mechanical model parameters. Hence, the purpose of this research is to develop and apply a method to systematically optimize the structure of a bipedal robot to maximize energy efficiency in different environments.

In the first step we consider an underactuated robot model which consists of five segments. Its rigid segments are additionally connected by elastic couplings such as torsion springs. Knowing that the robot’s periodic walking or running gaits can be analyzed as limit cycles of the controlled mechanical system, its resonance frequency can be adjusted to match the current step frequency by modifying the elastic couplings. A systematic approach to achieve this matching consists in the simultaneous optimization of the elastic couplings and the controlled motion. Consequently, the robot exploits the mechanical system's natural dynamics instead of wasting energy on its suppression.

Since the optimization of the parameters essentially depends on the conditions of the environment and the gait, the focus is on how the adaptation of the robot to the current operating state can be realized. Unlike the motion, which can be continuously updated by the controller based on the measured state variables, the elastic couplings need to be optimized to achieve the best characteristics by using "compliant smart mechanics" (COSM). Between the stance and swing phases of the robot’s legs, and especially during switching processes of the movement, an optimal (force-displacement and/or force-velocity) characteristic of COSM leads to an overall high energy efficiency for a wide range of walking scenarios.

This project is being carried out jointly by two research groups: The Compliant Systems Group (FG NSYS) from the Ilmenau University of Technology (TU Ilmenau) and the Institute of Engineering Mechanics (ITM) from the Karlsruhe Institute of Technology (KIT). While the team at TU Ilmenau investigates the compliant mechanical systems and their design and implementation, the team at KIT simulates and optimizes bipedal robots with those COSM mechanisms in different environments. It is financially supported by the German Research Foundation (DFG), grant FI 1761/4-1 | ZE 714/16-1.

Project partner: Prof. L. Zentner ,

Contact: Prof. A. Fidlin,

# Dynamics and Control of Hydraulic Systems

Hydraulic valves are known to show interesting dynamic behavior. Nevertheless they have not yet been investigated extensively from the viewpoint of nonlinear dynamics and are not suffciently understood. An elementary hydraulic pressure control valve can be described as a system of third order with a non-smooth nonlinearity.

The transition from an ideally impermeable valve to a valve allowing for leakage ﬂow uncovers an instability mechanism for for certain valve geometries. Leakage changes the character of the equilibrium position from a set-valued equilibrium position to a unique one. A loss of stability of the equilibrium position and the birth of a limit cycle due to leakage can be shown when increasing leakage flow or the working point pressure of the system. A bifurcation analysis reveals the different solution types for the system under external forcing, yielding evidence of period-doubling phenomena up to quasi-periodic solutions.

Building on the findings for the dynamics of the foundational element in hydraulics - the valve - a variable displacement vane pump is currently investigated. This type of pump is frequently used in automotive engineering in order to provide the required pressure for the actuation of a clutch mechanism. A subfunction of the pump is to provide the volume flow required by a cooling unit. The system under investigation shows many aspects which are characteristic for hydraulic systems. In steady state, the valve regulating the system pressure is lapped critically, therefore giving rise to non-smooth dynamics. Apart from analyzing the stability behavior of the pump system in a first step, in a second step control strategies shall be devised that result in a change of the working point of the system. Drawing on the control strategies identified, the task then is to synthesize hydraulic elements and their topology such that the control strategies can be implemented by means of hydraulic action.

The hydraulic consumer provided with volume flow from the variable displacement vane pump is the third field of interest in this research project. As pointed out, it is a clutch actuation mechanism. In order to simulate the dynamics of clutch systems adequately, reasonable estimates of the system parameters have to be known. By means of Kalman filtering, important parameters of the clutch actuation mechanism can be identified. To do so, the clutch actuation system is subjected to transient volume flow excitation. Measurements of the system responses to the transient excitation are then synchronized with a slave model of the consumer, resulting in good estimates of the true parameters to be identified.

Contact: Prof. A. Fidlin,

# Friction Induced Vibrations in Shift Gearboxes

A shift gearbox is a commonly used element in automotive transmissions, which is needed for two reasons. Firstly, it transmits the driving torque of the motor. Secondly, because a combustion engine works best at its nominal speed, it changes the gears. Experimental data reveal that undesired vibrations can arise during the clutch engagement process. The aim of this research is to explain the emergence of such vibrations in order to understand how to successfully prevent them.

Depending on the manifestation of the vibrations, several model approaches are thinkeable to explain the effect. A basic idea is to investigate the stability of the stationary behaviour and the origination of friction-induced vibrations, because the vibrations only occur while the disc contact is sliding. The physical modelling of the system is a flexible multibody system approach. Components taken into account are the clutch disc, gears, shafts and the actuation.

Well-known reasons for friction induced vibrations are clutch judder at low frequencies and the wobbling disc instability. They both cause vibrations perceiveable by customers, and which can be measured e.g. acoustically.

Another instability at medium frequencies arises because of the gear coupling, where the amplification of clutch friction forces induces an instability. In this case, strong translational vibrations of shafts occur with opening contacts in clutch and gears and stick-slip-transitions. Because of the translational movement, such vibrations can be measured in the actuation signal of the clutch.

Yet further instabilities can arise because of the interaction of elastic disc modes with rigid-body modes. The frequency of such instable behaviour is higher than the previous ones. Here too, translational vibrations occur in combination with elastic plate oscillations.

Contact: Prof. A. Fidlin

# Research Topic: Self-balancing of the planetary moving rotor

The phenomenon of self-balancing of rigid rotors is well known and investigated for rotors with fixed bearings. However, in some technical devices the rotor performs complex motions. An example of such system is a computed tomography scanner. Its anode rotates very fast in the housing of the X-ray tube. At the same time the X-ray tube itself rotates rather slowly around the patient’s body. It is very important for CT scanner to keep the minimal possible level of vibrations in order to obtain good image quality. The objective is to investigate how and to which extent the self-balancing devices can be used for reducing vibrations in a planetary moving rotor.

The model to consieder consists of the rotor of mass M, which is fixed on the end of the rigid carrier. The other end of the carrier is elastically suspended with radial spring-dampers of a certain stiffness c and damping b. The carrier rotates around it’s point of suspension with a constant velocity . At the same time the rotor rotates around its symmetry axis with a given velocity ω. Its centre of mass has an offset relative to the rotation axis. Two pendulum balancers of mass m, moment of inertia J and length r are placed on the rotation axis of the rotor.

The stationary solutions of the system and the passage through the resonanceare are investigated analytically using averaging technique for the strongly damped systems. Analytic results match very well with numeric simulations when the velocity of the planetary motion is sufficiently small.

Contact: Prof. A. Fidlin, O.Drozdetskaya

# Transient Behaviour of Coupled Exciters

Application of several unbalanced exciters instead of one is a widely used design for vibratory machines. This may have the benefit of distributing the excitation along the machine or decreasing the load of the exciter’s bearing, if several low power exciters are used instead of one powerful exciter. It also offers the possibility of coordinating rotors dynamics without kinematic connections between them. The self-synchronization phenomenon, which allows the automatic coordination of exciter rotations, is utilized in many vibratory machines for generating the required excitation force with constant or another required (for example elliptically changing) direction.

Even though stationary solutions of many self-synchronising systems, their condition of existence and stability has been extensively investigated, little is known about the transient behaviour about such systems. Attraction domains of already known stationray solutions and behaviour during passage through resonance are the two main topics of the investigation. Asymptotic methods, which are used to analyse systems with a single unbalanced exciter, will also be applied here.

Contact: Prof. A. Fidlin, T. Yüzbasioglu

# Dynamics of Piezo Actuated Journal Bearings

The dynamic behaviour of rotor-bearing systems represents an ongoing field of research.

By increasing the rotational speed of the rotor an instability can be detected which is often referred to as "oil-whirl" or "half-frequency-whirling" in literature. As the frequency of this "whirling" instability meets an eigenfrequency of the associated elastic rotor, its oscillation amplitudes increase tremendously which is also known as "oil-whip". These "oil-whirl" and "oil-whip" effects can be rated as rather critical and should be avoided during the operation of the rotor-bearing system.
Various modifications (compared to the "classical" cylindrical bearing) have been proposed in literature in order to suppress or at least to decrease these unwanted effects. By modifying the shape of the bearing sleeves improvements of the rotor's dynamic behaviour are expected. Starting from an initially circular shape, the bearing sleeve is elastically deformed by piezoelectric actuators which leads to a complex fluid-solid-interaction.

The effects of the piezo actuated shaping on the rotor-bearing system are studied by means of systematic stability and bifurcation analysis, focusing on time-efficient modelling methods.

Contact: Prof. W. Seemann, A. Bitner

# Contact Mechanics and Stochastic Dynamics

Systems with friction are widespread in all kinds of applications. Unfortunatelly the precise simulative prediction of the resulting friction force in frictional contacts is still an unsolved task which has led to the developement of many empirical friction laws. Despite their great value for many practical and theoretical applications, a deeper insight into the actual contact situation and the corresponding friction force with its dependencies is necessary to improve technical systems e.g. with regard to energy efficiency and wear.

Early works on this topic by Greenwood & Williamson and Archard for the purely elastic and Bowden & Tabor for the purely plastic deformation case have at least led to a justification for Coulombs friction law. Their research efforts indicate that the real contact area is almost proportional to the normal contact force whereat this relation can be attributed to surface roughness in both load cases. By the additional assumption that the friction force is proportional to the real contact area Coulombs friction law can be justified.

Further investigations on the dry contact of two sliding metallic bodies depending on various physical parameters and sliding speed are performed. For this purpose, a thermomechanical model is developed and evaluated for different contact configurations considering the surface roughness of both contact bodies in particular. Subsequent investigations on the consequences of the calculated friction coefficient in the context of friction-induced vibrations are carried out.

Contact: Prof. C. Proppe, L. Oestringer

# Stochastic Analysis of Geometric Mistuning in Radial Compressor

Radial compressor in turbochargers is often considered in theory as periodic system, but in fact it features inevitable small imperfections caused by material defects and manufacturing tolerances which break the cyclic periodicity. This is called mistuning. The loss of periodicity changes drastically the dynamic behavior of the compressor. Typically the forced response level of the mistuned bladed disk is larger than the tuned design. Because of the random nature of mistuning, the determination of the largest resonant response at any frequency has to be considered as a stochastic problem.

Mistuning receives significant attention from the research community since the late 1960s. Models using coupled lumped mass oscillators have allowed the fundamental phenomena of mistuning to be understood. More recently finite element model are used to explore with a better precision the behavior of mistuned compressor. To minimize the computational costs the finite element model has to be reduced first before performing a Monte Carlo Analysis. In the last decade several model reduction methods were developed. The way in which mistuning is implemented depends on the used reduction method and the vast part of them accounts only for a frequency mistuning model or a proportional mistuning model, in which the ideal tuned configuration is not modified.

Today a key topic is to combine this reduction methods with a more realistic introduction of mistuning, such as geometric mistuning. In other words, it will be tried to take directly random geometry modifications into consideration in a stochastic problem.

Contact: Prof. C. Proppe, M. Koebele

# Surrogate models for uncertainty quantification for the forecast of the West African Monsoon

Surrogate modeling is a method that can be applied if quantities of interest cannot be easily directly measured or simulated, e.g. if a simulation run or experiment is very expensive. In this case, a surrogate model for the outcome is obtained and used instead. Due to increasing complexity of models, surrogate modeling is playing an increasing role in various engineering, but also other scientific disciplines. In this work, a meteorological problem is analyzed in collaboration with the Institute of Meteorology and Climate Research - Department Troposphere Research at KIT.

The West African monsoon is a major wind system that affects regions between latitudes 9° and 20° N. The monsoon is the result of the seasonal shifts of the ITCZ (Intertropical Convergence Zone) and seasonal temperature and humidity differences between the Sahara and the equatorial Atlantic Ocean. The forecast of this monsoon has shown to suffer from remarkable uncertainties in several quantities. Some of these quantities are the local rainfall and the north-south shift which have a great impact on the inhabitants, particularly on the agriculture. In order to quantify the uncertainty in the forecast of the West African monsoon, a sensitivity analysis for a range of uncertain parameters in the weather model is conducted.

For this study the ICON model which is operationally used by the Deutscher Wetterdienst (DWD) is applied to carry out weather simulations. Since for the analysis a steady state for the monsoon quantities over simulated days is intended, the simulated time and thus the computational cost for one simulation run is very high. Therefore, surrogate models seem to be a promising opportunity. In this work, Gaussian Process Surrogates are used to achieve a relation between input parameters and monsoon quantities. The surrogate model is then used to carry out a global sensitivity analysis given defined ranges and probability density functions for all parameters. The results can offer an indication which parameter definitions should be specified more detailed by conducting further studies in order to reduce the uncertainty in forecasted monsoon quantities. Furthermore, the surrogate model can serve as a basis for parameter identification studies.

Contact: Prof. C. Proppe, M. Fischer

# Uncertainty Quantification for Lifetime Prediction of Metal Foams

Metal foams and their applications are of interest in the area of research and development. They benefit from high stiffness in combination with low density, high energy absorption capacity and good damping properties, which are optimal conditions for light weight constructions, crash elements or vibration damping. Fields of application are for instance: aerospace, automotive, battery technology and orthopedics.

Due to the manufacturing process, metal foams demonstrate imperfections and thus fluctuations in material properties that ultimately lead to computational challenges for precise failure and lifetime prediction. Therefore, the aim of this research is developing a suitable efficient method for structural reliability analysis with probability boxes (p-boxes) using model hierarchies, which combine cheap, approximate surrogate models with an expensive, accurate high-fidelity model.

Contact: Prof. C. Proppe, J. Kaupp