Matthew P. Cartmell

The use of finite element techniques for calculating the dynamic response of structures to moving loads

Abstract This paper presents a technique for using standard finite element packages for analysing the dynamic response of structures to time-variant moving loads. To illustrate the method and for validation purposes, the technique is first applied to a simply supported beam subject to a single load moving along the beam. Finally, it is applied to the problem that initiated the work: calculation of the effects of two-dimensional motion of the trolley on the response of the base structure of a mobile gantry crane model.

Using Motorized Tethers for Payload Orbital Transfer

The concept of symmetrical double-ended motorized spinning tethers for use as orbital transfer vehicles is introduced. The orbital elements of a payload released from above and below a hanging, prograde librating, and prograde spinning tether are derived and employed to evaluate the effectiveness of the three tether types along with their optimum configurations for payload transfer. A new ratio, the efficiency index, is defined as the altitude gain or loss half an orbit after tether release per tether length. The motorized tether is found to perform best and also most efficiently, improving by two orders of magnitude on the librating tether, which, in turn, improves on the hanging tether by a factor of two. A long motorized tether on a circular orbit can transfer an upper payload from a low to a geostationary Earth orbit by employing relatively high motor torque and a safety factor on the tether strength close to unity.

A methodology for the determination of dynamic instabilities in a car disc brake

The dynamics of a car disc brake system is investigated by a combined analytical and numerical method. The disc is rotated past the stationary pads and calliper in sliding friction at constant speed. The modal data of the disc are obtained by means of modal testing whereby a dynamic model for the disc is derived based on the thin plate theory. Then the pads, calliper and mounting are analysed by means of the finite element method. Finally the equations of motion for the whole disc brake system are established through the interfaces between the pads and the disc. The stability of the vibrating system is studied by the method of state space.

Regular and chaotic dynamics of a discontinuously nonlinear rotor system

Abstract The nonlinear vibrations are considered in a two-degree of freedom rotordynamic system subjected to a bearing clearance effect. The excitation is provided by an out-of-balance within the system, and the nonlinearity, in the form of a discontinuous stiffness, is effected by means of a radial clearance between the elastically supported rotor and the elastically supported outer ring. Different nonlinear dynamics analysis techniques are employed to unveil the global dynamics of the rotor system. In particular the system has been investigated with the help of time trajectories, phase portraits, bifurcation diagrams, Poincare maps, power spectrum analysis and the construction of basins of attraction. A numerical study is presented which encompasses the effects of different system parameters in order to demonstrate the severity of the vibrations. It is also shown that the response of the system can be extremely sensitive to changes in these parameters, and that chaos can exist over large regions of parameter space.

One-dimensional shape memory alloy models for use with reinforced composite structures

In this paper three models of the shape memory alloy behaviour have been presented and re-investigated. The models are attributed to Tanaka, Liang and Rogers, and Brinson, and have been used extensively in the literature for studying the static or dynamic performance of different composite material structures with embedded shape memory alloy components. The major differences and similarities between these models have been emphasised and examined in the paper. A simple experimental rig was designed and manufactured to gain additional insight into the main mechanics governing the shape memory alloy (SMA) mechanical properties. Data obtained from the experimental measurements on Ni-Ti wires have been used in the numerical simulation for validation purposes. It has been found that the three models all agree well in their predictions of the superelastic behaviour at higher temperatures, above the austenite finish temperature when shape memory alloys stay in the fully austenitic phase. However, at low temperatures, when the alloys stay in the fully martensitic phase, some difficulties may be encountered. The model developed by Brinson introduces two new state variables and therefore two different mechanisms for the instigation of stress-induced and temperature-induced martensite. This enables more accurate predictions of the superelastic behaviour. In general, it can be recommended that for investigations of the shape memory and superelastic behaviour of shape memory alloy components the Brinson model, or refinements based on the Brinson model, should be applied.

Friction-induced vibration of an elastic slider on a vibrating disc

The in-plane vibration of a slider-mass which is driven around the surface of a flexible disc, and the transverse vibration of the disc, are investigated. The disc is taken to be an elastic annular plate and the slider has flexibility and damping in the circumferential (in-plane) and transverse directions. The static friction coefficient is assumed to be higher than the dynamic friction. As a result of the friction force acting between the disc and the slider system, the slider will oscillate in the stick-slip mode in the plane of the disc. The transverse vibration induced by the slider will change the normal force on the disc, which in turn will change the in-plane oscillation of the slider. A numerical method is used to solve the two coupled equations of the motion. Results indicate that normal pressure and rotating speed can drive the system into instability. The rigidity and damping of the disc and transverse stiffness and damping of the slider tend to suppress the vibrations. The in-plane stiffness and damping of the slider do not always have a stabilizing effect. The motivation of this work is the understanding of instability and squeal in physical systems such as car brake discs where there are vibrations induced by non-smooth dry-friction forces.

Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems

This review article starts by addressing the mathematical principles of the perturbation method of multiple scales in the context of mechanical systems which are defined by weakly nonlinear ordinary differential equations. At this stage the paper investigates some different forms of typical nonlinearities which are frequently encountered in machine and structural dynamics. This leads to conclusions relating to the relevance and scope of this popular and versatile method, its strengths, its adaptability and potential for different variant forms, and also its weaknesses. Key examples from the literature are used to develop and consolidate these themes. In addition to this the paper examines the role of term-ordering, the integration of the so-called small (ie, perturbation) parameter within system constants, nondimensionalization and time-scaling, series truncation, inclusion and exclusion of higher order nonlinearities, and typical problems in the handling of secular terms. This general discussion is then applied to models of the dynamics of space tethers given that these systems are nonlinear and necessarily highly susceptible to modelling accuracy, thus offering a rigorous and testing applications case-study area for the multiple scales method. The paper concludes with comments on the use of variants of the multiple scales method, and also on the constraints that the method can bring to expectations of modelling accuracy.

Parametric resonances in an annular disc, with a rotating system of distributed mass and elasticity; and the effects of friction and damping

The parametric resonances that occur in a stationary annular disc under the action of a distributed mass–spring–damper system which rotates, with friction, around the disc at subcritical speeds is the main focus of this paper. The distributed system occupies an annular sector and the stiffness, mass, damping and friction are each expanded in a Fourier series. When the distributed system is rotated then a set of stiffness, mass, damping and friction terms are obtained that vary both spatially and temporally. Finite element discretization is applied to the disc and the distributed system and the parametric resonances are determined by a multiple time–scales analysis of the finite element equations. The equations that define the transition curves are generally complex, although they are strictly real when damping and friction are omitted. Numerical results show that friction has a destabilizing effect over the entire subcritical speed range. Disc damping and damping in the rotating system both tend to suppress the vibrations of the disc when the speeds are subcritical. The effects of the distributed mass and stiffness are found to be almost neutral at subcritical speeds, but active in the supercritical range where the findings of other researchers are available for comparison and found to be in agreement.

Parametric Resonances at Subcritical Speeds in Discs with Rotating Frictional Loads

This paper is concerned with the parametric resonances in a stationary classical annular disc when excited by a rotating mass-spring-damper system together with a frictional follower load. An analysis by the method of multiple scales is performed to reveal the existence of instabilities associated with subcritical parametric resonances, and other instabilities of the backward waves in modes with nodal diameters. The latter are shown to be driven by friction and not to be dependent upon the rotational speed. A state-space analysis, with truncated modes, is used to investigate the effect of varying the friction, stiffness, mass and damping prameters in a series of simulated problems. The results obtained from the state-space eigenvalue method tend to support the conclusions of the multiple scales analysis.

Dynamic responses of structures to moving bodies using combined finite element and analytical methods

This paper presents a technique using combined finite element and analytical methods for determining the dynamic responses of structures to moving bodies. In previous work (Comput. Struct., submitted), moving masses were treated as moving loads, ignoring inertia effects. This is not always reasonable and the technique described here allows inertia effects to be included in the analysis. In order to illustrate the methodology, and for validation purposes, the technique is first applied to a clamped-clamped beam subjected to a single mass moving along the beam. Finally, it is applied to the problem that initiated the work: to predict the dynamic response of an experimental mobile gantry crane structure due to the two-dimensional motion of the trolley.