John E. Mottershead

Finite Element Model Updating in Structural Dynamics

Preface. 1. Introduction. 2. Finite Element Modelling. 3. Vibration Testing. 4. Comparing Numerical Data with Test Results. 5. Estimation Techniques. 6. Parameters for Model Updating. 7. Direct Methods Using Modal Data. 8. Iterative Methods Using Modal Data. 9. Methods Using Frequency Domain Data. 10. Case Study: an Automobile Body M. Brughmans, J. Leuridan, K. Blauwkamp. 11. Discussion and Recommendations. Index.

FINITE ELEMENTS FOR DYNAMICAL ANALYSIS OF HELICAL RODS

Abstract Finite elements are presented for dynamical analysis of helical rods. The element stiffness and mass matrices are based on the exact differential equations governing static behaviour of an infinitesimal element. Natural frequencies obtained by use of the element, which allows for both shear deformation and rotary inertia, are compared to the frequency spectra of helical compression springs. The element performance is compared with that of other finite elements.

Receptance Method in Active Vibration Control

The pole/zero assignment problem is addressed using a method based on measured receptances. The approach, which is well known in passive structural modification, has not been used before in active vibration control. A number of significant advantages are claimed over the conventional state-space approach that uses the mass, damping, and stiffness matrices formed, for example, by finite elements. In fact, because the method is based solely upon measured vibration data, there is no need to evaluate or to know the M, C, and K matrices. It is demonstrated that all the poles may be assigned actively by the equivalent of a rank-1 modification to the dynamic stiffness matrix of the system. The assignment of zeros has a special significance in vibration suppression, because the vibration response at coordinatep vanishes completely when sinusoidal excitation is applied at coordinate q at the frequency of a zero of receptance H nn . A pole ofH pq may be eliminated by assigning a zero at the same frequency.

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.

Finite Element Modelling

In modern times the finite element method has become established as the universally accepted analysis method in structural design. The method leads to the construction of a discrete system of matrix equations to represent the mass and stiffness effects of a continuous structure. The matrices are usually banded and symmetric. No restriction is placed upon the geometrical complexity of the structure because the mass and stiffness matrices are assembled from the contributions of the individual finite elements with simple shapes. Thus, each finite element possesses a mathematical formula which is associated with a simple geometrical description, irrespective of the overall geometry of the structure. Accordingly, the structure is divided into discrete areas or volumes known as elements. Element boundaries are defined when nodal points are connected by a unique polynomial curve or surface. In the most popular (isoparametric, displacement type) elements, the same polynomial description is used to relate the internal, element displacements to the displacements of the nodes. This process is generally known as shape function interpolation. Since the boundary nodes are shared between neighbouring elements, the displacement field is usually continuous across the element boundaries. Figure 2.1 illustrates the geometric assembly of finite elements to form part of the mesh of a modelled structure.

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.

Vibration and squeal of a disc brake: Modelling and experimental results:

Abstract This paper presents a method for analysing the unstable vibration of a car disc brake, and numerical results are compared with squeal frequencies from an experimental test. The stationary components of the disc brake are modelled using many thousands of solid and special finite elements, and the contacts between the stationary components and between the pads and the disc are considered. The disc is modelled as a thin plate and its modes are obtained analytically. These two parts (stationary and rotating) of the disc brake are brought together with the contact conditions at the disc/pads interface in such a way that the friction-induced vibration of the disc brake is treated as a moving load problem. Predicted unstable frequencies are seen to be close to experimental squeal frequencies. The numerical simulation indicates that the stability can be improved by shifting the centre of the piston line pressure towards the trailing side of the pad.

Assignment of natural frequencies by an added mass and one or more springs

The problem of assigning natural frequencies to a multi-degree-of-freedom undamped system by an added mass connected by one or more springs is addressed. The added mass and stiffnesses are determined using receptances from the original system. The modifications required to assign a single natural frequency may be obtained by the non-unique solution of a polynomial equation. If more than one frequency is to be assigned, then a system of non-linear multivariate polynomial equations must be solved. Such a modification involves not only an added mass and one or more stiffness terms, but also an added coordinate. The paper presents a methodology, using Groebner bases for the solution of the multivariate polynomials, together with examples of natural frequency assignment. Realistic modifications are found to be bounded within certain frequency ranges. The effect of the modification on the natural frequencies not assigned and the antiresonances is explained.

A Moving-Load Model for Disc-Brake Stability Analysis

There are many elasto-mechanical systems that involve two components in moving contact where large-amplitude vibration and noise can be excited. This paper models the vibration and dynamic instability of a car disc brake as a moving load problem in which one component (the disc) is amenable to analytical treatment while the other component (the pads, calliper and mounting) has to be dealt with by the finite element method. A method is presented for solving the dynamic instability of the car disc brake as a nonlinear eigenvalue problem. The same approach can tackle other moving load problems.

Combining Subset Selection and Parameter Constraints in Model Updating

Model updating often produces sets of equations whose solution are ill-conditioned and extra information must be used to produce a well-conditioned estimation problem. One possibility is to change all the parameters, but to introduce extra constraints, for example by taking the minimum norm solution. This paper takes a different approach, by considering only a subset of the parameters to be in error. The critical decision is then the choice of parameters to include in the subset. The methods of subset selection are outlined and extended to the selection of groups of parameters. The incorporation of side constraints is considered and demonstrated using an experimental example.