School of Mechanical & Systems Engineering
Seminar Series: 2005/2006

"Inverse Problems in Vibration Suppression:
Passive Modification and Active Control"

(Dept of Engineering, University of Liverpool)

Wednesday 8 March 2006
3.30pm, Stephenson Building, room F1 (first floor)

The problem of vibration suppression has occupied engineering scientists for over a century. The classical vibration absorber was invented by Frahm in 1909 and the theory first appeared in the open literature (Ormondroyd and Den Hartog) in 1928. The problem of determining the dynamic behaviour of a compound system from two or more subsystems with known receptances and interconnection properties was solved (in theory) by WJ Duncan in the 1940s.

The effect of a stiffness, damping or mass modification on the dynamic behaviour of a structure may be readily determined, in theory, if the matrix of receptances is measured at the connection points. In practice, quite simple modifications such as beams and large masses present serious difficulties because of the need to measure rotational receptances.

 Theory is presented for the estimation of rotational receptances from force and linear accelerometer measurements. The forces are applied to an attachment in the form of a T- or X-block to which the accelerometers are attached. A finite element model of the attachment is used to overcome a problem of ill-conditioning in the resulting equations. The forward problem of estimating the dynamics of a helicopter tailcone when modified by a large mass (equivalent to the mass and inertia of the tail-rotor gearbox and hub) is described.

The inverse structural modification problem of assigning natural frequencies and antiresonances is explained. Applications include the modification of a structure by an added beam.

Physical modifications are limited by the requirements of symmetry, positive-definiteness etc of the system matrices and this of course places a restriction on the dynamics that can be assigned. This restriction is removed (and also the need to measure rotational receptances) when the problem is re-cast as an active control problem. The active-control inverse problem requires the measurement of receptances and there is no need for a K, C, M system as for example in the state-space method. The technique is demonstrated using simple examples.

John Mottershead is the Alexander Elder Professor of Applied Mechanics at Liverpool University. He was awarded the DEng degree for his work on Structural Dynamics by the University of Liverpool in 2002.