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The resonance frequency (and therefore the critical frequency ) can be adapted by changing the mass Because the reaction mass is attached to the structure by a spring, the proof-mass actuator has no authority over the rigid body modes (in contrast to reaction wheels). Operation at low frequency requires a large stroke, which is technically difficult. Medium to high frequency actuators (above 40 Hz) can be obtained with loudspeakers (slightly modified with an added mass ) or commercial electrodynamic shakers provided with an appropriate mass at the tip, to lower the natural frequency to the desired value.
For complicated geometries, it may become very large, especially with automated mesh generators. 2) by condensing the degrees of freedom with little or no inertia and which are not excited by external forces. 49), we find the reduced equation which involves only Note that in this case, the reduced equation has been obtained without approximation. 51) applies even if have some inertia (sub-matrix not identically 0) or applied forces. 52). If external loads are applied to , the reduced loads are obtained by equating the virtual work or Finally, the reduced equation of motion reads Usually, it is not necessary to consider the damping matrix in the reduction, because it is rarely known explicitly at this stage.
2), we find that the complex amplitudes of F and X are related by The matrix is the dynamic generalization of the flexibility matrix and is called the dynamic flexibility matrix. 2). 18), we obtain the modal expansion of the dynamic flexibility matrix: where the sum extends to all the modes. 20), we can transform the foregoing result in such a way that the high frequency modes do not appear explicitly in the expansion: The static contribution of the high frequency modes to the flexibility matrix is often called the residual mode; we shall denote it R.