Reducing multiple modes of vibration by digital filtering and input shaping

The residual vibration of flexible systems can be reduced by properly shaping the reference command. There has been substantial evidence presented that input shaping is better than notch filtering for shaping reference commands to suppress vibration in mechanical systems. Much of this evidence is empirical comparisons between traditional filters and robust input shapers. Recently, a proof showing that notch filters are always equal to or longer in duration than an input shaper with identical single-mode vibration suppression constraints was presented. This paper expands on that previous result by extending the proof to multi-mode systems. The important ramification of this proof is that multi-mode input shapers suppress vibration more quickly than multi-mode notch filters. Ease of design, computation, and implementation are also discussed. Simulations of an industrial bridge crane demonstrate the key differences between the two methods.