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The residual vibration of flexible systems can be reduced by shaping the reference command with notch filters, low-pass filters, and input shapers. Since the introduction of robust input shaping, there has been substantial evidence that input shaping is better than both notch and low-pass filtering for suppressing vibration in mechanical systems. Much of this evidence is empirical comparisons between traditional filters and robust input shapers. Given the large variety of filters and shapers and the large number of design strategies and parameters, there is still some uncertainty as to which approach is better. This paper seeks to end this debate by proving that notch and low-pass filters are never better than input shapers for suppressing mechanical vibration. This paper expands on previous efforts by presenting a proof showing that input shapers suppress vibration more quickly than notch or low-pass filters. The problem of suppressing multi-mode vibration is also examined. Experimental results from a portable bridge crane verify key theoretical results.