The emergence of high speed diesel engines for cars has emphasized the problems of mounting fuel injection pumps onto engines. Some engine manufacturers are using sheet steel pressings or thin cast brackets which allow the pump to vibrate in resonance in the engine firing frequency range. This increases the likelihood of failure of these brackets in service due to metal fatigue. A theoretical analysis of a simple bracket enables the essentials of the problem to be understood and it allows possible design solutions to be evaluated. This theoretical analysis is complemented by experimental evidence from actual brackets.