Why measuring the vibration of a machine is important?

In general, the vibration of rotating or reciprocating equipment under operation (pumps, compressors…etc.) obeys the following governing differential equation of motion M\ddot{x}+C\dot{x}+Kx=F_{Exciting \, Force}. The LHS of the equation represents the sum of the inertia force, damping force and spring force of the equipment. The RHS of the equation represents the excitation force due to faults in the running equipment. Faults like unbalance in the impeller, misalignment between motor and the pump, structural looseness, bearing failure, cavitation & flow turbulence in case of centrifugal pumps..etc induce F_{Excitation \, Force}. If RHS is greater than LHS, i.e. the magnitude of the exciting force is greater than the restoring force of the equipment, then the equipment is subjected to significant vibration. By analysing the frequency and the magnitude of the F_{Exciting \, Force}, the source of fault can be identified. Hence measuring the vibration of equipment indicates its health condition. By analysing the vibration, the incipient faults can be identified and equipment can be saved from a catastrophic failure.