If the running speed of the machine was changed so that it coincided with a natural frequency, how would the vibration amplitude change?

When the running speed of a machine reaches a point where it coincides with a natural frequency, a phenomenon known as resonance occurs. At this specific frequency, even small periodic forces can lead to significant vibrations within the system. This happens because the system's natural frequency aligns with the frequency of the external forces acting on it, causing the energy input from these forces to amplify the oscillations.

In this scenario, as the speed increases until it matches the natural frequency, the cumulative effect of the resonant vibration causes the amplitude of vibration to rise dramatically. This heightened amplitude results from the constructive interference of the forced vibrations with the natural oscillatory motion of the system, leading to potentially damaging levels of vibration if not controlled.

The other choices relate to conditions that do not account for the effects of resonance, which is a critical factor in understanding vibration behaviors in mechanical systems. Hence, the correct understanding of how vibration amplitudes respond to changes in machine speed in relation to natural frequencies clearly supports why resonance significantly increases vibration amplitude.