“Introduction:

The vibration characteristics of a machine can be markedly affected by the structural dynamic characteristics of the supporting system. If the structural supporting system is too flexible, then machine vibrations could be large. Even worse, if the structure is resonant, machine vibrations could be extreme.

Figure 1 contains a simple approximation for a structural-mechanical vibrating system that consists of the mass (m) of a machine supported by a structure approximated by two springs, each having stiffness (k). The total stiffness of the structure would then be 2k. This approximation is presented as an academic exercise in order to present the difference in the response of machine structures to static loading versus dynamic loading. Most structural-mechanical systems are too complex to be treated as such a simple vibrating system.

If the force (Fstatic) produced by the machine is statically applied to the structure, the resulting deflection is:

Deflection = Force/Stiffness = Fstatic/2k

The response of the structure to a dynamic load is more complex. It depends upon the ratio of the frequency of the dynamic force (Fdynamic) to the natural frequency (fn) of the structure. The vibration amplitude that will occur as the result of dynamic loading is:

Amplitude = MF(Fdynamic/2k)

This is similar to the calculation for static deflection except that it contains a magnification factor (MF) that accounts for the proximity of the frequency (fd) at which the dynamic force is applied to the natural frequency of the structure. For a simple single degree-of-freedom system with damping (ξ), the magnification factor is:

MF = 1/[{1-fd/fn)2}2 + {2ξ (fd/fn)}2]1/2

Most structural-mechanical systems are lightly damped. Neglecting damping further simplifies the magnification factor per the following:

MF = 1/[1-(fd/fn)2]

Figure 2 is a curve illustrating the effects of resonant amplification. The amplification increases as the frequency ratio increases from the origin toward 1.0. At a frequency ratio of 1.0, the magnification factor without damping is infinite. The magnification factor then begins to decrease as the frequency ratio increases beyond 1.0. At ratios above 1.414, vibration is attenuated instead of amplified.

Very stiff structures will have higher natural frequencies and lower frequency ratios. Very stiff structures will have higher natural frequencies and lower frequency ratios. Thus, stiff structures will be subject to some amount of resonant amplification, albeit in most cases the magnitude of the amplification is minimal. Also, since the amplitude of vibration without amplification is usually low due to the stiffness of the structure, the resultant amplified vibration is typically acceptable.”