##### Article

Modal Testing. Part 1: Introduction & Impact Testingby Robert J. Sayer, P.E.

### ABSTRACT

This first of three papers on the topic of modal analysis presents the theory and basic equations involved. Impact testing of equipment is described in detail and recommendations are given on how to perform it properly. Variables involved with impact testing such as hammer tip type, response sensor selection, frequency range, frequency resolution, force window type & transfer function selection are discussed in detail.

### PREVIEW

“Excitation of natural frequencies (resonance) in machinery and structural supporting systems is a common cause of excessive vibration. Modal testing techniques such as impact and coast-down testing can be used to identify natural frequencies. Experimental modal analysis (EMA) can be used to determine the mode shapes associated with each natural frequency. This information is crucial to successful modification of machinery and structures in order to change their natural frequencies. In some cases, operating deflection shape (ODS) testing can be used to develop the mode shapes of excited natural frequencies.

Resonance, which is the result of excitation of a natural frequency, is a common root cause of excessive machine and structural vibrations. Operating machinery that produces dynamic forces at frequencies close to natural frequencies can significantly amplify vibration.

Basic Theory

Figure 1 is a simple approximation of a structural-mechanical vibrating system. It 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 be 2k. This approximation is presented as an academic exercise in order to illustrate 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 Equation (1)

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) Equation (2)

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 (ζ), the magnification factor is:

MF = 1/ [{1-(fd/fn)2}2 + {2ζ(fd/fn)}2]1/2 Equation (3)

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

MF = 1/ [1-(fd/fn)2] Equation (4)

Figure 2 illustrates 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.”

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