THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

20.8 Vibration Measurements 611by proportional spacing so that the mesh is denser in areas of greater concern(particularly where sudden changes or discontinuities in geometry can lead tohigher values of stresses or steeper temperature gradients). Constraints and forcingfunctions are specified. The input data are converted into matrix format by the FEAcode. The form of matrix equation iswherema + Cv + kx = Fm, C, and k = the mass, damping, and stiffness matrices, respectively, forthe finite element representation of the structure being analyzeda, v, and x = the acceleration, velocity, and displacement vectors, respectivelyF = the forcing function vectorThe matrices are manipulated to determine natural frequencies, frequencies,mode shapes, response amplitudes, and/or dynamic stresses due to harmonic, random,or transient forcing functions. In the postprocessing phase, the FEA programarranges the output in a convenient format for engineering evaluation. The deformedstructure can be graphically portrayed as an overlay over the undeformedmesh diagram. In order to show the difference, mode shapes or displacement amplitudesare exaggerated in the display. In almost all situations, a great deal ofengineering time must be spent in defining even a simple problem by modelingit and setting up a mesh. But once a solution is obtained, redesign to improvevibrational characteristics can be done quickly and effectively by modifying thestored data. FEA software can be installed and used on personal computers, but formore elaborate designs it may be necessary to use minicomputers or mainframes inorder to deal with much larger matrices (cf. Brooks, 1986; Hughes, 1987; Grandin,1986; Huebner and Thornton, 1982).20.8 Vibration MeasurementsVibration displacement and vibrational velocity can be measured but the mostcommon measurements of vibration are those of acceleration. The basic transducerused to measure vibration and shock is the accelerometer. Most of the othercomponents in vibration measurement systems are similar to those used to measureairborne sound (cf. Chapter 9). Many instruments such as the fast Fouriertransform (FFT) analyzers are designed to be used for both acoustic and vibrationapplications.AccelerometersAccelerometers are usually mounted directly on a vibrating body, using a threadedstud, adhesive, or wax. Other versions mount the accelerometer in a probe thatis held against a vibrating body. A typical accelerometer houses one or more