Correction Factor for Longitudinal Vibrations

The relative motion transmissibility between the tip of the bar and the moving base can be extracted from Equation (2.58) as

where

is the correction factor for the lumped-parameter model of the rth mode for longitudinal vibrations (to predict the vibratory motion at x — L). Note that Kr is a function of Mt/mL and ar due to Equation (2.60), and ar is a function of Mt /mL from Equation (2.57). Therefore, for a given vibration mode, the correction factor Kr is a function of Mt/mL only. In the absence of a tip mass (Mt /mL — 0), the correction factor for the fundamental mode can explicitly be obtained from Equations (2.57) and (2.60) as k1 — 4/n = 1.273. However, in the presence of a tip mass, the transcendental equation given by Equation (2.57) should be solved numerically to obtain the correction factor. The variation of the correction factor of the fundamental mode (k1) with Mt/mL is given in Figure 2.14.

As in the transverse vibrations case, the correction factor tends to unity as the ratio of tip mass to bar mass increases, meaning that the uncorrected lumped-parameter model can be used only for bars with a tip mass that is much larger than the bar mass. Table 2.2 shows the correction factor K1 for lumped-parameter modeling of the fundamental longitudinal vibration mode for different Mt/mL ratios. Note that the error in the relative motion urel at the tip of the bar predicted by using the uncorrected lumped-parameter model is simply obtained from (1 — k1) /k1 x 100.

The following quadratic polynomial ratio (obtained by using the Curve Fitting Toolbox of MATLAB) represents the behavior of the correction factor shown in Figure 2.14 successfully with a maximum error less than 4.5 x 10—2% for all values of Mt/mL:

(Mt / mL)2 + 0.7664 (Mt / mL) + 0.2049 (Mt / mL)2 + 0.6005 (Mt / mL) + 0.161