In some cases, it is necessary to attach a tip mass (proof mass) to the beam in order to tune its fundamental natural frequency to the excitation frequency and to improve its dynamic flexibility. If the differential eigenvalue problem is solved for a uniform cantilevered beam
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with a tip mass of Mt rigidly attached at x — L, the eigenfunctions can be obtained as (Appendix C.1)[22]
where дг is obtained from
Mt
sin Хг — sinh Хг + Хг—— (cos Хг — cosh Хг)
________________ mL______________
Mt
cos Хг + cosh Хг — Хг—– (sin Хг — sinh Хг)
mL
and Ar is a modal amplitude constant which should be evaluated by normalizing the eigenfunctions according to the following orthogonality conditions:
L
j ф, (x )m фг (x) dx + ф, (L)Mt фг (L) –
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d 4фг (x) |
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d Зфг (x)’ |
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0
where it is sufficient to use the first expression (then the second one is automatically satisfied and vice versa). The natural frequency expression given by Equation (2.14) still holds but the dimensionless eigenvalues (Хг for the rth mode) should be obtained from
Mt
1 + cos Х cosh Х + Х—- (cos Х sinh Х
mL
where Mt/mL is a dimensionless parameter as it is the ratio of the tip mass to the beam mass. In the above equations, the rotary inertia of the tip mass is neglected for convenience; that is, the tip mass is assumed to be a point mass.
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.94Wrel(x, t) d5Wrel(x, t) YI—— —;——- + CsI— ————- + Ca |
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d x 4 -— [m + Mt 8(x — L)] |
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d x 4d t d2wb(x, t) d t2 |
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In addition to the modification of the eigenvalue problem in the presence of a tip mass, the forcing term due to base excitation also changes since the tip mass also contributes to the inertia of the structure. Equation (2.5) becomes
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d 2 h(t) |
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d2g(t) L d2h(t) dt2 + dt2 |
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where S(x) is the Dirac delta function and the forcing term due to external damping is neglected. The modal forcing function corresponding to the right hand side of Equation (2.45) is then
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As expected, the foregoing modification results in variation of the correction factor defined in the previous section. Since the base is assumed to be not rotating (i. e., h(t) = 0) in deriving the correction factor, one can extract the expression of the correction factor ii1 in the presence of a tip mass as
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The variation of the correction factor i1 of the fundamental transverse vibration mode given by Equation (2.47) with the ratio of tip mass (Mt) to beam mass (mL) is shown in Figure 2.7. Figure 2.7 illustrates that when there is no tip mass (Mt/mL = 0), i1 = 1.566 as previously obtained, whereas when Mt/mL becomes larger (Mt/mL ^ ro), i1 approaches unity. The important conclusion drawn from Figure 2.7 is that the uncorrected lumped-parameter model can be used safely only when the tip mass is sufficiently larger than the beam mass. From a physical point of view, if the tip mass is sufficiently large, the inertia of the tip mass dominates in the forcing function and the distributed inertia of the beam (as a component of excitation) becomes negligible. Table 2.1 shows the values that i1 takes for different Mt/mL ratios. It should be noted that, for the uncorrected lumped-parameter formulation, i1 = 1, and therefore the relative error in the motion at the tip of the beam predicted by the uncorrected lumped-parameter model is estimated from (1 — i1) /i1 x 100.
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Table 2.1 Correction factor for the fundamental transverse vibration mode and the error in the uncorrected lumped-parameter model for different values of tip mass – to – beam mass ratio |
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Mt/mL |
U1 |
Error in the uncorrected lumped-parameter model (%) |
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0 |
1.56598351 |
-36.14 |
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0.1 |
1.40764886 |
-28.96 |
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0.5 |
1.18922917 |
-15.91 |
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1 |
1.11285529 |
-10.14 |
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5 |
1.02662125 |
-2.59 |
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10 |
1.01361300 |
-1.34 |
The following quadratic polynomial ratio (obtained by using the Curve Fitting Toolbox of MATLAB) gives an estimate of д1 with an error less than 9 x 10-3% for all values of Mt/mL:
(Mt / mL)2 + 0.603 (Mt/mL) + 0.08955 (Mt/mL)2 + 0.4637 (Mt / mL) + 0.05718
