Verify the characteristic equation given in Table 10.4 for a pinned-free beam.

TABLE 10.4 Natural frequencies and mode shapes for beams Characteristic Five Lowest Natural Frequencies Kinetic Energy Scalar Product (x,(x), X, (x)) cos A}x a (sinhA}x sin A}x)] /x(x)X,(x)dx - End Conditions X=0 X=1 Equation Mode Shape Fixed-fixed cos A1/4 cosh 1/4=1 w =22.37 [cosh Ax = 61.66 cosh A1/4 cos A1/4 a w=120.9 sinh A sin A4 = 199-9 w=298.6 Pinned-pinned sin A1/4 o W = 9.870 Csin A/4x W = 39.48 = 88.83 S*x(x)x(x)dx Fixed-free cos 1/4 cosh 1/4 = -1 W 157-9 =246.7 3.51 - 22.03 w=61.70 120.9 w=199.9 Free-free cosh 1/4 cos/4=1 w = 0 Fixed-linear spring 3/4 (cosh 1/4 cos A4 + 1) - 22.37 w=61.66 [cosh Ax cos xa(sinhx - sin x)] S*x(x)x(x)dx cos A/4 + cosh A/4 sin A* + sinh)} 1,3x (k= = 1) C[coshA}x + cosA}x + a [sinhA}x + sinA}x)] cosh A1/4 - Cos A1/4 120.9 a sinh A4 sin A -199.9 *x(x)x(x)dx For B = 0.25 C cosA}x coshAx a (sinA x sinh A}x)] /X(x)X,(x)dx -B(cos A4 sin 4-cosh A4 sin 1/4) w 3.65 =0 = W = 22.08 cos A4 + cosh A sinA}/4 + sinhA}% - w = 61.70 120.9 W 199.9