1.An important engineering problem that arises in a lot of applications is the vibrations of a clamped beam where the other end is free. This problem can be analyzed analytically, but the calculations boil down to solving the following nonlinear algebraic equation: cosh β cos β = −1, where β is related to important beam parameters through β4 = ω2 A EI , where is the density of the beam, A is the area of the cross section, E is Young’s modulus, and I is the moment of the inertia of the cross section. The most important parameter of interest is ω, which is the frequency of the beam. We want to compute the frequencies of a vibrating steel beam with a rectangular cross section having width b = 25 mm and height h = 8 mm. The density of steel is 7850 kg/m3, and E = 2 × 1011 Pa. The moment of inertia of a rectangular cross section is I = bh3/12. a) Plot the equation to be solved so that one can inspect where the zero crossings occur.
