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ME4189_F12_9_18_Examples

# The goal is to obtain an upper bound on the first

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Unformatted text preview: ider a cantilever beam (not shown) of length with a force of applied to the end of the beam. The elastic modulus of the beam is and the beam’s area moment of inertia is . The goal is to obtain an upper bound on the first natural frequency of the beam using Rayleigh’s Quotient. Rayleigh’s method involves assuming a ‘mode’ of deflection. Though many different functions can be chosen for such a mode, we will use the static deflection of the beam, as the static deflection satisfies the geometrical boundary conditions (no deflection at the cantilevered end of the beam). The static deflection of a cantilevered beam with a force applied at the end of the beam is found from many mechanics of materials references to be 3 6 The magnitude of the deflection at the end of the beam, at 3 This deflection is used to normalize the deflection of the beam, 1 2 which can be rewritten as , is therefore: 3 : 3 2 Recall that the general form of the kinetic energy for an elastic bar is 1 2 Taking the derivative of yields 3 2 where the velocity at the end of the beam is designated becomes: 1 2 1 2 3 1 2 1 2 3 1 2 However, the density of the bar 33 140 /: is equal to 1 2 The effective mass of the beam, . Therefore, the kinetic energy 33 140 , is therefore 33 140 To find the effective stiffness of the beam, an expression for the potential energy stored in the beam must be found. A general expression for the potential energy is 1 2 where a spatial derivative with respect to is designated with the prime (‘) symbol. Taking the derivate of with respect to , and integrating, yields: 1 2 3 From which the equivalent stiffness is found to be 3 The natural frequency is thus 3 140 33 3.5675 From vibration of continuous systems theory, the exact value of is 3.5160 So, RQ method provides an upper bound on the natural frequency (and a fairly good estimate, at that)....
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