{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}


The goal is to obtain an upper bound on the first

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)....
View Full Document

{[ snackBarMessage ]}