For the fixed-fixed beam described in Problem 5.4, we can assume the mode shape to be expressed as..

For the fixed-fixed beam described in Problem 5.4, we can
assume the mode shape to be expressed as w(x) = a1Ψ1(x) + a2Ψ2(x),
where Ψ1(x) = (x4 − 2Lx3 + L2x2 ) and Ψ2(x) = (x5 − 3L2x3
+ 2L3 x2 ). Using these expressions for Ψ1(x) and Ψ2(x), approximate
the lowest two natural frequencies using the Ritz method.

Problem 5.4

An elastic material (E = 70 GPa, ν = 0.33) fills a
cavity in a rigid block. The dimensions of the cavity are a = 75 mm, b = 125
mm, and L = 300 mm. A rigid cap is placed on the material and a force P0 is
applied to the center of the cap as illustrated in Figure 5.22. The elastic
material is compressed an amount δ. Determine the general expressions for
the applied force P0 as well as the net forces that exist on the x and z faces
of the material (Px, Pz ). In addition, determine the explicit force in each
direction if the axial strain is 0.01%.