Question 18.P.3: The clamped-free bar shown in Figure P18.3 is vibrating in t...
The clamped-free bar shown in Figure P18.3 is vibrating in the axial direction under the action of a force \mathrm{P} acting on the free end. Section BC of the bar has a circular cross-section whose diameter varies linearly from 2 d at \mathrm{B} to d at \mathrm{C}. Section \mathrm{CD} has a constant cross section of diameter d. The mass density of the bar is \rho per unit volume. The length of each section is L. (a) Find the displacement of the bar when \mathrm{P} is a static load. Compare the exact displacement along the length of the bar with those obtained from the finite element solution. (b) Find the response u_{1}(t) and u_{2}(t) when \mathrm{P} is applied suddenly but remains constant afterward.
(a) u_{1}=\frac{3}{7} \frac{P L}{E A} and u_{2}=\frac{10}{7} \frac{P L}{E A}
where u_{1} is the axial displacement at \mathrm{C} and u_{2} that at \mathrm{D}.
The variation of displacements along the length of sections \mathrm{BC} and \mathrm{CD} are given by
\begin{aligned}&u(x)=u_{1} \frac{x}{L} \\&u(y)=u_{1}\left(1-\frac{y}{L}\right)+u_{2} \frac{y}{L}\end{aligned}where x is measured from \mathrm{B} and y from C.
The exact values of the displacement are determined from