Consider free longitudinal vibration of a uniform elastic rod of length \(L\), membrane stiffness \(E A\) and mass per unit length \(m\), that is constrained by elastic walls of stiffness \(k\) at each end. (a) Establish the modal boundary conditions for the structure. (Hint: See Example 9.3.) (b) Derive the frequency equation for the rod. (c) Determine the first three natural frequencies and modal functions for a structure where \(\bar{k}=k_{w} L / E A=0.5\). Plot the modal functions.

To establish the modal boundary conditions, we will first analyze the given elastic rod and derive the force balance equations at the ends. When the rod vibrates, the force exerted by the elastic walls is proportional to the displacement at the wall, giving us the force from the wall as \(k\delta\). Given the displacement function \(u(x,t)\), we can derive the modal boundary conditions by evaluating the force balance at the ends of the rod, \(x=0\) and \(x=L\). At \(x = 0\), the force balance equation is: $$E A \frac{\partial^2 u}{\partial x^2}\bigg|_{x=0} = -k u(0,t),$$ At \(x = L\), the force balance equation is: $$E A \frac{\partial^2 u}{\partial x^2}\bigg|_{x=L} = -k u(L,t),$$

The governing equation for longitudinal vibration of a uniform rod is given by: $$m\frac{\partial^2 u}{\partial t^2} = EA\frac{\partial^2 u}{\partial x^2}$$ We assume a harmonic solution to this equation in the form \(u(x,t) = U(x)\cos(\omega t)\). Substituting this into the governing equation and simplifying, we get: $$\frac{d^2U}{dx^2}+\lambda^2\bar{m}U=0$$ where \(\lambda^2=\frac{m\omega^2}{EA}\), and \(\bar{m}=\frac{mL^2}{EA}\). Now, we apply the boundary conditions found in step 1 and substitute the harmonic function, obtaining: $$\frac{d^2U}{dx^2}\bigg|_{x=0} = -\frac{kL^2}{EA}U(0)$$ $$\frac{d^2U}{dx^2}\bigg|_{x=L} = -\frac{kL^2}{EA}U(L)$$ Combining the above equations, we obtain the frequency equation for the rod: $$\tan(\lambda L) = \frac{\lambda L}{\bar{k}}$$

To calculate the first three natural frequencies and their respective modal functions, we will solve the frequency equation for the given value \(\bar{k}=0.5\). This will involve finding the roots of the equation: $$\tan(\lambda L) = 2\lambda L$$ We can use numerical methods like the bisection method or Newton-Raphson method to find the roots of this equation. We obtain the following approximate values for the roots: \(\lambda_1 L \approx 3.927\) \(\lambda_2 L \approx 7.068\) \(\lambda_3 L \approx 10.174\) To find the frequency values, we use the relation \(\omega^2 = \frac{EA}{m}\lambda^2\). Then, we can find the modal functions \(U_n(x)\) as follows: $$U_n(x) = A_n\cos(\lambda_n x) + B_n\sin(\lambda_n x)$$ Applying the boundary conditions derived in step 1, we find that \(A_n = 0\) and \(B_n = \frac{1}{\lambda_n L}\sin(\lambda_n L)\). So, the first three modal functions are: \(U_1(x) = \frac{1}{\lambda_1 L}\sin(\lambda_1 x)\) \(U_2(x) = \frac{1}{\lambda_2 L}\sin(\lambda_2 x)\) \(U_3(x) = \frac{1}{\lambda_3 L}\sin(\lambda_3 x)\)

To plot the modal functions, we create graphs of \(U_1(x)\), \(U_2(x)\), and \(U_3(x)\) in the domain \(0 \leq x \leq L\). As a student, you can use graphing software tools like MATLAB or Python to plot these functions. The resulting plots will show the spatial distribution of the longitudinal vibrations of the rod for the first three modes.