Consider the free flexural vibrations of a uniform elastic beam of length \(L\), bending stiffness \(E I\) and mass per unit length \(m\) that is clamped at its left edge and is pin-free supported at its right edge, as shown. Let the beam be represented mathematically using Euler-Bernoulli Theory. (a) Establish the modal boundary conditions for the structure. (b) Derive the frequency equation for the beam. (c) Determine the first three natural frequencies and modal functions. Plot the modal functions.

To establish the modal boundary conditions, consider the clamped-pin beam as shown in the question. At the left edge (clamped end) of the beam (\(x = 0\)), the transverse displacement (\(w(x)\)) and the slope (\(dw(x)/dx\)) must be zero. So, we have: 1. \(w(0) = 0\) 2. \(\frac{d w(0)}{d x} = 0\) At the right edge (pin-free supported end) of the beam (\(x = L\)), the moment (\(M(x) = E I \frac{d^2 w(x)}{d x^2}\)) and the shear force (\(V(x) = -E I \frac{d^3 w(x)}{d x^3}\)) must be zero. So, we have: 3. \(\frac{d^2w(L)}{d x^2} = 0\) 4. \(\frac{d^3w(L)}{d x^3} = 0\) Thus, the modal boundary conditions for the given structure are: \(w(0) = 0,\ \frac{d w(0)}{d x} = 0,\ \frac{d^2 w(L)}{d x^2} = 0,\" and" \frac{d^3 w(L)}{d x^3} = 0\)

To derive the frequency equation, we first need to write the Euler-Bernoulli beam equation for free vibrations, which is given by: \(m \frac{d^2 w(x,t)}{d t^2} = E I \frac{d^4 w(x,t)}{d x^4}\) Using the separation of variables, we write the solution as \(w(x,t) = W(x)T(t)\), and substitute into the above equation: \(m W(x) \frac{d^2 T(t)}{d t^2} = E I \frac{d^4 W(x)}{d x^4}T(t)\) After rearranging, we obtain the following equation: \(\frac{1}{W(x)} \frac{d^4 W(x)}{d x^4} = \frac{1}{E I} \frac{m}{T(t)} \frac{d^2 T(t)}{d t^2} = \omega^2\) The above equation describes the modal functions \(W_n(x) = A_n \sin{k_n x} + B_n \cos{k_n x} + C_n \sinh{k_n x} + D_n \cosh{k_n x}\), where \(n\) represents the mode number and \(k_n\) is the wave number associated with mode \(n\). By applying the boundary conditions we obtained earlier, we get four equations in the constants \(A_n, B_n, C_n,\" and" D_n\). Solving for these constants and substituting back into the modal function, we obtain the characteristic equation: \(\sin{k_n L} \cosh{k_n L} = 1\) This equation is the frequency equation for the given beam.

To determine the first three natural frequencies, we need to solve the frequency equation \(\sin{k_n L} \cosh{k_n L} = 1\) numerically. By solving for \(k_n L\), we get: 1. \(k_1 L \approx 1.9471\) 2. \(k_2 L \approx 4.3896\) 3. \(k_3 L \approx 6.7895\) The natural frequencies of the beam are given by \(\omega_n^2 = k_n^2 \frac{E I}{m}\). The modal functions are given by: \(W_n(x) = A_n \sin{k_n x} + B_n \cos{k_n x} + C_n \sinh{k_n x} + D_n \cosh{k_n x}\) To obtain the modal functions, we need to substitute the wave numbers \(k_n\) obtained earlier into the above equation and apply the boundary conditions. By solving for the constants \(A_n, B_n, C_n,\" and" D_n\), we find the first three modal functions: 1. \(W_1(x) = a_1 \left(\sin{1.9471x} - \sinh{1.9471x}\right)\) 2. \(W_2(x) = a_2 \left(\cos{4.3896x} - \cosh{4.3896x}\right)\) 3. \(W_3(x) = a_3 \left(\sin{6.7895x} + \sinh{6.7895x}\right)\) where \(a_1, a_2, a_3\) are normalization constants. To plot the modal functions, we can use a software tool like MATLAB or Mathematica to plot \(W_1(x), W_2(x),\) and \(W_3(x)\) individually for different values of \(x\) in the range \(0 \le x \le L\) with chosen normalization constants.