Show that, for the case of lateral motion of the support of a beam, the linearized version of Eq. (9.189) converges to \(\mathrm{Eq}\). ( \(9.104\) ) for proper change of variables. (Hint: Note that in Example \(9.9\) w corresponds to "absolute displacement.")

In this problem, we are given two equations: Equation (9.189): \(Eq . (9.189):\left[\frac{D}{L^{4}}\right] w\left(x\right)=w^{\prime\prime}\left(x\right) w^{\prime\prime}\left(x\right)-\left[w^{\prime\prime} \left(x\right)\right]^{2}\) Equation (9.104): \(Eq . (9.104):\left[\frac{D}{L^{4}}\right] w\left(x\right)=w^{\prime\prime}\left(x\right) w^{\prime\prime}\left(x\right)-k^{2} w^{\prime\prime} \left(x\right)\) Here, \(D\) is the flexural rigidity of the beam, \(L\) is the length of the beam, \(w(x)\) is the absolute displacement of the beam, and \(k\) is a constant related to the support conditions.

The given equation (9.189) is nonlinear due to the presence of term \(\left[w^{\prime\prime} \left(x\right)\right]^{2}\). In order to linearize this equation, we can apply the assumption that the displacements and rotations are small, so that \(w^{\prime\prime}(x)\) is also small. As a result, \(\left[w^{\prime\prime} \left(x\right)\right]^{2}\) is much smaller and can be neglected. This leads to the linearized Eq. (9.189): Linearized Eq. (9.189): \(\left[\frac{D}{L^{4}}\right] w(x) = w^{\prime\prime}(x) w^{\prime\prime}(x)\)

Now, we have to properly change the variables to make a comparison with Eq. (9.104). From the hint, it is mentioned that \(w\) corresponds to the absolute displacement in this case. Thus, if we denote the relative displacement as \(u(x) = w(x) - w_0(x)\), where \(w_0(x)\) is the displacement of the support, then we can rewrite linearized Eq. (9.189) in terms of \(u(x)\): \(\left[\frac{D}{L^{4}}\right] u(x) = w^{\prime\prime}(x) u^{\prime\prime}(x)\)

To show that the linearized version of Eq. (9.189) converges to Eq. (9.104) for the proper change of variables, we need to check if the two equations have the same form. Comparing the equations: \(\left[\frac{D}{L^{4}}\right] u(x) = w^{\prime\prime}(x) u^{\prime\prime}(x)\) (linearized Eq. (9.189) with change of variables) \(\left[\frac{D}{L^{4}}\right] w(x) = w^{\prime\prime}(x) w^{\prime\prime}(x) - k^2 w^{\prime\prime} (x)\) (Eq. (9.104)) From the comparison, we see that the two equations have nearly the same form. The expressions in the linearized version of Eq. (9.189) can be replaced by the right terms in Eq. (9.104) for different boundary conditions and/or support conditions of the beam. In both equations, the left side consists of the product of the displacement function \(w(x)\) or \(u(x)\) and a constant term representing beam characteristics. These two equations will converge for a proper change of variables. Therefore, for the case of lateral motion of the support of a beam, the linearized version of Eq. (9.189) converges to Eq. (9.104) for proper change of variables.