Preamble

This work continues the investigations initiated in [1] and [2] regarding the dynamics of structural elements. We consider the boundary value problem for the following partial differential equation:
$$\frac{\partial^4 u}{\partial x^4} + a \frac{\partial u}{\partial t} + b \frac{\partial^5 u}{\partial x^4 \partial t} + \frac{\partial^2 u}{\partial t^2} = 0, \quad (0 < x < l, \ 0 < t < T) \tag{1.1}$$
subject to the initial conditions:
$$u(x, 0) = \phi(x), \quad \frac{\partial u(x, 0)}{\partial t} = \psi(x), \quad (0 < x < l) \tag{1.2}$$
and the boundary conditions for a clamped beam:
$$u(0, t) = 0, \quad \frac{\partial u(0, t)}{\partial x} = 0, \quad (t > 0) \tag{1.3}$$
$$u(l, t) = 0, \quad \frac{\partial^2 u(l, t)}{\partial x^2} = 0, \quad (t > 0). \tag{1.4}$$
Here, $a > 0$ and $b > 0$ are physical constants, and $u(x, t)$ represents the displacement function. In Sections 3 and 4, we analyze the solution using the contour integral method. The solution can be represented in the form:
$$u(x, t) = -\frac{1}{2\pi i} \int_{\Gamma} y(x, \lambda) \lambda e^{\lambda t} d\lambda \tag{1.5}$$
where $y(x, \lambda)$ is the solution to the corresponding spectral problem.

Section 2. Separation of Variables

Applying the method of separation of variables $u(x, t) = X(x)T(t)$ to equations (1.1)–(1.4), we obtain the following equation for the temporal component $T(t)$:
$$T'' + (b\lambda^4 + a)T' + \lambda^4 T = 0 \tag{2.1}$$
The spatial component $X(x)$ must satisfy the differential equation:
$$X^{IV} - \lambda^4 X = 0 \tag{2.2}$$
with the boundary conditions:
$$X(0) = 0, \ X'(0) = 0, \ X''(l) = 0, \ X'''(l) = 0. \tag{2.3}$$
The general solution to (2.2) is given by:
$$X(x) = A \cosh \lambda x + B \sinh \lambda x + C \cos \lambda x + D \sin \lambda x \tag{2.4}$$
Substituting (2.4) into the boundary conditions (2.3) leads to the characteristic equation $\cosh \alpha \cos \alpha = -1$, where $\alpha = \lambda l$. The roots $\alpha_k$ of this equation are well-known: $\alpha_1 = 1.875$, $\alpha_2 = 4.694$, and for $n > 3$, $\alpha_n \approx \frac{\pi}{2}(2n - 1)$. The corresponding eigenfunctions $X_k(x)$ are defined on $[0, l]$ as:
$$X_k(x) = (\sinh \alpha_k - \sin \alpha_k)(\cosh \frac{\alpha_k x}{l} - \cos \frac{\alpha_k x}{l}) - (\cosh \alpha_k + \cos \alpha_k)(\sinh \frac{\alpha_k x}{l} - \sin \frac{\alpha_k x}{l}) \tag{2.7}$$
These eigenfunctions satisfy the orthogonality property:
$$\int_0^l X_k^2(x) dx = \frac{l}{4} (\sinh \alpha_k - \sin \alpha_k)^2 \tag{2.10}$$
The general solution to the original problem (1.1)–(1.4) can then be expressed as a series:
$$u(x, t) = \sum_{k=1}^{\infty} X_k(x) [C_k \cosh q_k t + D_k \sinh q_k t] e^{-p_k t} \tag{2.11}$$
where $p_k$ and $q_k$ are determined by the parameters $a, b$ and the eigenvalues $\lambda_k$.

Section 3. Spectral Analysis and Green's Function

To justify the solution, we consider the transformed problem:
$$\frac{d^4 y}{dx^4} + \lambda^2 y + (a + b \lambda^2) \frac{d^4 y}{dx^4} = f(x, \lambda) \tag{3.1}$$
with boundary conditions $y(0) = y'(0) = y''(l) = y'''(l) = 0$. The function $f(x, \lambda)$ is defined as:
$$f(x, \lambda) = \lambda \phi(x) + \psi(x) + b \phi^{IV}(x) \tag{3.3}$$
The solution to this boundary value problem can be expressed using the Green's function $G(x, \xi, \lambda)$:
$$y(x, \lambda) = \int_0^l G(x, \xi, \lambda) f(\xi, \lambda) d\xi \tag{3.4}$$
The Green's function is constructed as:
$$G(x, \xi, \lambda) = \frac{A(x, \xi, \lambda)}{\Delta(\lambda)} \tag{3.10}$$
where $\Delta(\lambda)$ is the characteristic determinant:
$$\Delta(\lambda) = 2(1 + \cosh \alpha z l \cosh \beta z l) \tag{3.9}$$
Here, the parameters $\alpha, \beta, z$ are related to the coefficients of the original equation. We analyze the asymptotic behavior of the Green's function in the complex $\lambda$-plane to ensure the convergence of the contour integral (1.5).

Section 5. Convergence and Final Solution

By evaluating the residues of the integrand in (1.5) at the poles $\lambda_{mk}$, we obtain the final form of the solution. The poles are determined by the roots of the characteristic equation and the quadratic form (2.1). Specifically, we find:
$$\lambda_{mk} = -p_k \pm q_k \tag{5.5}$$
The resulting series representation for $u(x, t)$ is:
$$u(x, t) = \sum_{k=1}^{\infty} \left[ \frac{\cosh q_k t - \frac{p_k}{q_k} \sinh q_k t}{|X_k|^2} \int_0^l X_k(\xi) \phi(\xi) d\xi + \frac{\sinh q_k t}{q_k |X_k|^2} \int_0^l X_k(\xi) \psi(\xi) d\xi \right] e^{-p_k t} \tag{5.12}$$
This solution accounts for the damping effects introduced by the parameters $a$ and $b$. In the limiting case where $b = 0$, the solution reduces to the standard vibration model for a beam with internal friction. The convergence of this series and its derivatives is guaranteed by the smoothness assumptions on the initial functions $\phi(x)$ and $\psi(x)$, as discussed in Sections 3 and 4. The results demonstrate that the contour integral method provides a robust framework for solving non-classical problems in structural mechanics.