# A Numerical Study on Computational Time Reversal for Structural Health Monitoring
## 2 Time Reversal Approaches for Structures
### Elastodynamic Reciprocity for Classical Beam Theory  
The basic assumption of the Euler-Bernoulli theory is that a plane cross-section, perpendicular to the axis of the beam, remains plane and perpendicular to the neutral axis during bending. The differential equation of motion for a beam, considering both materials' density and cross-section constant over the beam's axis \(x\) is given as  
\[EI\frac{\partial^{4}u}{\partial x^{4}}+\rho A\frac{\partial^{2}u}{\partial t^{2 }}=q(x,t) \tag{1}\]  
with the boundary conditions distinguished as the essential ones defined on \(x\in\Gamma_{u}\):  
\[u(x,t)=\tilde{u}(x,t), \tag{2a}\] \[\frac{\partial u(x,t)}{\partial x}=\tilde{\psi}(x,t) \tag{2b}\]  
and the natural ones defined on \(x\in\Gamma_{t}\):  
\[-EI\frac{\partial^{2}u(x,t)}{\partial x^{2}} =\bar{M}(x,t), \tag{2c}\] \[-EI\frac{\partial^{3}u(x,t)}{\partial x^{3}} =\bar{Q}(x,t). \tag{2d}\]  
with \(\bar{M}\) and \(\bar{Q}\) indicating the prescribed bending moment and shear force, respectively. The necessary initial condition to accompany the above system are:  
\[u(x,0)=u_{0}(x), \tag{3}\] \[\frac{\partial u(x,0)}{\partial t}=\dot{u}_{0}(x). \tag{4}\]  
We now consider a continuous function \(v(x,t)\) which is assumed to be the displacement field for some other elastodynamic configuration for the same beam under loading distribution \(w(x,t)\).  
We recall at this point the definition of Riemann convolution. Riemann convolution of functions \(f(t)\) and \(g(t)\) is defined as  
\[g(t)*h(t)=\begin{cases}\int\limits_{0}^{t}g(t-\tau)h(\tau)\,d\tau,&\text{for $t>0$}\\ 0,&\text{elsewhere.}\end{cases} \tag{5}\]  
Taking the Riemann convolution in time of function \(v(x,t)\) with equation of motion (1) and integrating over the \(x\)-axis we obtain:  
\[EI\int\limits_{0}^{L}\frac{\partial^{4}u}{\partial x^{4}}*v\,dx+\int\limits_{0 }^{L}\rho A\frac{\partial^{2}u}{\partial t^{2}}*v\,dx=\int\limits_{0}^{L}q(x, t)*v\,dx. \tag{6}\]  
The first integral of (6) after sequential integration by parts can be written as  
\[EI\int\limits_{0}^{L}\frac{\partial^{4}u}{\partial x^{4}}*v\,dx=EI\int\limits _{0}^{L}u*\frac{\partial^{4}v}{\partial x^{4}}\,dx+EI\bigg{[}\frac{\partial^ {3}u}{\partial x^{3}}*v-\frac{\partial^{2}u}{\partial x^{2}}*\frac{\partial v} {\partial x}+\frac{\partial u}{\partial x}*\frac{\partial^{2}v}{\partial x^{ 2}}-u*\frac{\partial^{3}v}{\partial x^{3}}\bigg{]}_{0}^{L}. \tag{7}\]The integral of (6) concerning inertial terms taking advantage of time convolution's identities can be written as  
\[\rho A\int\limits_{0}^{L}\frac{\partial^{2}u}{\partial t^{2}}*v\,dx=\rho A\int \limits_{0}^{L}u*\frac{\partial^{2}v}{\partial t^{2}}\,dx+\rho A\int\limits_{0} ^{L}\dot{v}_{0}u+v_{0}\dot{u}-\dot{u}_{0}v-u_{0}\dot{v}\,dx. \tag{8}\]  
Substitution of (7) and (8) into (6), taking under consideration that \(v(x,t)\) is the displacement field for the distributed loading \(w(x,t)\) results in the reciprocal relation between states of \((u,q)\) and \((v,w)\), that is:  
\[\int\limits_{0}^{L}w(x,t)*u(x,t)\,dx-\int\limits_{0}^{L}q(x,t)*v( x,t)\,dx=\] \[EI\left[\frac{\partial^{3}u(x,t)}{\partial x^{3}}*v(x,t)-\frac{ \partial^{2}u(x,t)}{\partial x^{2}}*\frac{\partial v(x,t)}{\partial x}+\frac{ \partial u(x,t)}{\partial x}*\frac{\partial^{2}v(x,t)}{\partial x^{2}}-u(x,t)* \frac{\partial^{3}v(x,t)}{\partial x^{3}}\right]_{0}^{L}\] \[+\rho A\int\limits_{0}^{L}\dot{v}_{0}(x)u(x,t)+v_{0}(x)\dot{u}(x, t)-\dot{u}_{0}(x)v(x,t)-u_{0}(x)\dot{v}(x,t)\,dx. \tag{9}\]  
As shown in what follows, the above reciprocal relation could serve as the starting point of time reversal approaches for the case of the Euler-Bernoulli beam. Furthermore, such an approach could be easily adapted in other cases of structural dynamics, e.g., axial waves, Timoshenko beam theory, or to the more general linear elastodynamic case.