## 3 Double-Beam System Traversed by a Moving Force
### Mathematical Formulation
#### 3.1.1 Fixed-Fixed and Fixed-Pinned Boundary Conditions  
It is found in Section 2.1 that the mode shapes of the double beam system for fixed-fixed boundary conditions have the same mathematical formulae as those for fixed-pinned boundary conditions except that the values of \(r_{1}l\) are different, so the solutions of Equation (52) would have the same formulae for the two kinds of boundary conditions except the values of \(r_{1}l\). The following derivations are based on double-beam systems with fixed-fixed boundary conditions.  
Substituting Equations (21) and (22) into Equation (53) and setting \(C\) to be one result in  
\[M_{nn}^{(i)}=\rho_{2}A_{2}l\bigg{[}1+s_{\rm m}\Big{(}A_{n}^{(i)}\Big{)}^{2} \bigg{]} \tag{54}\]  
where  
\[A_{n}^{(i)}=1+\frac{1-\bigg{(}\frac{\omega_{n}^{(i)}}{\omega_{2,n}^{(i)}} \bigg{)}^{2}}{s_{e}}\Big{(}r_{1}^{(n)}l\Big{)}^{4} \tag{55}\]  
and \(\omega_{n}^{(i)}\) and \(\widetilde{\omega}_{2,n}\) are obtained from Equations (9) and (11), respectively.  
Figure 11: Double-beam system with symmetric boundary conditions traversed by a moving force.  
Substituting Equation (22) into Equation (52) and setting \(C\) to be one result in  
\[\ddot{q}_{n}^{(i)}+\Big{(}\omega_{n}^{(i)}\Big{)}^{2}q_{n}^{(i)}=-\frac{fA_{n}^{( i)}}{M_{mn}^{(i)}}\Big{[}\cos\Big{(}r_{1}^{(n)}vt\Big{)}-\cosh\Big{(}r_{1}^{(n)}vt \Big{)}-B_{n}\Big{(}\sin\Big{(}r_{1}^{(n)}vt\Big{)}-\sinh\Big{(}r_{1}^{(n)}vt \Big{)}\Big{)}\Big{]} \tag{56}\]  
where  
\[B_{n}=\frac{\cos(r_{1}^{(n)}l)-\cosh(r_{1}^{(n)}l)}{\sin(r_{1}^{(n)}l)-\sinh( r_{1}^{(n)}l)} \tag{57}\]  
The solution of Equation (56) is  
\[q_{n}^{(i)}=p_{n}^{(i)}\Bigg{(}\frac{\cos\big{(}r_{1}^{(n)}vt\big{)}-B_{n} \sin\Big{(}r_{1}^{(n)}vt\Big{)}}{1-\Big{(}r_{1}^{(i)}\Big{)}^{2}}-\frac{\cosh \big{(}r_{1}^{(n)}vt\big{)}-B_{n}\sinh\Big{(}r_{1}^{(n)}vt\big{)}}{1+\Big{(} r_{1}^{(n)}\Big{)}^{2}}-\frac{2\big{(}r_{n}^{(i)}\big{)}^{2}\big{[}\cos\big{(} \omega_{n}^{(i)}l\big{)}-B_{n}\omega_{n}^{(i)}\sin\big{(}\omega_{n}^{(i)}l \big{)}\big{]}}{\big{[}1-\big{(}r_{n}^{(i)}\big{)}^{2}\big{]}\big{[}1+\big{(} r_{n}^{(i)}\big{)}^{2}\big{]}}\Bigg{)} \tag{58}\]  
where  
\[\left\{\begin{array}{c}s_{n}^{(i)}=\frac{r_{1}^{(n)}v}{\omega_{n}^{(i)}}\\ p_{n}^{(i)}=-\frac{fA_{n}^{(i)}}{M_{mn}^{(i)}\Big{(}\omega_{n}^{(i)}\Big{)}^{ 2}}\end{array}\right. \tag{59}\]  
The displacements of the primary beam and the secondary beam are  
\[w_{1}(x,t)=\sum_{n=1}^{m}\Big{[}\varphi_{1,n}^{(1)}q_{n}^{(1)}(t)+\varphi_{1, n}^{(2)}q_{n}^{(2)}(t)\Big{]} \tag{60}\]  
\[w_{2}(x,t)=\sum_{n=1}^{m}\varphi_{2,n}(x)\Big{[}q_{n}^{(1)}(t)+q_{n}^{(2)}(t) \Big{]} \tag{61}\]  
where \(m\) is the number of basic modes used in the MS method.  
To conduct parametric studies in Section 3.3, two dimensionless displacements for the primary beam and the secondary beam are defined as follows  
\[\overline{w}_{i}=\frac{w_{i}(x_{\max},\ \overline{t})}{w_{s}},\ i=1,\,2 \tag{62}\]  
where \(x_{\max}\) represents the location where maximum static deflection takes place; \(w_{s}\) is the maximum static deflection of a single secondary beam with the same boundary conditions as the double-beam loaded by a concentrated force \(f\) at \(x_{\max}\) of the single beam and \(\overline{t}\) is a dimensionless time  
\[\overline{t}=\frac{vt}{l} \tag{63}\]  
For a fixed-fixed beam, \(x_{\max}=0.5l\) and \(w_{s}\) is  
\[w_{s}=-\frac{fL^{3}}{192E_{2}I_{2}} \tag{64}\]  
For a fixed-pinned beam, \(x_{\max}=0.55l\) and \(w_{s}\) is  
\[w_{s}=-\frac{fL^{3}}{48\sqrt{5}E_{2}I_{2}} \tag{65}\]  
The speed ratio is defined as  
\[\alpha=\frac{v}{v_{\alpha}} \tag{66}\]where \(v_{cr}\) is the critical speed at which the single secondary beam is excited into resonance in the first mode by the moving force  
\[v_{cr}=\frac{\widetilde{\omega}_{2,1}L}{\pi} \tag{67}\]  
and the first frequency of the single secondary beam \(\widetilde{\omega}_{2,1}\) can be obtained by  
\[\widetilde{\omega}_{2,1}=\left(r_{1}^{(1)}\right)^{2}\sqrt{\frac{E_{2}I_{2}}{ \rho_{2}A_{2}}} \tag{68}\]