## 2 Dynamic Model
### Energy of System  
The energy of the crane includes the potential energy \(V\) and kinetic energy \(T\). The potential energy consists of the strain energy of the transmission shafts, the wire rope and two girders, and the gravitational potential energy of the trolley and the payload. The kinetic energy is composed of the kinetic energy of the two girders and the trolley due to the vibration, kinetic energy of the payload due to its translational movement in the vertical direction and kinetic energy of the rotational rigid bodies in lifting mechanism due to the rotation movement about their own axes. Figure 3: Physical model of electric overhead crane under braking of lifting mechanism. As shown in Figure 3, an orthogonal coordinate system \(OXYZ\) is defined in the crane workspace, and another orthogonal coordinate system \(O_{3}\chi_{3}\xi_{3}\eta_{3}\) is fixed to the trolley. The displacement \(w\) of a point in the two girders axis, whose \(Y\) coordinate is \(y\), relative to its position under the dead weight of the girder is a function of \(y\) and time \(t\):  
\[w(y,t)=\mathbf{N}(y)\mathbf{a}(t) \tag{1}\]  
where  
\[\mathbf{N}(y)=\left[\sin\!\left(\frac{\pi y}{L}\right),\ \sin\!\left(\frac{2\pi y }{L}\right),\cdots,\sin\!\left(\frac{n\pi y}{L}\right)\right],\ \mathbf{a}(t)=\left\{a_{1}(t),a_{2}(t),\cdots,a_{n}(t)\right\}^{\mathrm{T}}\]  
Here, \(\mathbf{a}(t)\) and \(n\) (\(n\in\mathrm{N}^{+}\)) are respectively the vector and the number of generalized coordinates used for describing elastic deformation of the girders along \(Z\) direction. If the static equilibrium position of girders under their dead weight is defined as the zero position of the strain energy of girders and plane \(XOY\) is defined as the zero position of the gravitational potential energy of the system, then the potential energy of the crane can be expressed as;  
\[ \begin{array}{l}V=\frac{1}{2}\theta^{\mathrm{T}}\mathbf{K}_{\mathrm{L}}\theta +\frac{1}{2}k_{\mathrm{w}}[l(t)+l_{\mathrm{d}}]\varepsilon^{2}(t)+\frac{1}{2 }\mathbf{a}^{\mathrm{T}}\mathbf{N}_{\mathrm{v1}}\mathbf{a}\ +\ m_{\mathrm{T} \mathrm{g}}(\mathrm{C}_{\mathrm{T}}+\mathbf{N}_{\mathrm{T}}\mathbf{a})\\ +m_{\mathrm{Q}\mathrm{g}}[\mathrm{C}_{\mathrm{d}}+\mathbf{N}_{\mathrm{d}} \mathbf{a}-l(t)[1+\varepsilon(t)]]\end{array}  \tag{2}\]  
with  
\[\theta^{\mathrm{T}}=[\theta_{1}(t),\theta_{2}(t),\theta_{3}(t),\ \theta_{4}(t), \theta_{5}(t),\theta_{6}(t)],\]  
\[\mathbf{N}_{\mathrm{v1}}=\pi^{\mathrm{d}}E_{\mathrm{x}}\mathrm{diag}\!\left( 1,2^{4},\cdots,n^{4}\right)\!/L^{3},\ \mathbf{N}_{\mathrm{T}}=[\chi_{\mathrm{T}}\mathbf{N}(y_{ \mathrm{v3}}+b)+(b-\chi_{\mathrm{T}})\mathbf{N}(y_{\mathrm{v3}})]/b,\]  
\[\mathbf{N}_{\mathrm{d}}=[\chi_{\mathrm{d}}\mathbf{N}(y_{\mathrm{v3}}+b)+(b- \chi_{\mathrm{d}})\mathbf{N}(y_{\mathrm{v3}})]/b,l_{\mathrm{d}}=R_{\mathrm{d }}/(2\mu m_{\mathrm{s}}),\]  
\[\mathbf{K}_{\mathrm{L}}=\left[ \begin{array}{ccccc}k_{1}&-k_{1}&0&0&0&0\\ -k_{1}&k_{1}+k_{2}&-k_{2}&0&0&0\\ 0&-k_{2}&k_{2}+k_{3}&-k_{3}&0&0\\ 0&0&-k_{3}&k_{3}+k_{4}&-k_{4}&0\\ 0&0&0&-k_{4}&k_{4}+k_{5}&-k_{5}\\ 0&0&0&0&-k_{5}&k_{5}\end{array} \right]\]  
Here, \(\varepsilon\), \(l\), and \(k_{\mathrm{w}}\) are used to describe respectively the tensile strain, the free length, and the total tensile force per unit strain of the wire rope segments hanging from the rope drum. \(m_{\mathrm{T}}\) and \(m_{\mathrm{Q}}\) are the masses of the trolley and lifted object (including the load-handling device) respectively. \(C_{i}\) (\(i=\mathrm{T}\), \(\mathrm{d}\)) are the constants involved in the elastic deformation due to the dead weight, the camber and the geometrical sizes of the girder. \(\theta_{i}\) (\(i=1,\ 2,\ldots,\ 6\)) is the angular displacement of rotating rigid body \(i\) in the lifting mechanism and is converted to quantities in reference to the rope drum axis. \(E\) and \(I_{x}\) are, respectively, the Young's modulus of the girder material and the moment of inertia of the girder cross section with respect to its neutral axis, which is parallel to the \(X\)-axis.