## 2 Materials and Methods
### Modeling of the Large-Deflection Deformation of a Multi-Pivot Flexible Nozzle
#### 2.1.3 Model Solving Based on the Variational Principle  
As shown in Figure 2, the initial length of each support rod is \(S_{0}\), the length of each beam segment is \(l_{1},\ l_{2},\ldots,\ l_{N}\), the width of the beam is \(b\), and the thickness is \(h\). The elongation of each support rod is \(S_{1}\), \(S_{2}\),..., \(S_{N}\), the support angle changes after the elongation of the rod is \(\theta_{1}\), \(\theta_{2}\),..., \(\theta_{N}\), and the coordinates of each support point are \((x_{1},y_{1})\)\(\ldots\)\((x_{N},y_{1})\)\(\ldots\)\((x_{N},y_{1})\)\(\ldots\)\((x_{N},y_{1})\). The boundary conditions for the multi-point-supported elastic beam problem are  
\[\left\{ \begin{array}{l}\int_{x_{i-1}}^{x_{i}}(\sqrt{1+w^{\prime 2}}dx-l_{i})=0 \\ x_{i}=\sum_{n=1}^{n}(l_{i}-(s_{i}+s_{0})\sin\theta_{i})\quad(i=1,2,3,\cdots,n) \\ y_{i}=(s_{0}+s_{i})\cos\theta_{i}-s_{0}\\ w(x_{i})=y_{i}\\ w^{\prime\prime}(x_{i})\
eq 0\end{array} \right., \tag{1}\]  
where \(w\) represents the deflection, while \(w^{\prime}\) and \(w^{\prime\prime}\) denote its first and second derivatives, respectively. As illustrated in Figure 2, \(s_{i}\) represents the increased length of the driving rod, and \(s_{0}\) indicates its initial length. The first line of Equation (1) expresses the conservation of the arc length during deformation. The final line of Equation (1) indicates that a non-zero bending moment exists at the support point. According to the energy principle, the true deformation of the beam will cause the deformation energy to take a very small value under the constraint that the beam is guaranteed to be inextensible. In turn, the solution is carried out by the variational principle. To obtain the shape function in the minimum energy state, the generalized function with constraints is  
\[\Pi=\int_{0}^{x_{N}}\frac{1}{2}EI\left[\frac{w^{\prime\prime}}{(1+w^{\prime 2})^{ \frac{2}{2}}}\right]^{2}dx+\sum_{i=1}^{N}\lambda_{i}\left(\int_{x_{i-1}}^{x_{ i}}\sqrt{1+w^{\prime 2}}dx-l_{i}\right), \tag{2}\]  
where \(\lambda_{i}\) (i = 1, 2) is the Raschner multiplier, whose exact value is determined by the boundary conditions and variational constraints. Variational differentiation of the generalized function of the above equation, i.e., \(\delta\Pi=0\), yields  
Figure 2: (**a**) The initial state of the elastic beam with multi-point supports. (**b–d**) The states of all elongation, elongation with shortening, and all contraction, respectively. \[EI\sum_{i=1}^{N}\biggl{[}\int_{\dot{x}_{i-1}}^{x_{i}}\biggl{(}\frac{3 \omega^{\prime\prime}w^{\prime 2}}{(1+w^{\prime 2})^{4}}-\frac{w^{\prime\prime \prime}}{(1+w^{\prime 2})^{3}}+\frac{\lambda_{i}}{EI}\frac{w^{\prime}}{\sqrt{1+w^{ \prime 2}}}\biggr{)}\delta w^{\prime}dx+\delta\lambda_{i}\Bigl{(}\int_{\dot{x} _{i-1}}^{x_{i}}\sqrt{1+w^{\prime 2}}dx-l_{i}\Bigr{)}\biggr{]} \tag{3}\] \[\qquad\qquad\qquad\qquad\qquad\qquad=\ 0\ (i\ =\ 1,\ 2,\ \ldots,\ N).\]  
This yields the Euler-Lagrange equations and constraints as  
\[ \begin{array}{l}\frac{3w^{\prime\prime}w^{\prime 2}}{(1+w^{\prime 2})^{4}}- \frac{w^{\prime\prime\prime}}{(1+w^{\prime 2})^{3}}+\frac{\lambda_{i}}{EI} \frac{w^{\prime}}{\sqrt{1+w^{\prime 2}}}=0\quad(i=1,2,\ldots,N)\\ \int_{\dot{x}_{i-1}}^{x_{i}}\sqrt{1+w^{\prime 2}}dx-l_{i}=0\quad(i=1,2,\ldots,N). \end{array}  \tag{4}\]  
Solving the Euler-Lagrange equation can obtain the general solution of the shape function, and the specific form of the shape function can be determined according to the boundary conditions and constraints. It is worth noting that the Euler equation here implies the influence of each segment of the strut on each other. Although it is solved segment by segment, it is also considered as a whole and the accuracy of the results is guaranteed. The Taylor series expansion uses the starting point of each beam section as the expansion point.