# Modeling and Analysis of Dynamics of Rigid-Flexible Coupled Parallel Robots
## 2 Modal Vector Bond Graph
### Flexible Connecting Rod Bond Graph Modeling  
Considering the flexibility of the rods, the flexible rods can be regarded as Euler-Bernoulli beams and modeled as bond graphs based on the assumed modal method. The longitudinal vibration of the connecting rod is neglected, and only the bending vibration of the connecting rod is considered. If the connecting rod is subjected to a concentrated force, \(F(t)\) at \(x=x_{1}\), as shown in Figure 2.  
The concentrated force \(F(t)\) can be transformed into a distributed force form \(F(t)\delta(x-x_{1})\), and then the vibration equation of the flexible rod can be expressed as:  
\[EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho A\frac{\partial^{2}w(x,t)}{ \partial t^{2}}=F(t)\delta(x-x_{1}) \tag{1}\]  
Figure 1: n-dimensional mode vector bond. where \(w\) is the deflection of the rod, \(E\) is the modulus of elasticity of the material, \(I\) is the moment of inertia of the cross-section, \(\rho\) is the density of the material, and \(A\) is the cross-sectional area. The response of a flexible rod, either forced or unforced, is expressed as a linear combination of the vibration patterns, i.e.,  
\[w(x,t)=\sum_{n=1}^{\infty}Y_{n}(x)\eta_{n}(t),\ n=1,2,\cdots \tag{2}\]  
Applying Equation (2) to the vibration Equation (1) by multiplying each term by \(Y_{m}(x)\) and taking the integrals from \(x=0\) to \(x=L\) for \(x\), and then utilizing the orthogonality of the modes, i.e., \(Y_{n}(x)=Y_{m}(x)\) only when \(n=m\) and \(0\) for the rest of the parts, then the following expression is obtained,  
\[(\int_{0}^{L}\rho AY_{n}^{2}dx)\ddot{\eta}_{n}+(\int_{0}^{L}\rho AY_{n}^{2}dx) \omega_{n}^{2}\eta_{n}=F(t)Y_{n}(x_{1}) \tag{3}\]  
where the modal mass is expressed as  
\[m_{n}=\int_{0}^{L}\rho AY_{n}^{2}dx,n=1,2,\cdots\]  
The modal stiffness is expressed as  
\[k_{n}=(\int_{0}^{L}\rho AY_{n}^{2}dx)\omega_{n}^{2}=m_{n}\omega_{n}^{2},n=1,2,\cdots\]  
At this point, Equation (3) can be expressed as  
\[m_{n}\ddot{\eta}_{n}+k_{n}\eta_{n}=F(t)Y_{n}(x_{1}),n=1,2,\cdots \tag{4}\]  
In addition, considering the existence of a rigid-body mode, i.e., translational motion of the whole flexible rod and rotational motion around the center of mass, in addition to the transverse vibration of the flexible rod under the action of an external force, it is obtained that  
\[m\ddot{\eta}_{00}=F(t) \tag{5}\]  
\[J\ddot{\eta}_{0}=F(t)(x_{1}-\frac{L}{2}) \tag{6}\]  
where Equation (5) represents the effect of external force on the acceleration of the center of mass of the flexible rod, \(\ddot{\eta}_{00}\) is the acceleration of the center of mass, and \(m\) is the mass of the flexible rod. Equation (6) represents the effect of the moment of the external force on the angular acceleration of the center of mass, and \(J\) is the rotational moment of inertia of the flexible rod around the center of mass. \[EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho A\frac{\partial^{2}w(x,t)}{ \partial t^{2}}=F(t)\delta(x-x_{1}) \tag{1}\]0 Uniform flexible linkage subjected to external force \(F\). Rewrite Equation (4) in the form of a bond graph energy variable:  
\[EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho A\frac{\partial^{2}w(x,t)}{ \partial t^{2}}=F(t)\delta(x-x_{1}) \tag{1}\]1  
where \(\mathbf{K}= \begin{bmatrix}k_{1}&&&\\ &k_{2}&&\\ &&\ldots&\\ &&&k_{n}\end{bmatrix} \) is the mode stiffness matrix; the form is a diagonal angular array with diagonal elements for each order of modal stiffness;  
\(\mathbf{Y}(x_{1})= \begin{bmatrix}Y_{1}(x_{1})&Y_{2}(x_{1})&\cdots&Y_{n}(x_{1}) \end{bmatrix} ^{T}\) is a vector of vibrational mode functions. According to Equations (5)-(7), the flexible rod modal vector bond graph is established, as shown in Figure 3.  
The modulus of the converter -MTF- in Figure 3 is a vector of vibrational functions, representing the system's conversion from a physical space variable to a modal space variable and its inverse transformation.