Statics analysis based on the reduced multibody system transfer matrix method

Abstract

The reduced multibody system transfer matrix method can efficiently solve the generalized accelerations of a system without establishing global dynamics equations with a system inertia matrix, given the generalized coordinates and generalized velocities of the system. The equilibrium position of a multibody system is significant in the dynamics analysis, which is difficult to obtain directly. In this paper, a statics analysis method is proposed based on the reduced multibody system transfer matrix method by applying the notion of direct differentiation. The partial derivatives of generalized accelerations with respect to generalized coordinates called Jacobian matrix can be obtained easily by differentiating the transfer equations in the reduced multibody system transfer matrix method. Let the generalized velocities be zero and solve the system of nonlinear equations with zero generalized accelerations to obtain the static equilibrium position of a multibody system. The formulation and solution procedure of the proposed method are presented. The numerical examples are compared with dynamic relaxation method and the iterative method based on the first kind of Lagrange’s equation, which demonstrates the proposed approach and show computational advantages. The proposed method is straightforward, highly programmable, universal and provides a powerful tool for solving static equilibrium positions of multibody systems while extending the application of the multibody system transfer matrix method.

Appendix:  Rigid body mechanics

Herein, the kinematics and dynamics equations of a rigid body are discussed. First consider the point $P$, which has no relative motion to the rigid body’s body-fixed coordinate system, as shown in Fig. 1. The position vector of the point $P$ can be expressed as

$$ {{{\overset{\scriptscriptstyle \rightharpoonup}{r}}_{OP}}={{ \overset{\scriptscriptstyle \rightharpoonup}{r}}_{OR}}+{{ \overset{\scriptscriptstyle \rightharpoonup}{r}}_{RP}}}. $$

(52)

The time derivative of Eq. (52) is the equation for the translational velocity vector at the point $P$, namely

$$ {{\dot{\overset{\scriptscriptstyle \rightharpoonup}{r}}_{OP}}={ \dot{\overset{\scriptscriptstyle \rightharpoonup}{r}}_{OR}}+{{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times {{ \overset{\scriptscriptstyle \rightharpoonup}{r}}_{RP}}}, $$

(53)

where $\overset{\scriptscriptstyle \rightharpoonup}{r }$ is the position vector and $\overset{\scriptscriptstyle \rightharpoonup}{\omega }$ is the angular velocity vector. The overhead dot (solid point) denotes the absolute derivatives with respect to the inertial coordinate system.

Since there is no relative motion between the point $P$ and the body-fixed coordinate system of the rigid body, the following equation is available:

$$ {{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OP}}={{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}. $$

(54)

The translational and angular acceleration vector of the point ${P}$ on the rigid body in the global inertial coordinate system could be obtained by taking the derivative of Eqs. (53) and (54) with respect to time, respectively [2, 24]:

$$ {{{\ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{OP}}={{ \ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{OR}}+{{ \dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}} \times {{\overset{\scriptscriptstyle \rightharpoonup}{r}}_{RP}}+{{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times ({{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times {{ \overset{\scriptscriptstyle \rightharpoonup}{r}}_{RP}})}, $$

(55)

$$ {{{\dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OP}}={{ \dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}}}. $$

(56)

The acceleration vector of the center of mass can be obtained by setting the point $P$ as the center of mass $C$, namely

$$ {{{\ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{OC}}={{ \ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{OR}}+{{ \dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}} \times {{\overset{\scriptscriptstyle \rightharpoonup}{r}}_{RC}}+{{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times ({{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times {{ \overset{\scriptscriptstyle \rightharpoonup}{r}}_{RC}})}, $$

(57)

$$ {{{\dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OC}}={{ \dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}}}. $$

(58)

The dynamics equations for a rigid body can be obtained from Newton’s law and the angular momentum theorem of a rigid body [27]:

$$ {\left \{ \textstyle\begin{array}{l} m{{{\ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}}_{OC}}={{{ \overset{\scriptscriptstyle \rightharpoonup}{F}}}{^{\mathrm{{Ex}}}}}, \\ m{{{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{RC}}\times {{{ \ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}}_{OR}}+{{{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}}\times ({{{ \overset{\scriptscriptstyle \rightharpoonup}{\overset{\scriptscriptstyle \rightharpoonup}{J}}}}_{R}} \cdot {{{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}})+{{{ \overset{\scriptscriptstyle \rightharpoonup}{\overset{\scriptscriptstyle \rightharpoonup}{J}}}}_{R}} \cdot {{{\dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}}_{OR}}= \overset{\scriptscriptstyle \rightharpoonup}{M}{_{R}^{\mathrm{{Ex}}}} , \end{array}\displaystyle \right .} $$

(59)

where $m$ is the mass of the rigid body and ${{ \overset{\scriptscriptstyle \rightharpoonup}{\overset{\scriptscriptstyle \rightharpoonup}{J}}}_{R}}$ is the inertia tensor of the spatially moving rigid body with respect to the root marker $R$. ${{\overset{\scriptscriptstyle \rightharpoonup}{F}}{^{\mathrm{{Ex}}}}}$ is the resultant external force acting on the body, and $\overset{\scriptscriptstyle \rightharpoonup}{M}{_{R}^{\mathrm{{Ex}}}}$ denotes the resultant external torque with respect to the root marker $R$. Substituting Eq. (57) into Eq. (59), the dynamics equations are obtained as follows:

$$ {\left \{ \textstyle\begin{array}{l} m({{\ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{OR}}+{{ \dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}} \times {{\overset{\scriptscriptstyle \rightharpoonup}{r}}_{RC}}+{{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times ({{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times {{ \overset{\scriptscriptstyle \rightharpoonup}{r}}_{RC}}))={{{ \overset{\scriptscriptstyle \rightharpoonup}{F}}}{^{\mathrm{{Ex}}}}} , \\ m{{{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{RC}}\times {{{ \ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}}_{OR}}+{{{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}}\times ({{{ \overset{\scriptscriptstyle \rightharpoonup}{\overset{\scriptscriptstyle \rightharpoonup}{J}}}}_{R}} \cdot {{{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}})+{{{ \overset{\scriptscriptstyle \rightharpoonup}{\overset{\scriptscriptstyle \rightharpoonup}{J}}}}_{R}} \cdot {{{\dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}}_{OR}}= \overset{\scriptscriptstyle \rightharpoonup}{M}{_{R}^{\mathrm{{Ex}}}} . \end{array}\displaystyle \right .} $$

(60)

Then the case when the point $P$ has relative motion to the body-fixed coordinate system of the rigid body is addressed. Equation (52) still holds. The motion of the point $P$ relative to the rigid body needs to be considered for the acceleration vector. The translational and angular acceleration vectors of the tip marker $T$ in the global inertial coordinate system are as shown below [10, 24, 27]:

$$ {{\ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{OP}}={{ \ddot{\overset{\scriptscriptstyle \rightharpoonup}{r}}}_{OR}}+{{ \dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}} \times {{\overset{\scriptscriptstyle \rightharpoonup}{r}}_{RP}}+{{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times ({{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times {{ \overset{\scriptscriptstyle \rightharpoonup}{r}}_{RP}})+{{ \overset{\circ \circ }{\mathop{{\overset{\scriptscriptstyle \rightharpoonup}{r}}}} \,}_{RP}}+2{{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}} \times {{ \overset{\circ }{\mathop{{\overset{\scriptscriptstyle \rightharpoonup}{r}}}} \,}_{RP}}, $$

(61)

$$ {{\dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OP}}={{ \dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{OR}}+{{ \dot{\overset{\scriptscriptstyle \rightharpoonup}{\omega }}}_{RP}}+ {{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{OR}}\times {{ \overset{\scriptscriptstyle \rightharpoonup}{\omega }}_{RP}}, $$

(62)

where the local derivative with respect to the body-fixed coordinate system $R{x}_{R}y_{R}z_{R}$ is expressed as a hollow point ∘ to distinguish it from the absolute derivative with respect to the inertial coordinate system expressed as a solid point.