### Dynamics of counter weight

Purpose is to derive an equation for $$\ddot{\beta}$$, the second derivate of the angle β w.r.t. time, see Fig.(A.1).

The coordinates of the center of mass of the counter weight (se black large dot) are easily given by : \begin{equation*} x_w = a - a \sin \alpha - L_w \sin \beta \tag{A.1a} \end{equation*} \begin{equation*} y_w = H - a \cos \alpha - L_w \cos \beta \tag{A.1b} \end{equation*} We take now twice the derivative w.r.t. time to arrive at equations for the accelerations in the x- and y-direction. \begin{equation*} \ddot{x}_w = a \sin(\alpha) \dot{\alpha}^2 - a \cos(\alpha) \ddot{\alpha} + L_w \sin(\beta) \dot{\beta}^2 - L_w \cos(\beta) \ddot{\beta} \tag{A.2a} \end{equation*} \begin{equation*} \ddot{y}_w = a \cos(\alpha) \dot{\alpha}^2 + a \sin(\alpha) \ddot{\alpha} + L_w \cos(\beta) \dot{\beta}^2 + L_w \sin(\beta) \ddot{\beta} \tag{A.2b} \end{equation*}

These accelerations are due to two forces action on the counter-weight, one at the hinge with an unknown x- and y-compnent and then gravity acting at the center of mass of the counter-weight, see Fig. (A.2)

\begin{equation*} F_x = M_w \ddot{x}_w \tag{A.3a} \end{equation*} \begin{equation*} F_y - M_w g = M_w \ddot{y}_w \tag{A.3b} \end{equation*}

Later on we will use Eqs.(A.2a) and (A.2b) for substitution. But first we want to consider the moment of all force w.r.t. the center of gravity of the counter-weight. To that end we need the components of Fx and Fy in perpendicular to the line from the center of mass to the hinge in order to determine the moment around the center of mass. \begin{equation*} (\;F_x cos\beta - F_y sin\beta\;) L_w = I_w \ddot{\beta}\tag{A.4} \end{equation*} Here Iw is the mass moment of inertia of the counter-weight w.r.t. its center of mass.

We now substitute Eqs.(A.3a) and (A.3b) into Eq.(A.4) and then replace the accelerations with the expressions on the right side of the Eqs.(A.2a) and (A.2b). After some tedious algebra we obtain : \begin{equation*} (I_w + M_w L^2_w) \ddot{\beta} = M_w L_w a \left( -\frac{g}{a}\sin\beta + \sin(\alpha{-}\beta) \dot{\alpha}^2 - \cos(\alpha{-}\beta) \ddot{\alpha} \right) \tag{A.5} \end{equation*} This equation is used in our computer code in file trebuchet/diffeqs.js to determine $$\ddot{\beta}$$ aka dY[3].

In preparation for investigating the dynamics of the beam itself we need an equation for the moment the counter-weight exerts on the beam. To this end we define the force Pw as the sum of those components of Fx and Fy perpendicular to the beam axis. The Pw, as shown in Fig. (A.3), is ment to indicate the action onto the counter-weight. The action onto the beam by the counter-weight is equal in value but in opposite direction leading to a clock-wise acceleration which is a positive $$\ddot{\alpha}$$.

Hence : \begin{equation*} P_w = F_x \cos\alpha - F_y \sin\alpha \end{equation*} We now substitute Eqs.(A.3a) and (A.3b) into this equation and then replace the accelerations with the expressions on the right side of the Eqs.(A.2a) and (A.2b). After some algebra we obtain : \begin{equation*} P_w = -M_w \left( g \sin\alpha + a \ddot{\alpha} + L_w \sin(\alpha{-}\beta) \dot{\beta}^2 + L_w \cos(\alpha{-}\beta) \ddot{\beta} \right) \tag{A.6} \end{equation*}

In order to solve our system of differential equation numerically we aim to develop a single differential equation for $$\ddot{\alpha}$$ in which terms depend only on α and β and their first derivative in addition to $$\ddot{\alpha}$$ itself. This differential equation will reflect the moment equation around the pivot for the beam in which Pw plays an essential role. Therefore, in Eq.(A.6) the term $$\ddot{\beta}$$ needs to be eliminated which can be done by substitution from Eq.(A.5).

\begin{eqnarray*} \frac{P_w}{M_w a}\; &=&\;\overbrace{ -\left\{ \frac{g}{a} \left(\sin\alpha-\rho_w \sin\beta \cos(\alpha{-}\beta) \right) + \rho_w \sin(\alpha{-}\beta) \cos(\alpha{-}\beta) \dot{\alpha}^2 + \frac{L_w}{a} \sin(\alpha{-}\beta) \dot{\beta}^2\right\} }^{=w_1} \\ &-& \underbrace{ \left(1 - \rho_w \cos^2(\alpha{-}\beta)\right) }_{=w_2} \;\ddot{\alpha} \tag{A.7a} \end{eqnarray*}

Here $$\rho_w$$ ( rhoW in the program ) is defined by :

\begin{equation*} \rho_w = \frac{M_w L_w^2}{I_w + M_w L_w^2} \tag{A.7b} \end{equation*}

Zig Herzog; hgnherzog@yahoo.com
Last revised: 03/10/20