Part C : Dynamics of Beam

Here we finally pull everything together to devœlope a differetial equation for \(\ddot{\alpha}\) which contains only constants reflecting the physical characteristics of the beam plus all independent integration variables which are :

\( \alpha \; \dot{\alpha} \; \beta \; \dot{\beta} \; \gamma \dot{\gamma} \)

Fig. (C.1)

In Fig.(C.1) the three forces, \(P_w\) from the counter-weight, \(P_p\) from the pumpkin and \(M_b g\) the weight of the beam act to rotate the beam about the pivot.

\begin{equation*} I_b \ddot{\alpha} = a P_w + b P_p - M_b\: g\: c\: \sin\alpha \tag{C.1} \end{equation*}

Here :

Mb = mass of the beam
c  = location of center of mass of beam, can have negative or positive value
Ib = mass moment of inertia of beam w.r.t. pivot
Pw = force of counter-weight onto beam, Eq.(A.7)
Pp = force of pumpkin rope onto beam, Eq.(B.5) or (B.7), respectively.

\begin{equation*} I_b \ddot{\alpha} = a^2 M_w ( w_1 - w_2 \ddot{\alpha} ) + b a M_p (p_1 - p_2 \ddot{\alpha} ) - M_b\: g\: c\: \sin\alpha \end{equation*}

In order to relate this equation more easily to what we use in the program, see file diffeqs.js, we divide through by Mb a2 and we utilize the abbreviations \(w_1, w_2, p_1, p_2\) as indicated in Eqs. (A.7a), (B.5) and (B.7), respectively .

\begin{equation*} \left( \frac{I_b}{a^2 M_b} + \frac{M_w}{M_b} w_2 + \frac{b}{a} \frac{M_p}{M_b} p_2 \right)\;\ddot{\alpha} = \frac{M_w}{M_b} w_1 + \frac{b}{a} \frac{M_p}{M_b} p_1 -\frac{c}{a} \frac{g}{a} \sin\alpha \tag{C.2} \end{equation*}

Trebuchet HomePage Trebuchet-Math/Physics Background Part A : Dynamics of counter-weight
Part B : Dynamics of pumpkin Part C : Dynamics of Beam Part D : Sling Finger Analysis
Part E : Stress Analysis Part F : A few Notes on Programming Aspects

Zig Herzog;
Last revised: xx