In a similar fashion as going from position to velocity, to calculate the accelerations of the point on the mechanism we simply take the time derivative of the result from the velocity analysis. Separating the mechanism into its two component parts we get the following two vector loops.
Part 1:
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$d_1(j\alpha_3-\omega_3^2)e^{j\theta_3}-L_2(j\alpha_2-\omega_2^2)e^{j\theta_2}-\ddot{h_1}j=0$ |
Part 2:
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$d_4(j\alpha_3-\omega_3^2)e^{j\theta_3} + L_6(j\alpha_6-\omega_6^2)e^{j\theta_6} + (h_2(j\alpha_6-\omega_6^2) + 2j\dot{h_2}\omega_6 + \ddot{h_2})e^{j(\theta_6-\frac{\pi}{2})} + L_5(j\alpha_5-\omega_5^2)e^{j\theta_5} = 0$ |
Now the solutions to these equations are quite long and a bit of a handful, but they will be shown nonetheless. These expressions were derived using MATLAB's symbolic solver to generate symbolic expressions, which were then directly implemented into the umbrella code.
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\begin{equation}
\alpha_2 = \frac{-L_2cos(\theta_2 - \theta_3)\omega_2^2 +d_1\omega_3^2 + \ddot{h_1}sin(\theta_3)}{L_2sin(\theta_2 - \theta_3)}
\end{equation} |
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\begin{equation}
\alpha_3 = \frac{-L_2\omega_2^2 + d_1cos(\theta_2 - \theta_3)\omega_3^2 + \ddot{h1}sin(\theta_2)}{d_1sin(\theta_2 - \theta_3)}
\end{equation} |
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\begin{equation}
\alpha_5 = -\frac{L_6^2\omega_6^2 - h_2\ddot{h_2} + h_2^2\omega_6^2 - 2L_6\dot{h_2}\omega_6cos(\theta_5 - \theta_6) + \alpha_3d_4h_2cos(\theta_3 - \theta_6) + L_5L_6\omega_5^2cos(\theta_5 - \theta_6) + L_6d_4w_3^2cos(\theta_3 - \theta_6) - L_5h_2\omega_5^2sin(\theta_5 - \theta_6) - d_4h_2\omega_3^2sin(\theta_3 - \theta_6)}{L_5(L_6sin(\theta_5 - \theta_6) + h_2cos(\theta_5 - \theta_6)}
\end{equation} |
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\begin{equation}
\alpha_6 = \frac{L_5\omega_5^2 + \ddot{h_2}sin(\theta_5 - \theta_6) - 2\dot{h_2}\omega_6cos(\theta_5 - \theta_6) + L_6\omega_6^2cos(\theta_5 - \theta_6) + d_4\omega_3^2cos(\theta_3 - \theta_5) - h_2\omega_6^2sin(\theta_5 - \theta_6) + \alpha_3d_4sin(\theta_3 - \theta_5)}{L_6sin(\theta_5 - \theta_6) + h_2cos(\theta_5 - \theta_6)}
\end{equation} |
As it was mentioned in the Project Overview, the acceleration profiles of the joints and links are not particularly important for the operation of the mechanism. Of more interest is the velocity profile and the mechanical advantage induced through the turning of the crank. These results have been included out of a desire for a thorough analysis of the mechanism and as such there is not much to discuss besides a simple presentation of the results.
We can see in the acceleration plots that swift acceleration of links 5 and 6 near to the end of the opening motion that was observed in the Velocity Analysis section.