November 1996  ICTCM IX (Reno)  13

Step 3 – Composite Solution

> V := V1 + (V2-V1)*Heaviside(t-td):
> X := X1 + (X2-X1)*Heaviside(t-td):

v(t) = \{ \begin{matrix}−\frac{mg}{k_{1}} (1 − e^{− \frac{tk_{1}}{m}} ) , & t < t_{d} \\ −\frac{mg}{k_{2}} + \frac{mg}{k_{1}k_{2}} (k_{1} − k_{2} (1 − e^{− \frac{t_{d}k_{1}}{m}} )) e^{− \frac{(t−t_{d})k_{2}}{m}} , & t ≥ t_{d} \end{matrix}

x(t) = \{ \begin{matrix}x_{0} − \frac{mg}{k_{1}} (t − \frac{m}{k_{1}} (1 − e^{− \frac{tk_{1}}{m}} )) , & t < t_{d} \\ what would you do with this if you had it?, & t ≥ t_{d} \end{matrix}