Numerical Starting Procedure

At time

t = 0

$t = 0$ , the displacement

u

$u$ and velocity

{\dot{u}}_{0}

${\dot{u}}_{0}$ are known from initial conditions. The acceleration

{\ddot{u}}_{0}

${\ddot{u}}_{0}$ and time step

h_{1}

$h_{1}$ are found from solving the equations of motion. The initial time step

h_{0}

$h_{0}$ is set to zero:(1)

h_{0} = 0; h_{\frac{1}{2}} = \frac{h_{1}}{2}

$h_{0} = 0; h_{\frac{1}{2}} = \frac{h_{1}}{2}$

(2)

{\dot{u}}_{- \frac{1}{2}} = {\dot{u}}_{0}

${\dot{u}}_{- \frac{1}{2}} = {\dot{u}}_{0}$

(3)

{\dot{u}}_{\frac{1}{2}} = {\dot{u}}_{0} + h_{\frac{1}{2}} {\ddot{u}}_{0}

${\dot{u}}_{\frac{1}{2}} = {\dot{u}}_{0} + h_{\frac{1}{2}} {\ddot{u}}_{0}$