Model Constrained Systems as DAEs 

Model the motion of a single pendulum.

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Derive the governing equations using Newton's second law of motion, and .

In[1]:=

X deqns = {m x''[t] == (\[Lambda][t]/l) x[t], m y''[t] == (\[Lambda][t]/l) y[t] - m g};

Express the fixed length of the pendulum rod as an algebraic constraint.

In[2]:=

X aeqns = {x[t]^2 + y[t]^2 == l^2};

The pendulum is released from the horizontal position with a vertical velocity of 1.

In[3]:=

X ics = {x[0] == 1, y[0] == 0, x'[0] == 0, y'[0] == 1};

Specify the physical parameters for the pendulum system.

In[4]:=

X params = {g -> 9.81, m -> 1, l -> 1};

Solve the high-index DAE and visualize the system.

In[5]:=

X pendulumSol = First[NDSolve[{deqns, aeqns, ics} /. params, {x, y, \[Lambda]}, {t, 0, 15}, Method -> {"IndexReduction" -> Automatic}]];

In[6]:=

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Out[6]=