Question

For the simple pendulum, a nonlinear time-invariant dynamical system, the equations of motion in the presence of some damping torque (due to friction) are the following: ml2θ¨+bθ˙+mglsinθ=0 (a) (10 pts) Analyze the stability of the origin equilibrium. What about in the absence of friction (b=0) ? Hint: Use the linearization method for stability analysis covered in lecture. (b) (10pts) For some initial conditions near the origin simulate the pendulum dynamics to confirm your stability analysis from part (a). Plot the results in a phase diagram. Use m=0.1 kg,l=0.3 m,b=0.1Nm⋅s, and g=9.81 m/s2. Hint: Use the MATLAB function ODE45() to simulate the system.

Answer 1

The equation of motion for a damped pendulum is ml^2θ¨ + bθ˙ + mglsinθ = 0. We can analyze the stability of the equilibrium using the linearization method. In the absence of friction, the equation simplifies to ml^2θ¨ + mglsinθ = 0.

Step-by-step explanation:

The simple pendulum is a nonlinear time-invariant dynamical system. The equation of motion for a damped pendulum is given by ml^2θ¨ + bθ˙ + mglsinθ = 0. (a) To analyze the stability of the equilibrium, we can use the linearization method for stability analysis. (b) In the absence of friction (b=0), the equation becomes ml^2θ¨ + mglsinθ = 0.