A rigid rod with uniform mass {and total mass equal to m} is hanging from a massless rope and undergoes oscillations under the action of a...

Modeling and Analysis of control systems questions

1. A rigid rod with uniform mass {and total mass equal to m} is hangingfrom a massless rope and undergoes oscillations under the action of ahorizontal foroe at its end [the direction of the force is constant].Gravity is included. Derive the equation{s} of motion using Lagrange’sequations. 2. Compute a_H equilibrium states of the following dynamical systems. For each problem, if the system is nonlinearlinearize it close to each equilibrium state and write the resulting linearized system in matrix form. {ilx'I1=31§“‘-2Iz=1'1"xz{ii]s+x+4x= =0{iiiJjE—-x+4x1=t]{iv}£+3x+x3 =0 {v}.if+3x—x3={l 3. Express each of the following dynamical systems in state form. If the system is linear write the state equations inmatrix form. {i}25'c' + i' — 23': =1+4x {iii ii + 51“? + 1'3 u 51! = 1'2 “Zr—1 {iii} j? + 101? -- yEt .. 1) = 3 sin 5: In these expressionsx = x[t) and y = fit}. 4. In class we derived the equations of motion ofthe double pendulum.{i} Verify that the double pendulum has the equilibrium positions (916,335) = {0,0}, (0,11), Err, D), (it, it}.{ii} Linearize the equations of motion of the double pendulum about each of these equilibrium positions.