ME 450 ME450 Homework 1 ANSWERS (Penn State University)

ME450 Homework 1 Solution (Penn State University)

Problem 1: A simple drawing and functional description of a hand-powered egg beater is shown. The goal is to construct a mathematical model of this mechanical system by making free-body diagram for each element. The input and output of the system are torque applied to the crown wheel ( Ti ) and and the angular velocity of the

beater ( b ), respectively. Use the following assumptions and notational conventions:

1. The mass-moments of inertia of the crown wheel, each bevel pinion and each beater are J c , J p and
   Jb , respectively. 
2. The ratio of crown wheel and bevel pinion diameters is n. 
3. The shafts which connect the bevel pinions to beaters are flexible. Use the torsional spring constant K s
   for each shaft to represent this effect. 
4. Assume that each beater experiences a linear viscous damping torque as it spins. Use the viscous
   damping coefficient B f to represent its effects.
   Questions: 

1. Derive basic equations of motion for the system using free-body diagrams. 
2. Identify all independent energy storing elements and types of energy stored by each of those elements. 
3. Select state variables, derive the state-variable equations and construct simulation diagram without using
   any differentiator. 
4. Obtain Input/Output differential equation. 

Problem 2: A nonlinear dynamic system has been modeled by the following state variable equations: q 1  q 2  q 3 q 2  u  q 2  0 . 5 q 1 q 1 q 3  6 q 3  3 q 1

1. Find the normal operating point (NOP) values of q1 , q2 and u when the NOP value q3  2 . 
2. Linearize state variable equations about NOP. 
3. Compare the responses of nonlinear and linearized systems using MATLAB: ODE23. Assume that the
   system is at NOP initially. Take uˆ  0.05u .