# 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:

- The mass-moments of inertia of the crown wheel, each bevel pinion and each beater are J c , J p and

Jb , respectively. - The ratio of crown wheel and bevel pinion diameters is n.
- 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. - 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:

- Derive basic equations of motion for the system using free-body diagrams.
- Identify all independent energy storing elements and types of energy stored by each of those elements.
- Select state variables, derive the state-variable equations and construct simulation diagram without using

any differentiator. - 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

- Find the normal operating point (NOP) values of q1 , q2 and u when the NOP value q3 2 .
- Linearize state variable equations about NOP.
- Compare the responses of nonlinear and linearized systems using MATLAB: ODE23. Assume that the

system is at NOP initially. Take uˆ 0.05u .