Dynamic Models: Mechanical Systems

1. Dynamic Models: Mechanical Systems • OBJECTIVES: • Examples of engineering systems • ODE modeling of dynamical systems • Example problems

2. Dynamic Systems • Examples: • Mechanical Systems • Systems that posses significant mass, inertia, spring, and energy dissipation components driven by forces, torques, and specified displacements. An automobile has a dynamic response as it speeds up, slows down or rounds a curve on the road. • Electrical Systems • They include circuits with resistive, capacitive, or inductive components excited by voltage or current.

3. Dynamic Systems • Examples: • Fluid Systems • They employ orifices, restrictions, control valves, accumulators, long tubes, and actuators that are excited by pressure or fluid flow. A city water tower has a dynamic response of the height of the water as a function of the amount of water pumped into the tower and the amount being used by the citizens. • Thermal Systems • They have components that provide resistance (conduction, convection, or radiation) and capacitance (mass and specific heat) when excited by temperature or heat flow. A boiling pot of water has a dynamic response to the temperature that depends on the material, the amount of water, etc..

4. Mixed Dynamic Systems • Examples: • Electrical-mechanical, fluid-mechanical, thermo-mechanical, etc.. Electro-mechanical Electro-fluid-mechanical Electro-fluid-mechanical Electro-thermal

5. System Models & ODEs Mass/spring system: Population of a single species: Electrical circuit:

6. System Models & ODEs Pendulum: Hanging cable: Loaded beam:

7. Modeling of Mechanical Systems • Mechanical Systems • Systems that posses significant mass, inertia, spring, and energy dissipation components driven by forces, torques, and specified displacements. • Modeling steps: • - Identify the discrete elements that compose the system and draw an idealized • representation for them • - Identify the end points of each element and label the connection points or nodes • - Draw the free-body diagrams of the connection nodes for each element • - Derive a governing differential equation by using the elements models and • Newton’s Laws of Motion.

8. Example 1: Machine Part

9. Example 2: Engine and Propeller Model

10. Example 3: Jammed Machine • Rotating Mechanical System • Suppose that a machine becomes jammed in its operation so that the left end of the shaft shown is fixed and there is a torque, T(t), applied to its right end. Assuming that the shaft inertia is very small compared to that of the disk, develop an equation of motion, or model, with respect to the angular position of the disk. The machine tries to override the malfunction, so the applied torque is periodic but has zero mean.