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# Lecture 4 ME 176 2 Mathematical Modeling

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### Lecture 4 ME 176 2 Mathematical Modeling

1. 1. ME 176 Control Systems Engineering Department of Mechanical Engineering Mathematical Modeling
2. 2. Mathematical Modeling: Mechanical Systems Translational Motion - Analogous to electrical variables and components. - Use of Newton's Law to form differential equation of motion. Assumptions: - A positive direction of motion. - Sum forces equal to Zero. - Initial conditions of Zero. Department of Mechanical Engineering Components: - fv is coefficient of viscous friction . - M is mass . - K is spring constant. - f(t) is force. - x(t) is displacement.
3. 3. Mathematical Modeling: Mechanical Systems Translational Motion Department of Mechanical Engineering for Analogy for Solving
4. 4. Mathematical Modeling: Mechanical Systems Translational Motion Case 1 : Force =>Voltage Velocityt =>Current Displacementt =>Charge Spring => Capacitor Mass => Inductor Damper => Resistor Summing forces in terms of velocity => Mesh equation Department of Mechanical Engineering
5. 5. Mathematical Modeling: Mechanical Systems Translational Motion Case 2 : Force =>Voltage Velocityt =>Current Displacementt =>Charge Spring => Inductor Mass => Capacitor Damper => Resistor Summing forces in terms of velocity => Nodal equation Department of Mechanical Engineering
6. 6. Mathematical Modeling: Mechanical Systems Translational Motion Department of Mechanical Engineering
7. 7. Mathematical Modeling: Mechanical Systems Translational Motion Department of Mechanical Engineering Mass Damper Spring
8. 8. Mathematical Modeling: Mechanical Systems Translational Motion 1. Number of equations of motion to describe system is equal to number of linearly independent motions. 2. Linearly independent motion is a point in a system that can move even if all other points are still. 3. Linearly independent motions is also called "degrees of freedom." Solving Equations with High Degrees of Freedom ( > 1): - Draw free body diagram for each point representing a linearly independent motion. - For each analysis, assume other points are still. - For each analysis, consider only forces related to motion of point. - For each analysis, use Newton's law and sum all forces to zero. Department of Mechanical Engineering
9. 9. Mathematical Modeling: Mechanical Systems Translational Motion : 2 Degrees of Freedom Department of Mechanical Engineering
10. 10. Mathematical Modeling: Mechanical Systems Translational Motion : 2 Degrees of Freedom Department of Mechanical Engineering
11. 11. Mathematical Modeling: Mechanical Systems Translational Motion : 2 Degrees of Freedom Department of Mechanical Engineering
12. 12. Mathematical Modeling: Mechanical Systems Translational Motion : 3 Degrees of Freedom Department of Mechanical Engineering
13. 13. Mathematical Modeling: Mechanical Systems Translational Motion : 3 Degrees of Freedom Department of Mechanical Engineering
14. 14. Mathematical Modeling: Mechanical Systems Translational Motion : 3 Degrees of Freedom Department of Mechanical Engineering
15. 15. Mathematical Modeling: Mechanical Systems Translational Motion : 3 Degrees of Freedom Department of Mechanical Engineering
16. 16. Mathematical Modeling: Mechanical Systems Rotational Motion - Same as rotational just that torque, replaces force; and displacement is measured angular. - Analogous to electrical variables and components. - Use of Newton's Law to form differential equation of motion. Assumptions: - A positive direction of motion. - Sum forces equal to Zero. - Initial conditions of Zero. Department of Mechanical Engineering Components: - D is coefficient of viscous friction . - J is moment of inertia . - K is spring constant. - T is Torque. - is angular displacement.
17. 17. Mathematical Modeling: Mechanical Systems Rotational Motion Department of Mechanical Engineering for Analogy for Solving
18. 18. Mathematical Modeling: Mechanical Systems Rotational Motion Case 1 : Torque =>Voltage Velocitya =>Current Displacementa =>Charge Spring => Capacitor Inertia => Inductor Damper => Resistor Summing forces in terms of velocity => Mesh equation Department of Mechanical Engineering
19. 19. Mathematical Modeling: Mechanical Systems Rotational Motion Case 2 : Torque =>Voltage Velocitya =>Current Displacementa =>Charge Spring => Inductor Mass => Capacitor Damper => Resistor Summing forces in terms of velocity => Nodal equation Department of Mechanical Engineering
20. 20. Mathematical Modeling: Mechanical Systems Rotational Motion : 2 Degrees of Freedom Department of Mechanical Engineering
21. 21. Mathematical Modeling: Mechanical Systems Rotational Motion : 2 Degrees of Freedom Department of Mechanical Engineering
22. 22. Mathematical Modeling: Mechanical Systems Rotational Motion : 2 Degrees of Freedom Department of Mechanical Engineering
23. 23. Mathematical Modeling: Mechanical Systems Rotational Motion : 3 Degrees of Freedom Department of Mechanical Engineering
24. 24. Mathematical Modeling: Mechanical Systems Rotational Motion : 3 Degrees of Freedom Department of Mechanical Engineering
25. 25. Mathematical Modeling: Mechanical Systems Rotational Motion : 3 Degrees of Freedom Department of Mechanical Engineering
26. 26. Mathematical Modeling: Mechanical Systems Rotational Motion : 3 Degrees of Freedom Department of Mechanical Engineering