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# Lecture 2 ME 176 2 Mathematical Modeling

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### Lecture 2 ME 176 2 Mathematical Modeling

1. 1. ME 176 Control Systems Engineering Department of Mechanical Engineering Mathematical Modeling
2. 2. Mathematical Modeling: Introduction Mathematical Models are representation of a system's schematics, which in turn is a representation of a system simplified using assumptions in order to keep the model manageable and still an approximation of reality. 1. Transfer Functions (Frequency Domain) 2. State Equations (Time Domain) First step in creating a mathematical model is applying the fundamental laws of physics and engineering: Electrical Networks - Ohm's law and Kirchhoff's laws Mechanical Systems - Newton's laws. Department of Mechanical Engineering
3. 3. Mathematical Modeling: Transfer Functions Differential Equation Representation: Transfer Function : - Distinct and Separate - Cascaded Interconnection Department of Mechanical Engineering
4. 4. Mathematical Modeling: Laplace Transform Represents relations of subsystems as separate entities. Interrelationship are algebraic. Laplace Transform: Inverse Laplace Transform: Department of Mechanical Engineering
5. 5. Mathematical Modeling: Laplace Transform Department of Mechanical Engineering Laplace Transform Table Unit Impulse Unit Step Ramp Exponential Decay Sine Cosine Laplace Transform Theorems Linearity Frequency Shift Time Shift Scaling Differentiation Integration Initial Value Final Value
6. 6. Mathematical Modeling: Laplace Transform Partial Fraction Expansion, where roots of the Denominator of F(s) are: Note: N(s) must be less order that D(s) . 1. Real and Distinct where, Department of Mechanical Engineering
7. 7. Mathematical Modeling: Laplace Transform Partial Fraction Expansion, where roots of the Denominator of F(s) are: 2. Real and Repeated where, Department of Mechanical Engineering
8. 8. Mathematical Modeling: Laplace Transform Partial Fraction Expansion, where roots of the Denominator of F(s) are: 3. Complex or Imaginary (Part 1) - compute for residues Steps: a. Solve for known residues. b. Multiply by lowest common denominator and clearing fractions. c. Balancing coefficients. Department of Mechanical Engineering
9. 9. Mathematical Modeling: Laplace Transform Partial Fraction Expansion, where roots of the Denominator of F(s) are: 3. Complex or Imaginary (Part 2) - put into proper form Steps: a. Complete squares of denominator. b. Adjust numerator. Department of Mechanical Engineering
10. 10. Mathematical Modeling: Transfer Functions c(t) - Output r(t) - Input Systems that can be represented by linear time invariant differential equations. Initial conditions are assumed to be zero . Permits interconnections of physical systems which can be algebraically manipulated. Department of Mechanical Engineering
11. 11. Mathematical Modeling: Transfer Functions Department of Mechanical Engineering