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Unformatted text preview: K G(s) +  MECH 431 Mathematical modeling Slide 1 Mathematical modeling Chapter 3 K G(s) +  MECH 431 Mathematical modeling Slide 2 Introduction • A mathematical model is a set of equations that represent the dynamics of the system accurately. • A system may be represented differently, via different mathematical models. • The dynamics of many systems may be modeled in terms of differential equations . m k u y K G(s) +  MECH 431 Mathematical modeling Slide 3 Introduction • We usually resort to physical laws governing particular systems to develop such differential equations: – Newton’s law for mechanical systems. – Kirchhoff’s law for electrical systems. K G(s) +  MECH 431 Mathematical modeling Slide 4 Mathematical models • Assume different forms of mathematical models based on the particular system and particular circumstances. • Options include using: – Transfer function representation (SISO linear timeinvariant) – Statespace representation (optimal control) K G(s) +  MECH 431 Mathematical modeling Slide 5 Simplicity vs. Accuracy • In developing mathematical models, we make a compromise between: – Simplicity of the model, – Accuracy of the results. • To simplify a model we have to ignore certain physical models of the system ( e.g., mass of spring). • For example, if our model is assumed linear, we ignore nonlinearities . If the effect of these assumptions are small, we get an accurate model ( i.e., agrees with physical system). m k u y K G(s) +  MECH 431 Mathematical modeling Slide 6 Simplicity vs. Accuracy • In general, start with a simple model. • Once you understand the system, make a more complete mathematical model. • The frequency range of operation of the system can also dictate the validity of a simple model: – For example, the mass of a spring might be neglected at low frequencies but requires modeling at high frequencies. m k K G(s) +  MECH 431 Mathematical modeling Slide 7 Linear systems • A system is linear if the principle of superposition holds . • This principle states that the response produced by the simultaneous application of two forcing functions is the sum of the two individual responses. k F 1 y F 2 K G(s) +  MECH 431 Mathematical modeling Slide 8 Linear systems • By this principle, one can build complicated solutions to linear differential equations from simple solutions. • Experimentally, if input and output are proportional → superposition holds → the system is considered linear. K G(s) +  MECH 431 Mathematical modeling Slide 9 Linear timeinvariant system • A differential equation is linear if the coefficients are constant or functions only of the independent variable....
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This note was uploaded on 02/07/2011 for the course MECH 433 taught by Professor Danielasmar during the Spring '09 term at American University of Beirut.
 Spring '09
 DanielAsmar

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