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Mechanical Engineer´s Handbook P37

CHAPTER 2 7 MATHEMATICAL MODELS OF DYNAMIC PHYSICAL SYSTEMS K. Preston White, Jr. Department of Systems Engineering University of Virginia Charlottesville, Virginia 27.1 RATIONALE 27.2 IDEAL ELEMENTS 27.2.1 Physical Variables 27.2.2 Power and Energy 27.2.3 One-Port Element Laws 27.2.4 Multiport Elements 27.3 SYSTEM STRUCTURE AND INTERCONNECTION LAWS 27.3.1 A Simple Example 27.3.2 Structure and Graphs 27.3.3 System Relations 27.3.4 Analogs and Duals 27.4 795 796 796 797 798 799 802 802 804 806 807 27.5 APPROACHES TO LINEAR SYSTEMS ANALYSIS 27.5.1 Transform Methods 27.5.2 Transient Analysis Using Transform Methods 27.5.3 Response to Periodic Inputs Using Transform Methods 27.6 STATE-VARIABLE METHODS 27.6.1 Solution of. | CHAPTER 27__ MATHEMATICAL MODELS OF DYNAMIC PHYSICAL SYSTEMS K. Preston White Jr. Department of Systems Engineering University of Virginia Charlottesville Virginia 27.1 RATIONALE 795 27.2 IDEAL ELEMENTS 796 27.2.1 Physical Variables 796 27.2.2 Power and Energy 797 27.2.3 One-Port Element Laws 798 27.2.4 Multiport Elements 799 27.3 SYSTEM STRUCTURE AND INTERCONNECTION LAWS 802 27.3.1 A Simple Example 802 27.3.2 Structure and Graphs 804 27.3.3 System Relations 806 27.3.4 Analogs and Duals 807 27.4 STANDARD FORMS FOR LINEAR MODELS 807 27.4.1 I O Form 27.4.2 Deriving the I O Form 808 An Example 808 27.4.3 State-Variable Form 27.4.4 Deriving the Natural State Variables A 810 Procedure 27.4.5 Deriving the Natural State Variables An 811 Example 27.4.6 Converting from I O to 812 Phase-Variable Form 812 27.5 APPROACHES TO LINEAR SYSTEMS ANALYSIS 813 27.5.1 Transform Methods 813 27.5.2 Transient Analysis Using Transform Methods 818 27.5.3 Response to Periodic Inputs Using Transform Methods 827 27.6 STATE-VARIABLE METHODS 829 27.6.1 Solution of the State Equation 829 27.6.2 Eigenstructure 831 27.7 SIMULATION 840 27.7.1 Simulation Experimental Analysis of Model Behavior 840 Tin.2 Digital Simulation 841 27.8 MODEL CLASSIFICATIONS 846 27.8.1 Stochastic Systems 846 27.8.2 Distributed-Parameter Models 850 27.8.3 Time-Varying Systems 851 27.8.4 Nonlinear Systems 852 27.8.5 Discrete and Hybrid Systems 861 27.1 RATIONALE The design of modem control systems relies on the formulation and analysis of mathematical models of dynamic physical systems. This is simply because a model is more accessible to study than the physical system the model represents. Models typically are less costly and less time consuming to construct and test. Changes in the structure of a model are easier to implement and changes in the behavior of a model are easier to isolate and understand. A model often can be used to achieve insight when the corresponding physical system cannot because