Lumped Parameter Systems

1. Lumped Parameter Systems Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo

2. Outline of Today’s Lecture • Review • Engineering Modeling Procedure • State Space Models • Lumped Parameter Systems • DC Armature control motor • Balance Systems

3. Models REAL WORLD OBSERVATIONS SENSE FORMULATE TEST EXPLANATION/ PREDICTION MATHEMATICAL MODEL INTERPRET

4. Engineering Modeling Procedure • Understand the problem • What are the factors and relevant relationships? • What assumptions can be made? • What equilibrium conditions exist? • What should the result look like? • Draw and label an engineering sketch • Free body diagram • Hydraulic schematic • Electrical schematic • Write the equilibrium equations (usually differential or difference) • Newton 2nd Law • Kirchoff Laws for current and voltages • Flow continuity laws • Solve the equations for the desired result • Check the validity of the results

5. Modeling is an Iterative Process Can you formulate a model? Mathematical Model Understand the Problem Sketch YES NO Can you solve the model? NO YES NO Do the results represent reality? YES Validate the Results Solve the Model Use the Model

6. Modeling Terms • System: a functional group of interrelated things • State: A condition (which may or may not be physical) of the system regarding form, structure, location, thermodynamics or composition • State vector: a collection of variables that fully describe the object over time • Input: an external object provide to the system • Output: a dependent variable (often a state) from within the system that can be measured or quantified • Dynamics: a chance process of the state variables over time

7. State Space FormulationContinuous Models • Let x be a vector formed of the state variables • The number of components of the state vector is called the order • Formulate the system as • The matrices A, B, C and D have constant elements • The matrix A is the called the State Dynamics Matrix • The matrix B is called the Input or Control Matrix • The matrix C is called the Output or Sensor Matrix • The matrix D is called the Pass Through or Direct term

8. State Space FormulationDiscrete Models • Let x be a vector formed of the state variables • The number of components of the state vector is called the order • Formulate the system as • The matrices A, B, C and D have constant elements • The matrix A is the called the State Dynamics Matrix • The matrix B is called the Input or Control Matrix • The matrix C is called the Output or Sensor Matrix • The matrix D is called the Pass Through or Direct term

9. State Space Formulation • Procedure: • Develop the equations of equilibrium • Put the equilibrium equations in the form of the highest derivative equal the remainder of the terms • Make a choice of states, the input and the outputs • Write the equilibrium equations in terms of the state variables • Construct the dynamics, the input, the output and the pass through matrices • Write the state space formulation

10. Distributed vs. Lumped Parameters • Distributed parameter • Analysis is at the material element level • Partial differential equations describe the transfer of force from the constitutive equations • FEM/BEM often used • Lumped parameter • Analysis is at the component level • Component properties are self contained and complete • ODE/Diff E based on linking component parameters • Equations solved analytically or numerically

11. Distributed vs. Lumped Parameters • Distributed parameter systems • physically better descriptions • more accurate results when done correctly • Lumped parameter systems • simpler • quicker results • Both can be used in building controls • Lumped parameter descriptions are appropriate when the property being examined is of much greater magnitude than the added accuracy that would be gained using a distributed parameter model ?

12. Lumped Parameter Variables From Richard C. Dorf, Modern Control Systems, 6 ed.

13. Mechanical Systems What are the noises from wheel speed? Determine the number of equations need form the number of inertial coordinates (qe,qd,qa,andqw) and their linkages Equilibrium Equations Needed: Engine to clutch Clutch to transmission Transmission to wheel Wheel to ground

14. Mechanical Systems What are the noises from wheel speed?

15. Lumped Parameter Model of an Armature Controlled DC Motor What is the speed?

16. Lumped Parameter Model of an Armature Controlled DC Motor Note how the mechanicaland the electrical domains were put together here: KVL for the electrical NSL for the mechanical Relationship or couplingequation between the two In a controls problem, sometimes called Mechatronics, this is often necessary What is the speed? Is this a good modelfor motor angle?

17. Lumped Parameter Model of an Armature Controlled DC Motor What is the motor angle? Same process, different question, different formulation

18. Lumped Parameter Model of an Armature Controlled DC Motor What is the motor angle? If the inductance La is small such that it can be neglected, then another simpler formulation is

19. Balance Systems A large number of control problems are called balance systems where an object must be maintained in technically an unstable position

20. Balance Dynamics General Dynamics Equation form is This equation is usually nonlinear External Forcing terms Energy Dissipating (Rayleigh) Terms Energy Conserving Terms

21. Example: Inverted Pendulum Clearly Nonlinear

22. Example: Inverted Pendulum

23. Example: Inverted Pendulum

24. Summary • Lumped Parameter vs. Distributed Parameter Systems • Distributed parameter systems: • Material element level • Partial differential equations describe the transfer of force from the constitutive equations • Lumped Parameter Systems • Component level • Component properties are self contained and complete with ODE/Diff E based on linking component parameters for equilibrium equations • Mechanical system equations • Electric Motor • Balance systems • Next Class: Matlab and Simulink