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Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoLumped Parameter Systems1Outline of Todays LectureReviewEngineering Modeling ProcedureState Space ModelsLumped Parameter SystemsDC Armature control motorBalance Systems

ModelsREAL WORLDOBSERVATIONSMATHEMATICALMODELEXPLANATION/PREDICTIONSENSEFORMULATEINTERPRETTESTEngineering Modeling ProcedureUnderstand the problemWhat 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 sketchFree body diagramHydraulic schematicElectrical schematicWrite the equilibrium equations (usually differential or difference)Newton 2nd LawKirchoff Laws for current and voltagesFlow continuity lawsSolve the equations for the desired resultCheck the validity of the results

Modeling is an Iterative ProcessUnderstandthe ProblemSketchMathematicalModelSolvethe ModelValidate the ResultsUse theModelCan you formulate a model?Can you solve the model?Do the resultsrepresentreality?YESYESYESNONONOModeling TermsSystem: a functional group of interrelated thingsState: A condition (which may or may not be physical) of the system regarding form, structure, location, thermodynamics or compositionState vector: a collection of variables that fully describe the object over timeInput: an external object provide to the systemOutput: a dependent variable (often a state) from within the system that can be measured or quantifiedDynamics: a chance process of the state variables over timeState Space FormulationContinuous ModelsLet 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 elementsThe matrix A is the called the State Dynamics MatrixThe matrix B is called the Input or Control MatrixThe matrix C is called the Output or Sensor MatrixThe matrix D is called the Pass Through or Direct term

State Space FormulationDiscrete ModelsLet 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 elementsThe matrix A is the called the State Dynamics MatrixThe matrix B is called the Input or Control MatrixThe matrix C is called the Output or Sensor MatrixThe matrix D is called the Pass Through or Direct term

State Space Formulation Procedure:Develop the equations of equilibriumPut the equilibrium equations in the form of the highest derivative equal the remainder of the termsMake a choice of states, the input and the outputs Write the equilibrium equations in terms of the state variablesConstruct the dynamics, the input, the output and the pass through matricesWrite the state space formulation

Distributed vs. Lumped ParametersDistributed parameterAnalysis is at the material element levelPartial differential equations describe the transfer of force from the constitutive equationsFEM/BEM often usedLumped parameterAnalysis is at the component levelComponent properties are self contained and completeODE/Diff E based on linking component parameters Equations solved analytically or numerically

Distributed vs. Lumped ParametersDistributed parameter systemsphysically better descriptions more accurate results when done correctlyLumped parameter systemssimplerquicker resultsBoth can be used in building controlsLumped 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

?Lumped Parameter Variables

From Richard C. Dorf, Modern Control Systems, 6 ed.Mechanical SystemsWhat are the noises from wheel speed?

Equilibrium Equations Needed:

Engine to clutch

Clutch to transmission

Transmission to wheel

Wheel to groundDetermine the number of equationsneed form the number of inertialcoordinates (qe,qd,qa,and qw)and their linkages

Mechanical Systems

What are the noises from wheel speed?

Lumped Parameter Model of an Armature Controlled DC Motor

What is the speed?

Lumped Parameter Model of an Armature Controlled DC Motor

What is the speed?Note how the mechanicaland the electrical domainswere put together here:

KVL for the electricalNSL for the mechanicalRelationship or couplingequation between the two

In a controls problem, sometimes called Mechatronics, this is oftennecessaryIs this a good modelfor motor angle?

Lumped Parameter Model of an Armature Controlled DC Motor

What is the motor angle?

Same process, different question,different formulationLumped Parameter Model of an Armature Controlled DC Motor

What is the motor angle?

If the inductance La is small suchthat it can be neglected, then another simpler formulation is

Balance Systems A large number of control problems are called balance systems where an object must be maintained in technically an unstable positionBalance DynamicsGeneral Dynamics Equation form is

This equation is usually nonlinear

Energy Conserving TermsEnergy Dissipating (Rayleigh) TermsExternal Forcing termsExample: Inverted Pendulum

Clearly NonlinearExample: Inverted Pendulum

Example: Inverted Pendulum

SummaryLumped Parameter vs. Distributed Parameter SystemsDistributed parameter systems: Material element levelPartial differential equations describe the transfer of force from the constitutive equationsLumped Parameter SystemsComponent levelComponent properties are self contained and complete with ODE/Diff E based on linking component parameters for equilibrium equationsMechanical system equationsElectric MotorBalance systemsNext Class: Matlab and Simulink