ET3-7: Modelling I(V) Introduction and Objectives Electrical, Mechanical and Thermal Systems
Objectives analyse and model basic linear dynamic systems -Electrical -Mechanical -Thermal Recognise the analogies between these systems. Apply block diagram representations reduce them. Learn about non-linear systems - determine working points and linearisation. Work with Matlab/Simulink.
Agenda of the Day 1. Objectives and goals of modelling of dynamic systems. 2. Introduction to Dynamic Systems and Modelling 3. Models of basic mechanical system elements 4. State-Space equations 5. Application of Matlab/Simulink to the simulation of the behaviour of dynamic systems
1.1 Objectives and goals of modelling of dynamic systems. What do we want to model? Why do we want to model it? How can we construct models? How can we use the models?
2 Introduction to Dynamic Systems and Modelling
2.1(7) An example dynamic system An electrical dynamic system Moves the coil and cone A mechanical system Which drives the air A thermodynamic system We want a method to predict what will happen during operation
2.2(7) Sometimes we will meet a computer controlled process The Computer processes discrete data The plant may be discrete or continuous The algorithm needs to predict how the plant will behave In order to take appropriate control action
2.3(7) Some advantages and possibilities of computer control Energy saving High efficiency Modelling and Simulations Precise, high speed actuators Tool centre control Off-road vehicles Electrically and hydraulically powered systems Measurement of electrical signals Active damping
2.4(7) What do we need? Theory that works for many different physical processes Solves electrical, magnetic, mechanical and thermal problems Steady state characteristics; capacity; efficiency; losses; sizing Transient response; steady state errors; stability; settling time; etc. System behaviour during faults? Theory must account for the past history of the system and enable us to predict future behaviour under known applied conditions
2.5(7) Example - Concept of a cruise control system for a car A cruise control system in a car comprises several sub-systems.
2.6(7) Example - Coupling between mechanical and electrical systems concept of a battery powered vehicle
2.7(7) Electrical Analogies for Mechanical Systems Electrical engineers are used to carrying out dynamic circuit analysis It would be easier for them, if they could use the same techniques on mechanical arrangements can we find an analogy enabling mechanical dynamic events to be represented in the form of electric circuits?
3 Models of basic mechanical system elements
3.1(11) Mechanical work W = F dx mech e F e dx F e is of electrical origin F e is opposed by mechanical forces A force balance equation arises
3.2(11) Force Balance The equation of motion Various mechanical forces oppose movement Acceleration of the mass Various frictional forces Gravity (not shown) Wind (not shown) F = m x + B x + f e F e Bx + f mx dx m
3.3(11) Simple Linear Mechanical system A mass x is the distance from the reference F is the driving force f is the dry friction force B is the coefficient of viscous friction
3.4(11) Simple Linear Mechanical system = + + F mx Bx f
3.5(11) Simple Rotating Mechanical system J The moment of inertia Θ is the angle measured to the reference T is the driving torque f is the torque due to dry friction B is the coefficient of viscous friction
3.6(11) Simple Rotating Mechanical system T = J + θ Bθ + f T = J ω + Bω + f
3.7(11) More Complicated Mechanical system Break it down to a set of free body diagrams Write the differential equations And solve them (integrate them)
3.8(11) Simple Linear Mechanical system The conceptual model Is reduced to a Free Body Diagram Which leads to the differential equation of motion ( ) f t ( M v + B v + K x) = 0 l ( ) M x + B x + K x = f t l
3.9(11) Our Complicated Mechanical system Break it down to a set of free body diagrams Write the differential equations of motion And solve them (integrate them)
3.10(11) Needs a Free Body Diagram for Each Mass We arrive at a set of simultaneous differential equations
3.11(11) Needs a Free Body Diagram for Each Mass In this way, we arrive at a set of simultaneous differential equations ( ) ( ) 2 1 2 2 1 1 1 1 1 ( ) ( ) ( ) B x x + K x x Mx K x = f t Mx B x x K x x = a 2 2 2 1 2 2 1 0 0
4 State-Space Equations
4.1(2) The State-Space formulation Convenient to solve using Matlab/Simulink Manipulate the differential equations a set of first order ordinary differential equations Isolate the differential term The standard form q = aq+ aq + aq + bu+ bu 1 11 1 12 2 13 3 11 1 12 2 q = a q + a q + a q + b u + b u 2 21 1 22 2 23 3 21 1 22 2 q = a q + a q + a q + b u + b u 3 31 1 32 2 33 3 31 1 32 2 y = cq+ cq + cq + du+ du 1 11 1 12 2 13 3 11 1 12 2 y = c q + c q + c q + d u + d u 2 21 1 22 2 23 3 21 1 22 2
4.2(2) The State-Space formulation in Matrix Form The standard form In Matrix Form Q = + A Q B U = + Y C Q D U
5 Application of Matlab/Simulink to the simulation of dynamic systems behaviour
5.1(9) Steps When Modelling A System Physics and topology Select the model Simplifying assumptions Draw the Free Body Diagram(s) Write the equations of motion Differential equations Values of system coefficients Mass; moment of inertia; inductance; resistance etc. Solve the equations of motion Integrate them
5.2(9) Example System Cruise Control for a Car Physics and topology Make simplifying assumptions e.g. Neglect wheels Linear friction Draw the Free-Body Diagram
5.3(9) Cruise Control for a Car From the Free-Body Diagram write the differential equations The equations of motion For a first order system only one equation u bx = mx b x + x = m u m
5.4(9) Cruise Control for a Car Rewrite the equation of motion in state-space form A set of first order differential equations First choose the state variables We choose x & v x = v b 1 v = v + u m m
5.5(9) Cruise Control for a Car Write them in the standard form 0 1 0 x x = + b 1 u v 0 v m m Q = AQ + Bu
5.6(9) Cruise Control for a Car If the required output is the car position y = 1 0 x v Y = CQ + Du
5.7(9) Cruise Control for a Car So the State-Space Matrices are 0 1 0 A = = = = b, B 1, C 1 0, D 0 0 m m
5.8(9) Cruise Control for a Car Values At time t=0, the input u jumps from u=0 to u=500n The mass of the car is 1000kg and b=50n.sec/m Write a Matlab script to simulate and plot the motion of the car
5.9(9) Cruise Control for a Car Demo Matlabscript CarCruiseControl.m
Exercises Write a Matlab script to solve Problem 2.1 Problem 2.4 In Lit. [1] In each case plot all relevant outputs in the form of a step response
Lit. [1] Problem 2.1
Lit. [1] Problem 2.4