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Mechatronikai modellezés

Dr. Szabó Tamás

Copyright © 2014 Miskolci Egyetem

A tananyag a TÁMOP-4.1.2.A/1-11/1-2011-0042 azonosító számú

„ Mechatronikai mérnök MSc tananyagfejlesztés ” projekt keretében készült. A tananyagfejlesztés az Európai Unió támogatásával és az Európai Szociális Alap társfinanszírozásával valósult meg.

Kézirat lezárva: 2014 február

A kiadásért felel a(z): Miskolci Egyetem Felelős szerkesztő: Miskolci Egyetem 2014

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Mechatronical Modelling

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Table of Contents

1 Introduction 1

1.1 Introductive Examples... 2

1.2 The Dynamical System ... 4

1.2.1 Definitions ... 5

1.2.2 System Classification ... 10

1.3 System Models ... 11

1.3.1 Model Types ... 11

1.3.2 System Analysis ... 12

1.3.3 System Simulation ... 13

1.3.4 System Identification ... 14

1.3.5 System Optimization (cp. Chapter 10) ... 15

1.3.6 Model Schemes and Block Diagrams ... 16

1.4 Task of System Dynamics ... 20

1.5 Summary ... 22

2 Mathematical Description of Dynamical Systems 24 2.1 Differential Equations ... 24

2.2 State Equations ... 28

2.3 Stationary Solutions and Equilibrium Positions ... 31

2.4 Linear State Equations ... 33

2.5 Analysis of Typical Problems of System Dynamics by Means of State Equations ... 39

3 Differential-Algebraic-Equation-Systems and Multiport Method 42 3.1 Differential-Algebraic-Equation-Systems (DAE-Systems) ... 42

3.2 The “Cut-Set“ or “Multiport“-Method ... 47

3.2.1 Basic Idea ... 47

3.2.2 Equation Structure ... 49

3.2.3 Examples ... 52

4 Solution of State Space Equations 61 4.1 Existence and Uniqueness of Solutions of Ordinary Differential Equations ... 61

4.2 Solution Design in the Phase Plane ... 64

4.3 Solution Methods for Linear State Equations ... 66

4.3.1 Solution of homogenous Systems, Fundamental Matrix ... 66

4.3.2 Solution of the inhomogeneous State Equation ... 70

5 State Space Equations with Normal Coordinates 73 5.1 Normal Coordinates ... 73

5.2 Eigenbehaviour of Systems with Multiple Eigenvalues ... 83

5.2.1 Effect of Multiple Eigenvalues ... 83

5.2.2 Jordan’s Normal Form ... 85

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6 Numerical Methods with Dynamical Systems 91

6.1 Introduction ... 91

6.1.1 Taylor Expansion ... 91

6.1.2 Numerical Algorithms ... 92

6.1.3 Rounding Error and Error Propagation ... 93

7 Numerical Methods for Initial Value Problems 97 7.1 Numerical Solution of Initial Value Problems ... 97

7.1.1 Explicit Euler Method (Forward Euler Method) ... 97

7.1.2 Numerical Stability and Stability Domain ... 101

7.1.3 Modified Euler Method (Trapezoidal Rule) ... 102

7.1.4 Implicit Euler method ... 106

7.1.5 Summary of Euler’s Method ... 107

7.1.6 General One-Step Procedures ... 107

7.1.7 Step Size Control ... 115

7.1.8 Linear Multi-Step Methods ... 118

7.1.9 Activation of Linear Multi-Step Procedures ... 122

7.1.10 System of Differential Equations ... 123

7.1.11 BDF Methods ... 123

7.1.12 Remarks on Stiff Differential Equations ... 125

7.1.13 Implicit Runge-Kutta Methods ... 133

7.1.14 Comparison of Methods for Numerical Solution of Initial Value Problems (IVP) ... 135

8 Integration of Discontinuous Systems and DAEs 137 8.1 Integration of Discontinuous Systems ... 137

8.2 Differential Algebraic Equations (DAEs) ... 138

9 Numerical Solution of Non-linear Sysem Equations 143 9.1 Nonlinear Equations ... 143

9.2 Solution with Numerical Integration... 147

9.3 Fixed Point Iteration ... 148

9.4 Newton-Raphson Iteration ... 150

10Identification and Optimisation 153 10.1 Linear Compensation Problem ... 154

10.2 Non-linear Parameter Dependency ... 157

10.3 Stability of Dynamical Systems ... 159

10.4 Stability Criteria for Linear Systems ... 161

10.4.1 Stability Criteria based on Stodola ... 161

10.4.2 Hurwitz Criteria ... 162

10.4.3 Stability Criteria based on Routh ... 164

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11Modelling of Mechanical Systems 167

11.1 Fundamental Terms ... 167

11.1.1 Modelling: Mass, Elasticity and Damping ... 167

11.1.2 Forces, System Boundary, Method of Sections ... 169

11.1.3 Constraints ... 170

11.1.4 Virtual Displacements ... 171

11.1.5 Kinematics ... 173

11.2 Principle of Linear and Angular Momentum ... 179

11.3 Consideration of Constraints and the Principle of d’Alembert... 180

11.4 Equations of Motion ... 181

11.5 Equations of Motion of a Double Pendulum ... 181

11.6 Linear Equations of Motion ... 186

11.7 State Equations ... 189

12Lagrange’s Equations of Motion of Second Kind 191 13Nonlinear Single Track Modell (based on [9] 196 13.1 Equations of Motion of the chassis ... 197

13.2 Tyre Model ... 198

13.2.1 Stationary Tyre Model ... 198

13.2.2 Dynamic Tyre Model ... 201

14Dynamic Wheel Rotation 203 14.1 Driving Torques ... 203

14.2 Breaking Torques ... 204

15The Overall Modell 205 15.1 Simulation Results ... 207

15.2 Animations ... 208

15.3 Videos ... 209

16Mathematical Basics 211 16.1 Matrix Calculations ... 211

16.1.1 Matrix Operations ... 213

16.1.2 Determinants ... 218

16.1.3 Norms ... 219

17Worksheets 221

18Exercises 237

19References 240

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List of Figures

Fig. 1.1: System descriptions in different fields of study [1]. ... 1

Fig. 1.2: ABS- magnetic valves, interfaces. ... 3

Fig. 1.3: Above: Percussion hammer [2]; Below: Simulation model of a percussion hammer. ... 4

Fig. 1.4: Representation of a system as a block. ... 6

Fig. 1.5: System of a motor vehicle [5]. ... 7

Fig. 1.6: Degrees of freedom of the spatial twin-track model [6]. ... 8

Fig. 1.7: Principle process of system simulation... 13

Fig. 1.8: Process of parameter identification. ... 15

Fig. 1.9: Process of system optimization. ... 16

Fig. 1.10: Example of a block diagram (single mass pendulum). ... 17

Fig. 1.11: Simple model of a wheel suspension. ... 18

Fig. 1.12: Block diagram for suspension, possible refinement levels. ... 19

Fig. 1.13: Electrical low pass-filter. ... 19

Fig. 1.14: Block diagram for electrical low-pass filter, possible refinement levels. ... 20

Fig. 1.15: Signal flow diagrams for a) suspension and b) low-pass filter... 20

Fig. 2.1: Mathematical pendulum. ... 24

Fig. 2.2: Longitudinal beam oscillations. ... 25

Fig. 2.3: Discrete model of the description of longitudinal oscillations. ... 26

Fig. 2.4: Simple frequency response system with concentrated parameters (multybody system). ... 28

Fig. 2.5: Block diagram of the non-linear state equation. ... 29

Fig. 2.6: Block diagram: Logistical growth at constant crop. ... 31

Fig. 2.7: Linearisation of a scalar function of a variable. ... 33

Fig. 2.8: Block diagram of additivity of a linear relation. ... 35

Fig. 2.9: Non-linear force characteristic (in combination with play). ... 35

Fig. 2.10: Block diagram of linear state equations. ... 38

Fig. 3.1: Non-linear simple pendulum. ... 43

Fig. 3.2: Simulation on the basis of state equations. ... 47

Fig. 3.3: Object-oriented modelling/simulation. ... 47

Fig. 3.4: Block diagram. ... 48

Fig. 3.5: Difference measurement (across). ... 48

Fig. 3.6: Flow measurement (through). ... 48

Fig. 3.7: Linkage of objects on the basis of the Cut-Set method. ... 49

Fig. 3.8: Mathematical pendulum example. ... 50

Fig. 3.9: Graphical representation. ... 51

Fig. 3.10: Sign convention. ... 51

Fig. 3.11: Ohmic resistance. ... 52

Fig. 3.12: Inductive reactance. ... 52

Fig. 3.13: Capacity... 53

Fig. 3.14: Simple electrical network. ... 53

Fig. 3.15: Spring. ... 55

Fig. 3.16: Multiport model spring. ... 56

Fig. 3.17: Mass. ... 56

Fig. 3.18: Multiport model mass ... 56

Fig. 3.19: Spring-mass system. ... 57

Fig. 3.20: Multiport. ... 58

Fig. 4.1: Solution of the equation for logistic growth (qualitative). ... 64

Fig. 4.2: Phase curves of the mathematical pendulum. ... 66

Fig. 5.1: Decaying behaviour with real eigenvalues. ... 78

Fig. 5.2: Decaying behaviour with complex eigenvalues. ... 79

Fig. 5.3: Behaviour of linear systems in dependence of the position of the complex plane. ... 80

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Fig. 5.4: Quarter-car. ... 80

Fig. 5.5: Eigenvalues of the quarter-car where 1500 Ns/m ≤ dA ≤ 6500 Ns/m... 83

Fig. 7.1: Forward Euler Method. ... 98

Fig. 7.2: Application of forward Euler method. ... 100

Fig. 7.3: Forward Euler method; instability. ... 100

Fig. 7.4: Stability domain of the explicit Euler’s method. ... 102

Fig. 7.5: Derivation of the Trapezoidal Rule. ... 103

Fig. 7.6: Stability domain after applying the fixed-point iteration. ... 105

Fig. 7.7: The implicit Euler method... 107

Fig. 7.8: Instability area in the Runga-Kutta method of 4th order. ... 112

Fig. 7.9: Transition points in the Runge-Kutta method. ... 113

Fig. 7.10: Stability area of the Runge-Kutta method of order 1-4. ... 115

Fig. 7.11: Step size control depicted as a (feedback) control problem ... 118

Fig. 7.12: Integration of one-mass pendulum by means of the explicit Euler method. ... 128

Fig. 7.13: Integration of one-mass oscillator by means of the implicit Euler method. ... 131

Fig. 7.14: Energy behaviour of a plane oscillator. ... 132

Fig. 7.15: A(0)-stability. ... 133

Fig. 8.1: Bouncing ball. ... 137

Fig. 8.2: Bouncing ball (simulation result). ... 138

Fig. 9.1: Planar fourbar mechanism. ... 144

Fig. 9.2: Solutions of the planar four-bar mechanism. ... 145

Fig. 9.3: Five-link wheel suspension. ... 145

Fig. 9.4: Vectors of the five-point wheel suspension. ... 147

Fig. 9.5: Wheel centre trajectory of the five-point real axle wheel suspension. ... 147

Fig. 9.6: Convergent fix-point iteration; gradient in fix point. ... 149

Fig. 9.7: Divergent fix point iteration ... 150

Fig. 9.8: Newton-Raphson iteration steps. ... 151

Fig. 10.1: Parameter determined system. ... 153

Fig. 10.2: Target system... 156

Fig. 10.3: Geometric illustration of the pseudoinverse. ... 157

Fig. 10.4: Simulator model ... 157

Fig. 10.5: Scan of y(t) ... 158

Fig. 11.1: Examples for elements of a multi-body system. ... 169

Fig. 11.2: Method of section in mechanics. ... 170

Fig. 11.3: Examples for constraints... 171

Fig. 11.4: Sphere pendulum. ... 173

Fig. 11.5: Virtual Displacement. ... 173

Fig. 11.6: Coordinate transformation by rotation around the z-axis. ... 175

Fig. 11.7: Double pendulum. ... 182

Fig. 13.1: Nonlinear single track model (bicycle model), top view ... 196

Fig. 13.2: Nonlinear single track model, side view ... 196

Fig. 14.1: Dynamic wheel rotation. ... 203

Fig. 14.2: Engine characteristics. ... 204

Fig. 14.3: Braking torque characteristics. ... 205

Fig. 15.1: Nonlinear single track model: different steering wheel angle jumps at v = 100km/h. ... 208

List of Tables

Table 1.1: Examples of state variables in different disciplines. ... 7

Table 1.2: Examples of system elements from different disciplines. ... 8

Table 3.1: Classification of across and through qualities in various fields. ... 48

Table 7.1: Accumulation of the amount of transition values and achieved order. ... 114

Table 7.2: Opening angle α for different k ... 133

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1 Introduction

This manuscript contains descriptions of modelling and simulation methods and aims at understanding the characteristics of complex (and other) dynamical systems in order to influence them effectively.

Consequently, it is required to know how to apply methods of modelling and analysis of such systems, their subsystems and components reasonably. This lecture deals with:

• mathematical formulations of dynamical systems,

• modelling techniques,

• simulation,

• numerical methods to integrate differential equations and to solve non-linear equation systems,

• identification and estimation of system parameters,

• stabilizing characteristics of dynamical systems.

In this context it is important to emphasize that the subject systems are not part of one particular discipline (e.g. mechanics), but that in order to describe such systems academic collaborations between different methods of branches of study have to be made, cp. Fig.

1.1.

Fig. 1.1: System descriptions in different fields of study [1].

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1.1 Introductive Examples

The following examples will be covered in lecture in more detail.

Example 1.1: ABS – Magnetic Valve, Fig. 1.2 System: ABS

Subsystem (components): Magnetic valve Potential objectives:

a) Describe the time-dependent process x(t) of the armature .

b) Where and how one can exert influence on the dynamic behaviour (e.g. a faster build- up of x to 90% of the total travel)?

c) What other objectives of non-dynamic nature exist in this context?

d) What other opportunities exist to influence the dynamic characteristics in a positive way (what is the meaning of “positive” in this context?)?

The construction of the ABS involves different disciplines of engineering, e.g.:

• mechanics (brake mechanism, driving dynamics),

• hydraulics (valves, pumps, pipes, ...),

• electrotechnology (controller, power electronics, …),

• control theory (ABS-algorithm).

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Fig. 1.2: ABS- magnetic valves, interfaces.

Example 1.2: Percussion Hammer, Fig. 1.3 System: Percussion hammer

Subsystem (component): Crank mechanism Potential objectives:

a) Maximize the impulse velocity of the percussion.

b) Minimize the vibrations of the hand grip while keeping the percussion power at a constant level.

c) How must the components of the crank mechanism be designed in order to achieve good results of drilling?

d) Is there a potential risk of breakage caused by heavy dynamic (working voltage) loadings on the components of the crank mechanism?

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Fig. 1.3: Above: Percussion hammer [2]; Below: Simulation model of a percussion hammer.

Worksheet 1: Simple simulation model of a percussion hammer [3]

1.2 The Dynamical System

Examples of the colloquial usage of the term system are solar system, transportation system, computer system, cardiovascular system, nervous system, etc.

In this context and in the fields of engineering and science, the term system is used in connection with complex and in transparent operations and processes. In general, we can observe a collaboration of different disciplines.

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1.2.1 Definitions

Definition 1.1 [3]:

A system is a set of elements (parts, components, objects),

• Which mutually influence each other (interaction),

• Which are subjected to external influences and affection (input) and

• Which effect to the external (output).

Annotation 1.1:

Science provides access to various definitions of the term system, e.g.:

Definition 1.2 [4]:

An object is identified as a system as long as all particular elements and all attributes with their interdependencies (to outside the system also) are seen to be components of that whole in some logical sense. Additionally, there are axioms.

Elements are e.g. modules, components, objects, fractions.

Attributes are properties, qualities, features, characteristics and also interfaces between the system and its environment. The term state characterizes the constitution of the system at any given time.

Definition 1.3 [4]:

A system is a well-defined assembly which consists of interacting entities. This assembly is delimited by a cladding plain or boundary (a set of system elements with interfaces to outside the system). The cladding plain provides for an interface between the system and its environment. The relationships between attributes and states, transported by the interfaces mentioned above, are variables which describe the peculiar characteristics of the system.

A block is a way of illustrating a system. The borderline represents the cladding plain.

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Fig. 1.4: Representation of a system as a block.

Interfaces are

r input quantities ui,

m output quantities yi and

q interfering quantities zi, if applicable.

Closed systems are self-contained and maintain no connections with the environment, i.e.

outside events have no influence on the system. The total state of the system is defined by n state quantities xi.

Definition 1.4:

State variables are those variables in a system, which completely describe the system’s behaviour. State variables are time-dependent. The term system state refers to the entirety of all values of the state variables.

Annotation 1.2:

• State variables are “internal variables” of a system.

• There is no definite choice of state variables. But characteristics can be assigned to the chosen state variables, such as uniqueness, independence and freedom of redundancy.

From the latter characteristic it follows that the choice of state variables is absolutely arbitrary; however, the values are fixed.

• In practice, one is not interested in the entirety of all state variables, but only chooses the variables needed for the application; the so-called output quantities. Output quantities can also result from a combination of state variables.

Examples of state variables from different disciplines; cp Table 1.1.

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Table 1.1: Examples of state variables in different disciplines.

Electrotechnology Mechanics Process engineering Environmental engineering

Current Position Temperature Population

Voltage Velocity Mass fraction Environmental state

Load Acceleration CO2 production

Kinetic energy Ozone value

Potential energy Example 1.3: Motor Vehicle System

In a motor vehicle system, the variables which emanate from the driver are e.g. the steering wheel angle and the brake and acceleration pedal travels which represent the input quantities. The variables describing the motion of the vehicle are defined as the state variables. Combinations or subsets of the quantities mentioned above are output quantities and the impulses resulting from the road are identified as intervening quantities (cp. Fig.

1.5).

Fig. 1.5: System of a motor vehicle [5].

A system which describes the vehicle, for example for vehicle dynamics analysis, is shown in Fig. 1.6.

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Fig. 1.6: Degrees of freedom of the spatial twin-track model [6].

Example 1.4: Framework System

Elements of the system are specific bars of the framework. Internal interactions between the bars are nodal forces. Input quantities are external loads and reaction forces in the bearings can be defined as output quantities.

Example 1.5: Road Traffic System

Specific motor vehicles are the system elements in a delimited road traffic network. The driver and his motor vehicle can represent the subsystems. Internal interactions can be comparative distances and relative velocities of the vehicles to each other.

Definition 1.5:

A system with more than one input quantity and more than one output quantity is defined as a multi-quantities system (multi-input-multi-output system “MIMO”).

System elements from different technical disciplines are listed in Table 1.2.

Table 1.2: Examples of system elements from different disciplines.

Electrotechnology Mechanics Process engineering Population development

Resistance Mass Vessel Birth

Capacitor Spring Valve Death

Coil Damper Pipe line Disease

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Transistor Beam Stirrer vessel Consumption

Amplifier Bearing Filter

Filter Guidance Reactor

Dead stop Position actuator

Force actuator

The form of interaction between the system elements defines the system structure. A system can be structured into subsystems (components).

The structuring process into subsystems has the advantage that we obtain a better overview of the behaviour of the system and that the subsystems can be constructed by different people. The structuring process should be carried out in such way that the subsystems can be reintegrated into the total system. This makes the clear definition of the interfaces inevitable, which also represents one of the major problems of industrial treatment of complex dynamical systems.

The system boundary (cladding plain) is arbitrary. But in general, limitations are introduced where a clear distinction to the environment exists, with a few uniquely defined relationships that can be observed and where the system is reactionless to its environment.

Definition 1.6:

The motion of dynamical systems is a change of the variables of the system with respect to time. Mechanical motion (changes referring to place) represents a special case of all general types of motions.

Furthermore, the term motion appears in literature a number of times, [7]:

System parameters are quantities which remain constant for the system during the observation period. Examples are given by natural constants, spring constants and damping factors, resistance, etc.

Environmental influences are quantities which affect the system from the outside, but which are not influenced by the system in return (reactionless). Examples are changes in temperature in big rooms, forced movements, etc.

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Initial values of the state quantities.

Rate of change of a state quantity refers to the rate at which the value of the state quantity changes in course of time. Examples are given by velocity, acceleration, rate of birth and death of a population, mass flows, pressure changes, etc. According to the first example (velocity), state variables can be both rate of change of state quantities and state quantities themselves. This becomes important in mechanics.

In mathematics, dynamical systems are described by differential equations, algebraic equations of n-th order and differential-algebraic equations (cp. Chapter 3).

1.2.2 System Classification

Definition 1.7:

A system is

static, if the conditions inside the system boundary (cladding plain) do not change in runtime. The model of a static system bases upon algebraic equations.

dynamic, if the present state of x t

( )

1 definitely depends on its initial state x t

( )

0 and the input quantity u t

( )

in the time slice

[ ]

t t0,1 , with t1>t0. Consequently, dynamical systems always comprise of storage elements like for e.g. energy, mass, information, etc. The characteristics of dynamical elements are therefore described by differential equations or differential-algebraic-equations.

a system with concentrated parameters, if system quantities such as masses, capacitors, springs, damper, etc. can be represented by the integral of particular constants; i.e. if the system quantities are not locally dispersed (e.g. spring-mass- pendulum). In dynamics, such systems are illustrated by common differential equations. If such a simplification is not permitted (e.g. if we have to take the local dispersion of the elements into consideration), the system is referred to as a system with distributed parameters. In this case system description takes place by means of partial differential equation (e.g. oscillating beam).

time invariant, if its parameters remain constant in runtime. A system is time variant when the parameters do change in course of time (e.g. rockets which change their masses due to fuel consumption or plastic parts whose modulus of elasticity constantly drops due to increasing temperature)

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causal, if its output signal at an arbitrary point in time only depends on the value of the input signal at that time and before it (referred to as principle of causality = principle of cause and effect). This lecture exclusively deals with causal systems.

deterministic, if the system’s behaviour can be predicted with hundred percent certainty using equations. If a system is stochastic, a system’s behaviour can only be predicted by means of probability calculation and statistics.

This lecture concentrates on the dynamic behaviour of (mainly technical) systems. In this context, the term dynamics must be defined in more detail. In historical context, the term dynamics (Greek: dynamis = force) originally derives from the study of changes in motions as a result of forces acting on the mechanical systems, and vice versa (inverse dynamics). In general, dynamics refers to the analysis and description of changes in time and, thereby, the phenomena which appear and how they proceed, under influence of general stimuli and incidents (e.g. forces, mass flows, voltages, death, birth, etc.). In fact, dynamics means motion, change.

1.3 System Models

A model is a simplified version of the complex reality which aims at analysing specific target functions. Models are usually not unique. Models are for e.g. the basis of system simulation.

There is no model which represents an exact copy of a real world system. Behaviour of the real system and its model always diverge. This divergence is called modelling mistakes (error).

1.3.1 Model Types

Different types of investigation (especially in case of different depths of investigation) require different types of models:

• object models: copy of an object on a smaller scale; e.g. reduced version of vehicle and aeroplane models for researches on wind tunnels,

• conceptual models;

• mathematical and physical models.

This lecture mainly concentrates on the mathematical and physical model types.

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Example 1.6:

Scale reduced models of vehicles, aeroplanes and ships allow us to conduct experiments on their dynamical characteristics in wind channels and water tunnels.

Here, similarity laws of fluid mechanics are used: Even if the fluid or the measurements do diverge, at a constant Reynolds number, surge flow similarity will always be observed.

• Analogous computer models, with electrical circuits which serve as reference systems.

This procedure is based on analogies between mechanical and electrical systems, e.g.

let the differential equations of single mass pendulums in mechanics be ( )

mx+dx+cx=F t

with deflection x, mass m, damper d, rigidity c and imposed force F t( ).

• If we equate the stored loads of a capacitor with the deflection x, the inductance L with the mass m, the resistance R with the damper d, the capacity C with the reciprocal spring stiffness 1 c and the adjacent/ fitting voltage U t

( )

with the imposed force F t( ). Subsequently, we obtain the equation:

1 ( )

Lq Rq q U t

+ +C =

 

• Hardware-in-the-Loop simulation (HIL) with mathematical models which are coupled with subsystems of the target system, e.g. electrical controllers. In fact, the physical component is not a model, but an essential part of the target system.

1.3.2 System Analysis

The analysis of a target system aims at identifying input and output characteristics of the system in order to understand the system and, where necessary, to develop methods to exert an influence on it.

Basically, there exists the opportunity to stimulate the system, which means to memorise the input quantities for the time elapsed and to measure the output quantities.

This procedure frequently proves to be difficult because

• in many cases the target system does not exist (innovation of new products) or does not exist anymore (reconstruction of casualties).

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• geometric admeasurements of the system are either too small or too large to perform measurements

• adequate sensors to measure the characteristics of the system are not available or are too expensive

• appropriate experiments take too long because the system is too languid

• the experiment with its target system is too dangerous, too expensive or not justifiable for other reasons (bio-mechanics, crash tests, threshold driving tests)

• the system is unique and should not be, is not allowed to be or cannot be stimulated (economic systems, biology, weather, …)

1.3.3 System Simulation

In many cases, the procedure described in paragraph 1.3.2 is not often exercised on the actual target system (on grounds which were mentioned above or for other practical reasons). Instead, it is applied to an appropriate model.

This procedure is also called system simulation, which de facto means that experiments are conducted on the basis of system models instead of real world systems. The following Fig.

1.7 shows a basic schema on which such an analysis can be regulated:

Fig. 1.7: Principle process of system simulation.

The process of simulation generally refers to the conduction of experiments on the basis of mathematical models of dynamical system instead of real world systems.

Simulations have the following advantages, i.e.

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• the understanding of the functions of the system can be deepened

• in some cases, system simulation is faster and cheaper than conducting and evaluating experiments

• simulations are reproducible and are normally also comparable

In order to conduct experiments on mathematical simulation models, first of all equations have to be formulated and solved. To set up an equation, fundamental physical laws from different disciplines are required, e.g.

• Euler’s Equations and Newton’s Law,

• the Law of Conservation of Energy,

• the Material Law,

• d’ Alembert’s Principle,

• the Maxwell Equations

• Navier-Stokes Equations

In Chapter 11 of this script this topic with respect to mechanical systems will be dealt with.

The solutions of the equation systems will be part of Chapters 4-6.

In order to develop a simulation system (which means: formulation and solving of equations), the model requires basic simplifications, e.g.:

• substitution of the elastic bodies with distributed masses by rigid bodies with concentrated mass quantities (mass and moment of inertia),

• replacement of non-linear material laws by linear ones,

• use of simple, linear kinetic relations instead of the actual non-linear ones,

• Negligence or highly simplified description of friction effects.

1.3.4 System Identification

(cp. Chapter 10)

A systematic, computerised method to approximately identify system parameters is the process of system identification (or parameter identification), Fig. 1.8.

An appropriate system model and available measurements of the input and output characteristics of the target system are prerequisites. In the following, we will identify a system model with a physical system model, which was constructed on the basis of physical laws. In situations involving complex issues (e.g. description of the flow ratio of air in the passenger compartment of a motor vehicle in order to interpret the climate control), we

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often employ other descriptions, e.g. neural networks which extensively abandon physical approaches.

On the basis of measurements the unknown parameters

[

p p1, 2, ,pn

]

T

=

p

are varied in such a way that we obtain the best possible conformity between the real system and the model. Adequate mathematical optimisation methods are used in order to identify the unknown parameters (cp. Chapter 10).

This method is particularly is used when parameters are difficult to measure, e.g.

• friction factors and material damping,

• heat transfer coefficients,

• flow resistance, etc.

Further physical quantities such as masses, spring stiffness, geometric data, material properties (elasticity modules, viscosity, etc.) are taken from tables, data sheets, CAD drawings and models.

In general, as many parameters as possible are measured directly and the rest will be identified by parameter identification.

Parameter identification can be applied with limited success. In most cases, it depends on the appropriate choice of the models and the availability and quality of the measurements.

Fig. 1.8: Process of parameter identification.

1.3.5 System Optimization (cp. Chapter 10)

System optimization aims at creating a plan of a new system with characteristics which are optimised in respect to fixed criteria (cp. Fig. 1.9).

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A mathematical model will be parameterised in such a way that the parameterised model definitely satisfies the criteria of the target behaviour. The target behaviour will be described by target solutions of given input and output signals.

Deviations between the actual behaviour of the model and the target behaviour will be determined by simulating with test signals. A performance function is defined as an admeasurement of the difference between the actual solution and the target solution, which is identified as target function too.

In fact, we can observe similarities between the process of system identification and the process of parameter identification. Both aims at defining the parameter set p which guarantees an approximate conformity of the actual characteristics yact with the measured value ymeas (parameter identification) and the target value ydes (system optimization).

The process of system optimization will be discussed in Chapter 10.

Fig. 1.9: Process of system optimization.

1.3.6 Model Schemes and Block Diagrams

Model schemes are detailed descriptions of systems which (e.g. in mechanics) are strongly oriented on representation. The way of representing the model plan depends on the correlation between the problem and the discipline of engineering. Because of the increasing importance of interdisciplinary applications (“mechatronics”), these schemes should, if possible, be substituted by universal diagrams (e.g. block diagrams). However, one faces considerable difficulties by doing so with complex systems (especially those deriving from the field of mechatronics).

Block diagrams are descriptive representations of model equations. This consequently means that they provide an allegorical representation of response relationships between two quantities (signals) of a system.

Following assignments are essential:

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• Signal ↔ arrow of action

• operation ↔ block

A block diagram consists of

• the blocks,

• directed arrows of actions (lines of actions) representing outgoing and arriving signals to the blocks. The arrowhead, directing to the particular direction of action, indicate whether it deals with input or output signals,

• the title of input and output quantities,

• summation points represented by little circles,

• signal branch points represented by little points.

Further details of the structure and the elements of block diagrams are standardized in DIN19226.

Fig. 1.10: Example of a block diagram (single mass pendulum).

Particularly in the fields of control theory, block diagrams are wide spread. Draft programmes and simulation programmes support the graphical direct input of block diagrams. System equations will be put together by the programme of the block structure automatically, Fig. 1.10.

Block diagrams can also be divided hierarchically.

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Example 1.7: Wheel Suspension Fig. 1.11

Fig. 1.11: Simple model of a wheel suspension.

Model Equations

Newton’s equation for the physical construction:

(

0

) ( )

A A

m y= −c y− −s ld y −sm g

( )

0 0

( )

1 .

A A A

A h t

m y dy cy cs ds cl m g y h m g dy cy cl

+ + = + + − ⇒ =m − − − +

    

By means of the equilibrium condition where s=0 we obtain:

0 0 0

0 A mA .

cy cl m g y l g

= − + − ⇒ = − c

Introducing a new model variable,

0 0 A ,

A

y y y y l m g

= − = − + c

we obtain the linear equation of motion,

( )

1

A A A

A

y h dy cy

= m − −

 

and out of it the block diagram in Fig. 1.12.

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Fig. 1.12: Block diagram for suspension, possible refinement levels.

Example 1.8: Electrical Circuits , Fig. 1.13

Fig. 1.13: Electrical low pass-filter.

Model Equations

Kirchhoff’s loop rule:

u ti

( )

= ⇒0 u t1

( )

Ri t

( )

u t2

( )

=0

Capacitor: 2

( ) ( ) ( )

0

1 1 t

u t q t i t dt

C C

= =

Connection between current and load: q t

( ) ( )

=i t .

As a result, we obtain:

( )

1

1 1

q t q u

RC R

+ =

 , 2

( )

u t 1 q

=C ,

thus, 2 2 1 2

(

1 2

)

1 1 1

.

Cu u u u u u

R R RC

+ = ⇒ = −

 

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Block Diagram

Fig. 1.14: Block diagram for electrical low-pass filter, possible refinement levels.

Signal Flow Diagrams are mainly preferred in English-speaking countries. They have a very simple structure and can be easily drawn. The disadvantage is that the representation of these diagrams is highly limited to linear correlations.

Elements of signal flow diagrams are

• connections with signed amplification factor

• coupling points represented by system variables

Fig. 1.15: Signal flow diagrams for a) suspension and b) low-pass filter.

Wroksheet 2: Mathematical Pendelum

1.4 Task of System Dynamics

Task 1: Modelling

Formulation of mathematical relationships which describe the system behaviour: Modelling is always connected with abstraction and idealisation.

Examples: mass point, rigid body, mass-free spring, “free of time lag” position actuator …

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Depending on the given task, the same system requires different modelling depths and therefore also different idealisations.

Example 1.9: Vehicle Model

Analysis of driving dynamics: Complex multibody system (degree of freedom, e.g. roll, pitch, yaw, x− − −,y ,z translation). The drive train and with it also the longitudinal dynamics are frequently neglected, (substituted by a default longitudinal vehicle motion)

Analysis of jerking: Rigid body (structure) with four wheels, suspension is neglected, but a detailed description of the drive train.

Vehicle simulation: Detailed model of the whole vehicle,

Crash simulation: Detailed FEM-model of the car body and the load-bearing components.

Task 2: Model Research

Research on system characteristics

Examples: stability, response time, functionality survey … Task 3: Design of Controlling Inputs

The inputs of the system must be designed in such a way that the desired target system characteristics are achieved.

Examples:

• In order to set weld points correctly, motor currents of welding robots have to be regulated

• In order to achieve a sufficient impulse velocity of the percussion piston, the drive piston of a hammer drill must be set in motion.

Task 4: Simulation of System Behaviour

Conducting experiments in reality is frequently not possible due to various reasons (e.g. new vehicle constructions (no prototypes), a high level of danger (crash tests, possible environmental damage), long-term duration (population development)),

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1.5 Summary

The systems discussed in modelling and simulation originally stem from different fields:

Technical areas:

• mechanics

• electrotechnology

• hydraulics

• pneumatics

• informatics

• etc.

Non-technical areas:

• social sciences

• business administration

• economics

• biology

• mathematics

• meteorology

• etc.

In system dynamics, the systems will be analysed with similar methods and standardized criteria.

Consequences:

• System dynamics is interdisciplinary.

• Similarities between systems from different disciplines are utilized in order to make use of uniformed analysis techniques.

• In order to represent uniformed systems from different disciplines, general description techniques are preferred.

• But: In course of time different disciplines (e.g. mechanics, electrotechnology) developed own description techniques (and software packages) which are irreconcilable to each other: Exemplifying a complex mechanical system which is difficult to be represented as a block diagram. Therefore, different forms of description on mathematical system level have to be retained in order to link these systems.

• We need description languages which are easy to learn and which are unique.

• Trend: Substitution of mathematical description forms by diagrams:

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o block diagrams, o program plan,

o Bond-graph, Rosenberg and Karnopp [8]

o Multiport and Cut Set Method o etc.

The knowledge of basic equations and solution formalism is fundamental, to guarantee a thorough understanding of the application possibilities and the limits of programmes for system analysis.

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2 Mathematical Description of Dynamical Systems

2.1 Differential Equations

The time-dependent behaviour of a dynamical system can be described by differential equations (DE) or, in more general cases, by differential-algebraic equations (DAE). Our approach is limited to such systems which can only be analysed by differential equations.

Systems, which are only describable by differential-algebraic equations, will be dealt with in Section 3.1 in more detail.

Definition 2.1:

An ordinary differential equation is a relation which contains a function of a conditional equation with only one independent variable, and one or more of its derivatives with respect to that variable.

Systems with concentrated parameters (e.g. multi-body-systems) are described by ODEs.

Example 2.1: DE (motion equation) of a Mathematical Pendulum, Fig. 2.1

Fig. 2.1: Mathematical pendulum.

From the principle of conservation of angular momentum with mounting point 0 we first obtain by substitution:

1 , 2 ,

xx =ϕ

2 sin

ml ϕ= −mgl ϕ (2.1)

2 1

1

2 glsin

x x

x x

 

=    = −  x

  (2.2)

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The expressions (2.1) and (2.2) are also called minimal representations of the equations of motion, or state equation (s. passage 2.2) of this system.

Definition 2.2:

A partial differential equation involves a relation which contains a function (of a conditional equation) with several independent variables and its partial derivatives with respect to those variables.

Systems with (locally) distributed parameters (e.g. continuous mechanical systems) are described by partial DE.

Example 2.2: Longitudinal Oscillations of a Continuous Beam, Fig. 2.2

Fig. 2.2: Longitudinal beam oscillations.

Because the relevant physical characteristics of the beam (rigidity, mass and, if necessary, variable cross section) are distributed over the whole beam length, a partial differential equation is necessary to describe the dynamics of the system.

The equation to describe the longitudinal waves of a slim beam is

( )

, 2u2

( )

,

Au x t EA x t

ρ ∂x

= ∂



With

ρ: density

E: elasticity modulus A: beam cross section L : beam length MAL : total mass

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Next to the derivative of the spatiotemporal deflection u x t

( )

, with respect to time, derivatives with respect to the position coordinate x also appear. The exact solution of such equations would go beyond the scope of this lecture and will therefore not be discussed for the time being.

Worksheet 3: Longitudinal oscillations of a continuous beam Annotations 2.1:

We will approach a system with distributed parameters with an appropriate spacious discretisation of a system with concentrated parameters (Fig. 2.3). In this case, instead of a partial differential equation, we obtain a system with ODEs.

Fig. 2.3: Discrete model of the description of longitudinal oscillations.

1 1 2

2 1 2 3

3 2 3 4

4 3 4 5

5 4 5 6

6 5 6

3

2

2

2

2

mu cu cu

mu cu cu cu

mu cu cu cu

mu cu cu cu

mu cu cu cu

mu cu cu

= − +

= − +

= − +

= − +

= − +

= −













with

6

m= ρAL and

/ 6 c AE

= L . In matrix form we obtain:

1 1

2 2

3 3

2

4 4

5 5

6 6

3 1 0 0 0 0

1 2 1 0 0 0

0 1 2 1 0 0

36

0 0 1 2 1 0

0 0 0 1 2 1

0 0 0 0 1 1

u u

u u

u E u

u L u

u u

u u

ρ

  −  

   −  

    

   −  

=  =  −   =

   −  

   −  

   

u Au





 







The Matrix A is called the stiffness matrix of a discrete system.

(35)

This lecture mainly deals with systems with concentrated parameters. In this case, the dynamical system can be described by ODEs:

( ) ( )

( )

( )

( )

Implicit form: F t y ti , , 'y t ,,yn t =0, (2.3) or resolved in respect to y( )n

( )

t ,

( )

( ) ( ( ) ( )

( 1)

( ) )

Explicit form: yn t =Fe t y t, , 'y t ,,yn t =0 (2.4) Annotation 2.2:

• In some cases the transition from implicit to explicit equation form is not possible analytically.

• In practice, there exist some systems which can only be described by intercoupled differential and algebraic equations. Examples are given by multibody systems with kinematical loops.

• The highest number of derivatives a differential equation contains is called its order.

• To solve a differential equation of n-th order, we have to determine all the continuously differentiable functions which together with their derivatives satisfy the DE.

• In system dynamics, the unknown function y depends on time. The derivative of y is therefore, in the following, marked by a dot over the variable (y) instead of an inverted comma (y').

Example 2.3: Spring-Mass Oscillator with Viscous Damping, Fig. 2.4

(

, ,

) ( )

Implicit equation: Fi t y y y , =my+dy+cyP t =0 (2.5)

( ) ( ) ( ( ) )

1

Explicit equati n: , 1

o y t Fe t y y, dy cy P t m

 

= = − + − 

 

   (2.6)

(36)

Fig. 2.4: Simple frequency response system with concentrated parameters (multybody system).

2.2 State Equations

By introducing the state variables x1,,xn and the substitutions

( 1)

1 , 2 ', n n

x = y x = y x = y (2.7)

equation (2.4) can be transformed to n DE of first order.

Example 2.4: One Dimensional Oscillator

The motion equation of a one-dimensional oscillator with the undamped eigenfrequency v0

,the damping δ and the external excitation h t

( )

is as follows:

( )

2

2 0

y+ δyy=h t

  By substitution

x1= y, x2 = y,

we obtain a DE-system of first order:

1 2

x =x ,

( )

2

2 2 2 0 1

x =h t − δx −ν x

(37)

It is useful to generally use this transformation in order to carry further research methods on DE systems of first order.

Definition 2.3

The general state equation for a finite dimensional, non-linear and time continuous dynamical system with n state variables and m output quantities is:

State equation

( ) ( )

(

, ,

)

d t t t

= dtx =

xf x u (2.8)

Output equation

( ) ( )

( )

( )t = t , t t,

y g x u (2.9)

Where :n×1

x – state vector :m×1

y – output vector :n×1

u – control vector , :

f g non-linear n×1 - respectively m×1 - vector function

Fig. 2.5: Block diagram of the non-linear state equation.

(38)

Example 2.5: Point Mass

Principle of linear momentum: my= F Substitution: x1 = y, x2 = y

State equation: 1 2 1

2 2

, ,

F m

x

x F x

x u m x

 

 =  = = 

   

    x  

Example 2.6: Spring-Mass Oscillator with Viscose Damping, Fig. 2.4 Substitution: x1 = y, x2 = y, x1= y, x2 =y

State equation: 12 m1

(

1 2 2

( ) )

x x

cx dx P t x

 

=    = − + −  x

 

Here, the force excitation P t

( )

represents the control factor u t

( )

. Example 2.7: Mathematical Pendulum, Fig. 2.1

State quantities:

x1=ϕ (position) x2 =ϕ (velocity) State equation:

1 2

x =x

2 gsin 1

x x

= − l

Introduction 2.8: Logistical Growth at Constant Crop, [7]

State quantity: x (crop, e.g. fishes)

Control factor: u= =h const (removal, e.g. fishing)

State equation: 1 x ,

x a x h k

k

 

=  −  −

 : capacity limit

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Fig. 2.6: Block diagram: Logistical growth at constant crop.

Annotation 2.3:

• The non linearity of the state equation is predetermined by the structure of the dynamical system, e.g. the powers of the state variables or non-linear characteristic lines.

• Analytical solutions of non-linear state equations are not possible for most parts. A numerical approach is normally necessary (simulation).

Worksheet 4: Predator-Prey-Models

2.3 Stationary Solutions and Equilibrium Positions

Definition 2.4:

The solution x0

( )

t of a dynamical system is called a stationary solution, if for the excitation

( )

0 t

u it is found that

( ) ( )

(

0 t , 0 t

)

= ,

f x u 0 (2.10)

i.e.x0

( )

t0.

Where u0

( )

t0, x0

( )

t represents an equilibrium position of the free (non excited, uncontrolled) system.

Annotation 2.4:

In general, the equation (2.10) is non-linear. The solution mostly results from numerical approaches, s. Chapter 9.

(40)

Example 2.9: Mass Point

( )

2 2 0

F m

, , =  x = ⇒x =

f x u t   0 and F =0

i.e. an equilibrium position is achieved when the excitation force vanishes and the point originally rests in equilibrium.

Example 2.10: Spring-Mass Oscillator with Viscous Damping

(

, ,

)

=m1

(

cx1+dx2x2mgP t

( ) )

=

f x u t 0, where P t

( )

=0

2 0 1 mg

x x

⇒ = ⇒ = c The quantity mg

c is the tendency by which the mass drops as a result of only the gravitational force.

Example 2.11: Mathematical Pendulum

( )

2

1

, , glsin x

x

 

=− =

f x u t 0

x2 =0, sinx1=0, i.e. x1=0, 2 ,π 

This example shows that the equilibrium positions have different characteristics, if for e.g.

the equilibrium positions x1 =0 and x1=π are compared to each other.

Example 2.12: Logistical Growth at Constant Crop The equilibrium positions are defined as follows:

(

, ,

)

1 x 0 a 2 0

f x u t a x h ax x h

k k

 

=  −  − = ⇒ − − =

 

2 1,2

1 4

2

x k k kh

a

 

⇒ =  ± − 

 

Especially when h=0 we obtain:

(41)

1 0, 2

x = x =k.

2.4 Linear State Equations

In reality, we are more often than not confronted with non-linear systems. In non trivial cases such systems mostly imply considerably difficulties. This is why one is interested in investing high efforts to approximate non-linear relations by linear ones and to linearize such equations. Then we obtain linear approximations which reflect the behaviour of such systems, except for certain errors. But these approximations are significantly easier, i.e. in the first place manageable and resolvable. Generally, we observe that linear approximation relations are valid only to a certain extend, and that the error grows with an increasing degree as much as the scope of validity is transgressed.

The principal of linearization can easily be described by a sufficiently smooth function of a variable f x

( )

.

Fig. 2.7: Linearisation of a scalar function of a variable.

This function can be approximated by a straight line in the environment of the point. The environment of the point shall be analysed with respect to its system behaviour.

( )

0 0 .

y= y +k xx (2.11)

By the use of coordinates transformation

0, 0.

x x x y y y

∆ = − ∆ = − (2.12)

we finally obtain the homogenous and linear relation

(42)

( )

x x0

y f x x k x

x =

 

∆ = ∂  ∆ = ∆ (2.13)

A general method to linearize functions (more than one variable) is the Taylor series expansion (will be discussed later in more detail).

Example 2.13: Sinus Function

The function f

( )

α =sin

( )

α should be linearized around α α= 0 where α α− 0 01. We proceed as described above and obtain

( )

( ) ( )

( )

0

0 0

0 0

' cos

sin sin sin

f

f

α α

α α

α α α α α α

α =

=

∂ 

= ≈ + ∂  −

((((

After transformation ∆ = −α α α: 0, we obtain a linear relation with the constant

proportionality factor cosα0: .

( )

0

ˆ cos

f f α α α

∆ = ∆ = ∆ .

Example 2.14: Mathematical Pendulum

For small angle ϕ we obtain a linear pendulum equation:

g ϕ= − l ϕ

In order to specify the term “linear” we use the following Definition 2.5:

A function f x

( )

is linear, if the following characteristics are applicable:

(

1 2

) ( )

1

( )

2

Additivity: f x +x = f x + f x (2.14)

( ) ( )

Homogeneous: f λxf x , λ∈R (2.15)

This definition is also applicable, if x1 and x2are vectors and f is a vector function.

Description of additive characteristics in a block diagram:

(43)

Fig. 2.8: Block diagram of additivity of a linear relation.

Example 2.15: Non-linear Functions

( )

f x =ax+b

( )

2

f x =x

( )

sign

( )

f x =a x

An example of a piecewise linear but on the whole a non-linear relation is illustrated in Fig.

2.9.

Fig. 2.9: Non-linear force characteristic (in combination with play).

The linearization of the state equations can be accomplished in the proximity of any time- dependent target processes x0 and u0

( )

t (not necessarily constant!). Depending on the problem, it however makes sense to linearize in the proximity of reference processes which are in equilibrium positions i.e. steady solutions of a system.

= 0 + x x Δx,

= 0+

u u Δu with Δx 0 a0 and Δu 0 b0 ,

Ábra

Fig. 1.2: ABS- magnetic valves, interfaces.
Fig. 1.3: Above: Percussion hammer [2]; Below: Simulation model of a percussion hammer
Fig. 1.4: Representation of a system as a block.
Fig. 1.6: Degrees of freedom of the spatial twin-track model [6].
+7

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