ch8_1 BasicMechanicalSystemModel (1).pdf

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Slide 10.1

Bolton, Mechatronics PowerPoints , 4th Edition, © Pearson Education Limited 2008

Basic system Models• Objectives:

• Devise Models from basic buildingblocks of mechanical, electrical, fluid

and thermal systems• Recognize analogies between

mechanical, electrical, fluid and thermal

systems



Slide 10.2


• Mathematical Models

• Mechanical system building blocks – Rotational systems

– Building up a mechanical system

Electrical system building blocks- Building up a model for electrical systems

- Electrical and mechanical analogyies

Fluid system building blocksThermal system building blocks

Basic system Models



Slide 10.3


Mathematical Models

• In order to understand the behavior ofsystems, mathematical models are needed.

Such a model is created using equations andcan be used to enable predictions to bemade of the behavior of a system underspecific conditions.

• The basics for any mathematical model isprovided by the fundamental physical lawsthat govern the behavior of the system.

• This chapter deals with basic buildingblocks and how to combine such blocks tobuild a mathematical system model.



Slide 10.4


Figure 10.1 Mechanical systems: (a) spring, (b) dashpot, (c) mass

Mechanical system building blocksThe models used to represent mechanical systems have the basic building blocks of:

Springs: represent the stiffness of a system

Dashpots: dashpots are the forces opposing motion, i.e. friction or damping

Masses: the inertia or resistance to acceleration

All these building blocks can be considered to have a force as an inputand a displacement as an output



Slide 10.5


• The stiffness of aspring is described by:

F=k.x

The object applying theforce to stretch the

spring is also acted onby a force (Newton’sthird law), this force

will be in the oppositedirection and equal insize to the force usedto stretch the spring

Mech. sys blocks: Spring

k is thestiffnessconstant



Slide 10.6


c : speed of the body

It is a type of forces when wepush an object through a fluid

or move an object againstfriction forces.

Thus the relation between thedisplacement x of the piston,i.e. the output and the forceas input is a relationshipdepending on the rate ofchange of the output

Mech. sys blocks: Dashpots



Slide 10.7


• F=ma

m: mass, a: acceleration

Mech. sys blocks: Masses



Slide 10.8


Energy in basic mechanical blocks• The spring when stretched stores energy, the

energy being released when the spring springsback to its original length.

The energy stored when there is an extensionx is:E= kx2 /2=

Energy stored in the mass when its moving with avelocity v, its called kinetic energy, and releasedwhen it stops moving:

E=mv2 /2

No stored energy in dashpot, it dissipates

energy=cv2



Slide 10.9


Basic Blocks or Rotational System• For rotational system, the equivalent three building blocks are:

a Torsion spring, a rotary damper, and the moment of inertia

With such building blocks, the inputs are torque and the

outputs angle rotated

With a torsional spring

With a rotary damper a disc is rotated in a fluid and

the resistive torque T is:

The moment of inertia has the property that the greaterthe moment of inertia I, the greater the torque neededto produce an angular acceleration



Slide 10.10


• The stored energy in rotary system:

• For torsional spring:

• Energy stored in mass rotating is :

• The power dissipated by rotary damper

when rotating with angular velocity ωωωω is:

Energy in rotary system



Slide 10.11

Bolton, Mechatronics PowerPoints , 4th Edition, © Pearson Education Limited 2008Table 10.1 Mechanical building blocks

Summary of Mechanical building blocks



Slide 10.12


Figure 10.2 (a) Spring–dashpot–mass, (b) system, (c) free-body diagram

Building up a mechanical systemMany systems can be considered to be a mass, a springand dashpot combined in the way shown below



Slide 10.13


Building up a mechanical system• The net forced applied to

the mass m is F-kx-cv

V: is the velocity with whichthe piston (mass) ismoving

The net fore is the force

applied to the mass to

cause it to accelerate thus:

net force applied to mass

=ma =

F kxdt

dxcdt

xd mor

dt

xd m

dt

dxckxF

=++

=−−

2

2

2

2

2nd order differential equationdescribes the relationshipbetween the input of force F tothe system and the output ofdisplacement x



Slide 10.14


Figure 10.3 Model for (a) a machine mounted on the ground, (b) the chassis of a car

as a result of a wheel moving along a road, (c) the driver of a car as it is drivenalong a road

Example of mechanical systems

The model in b can beused for the study of thebehavior that could be

expected of the vehiclewhen driven over a roughroad and hence as a basisfor the design of thevehicle suspension model

The model in C can beused as a part of a largermodel to predict how the

driver might feel whendriving along a road



Slide 10.15


Figure 10.4 Example

Analysis of mechanical systems

The analysis of such systems is carried out by drawing afree-body diagram for each mass in the system, thereafterthe system equations can be derived



Slide 10.16


• Procedure to obtain the differential

equation relating the inputs to the outputsfor a mechanical system consisting of anumber of components can be written as

follows



Slide 10.17


Figure 10.5 Mass–spring system

Example: derive the differential

equations for the system in Figure

Consider the free body diagram

For the mass m2 we can write

For the free body diagram ofmass m1 we can write



Slide 10.18


Figure 10.6 Rotating a mass on the end of a shaft: (a) physical situation,(b) building block model

Rotary system analysisThe same analysis procedures can also be applied to rotary system,

so just one rotational mass block and just the torque acting on thebody are considered

Spring



Slide 10.19


Figure 10.7 Electrical building blocks

Electrical system building blocks



Slide 10.20


Table 10.2 Electrical building blocks



Slide 10.21


Figure 10.8 Resistor–capacitor system



Slide 10.22


Figure 10.9 Resistor–inductor–capacitor system



Slide 10.23


Figure 10.10 Resistor–inductor system

Slid 10 24



Slide 10.24


Electrical System Model

Resistor–capacitor–inductor system

Slide 10 25



Slide 10.25

Bolton, Mechatronics PowerPoints , 4th Edition, © Pearson Education Limited 2008Figure 10.12 Analogous systems

Electrical and Mechanical Analogy

F I

Velocity Volt

C dashpot 1/R

Spring inductor

Mass capacitor

Slide 10 26



Slide 10.26


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