Me 2401 Mechatronics

ME 2401 MECHATRONICS UNIT III SYSTEM MODELS AND CONTROLLERS Building blocks of Mechanical, Electrical, Fluid and Thermal Systems, Rotational – Transnational Systems , Electromechanical Systems – Hydraulic – Mechanical Systems. Continuous and discrete process Controllers – Control Mode – Two – Step mode –Proportional Mode – Derivative Mode – Integral Mode – PID Controllers – Digital Controllers Velocity Control – Adaptive Control – Digital Logic Control – Micro Processors Control.

System Models Mathematical Models Mechanical System Building Blocks Electrical System Building Blocks Fluid System Building Blocks Thermal Systems Building Blocks

Mathematical Models • Think how systems behave with time when subject to some disturbances.

• In order to understand the behaviour of systems, mathematical models are required. • Mathematical models are equations which describe the relationship between the input and output of a system. • The basis for any mathematical model is provided by the fundamental physical laws that govern the behaviour of the system.

Building Blocks • Systems can be made up from a range of building blocks. • Each building block is considered to have a single property or function. • Example: an electric circuit system which is made up from blocks which represent the behaviour of resistance, capacitance, and inductor, respectively. • By combining these building blocks a variety of electrical circuit systems can be built up and the overall input-output relationship can be obtained. • A system built in this way is called a lumped parameter system.

BUILDING BLOCKS - MECHANICAL SYSTEM • Basic building block: spring, dashpots, and masses. • Springs represent the stiffness of a system • Dashpots represent the forces opposing motion, for example frictional or damping effects. • Masses represent the inertia or resistance to acceleration. • Mechanical systems does not have to be really made up of springs, dashpots, and masses but have the properties of stiffness, damping, and inertia. • All these building blocks may be considered to have a force as an input and displacement as an output.

Stiffness of a Spring • Stiffness of a spring is described as the relationship between the force F used to extend or compress a spring and the resulting extension or compression x. • In the case of spring where the extension or compression is proportional to the force (linear spring): F = kx, where k is a constant, the bigger the value of k the greater the forces have to be to stretch or compress the spring and so the greater the stiffness. F

x Spring

Translational Spring, k (N) Appied force Fa (t ) in Newton Linear velocity v(t ) (m/sec) Linear position x(t ) (m) Fa (t )  k s x(t ) 1 x(t )  Fa (t ) ks dx(t ) 1 dFa (t ) v(t )   dt k s dt t

Fa (t )  k s  v(t )dt t0

x(t)

Fa(t)

Rotational Spring, ks (N-m-sec/rad) Appied torque Ta (t ) (N - m) Angular velocity  (t ) (rad/sec) Angular displaceme nt  (t ) (rad) Ta (t )  Bm (t ) 1  (t )  Ta (t ) ks d (t ) 1 dTa (t )  (t )   dt k s dt t

Ta (t )  k s   (t )dt t0

 (t)  (t)

Fa(t)

ks

Dashpot • The dashpot block represents the types of forces experienced when pushing an object through a fluid or move an object against frictional forces. The faster the object is pushed the greater becomes the opposing forces. • The dashpot which represents these damping forces that slow down moving objects consists of a piston moving in a closed cylinder. • Movement of the piston requires the fluid on one side of the piston to flow through or past the piston. This flow produces a resistive force. The damping or resistive force is proportional to the velocity v of the piston: F = cv or F = c dv/dt.

Translational Damper, Bv (N-sec) Appied force Fa (t ) in Newton Linear velocity v(t ) (m/sec) Linear position x (t ) (m) x(t)

Fa (t )  Bm v (t ) 1 v (t )  Fa (t ) Bm dx(t ) Fa (t )  Bm v (t )  Bm dt 1 t x (t )  Fa (t )dt  Bv t0

Fa(t)

Bm

Rotational Damper, Bm (N-m-sec/rad) Appied torque Ta (t ) (N - m) Angular velocity  (t ) (rad/sec) Angular displaceme nt  (t ) (rad)

 (t)  (t)

Ta (t )  Bm (t ) 1  (t )  Ta (t ) Bm d (t ) Ta (t )  Bm (t )  Bm dt 1 t  (t )  Ta (t )dt  Bm t 0

Fa(t)

Bm

Mass • The mass exhibits the property that the bigger the mass the greater the force required to give it a specific acceleration. • The relationship between the force F and acceleration a is Newton’s second law as shown below. • Energy is needed to stretch the spring, accelerate the mass and move the piston in the dashpot. In the case of spring and mass we can get the energy back but with the dashpot we cannot. Force

Acceleration Mass

dv d 2x F  ma  m m 2 dt dt

Mechanical Building Blocks Building Block

Spring Dashpot

Mass Spring

Damper Moment of inertia

Equation

Translational F = kx F = c dx/dt F = m d2x/dt2 Rotational T = k T = c d/dt T = J d2/dt2

Energy representation

E = 0.5 F2/k P = cv2 E = 0.5 mv2 E = 0.5 T2/k P = c2 P = 0.5 J2

Building Mechanical Blocks Output, displacement

• Mathematical model of a machine mounted on the ground

Mass

d 2x

dx m  c  kx  F 2 dt dt

Ground

Input, force

Building Mechanical Blocks Moment of inertia

Torque

• Mathematical model of a rotating a mass

d 

d J c  k  T dt dt 2 2

Torsional resistance

Shaft Block model

Physical situation

BUILDING BLOCKS - ELECTRICAL SYSTEM • From Newton’s law or using Lagrange equations of motions, the second-order differential equations of translationaldynamics and torsional-dynamics are found as m j

d 2x dt 2 d 2 dt 2

 Bv

dx  k s x  Fa (t ) (Translational dynamics) dt

 Bm

d  k s  Ta (t ) (Torsionaldynamics) dt

Electrical System Building Blocks • The basic building blocks of electrical systems are resistance, inductance and capacitance.

2

Resistor : v  iR; P  i R 1 1 2 Inductor: i   vdt; E  Li L 2 dv 1 2 Capacitor : i  C ; E  Cv dt 2

Resistance, R (ohm) Appied voltage v(t ) Current i (t ) v(t )  Ri(t ) 1 i (t )  v(t ) R

i(t)

v(t)

R

Inductance, L (H)

Appied voltage v(t ) Current i (t ) di(t ) v(t )  L dt t 1 i (t )   v(t )dt L t0

i(t)

v(t)

L

Capacitance, C (F) Appied voltage v(t ) Current i (t ) 1 t v(t )   i (t )dt C t0 dv(t ) i (t )  C dt

i(t)

v(t)

C

For a series RLC circuit, find the characteristic equation and define the analytical relationships between the characteristic roots and circuitry parameters.

d 2i

R di 1 1 dva   i 2 L dt LC L dt dt R 1 2 s  s 0 L LC The characteristic roots are 2

R 1  R  s1       2L LC  2L  2

R 1  R  s2       2L LC  2L 

BUILDING BLOCKS – FLUID SYSTEM •

The basic building blocks of fluid systems are the volumetric rate of flow q and the pressure difference. Input

Output

Pressure difference

Volumetric rate of flow

Fluid system can be divided into two types: hydraulic and pneumatic. Hydraulic resistance is the resistance to flow of liquid as the liquid flow through valves or changes in pipe diameter takes place.

p1  p2  Rq p1 - p2 is pressure difference R is the hydraulic resistance q is the volumetric rate of flow

Hydraulic capacitance is the term used to describe energy storage with a liquid where it is stored in the form of potential energy. A height of liquid in a container is one form of such a storage. For such capacitance, the rate of change of volume V in the container (dV / dt) is equal to the difference between the volumetric rate at which liquid enters the container q1 and the rate at which it leaves q2.

dV q1  q2  ; V  Ah dt dh q1  q2  A dt A dp q1  q2  pg dt ( p is liquid density; g is the acceleration due to gravity) A dp C ; q1  q2  C pg dt

Hydraulic inertance is the equivalent of inductance in electrical systems or a spring in mechanical systems. To accelerate a fluid and so increase its velocity a force is required.

F1=p1A

Mass m

F1  F2  p1 A  p2 A  ( p1  p2 ) A ( p1  p2 ) A  ma ( p1  p2 ) A  m

dv dv dq  ALp  Lp dt dt dt

dq Lg ; I is the hydraulic inertance  dt A L is the length of the block and g is the density p1  p2  I

F2=p2A

L

With pneumatic systems the three basic buildings blocks are as with hydraulic systems, resistance, capacitance, and inertance. However, gasses differ from liquids in being compressible.

dm p1  p2 Resistance  dt R dm d ( p1  p2 ) Capacitance C dt dt dm 1 Inertance   ( p1  p2 )dt dt L

A fluid system p is the liquid density g is the acceleration due to gravity q1

q is the volumetric rate of flow dp q1  q2  C (Capacitor) dt p1  p2  Rq2 (Resistance)

h

q2

hpg p1-p2  hpg; q2  R hpg d (hpg ) dh pgh q1  C A  R dt dt R

BUILDING BLOCKS - THERMAL SYSTEM • There are only two basic building blocks for thermal systems: resistance and capacitance. • There is a net flow of heat between two points if there is a temperature difference between them. • The value of the resistance depends on the mode of heat transfer. T2  T1 T T  Ak 2 1 R L A : Cross sectional area of the material through which the heat is being conducted L is the length of material between the points at which the temperature are T1 and T2 . q

k is the thermalconductivity

Thermal System q is the net rate of heat flow C is the capacitance R is the thermal resistance

T TL

q

TL  T q R dT dT q1  q2  C ;q  C dt dt dT TL  T C  dt R dT RC  T  TL dt

Rotational Systems • The mass, spring, and dashpot are the basic building blocks for mechanical systems where forces and straight line displacements are involved without any rotation. • If rotation is involved, then the equivalent three building blocks are a torsional spring, a rotary damper and the moment of inertia (i.e. the inertia of a rotating mass). • With a torsional spring the angle  rotated is proportional to the torque: T = k. • With a rotary damper a disc is rotated in a fluid and the resistive torque T is proportional to the angular velocity . • The moment of inertia block exhibit the property that the greater the moment of inertia J the greater the torque needed to produce an angular acceleration

d T  c  c ; T  Ja dt

TRANSNATIONAL SYSTEM

ELECTROMECHANICAL SYSTEM

HYDRAULIC SYSTEM

MECHANICAL SYSTEM

CONTINUOUS PROCESS CONTROLLERS

DISCRETE PROCESS CONTROLLERS

CONTROL MODE

TWO STEP MODE

PROPORTIONAL MODE

DERIVATIVE MODE

INTEGRAL MODE

PID CONTROLLERS

DIGITAL CONTROLLERS

VELOCITY CONTROL

ADAPTIVE CONTROL

DIGITAL LOGIC CONTROL

MICRO PROCESSORS CONTROL.