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SYSTEM MODELS
◦ In order to understand the behaviour of systems, mathematical models are needed.
◦ Eg1:A microprocessor switches on a motor. How will the rotation of the motor shaft vary with
time?
◦ Eg2: A hydraulic system is used to open a valve which allows water into a tank. How will the
water level vary with time?
◦ These are simplified representations of certain aspects of a real system.
◦ Such a model is created using equations to describe the relationship between the input and
output of a system.
◦ Note: Real systems often exhibit non-linear characteristics and can depart from the ideal models
developed (Eg: F=kx).
TYPES OF SYSTEMS
◦MECHANICAL SYSTEMS
◦ELECTRICAL SYSTEMS
◦THERMAL SYSTEMS
◦FLUID SYSTEMS
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Mechanical system building blocks
◦ The basic building blocks of the models used to represent mechanical systems are
◦ 1) Springs
◦ Springs represent the stiffness of the system.
◦ 2) Dashpots
◦ Dashpots represent the forces opposing motion, i.e. frictional or damping effects
◦ 3) Masses
◦ Masses represent the inertia or resistance to acceleration
Spring
◦ Springs represents the stiffness of the system.
◦ In case of spring the extension (or) compression is proportional to the
applied forces.
F = k.x
◦ F – Applied force x – extension k – a constant
◦ The spring when stretched stores energy, the energy being released when
the spring back to its original length. The energy stored,
1 F2
◦ E= kx 2 =
2 2k
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Dashpot
◦ Dashpots building blocks represent the types of forces experienced
when we push the object through a fluid or move an object against
frictional forces.
◦ In ideal case damping or resisting force F is proportional to the
velocity of the piston.
F=cv
v – Velocity of piston c – a constant
𝒅𝒙
𝑭=𝒄
𝒅𝒕
◦ The faster the object is pushed, the greater the opposing forces become.
Mass
◦ Masses represent the inertia or resistance to acceleration.
◦ According to Newton’s II law F = ma
◦ There is also energy stored in the mass when it is moving with a velocity v, the
energy being referred to as kinetic energy, and released when it stops moving:
𝟏
𝑬 = 𝒎𝒗𝟐
𝟐
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◦ Energy is needed to stretch the spring, accelerate the mass and move the piston in the dashpot.
◦ The spring when stretched stores energy, the energy being released when the spring springs back to its
original length.
1 F2
◦ E= kx 2 =
2 2k
◦ There is also energy stored in the mass when it is moving with a velocity v, the energy being referred to
as kinetic energy, and released when it stops moving.
𝟏
◦ 𝑬 = 𝒎𝒗𝟐
𝟐
◦ However, there is no energy stored in the dashpot. It does not return to its original position when there
is no force input. The dashpot dissipates energy rather than storing it, the power P dissipated depending
on the velocity v
◦ 𝑃 = 𝑐𝑣
Rotational Systems
◦ If there is rotation then the equivalent three building blocks are
◦ 1. Torsional spring
◦ 2.Rotary damper
◦ 3.The moment of inertia i.e. the inertia of a rotating mass.
◦ With such building blocks the inputs are torque and the outputs angle rotated.
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◦ With a torsional spring the angle θ rotated is proportional to the toque T.
◦ With the rotary damper a disc is rotated in a fluid and the resistive toque T is proportional to the
angular velocity ω
◦ The moment of inertia building block exhibits the property that the greater the moment of inertia,
the greater the torque needed to produce an angular acceleration α.
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Comparison with translational and
rotational elements
Building up a mechanical system
◦ A spring mass damper system is shown in fig.
◦ The system is fixed at one end and the mass is supported by a spring and damper. The mass is
excited by force and free to oscillate. The equation of motion related to horizontal motion x of
mass to applied force can be developed with of a free body diagram.
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◦ If we consider just the forces acting on that block then the net force applied to the mass is the applied
force F minus the force resulting from the stretching or compressing of the spring and minus the force
from the damper.
This net force is the force applied to the mass to cause it to accelerate.
◦ This equation, called a differential equation, describes the relationship between the input of force F to
the system and the output of displacement x.
Derivation of DE
◦ The net force applied to the mass is F minus the resisting forces exerted by each of
the springs.
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Derivation of DE
◦ For mass m2 these are the force F and the force exerted by the upper spring.
◦ For the free-body diagram for mass m1, the force exerted by the upper spring is
k2(x3-x2) and that by the lower spring is k1(x2-x1).
Procedure for obtaining the differential
equation
◦ The procedure for obtaining the differential equation relating the inputs and outputs for a
mechanical system consisting of a number of components can be summarised as:
◦ 1. isolate the various components in the system and draw free-body diagrams for each;
◦ 2 hence, with the forces identified for a component, write the modelling equation for it;
◦ 3 combine the equations for the various system components to obtain the system differential
equation.
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Derivation of DE (Rotational motion)
◦ A system involving a torque being used to rotate a mass on the end of a shaft can be considered to be
represented by the rotational building blocks.
MATHEMATICAL MODEL FOR A MACHINE MOUNTED ON THE GROUND
◦ This model can be used as a basis for studying the effects of ground
disturbances on the displacements of a machine bed.
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A model for the wheel and its suspension for a car
A larger model to predict how the driver might feel when driven
along a road.
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Electrical system building blocks
◦ The basic building blocks of electrical building blocks are
◦ 1.Inductors
◦ 2.Capacitors
◦ 3.Resistors.
Inductors
◦ It consists of a coil wire. When current flows through the wire, a magnetic
field surrounding the wire is produced. Any attempt to change the density of
this magnetic field leads to the induction of voltage.
◦ For an inductor the potential difference v across it at any instant depends on the
rate of change of current di/dt through it:
◦ where L is the inductance.
◦ The direction of the potential difference is in the opposite direction to the
potential difference used to drive the current through the inductor, hence the
termed as back e.m.f.
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Capacitors
◦ Capacitors are used to store charge . A capacitor consists of two
parallel plates separated by insulating material.
◦ For a capacitor, the potential difference across it depends on the
charge q on the capacitor plates at the instant concerned:
where C is the capacitance. Since the current i to or from the capacitor
is the rate at which charge moves to or from the capacitor plates, i.e.
i= dq/dt
Resistors
◦ An electric resistor opposes the flow of current.
◦ For a resistor, the potential difference v across it at any instant depends on the
current i through it
◦ where R is the resistance.
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◦ The energy stored by an inductor when there is a current i is
◦ The energy stored by a capacitor when there is a potential difference v across it is
◦ The power P dissipated by a resistor when there is a potential difference v across it is
◦ NB: Both the inductor and capacitor store energy which can then be released at a later time. A resistor
does not store energy but just dissipates it.
Electrical building blocks
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Building up a model for an electrical system
◦ The equations describing how the electrical building blocks can be combined are Kirchhoff ’s laws:
◦ Law 1:The total current flowing towards a junction is equal to the total current flowing from that
junction, i.e. the algebraic sum of the currents at the junction is zero.
◦ Law 2: In a closed circuit or loop, the algebraic sum of the potential differences across each part of the
circuit is equal to the applied e.m.f.
Example: resistor–capacitor system
◦ Applying Kirchhoff’s second law to the circuit loop gives
◦ Since this is just a single loop, the current i through all the circuit elements will
be the same. If the output from the circuit is the potential difference across
the capacitor vC, then since vR = iR and i = C(dvC /dt ),
◦ This gives the relationship between the output vC and the input v and is a first-
order differential equation.
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Example: resistor–inductor–capacitor system
◦ If Kirchhoff ’s second law is applied to this circuit loop,
◦ Since there is just a single loop, the current i will be the same through all
circuit elements. If the output from the circuit is the potential difference
across the capacitor is vC, then since vR = iR and vL = L(di/dt).
FLUID SYSTEMS
◦ ELECTRICAL SYSTEMS
◦ CURRENT»POTENTIAL
DIFFERENCE
◦ FLUID SYSTEMS
◦ VOLUMETRIC FLOW
RATE» PRESSURE
DIFFERNECE
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Fluid system building blocks
◦ Fluid systems can be considered to fall into two categories:
◦ 1. Hydraulic, where the fluid is a liquid and is deemed to be incompressible;
◦ 2. Pneumatic, where it is a gas which can be compressed and consequently shows a density change.
◦ The basic building blocks of fluid building blocks are:
◦ Hydraulic resistance / Pneumatic resistance
◦ Hydraulic capacitance / Pneumatic capacitance
◦ Hydraulic inertance / Pneumatic inertance
Hydraulic Resistance
◦ Hydraulic resistance is the resistance to flow which occurs as a result of a liquid flowing through valves
or changes in a pipe diameter.
◦ The relationship between the volume rate of flow of liquid q through the resistance element and the
resulting pressure difference (p1 - p2) is
◦ R is a constant called the hydraulic resistance.
◦ A fluid resistor dissipates energy. A flow of fluid through a fluid system
Gives rise to a pressure drop analogous to voltage drop across a
electrical resistance element
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Hydraulic Capacitance
◦ 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 i.e. a so-called pressure head, is one form of
such a storage.
◦ The rate of change of volume V in the container, i.e. 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
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Hydraulic Inertance
◦ 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.
◦ Consider a block of liquid of mass m . The net force acting on the liquid is
This net force causes the mass to accelerate with an acceleration a,
◦ If the liquid has a density r then m=ALρ and so
◦ Volume flow rate q= A.v
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Pneumatic Resistance
◦ Pneumatic resistance R is defined in terms of the mass rate of flow dm/dt and the pressure difference
(p1-p2) as
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Pneumatic Capacitance
◦ Pneumatic capacitance C is due to the compressibility of the gas, and is comparable with the way in
which the compression of a spring stores energy. If there is a mass rate of flow dm1/dt entering a
container of volume V and a mass rate of flow of dm2/dt leaving it, then the rate at which the mass in
the container is changing is (dm1/dt - dm2/dt) .
◦ If the gas in the container has a density ρ then the rate of change of mass in the container is
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Pneumatic Inertance
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Hydraulic and pneumatic building blocks
THERMAL SYSTEMS
◦ There are only two basic building blocks for thermal systems:
◦ Thermal resistance
◦ Thermal capacitance
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THERMAL RESISTANCE
◦ There is a net flow of heat between two points if there is a temperature difference between them.
(Similar to electrical resistance V=IR).
◦ If q is the rate of flow of heat and (T1 - T2) the temperature difference, then
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THERMAL CAPACITANCE
◦ Thermal capacitance is a measure of the store of internal energy in a system. Thus, if the rate of flow of
heat into a system is q1 and the rate of flow out is q2, then
◦ C is the thermal capacitance and so C =mc.
Thermal building blocks.
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Example- Thermal System
◦ Consider a thermometer at temperature T which has just been inserted into a liquid at temperature TL
◦ This first-order differential equation, describes how the temperature indicated by the thermometer T will vary
with time when the thermometer is inserted into a hot liquid
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