Eng. R. L. NKUMBWA
Copperebelt University
School of Technology
2010
1.0 INTRODUCTION
2.0 TRANSLATIONAL SYSTEMS
2.3
Electrical Systems
3.0 ROTATIONAL SYSTEMS
4.0 HYDRAULIC SYSTEMS
5.0 PNEUMATIC SYSTEMS
6.0 CONCLUSION
7.0 REFERENCES
 In
analyzing and designing of control systems, we
need to formulate a mathematical description of the
system which is commonly referred to as Modeling.
 Modeling is the process of obtaining the desired
mathematical description of the control system.
 The basic models of the dynamic physical systems
are represented by differential equations.
 Analysis of a dynamic system requires the ability to
predict its control system performance.
 A model
in control system is defined as representation of
essential aspects of a system illustrating how it operates.
 These may include inputs, the system, outputs and
feedback.
 The components of a control system are diverse in nature
and may include mechanical and electrical devices.
 The
basic models of the dynamic physical systems are
differential equations obtained by the application of the
appropriate laws of nature.
 These equations may be linear or nonlinear depending on the
phenomena being modeled.
 The differential equations are inconvenient for the analysis
and design manipulations and so the use of Laplace
Transforms which converts the differential equations into
algebraic equations is recommended.
 The algebraic equations may be put in transfer function form,
and the system modeled graphically as transfer function block
diagram.
 This
Chapter is mostly concerned with differential
equations, transfer functions of different physical
system such as; Electrical, Hydraulic, Pneumatic,
Translational and Rotational systems.
 This ability and the precision of the results depend on
how well the characteristics of each component can be
expressed mathematically.
 The
Mechanical Translational Systems basic law is
that the sum of forces must be equal to zero.
 It applies Newton’s law which states that the sum of
the applied forces must be equal to the sum of the
reactive forces.
 The basic characterizing elements in a Mechanical
Translational System are; The Mass (M), Damper (B)
and the Spring (K).
 The Mass is an inertial element; a force applied to the
mass which produces acceleration.
The reaction force Fm equals the product of mass and acceleration
and is opposite in the direction to the applied force.

Fm = -Ma
 In terms of displacement x, velocity v, and acceleration a, the
force equation is given by;

F(t) =Ma = M.d2x/dt2

The damping force is proportional to the difference in velocity of
two bodies.
 The reaction damping force fB equals the product of damping B
and the relative of the two end of the dashpot.
 The damping force fB is give by:
F(t) = B (v1-v2) = Bv

 In
terms of displacement it is given by:

F(t) = B (x1-x2)
 The spring element, the force equation in accordance
with Hooke’s Law is given by:

Fk = K (xe-xf)
 Where xe and xf are the end displacements at e and f
respectively.
 If the end is stationary, the equation is given by:
Fk = Kxe
x
X
Reference
B
K
f(t)
Figure 1: Mechanical System
Rotational Systems are similar to Translational Systems except
for the difference that the torque equations are written in place of
force equations and that the displacement, velocity and
acceleration terms are angular quantities.
 Meaning that the applied torque is equal to the sum of the
reaction torques.
 Elements of the Rotational System are;

1.Inertia element
Tj = Jdω/dt =J.d2ϴ/dt2 = J.D2ϴ
2.Torsional spring element
Tk = K (ϴ1 - ϴ2) = Kϴ
3.Damper element
TB = B (ω1 – ω2) = Bω
θ
Disc
T
Shaft
Stiffness
Viscous friction coefficient
(B)
Figure 2: Rotational Mechanical system
θ
B
T(t)
K
Reference node
Figure 3: Electrical Equivalent of Mechanical System
J d2θ/dt2 + B dθ/dt + Kθ = T (t)
 In terms of D operator,
JD2θ + BDθ + Kθ = T(t)
 In Laplace transform the equation will be
J s2θ(s) + Bsθ(s) + Kθ(s) = T(s)
[J s2 + BS + K] θ(s) = T(s)
θ(s)/T(s) = 1/Js2 + Bs + K
 This equation is a transfer function of the Rotational
Mechanical System.
 Hydraulics
is the study of incompressible fluids such as
oil and water.
 Incompressible fluids means that the fluid’s density
remains constant despite changes in fluid pressure.
 The variables for fluid systems are pressure, mass and
mass flow rate.
P
h
R
A
Q
Figure 4: Fluid Flow System
 Where,
 A is
the surface area of the tanks bottom.
 P is fluid in, Q the fluid out, R is the fluid resistance,
and m is mass.
 If the tank’s side is vertical, the liquid height h is
related by;
m = ρ Ah
 In
hydraulic actuators as the brake on an automobile.
 The control flaps of airplanes are actuated by similar
hydraulic systems.
 Hydraulic jacks and lifts are used for raising vehicles
in service stations and for lifting heavy loads in the
construction and mining industry.
 For
incompressible fluids, conservation of mass is
equivalent to conservation of volume, because the
fluid density is constant.
 If qm and q are the mass and the volume flow rates
and ρ is the fluid density, the equation is give as:

qm = ρ q
The conservation of the mass of a container holding mass
of fluid m, is given by the following equation;
m = qin – qout
 Where qin is the inflow rate, qout is the outflow rate and
fluid mass is related to the container volume V by
m=ρV
 For incompressible fluid ρ is a constant and m = ρV.
 If q1 and q2 are the total volume inflow and outflow rates,
qin = ρ q1 and qout= ρ q2.
 Substituting this relationships into equation; it gives the
following
ρ V = ρ q 1 - ρ q2
V = q1 – q2

 The
working medium in a pneumatic systems or device
is a compressible fluid.
 Industrial Control Systems most frequently use
pneumatic to provide forces greater than those available
from electrical devices.
 The response of pneumatic systems is slower than that
of hydraulic systems because of the compressibility of
the working fluid.
 The
following are the quantities of Pneumatic
Systems; Mass, Volume, Pressure and Temperature.
 This relationship is called the “Perfect Gas Law”
which states that, pV = mRgT
 Where p is the absolute pressure of the gas with
volume V, m is the mass, T its absolute temperature,
and Rg the gas constant that depends on the particular
type of gas.
 The compressible “flow resistance” of pneumatic
components is modeled in the form of the turbulent
resistance written here as, Rpq2m = ∆ p
 Where ∆ p is the pressure drop across the component
and Rp is the “pneumatic resistance”.
p, q
Q
P
valve
container
Figure 5: Pneumatic System
 In
this chapter we have defined what Translational,
Rotational, Fluid and Pneumatic systems are.
 There are differences that have been outlined according
to mathematical modeling and natural laws
 Translation uses force equations while Rotational uses
torque equations.
 Hydraulic system is the study of incompressible liquids
while Pneumatic uses compressible liquids
 These equations maybe linear or nonlinear depending
on the systems being modeled.





Chesmond, C.J, (1990), Basic Control System
Technology, J W Arrow Smith Ltd, Bristol,
United Kingdom.
Nkumbwa, R.L, (2009), Control Systems
engineering for 21st Century Engineers and
Technologists, Lusaka, Zambia.
Parr, E.A, (1996), Control Engineering, Hartnolls
Ltd, London, England.
Chand, S (1999), Principles of` Control Systems