1. GUJARAT TECHNOLOGICAL UNIVERSITY BIRLA VISHVAKARMA MAHAVIDYALAYA
(ENGINEERING COLLEGE)
VALLABH VIDYANAGAR
MECHANICAL ENGINEERING
SUB : CONTROL ENGINEERING
TOPIC : MATHEMATICAL MODELLING
DIV- 7 BATCH: B-7
CREATED BY:-
GUIDED BY:-
Prof. H.P.PATOLIA

2. What is Mathematical Model?
A set of mathematical equations (e.g., differential eqs.) that describes the input-output behavior of a system.
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.

3. Basic Types of Mechanical Systems
Translational
Linear Motion
Rotational
Rotational Motion

4. Basic Elements of Translational Mechanical Systems
Translational Spring i)
Translational Mass
ii)
Translational Damper
iii)

5. Translational Spring
A translational spring is a mechanical element that can be deformed by an external force such that the deformation is directly proportional to the force applied to it.
Translational Spring i)
Circuit Symbols
Translational Spring

6. Translational Spring If F is the applied force
Then is the deformation if
Or is the deformation.
The equation of motion is given as
Where is stiffness of spring expressed in N/m

7. Translational Spring
Given two springs with spring constant k1 and k2, obtain the equivalent spring constant keq for the two springs connected in:
(1) Parallel
(2) Series

8. Translational Spring The two springs have same displacement therefore:
(1) Parallel
If n springs are connected in parallel then:

9. Translational Spring
The forces on two springs are same, F, however displacements are different therefore:
(2) Series
Since the total displacement is , and we have

10. Translational Spring Then we can obtain
If n springs are connected in series then:

11. Translational Mass Translational Mass is an inertia element.
A mechanical system without mass does not exist.
If a force F is applied to a mass and it is displaced to x meters then the relation between force and displacements is given by Newton’s law.
Translational Mass
ii) M

12. Translational Damper Damper opposes the rate of change of motion.
All the materials exhibit the property of damping to some extent.
If damping in the system is not enough then extra elements (e.g. Dashpot) are added to increase damping.
Translational Damper
iii)

13. Translational Damper
Where C is damping coefficient (N/ms-1).

14. Translational Damper
Translational Dampers in series and parallel.

15. Common Uses of Dashpots
Door Stoppers
Vehicle Suspension
Bridge Suspension
Flyover Suspension

16. Example-1 M Consider the following system (friction is negligible)
Free Body Diagram M
Where and are force applied by the spring and inertial force respectively.

17. Example-1 M Then the differential equation of the system is:
Taking the Laplace Transform of both sides and ignoring initial conditions we get

18. Example-2
Consider the following system
Free Body Diagram M

19. Differential equation of the system is:
Taking the Laplace Transform of both sides and ignoring
Initial conditions we get

20. Find the transfer function of the mechanical translational system given in Figure-1.
Free Body Diagram M
Figure-1

21. Basic Elements of Rotational Mechanical Systems
Rotational Spring
Rotational Damper
Moment of Inertia

22. Mathematical model of a rotating a mass
Moment of inertia
Mathematical model of a rotating a mass
Torque
Torsional resistance
Block model

23. Electrical System
The basic elements of electrical systems are resistance, inductance and capacitance.

24. Circuit Systems
Inductor
Capacitor
Resistor

25. A second-order Circuit
From Kirchihoff Voltage Law, we obtain
(Transfer Function from )

26. Hydraulic system Qi=Qo ΔQo = ΔH/R Qi = steady state inflow rate
Qo = steady state outflow rate
H = net head in tank
ΔH = small deviation in head
R = resistant of tank
R = change in level difference(m)/change in flow rate (m3/s)
C = capacitance of tank
C = change in liquid stored(m3/s)/change in head(m)
Under steady state condition
Qi=Qo
Let ΔQi be a small increase in liquid inflow rate from its steady state value this increased liquid inflow rate causes increase of head of liquid in tank by ΔH .Resulting increase of liquid outflow by
ΔQo = ΔH/R

27. C(dΔH/dt) = ΔQi – (ΔH/R) RC(dΔH/dt) + ΔH = RΔQi
Rate of liquid storage in tank = Rate of liquid Inflow- Rate of liquid Outflow
C(dΔH/dt) = ΔQi - ΔQo
C(dΔH/dt) = ΔQi – (ΔH/R)
RC(dΔH/dt) + ΔH = RΔQi
By taking Laplace transform
RCs H(s) + H(s) = RQi(s)
Transfer Function:
H(s)/Qi(s) = R/(RCs+1)
And
Qo(s)/Qi(s) = 1/(RCs+1)

28. Pneumatic system R = (Pi-Po)/q C = dm/dP m = mass of gas in vessel
V = volume of vessel
R = resistance of vessel
R = change in gas pressure difference/change in gas flow rate
C = capacitance
C = change in gas stored/change in gas pressure
P = gas pressure
Pi = small change in inflow gas pressure
Po = small change in outflow gas pressure
q = gas flow rate
D = density of gas
For small values of Po and Pi, the resistance R are given by
R = (Pi-Po)/q
The capacitance C is given by,
C = dm/dP

29. C dPo = q dt C (dpo/dt) = (Pi – Po)R RC(dpo/dt) + Po = Pi
Since the pressure change dPo times the capacitance C is equal to the gas added to the vessel during dt seconds, we obtain
C dPo = q dt
C (dpo/dt) = (Pi – Po)R
Which can be written as,
RC(dpo/dt) + Po = Pi
If Pi and Po are considered the input and output, respectively, then the transfer function of the system is
Po(s)/Pi(s) = 1/(RCs + 1)
where RC has the dimension of time and is the time constant of the system.