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:- 130070119079 130070119080 130070119081 130070119082 GUIDED BY:- Prof. H.P.PATOLIA

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.

Basic Types of Mechanical Systems Translational Linear Motion Rotational Rotational Motion

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

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

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

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

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

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

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

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

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)

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

Translational Damper Translational Dampers in series and parallel.

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

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.

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

Example-2 Consider the following system Free Body Diagram M

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

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

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

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

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

Circuit Systems Inductor Capacitor Resistor

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

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

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)

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

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.