Automatic Control Theory CSE 322
Lec. 4 Mathematical Modeling of Dynamic System Dr. Tamer Samy Gaafar Course Web Site www.tsgaafar.faculty.zu.edu.eg Email:
2. Mechanical Systems Mechanical Systems Part-I: Translational Mechanical System
Part-II: Rotational Mechanical System Part-III: Mechanical Linkages 2. Rotational Mechanical Systems Basic Elements of Rotational Mechanical Systems
Rotational Spring Basic Elements of Rotational Mechanical Systems
Rotational Damper Basic Elements of Rotational Mechanical Systems
Moment of Inertia Example-1 J1 J2 Example-2 J1 J2 3. Mechanical Linkages Gear Gear is a toothed machinepart, such as a wheel orcylinder, that meshes withanother toothed part totransmit motion or to changespeed or direction. Fundamental Properties
The two gears turn in opposite directions: oneclockwise and the other counterclockwise. Two gears revolve at different speeds when numberof teeth on each gear are different. Gearing Up and Down Gearing up is able to convert torque to velocity.
The more velocity gained, the moretorque sacrifice. The ratio is exactly the same: if youget three times your original angularvelocity, you reduce the resultingtorque to one third. This conversion is symmetric: we canalso convert velocity to torque at thesame ratio. The price of the conversion is powerloss due to friction. Why Gearing is necessary?
A typical DC motor operates at speeds that are far too high to be useful, and at torques that are far too low. Gear reduction is the standard method by which a motor is made useful. Gear Trains Gear Ratio Gear Ratio = # teeth input gear / # teeth output gear
You can calculate the gear ratioby using the number of teeth ofthe driver divided by the numberof teeth of the follower. We gear up when we increasevelocity and decrease torque. Ratio: 3:1 We gear down when we increasetorque and reduce velocity. Ratio: 1:3 Follower Driver Gear Ratio = # teeth input gear / # teeth output gear = torque in / torque out = speed out / speed in Example of Gear Trains A most commonly used example of gear trains isthe gears of an automobile. Manual Transmission (Gear Box) Mathematical Modeling of Gear Trains
Gears increase or reduce angular velocity(while simultaneously decreasing or increasingtorque, such that energy is conserved). Energy of Driving Gear = Energy of Following Gear Number of Teeth of Driving Gear Angular Movement of DrivingGear Number of Teeth ofFollowing Gear Angular Movement of Following Gear Mathematical Modeling of Gear Trains
In the system below, a torque,a, is applied to gear 1(with number of teeth N1, moment of inertia J1 and arotational friction B1). It, in turn, is connected to gear 2 (with number ofteeth N2, moment of inertia J2 and a rotational frictionB2). The angle 1is defined positive clockwise, 2isdefined positive clockwise. The torque acts in thedirection of 1. Assume that TL is the load torque applied by the loadconnected to Gear-2. B1 B2 N1 N2 Mathematical Modeling of Gear Trains
For Gear-1 For Gear-2 Since therefore Eq (1) B1 B2 N1 N2 Eq (2) Eq (3) Mathematical Modeling of Gear Trains
Gear Ratio is calculated as Put this value in eq (1) Put T2 from eq (2) Substitute 2 from eq (3) B1 B2 N1 N2 Mathematical Modeling of Gear Trains
After simplification Mathematical Modeling of Gear Trains
For three gears connected together Electro-Mechanical Systems.
Self Study Electro-Mechanical Systems. Liquid Level Systems. Laminar vs Turbulent Flow
Laminar Flow Flow dominated by viscosityforces is called laminar flowand is characterized by asmooth, parallel line motion ofthe fluid Turbulent Flow When inertia forces dominate,the flow is called turbulent flowand is characterized by anirregular motion of the fluid. Resistance of Liquid-Level Systems
Consider the flow through a short pipeconnecting two tanks as shown in Figure. Where H1 is the height (or level) of first tank, H2is the height of second tank, R is the resistancein flow of liquid and Q is the flow rate. Resistance of Liquid-Level Systems
The resistance for liquid flow in such a pipe is defined asthe change in the level difference necessary to cause aunit change inflow rate. Resistance in Laminar Flow
For laminar flow, the relationship between the steady- state flow rate and steady state height at the restrictionis given by: Where Q = steady-state liquid flow rate in m/s3 Kl = constant in m/s2 and H = steady-state height in m. The resistance Rl is Capacitance of Liquid-Level Systems
The capacitance of a tank is defined to be the change inquantity of stored liquid necessary to cause a unitychange in the height. Capacitance (C) is cross sectional area (A) of the tank. h Capacitance of Liquid-Level Systems
h Capacitance of Liquid-Level Systems
h Modelling Example-1 Modelling Example -1 The rate of change in liquid stored in the tank is equal to the flow in minus flow out. The resistance R may be written as Rearranging equation(2) (1) (2) (3) Modelling Example -1 Substitute qo in equation (3)
After simplifying above equation Taking Laplace transform considering initial conditions to zero (4) (1) Modelling Example -1 The transfer function can be obtained as Modelling Example -1 The liquid level system considered here is analogous to the electrical and mechanical systems shown below. Modelling Example -2 Consider the liquid level system shown in following Figure. In this system, two tanks interact. Find transfer function Q2(s)/Q(s). Modelling Example - 2 Tank Pipe 1 Tank Pipe 2 Modelling Example - 2 Tank 1 Pipe 1 Tank 2 Pipe 2
Re-arranging above equation Modelling Example - 2 Taking LT of both equations considering initial conditions to zero [i.e. h1(0)=h2(0)=0]. (1) (2) Modelling Example - 2 (1) (2) From Equation (1)
Substitute the expression of H1(s) into Equation (2), we get Modelling Example - 2 Using H2(s) = R2Q2 (s) in the above equation Modelling Example -3 Write down the system differential equations.