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Feedback Control Systems (FCS) Lecture-3-4-5 Introduction Mathematical Modeling Mathematical Modeling of Mechanical Systems Dr. Imtiaz Hussain email: [email protected] URL :http://imtiazhussainkalwar.weebly.com/ 1 Lecture Outline • Introduction to Modeling – Ways to Study System – Modeling Classification • Mathematical Modeling of Mechanical Systems – Translational Mechanical Systems – Rotational Mechanical Systems – Mechanical Linkages 2 Model • A model is a simplified representation or abstraction of reality. • Reality is generally too complex to copy exactly. • Much of the complexity is actually irrelevant in problem solving. 3 What is Mathematical Model? A set of mathematical equations (e.g., differential eqs.) that describes the input-output behavior of a system. What is a model used for? • Simulation • Prediction/Forecasting • Prognostics/Diagnostics • Design/Performance Evaluation • Control System Design Ways to Study a System System Experiment with a model of the System Experiment with actual System Mathematical Model Physical Model Analytical Solution Simulation Frequency Domain Time Domain Hybrid Domain 5 Mathematical Models • Black box • Gray box • White box 6 Black Box Model • When only input and output are known. • Internal dynamics are either too complex or unknown. Input Output • Easy to Model 7 Grey Box Model • When input and output and some information about the internal dynamics of the system are known. u(t) y(t) y[u(t), t] • Easier than white box Modelling. 8 White Box Model • When input and output and internal dynamics of the system are known. u(t) dy(t ) du(t ) d 2 y(t ) 3 dt dt dt 2 y(t) • One should have complete knowledge of the system to derive a white box model. 9 MATHEMATICAL MODELING OF MECHANICAL SYSTEMS 10 Basic Types of Mechanical Systems • Translational – Linear Motion • Rotational – Rotational Motion 11 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 x2 x1 • Then x1 is the deformation if x2 0 • Or ( x1 x2 ) is the deformation. • The equation of motion is given as F k ( x1 x2 ) • Where k is stiffness of spring expressed in N/m F F Translational Mass • Translational Mass is an inertia element. Translational Mass ii) • 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 b/w force and displacements is given by Newton’s law. F Mx x(t ) F (t ) 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) Common Uses of Dashpots Door Stoppers Bridge Suspension Vehicle Suspension Flyover Suspension Translational Damper F Cx F C ( x1 x 2 ) • Where C is damping coefficient (N/ms-1). Example-1 • Consider the following system (friction is negligible) k x M F • Free Body Diagram fk F M fM • Where f k and f M are force applied by the spring and inertial force respectively. 19 Example-1 fk F M fM F fk fM • Then the differential equation of the system is: F kx Mx • Taking the Laplace Transform of both sides and ignoring initial conditions we get F ( s ) Ms 2 X ( s ) kX( s ) 20 Example-1 F ( s ) Ms 2 X ( s ) kX( s ) • The transfer function of the system is X (s) 1 F(s) Ms 2 k • if M 1000 kg k 2000 Nm 1 X (s) 0.001 2 F(s) s 2 21 Example-2 X (s) 0.001 2 F(s) s 2 • The pole-zero map of the system is Pole-Zero Map Imaginary Axis 𝑗 2 0 −𝑗 2 -1 -0.5 0 Real Axis 0.5 1 22 Example-2 • Consider the following system k F x M C • Free Body Diagram fk F M fC fM F f k f M fC 23 Example-3 Differential equation of the system is: F Mx Cx kx Taking the Laplace Transform of both sides and ignoring Initial conditions we get F ( s ) Ms 2 X ( s ) CsX ( s ) kX( s ) X (s) 1 F(s) Ms 2 Cs k 24 Example-3 X (s) 1 F(s) Ms 2 Cs k • if Pole-Zero Map 2 1.5 M 1000 kg k 2000 Nm 1 C 1000 N / ms 1 Imaginary Axis 1 0.5 0 -0.5 -1 X (s) 0.001 2 F(s) s s 1000 -1.5 -2 -1 -0.5 0 0.5 1 Real Axis 25 Example-4 • Consider the following system x2 x1 k B F M • Mechanical Network x1 F ↑ k x2 M B 26 Example-4 • Mechanical Network x1 F At node ↑ k x2 M B x1 F k ( x1 x 2 ) At node x2 2 0 k ( x2 x1 ) Mx2 Bx 27 Example-5 • Find the transfer function X2(s)/F(s) of the following system. k M1 B M2 Example-6 x1 x2 B3 k f (t ) B4 M1 M2 B1 B3 x1 f (t ) ↑ k M1 B2 B1 B2 x2 M2 B4 29 Example-7 • Find the transfer function of the mechanical translational system given in Figure-1. Free Body Diagram fk fB Figure-1 M f (t ) f (t ) f k f M f B fM X (s) 1 F(s) Ms 2 Bs k 30 Example-8 • Restaurant plate dispenser 31 Example-9 • Find the transfer function X2(s)/F(s) of the following system. Free Body Diagram f k1 f k f B 2 k2 f k1 fB M2 M1 F (t ) f M 2 f M1 F (t ) f k1 f k2 f M 2 f B 0 f k1 f M1 f B 32 Example-10 x1 u(t ) x2 k1 B1 x3 B4 B3 M1 k2 B2 M2 k3 B5 33 Basic Elements of Rotational Mechanical Systems Rotational Spring 2 1 T k (1 2 ) Basic Elements of Rotational Mechanical Systems Rotational Damper C 2 1 T C(1 2 ) T Basic Elements of Rotational Mechanical Systems Moment of Inertia J T J T Example-11 1 k1 B1 2 3 T J1 1 T ↑ k1 2 J1 k2 J2 B1 3 J2 k2 Example-12 1 k1 B2 2 T J1 3 J2 B3 B1 1 T ↑ k1 2 J1 B4 B2 B1 B3 3 J2 B4 Example-13 1 k1 J1 B2 2 T J2 k2 Example-14 MECHANICAL LINKAGES 41 Gear • Gear is a toothed machine part, such as a wheel or cylinder, that meshes with another toothed part to transmit motion or to change speed or direction. 42 Fundamental Properties • The two gears turn in opposite directions: one clockwise and the other counterclockwise. • Two gears revolve at different speeds when number of teeth on each gear are different. Gearing Up and Down • Gearing up is able to convert torque to velocity. • The more velocity gained, the more torque sacrifice. • The ratio is exactly the same: if you get three times your original angular velocity, you reduce the resulting torque to one third. • This conversion is symmetric: we can also convert velocity to torque at the same ratio. • The price of the conversion is power loss 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. 45 Gear Trains 46 Gear Ratio • You can calculate the gear ratio by using the number of teeth of the driver divided by the number of teeth of the follower. • We gear up when we increase velocity and decrease torque. Ratio: 3:1 Driver Follower • We gear down when we increase torque and reduce velocity. Ratio: 1:3 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑒𝑡ℎ 𝑜𝑓 𝑖𝑛𝑝𝑢𝑡 𝑔𝑒𝑎𝑟 𝐼𝑛𝑝𝑢𝑡 𝑇𝑜𝑟𝑞𝑢𝑒 𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑝𝑒𝑒𝑑 𝐺𝑒𝑎𝑟 𝑟𝑎𝑡𝑖𝑜 = = = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑒𝑡ℎ 𝑜𝑓 𝑜𝑢𝑝𝑢𝑡 𝑔𝑒𝑎𝑟 𝑂𝑢𝑝𝑢𝑡 𝑇𝑜𝑟𝑞𝑢𝑒 𝐼𝑛𝑝𝑢𝑡 𝑆𝑝𝑒𝑒𝑑 Example of Gear Trains • A most commonly used example of gear trains is the gears of an automobile. 48 Mathematical Modeling of Gear Trains • Gears increase or descrease angular velocity (while simultaneously decreasing or increasing torque, such that energy is conserved). Energy of Driving Gear = Energy of Following Gear N11 N1 N 2 2 Number of Teeth of Driving Gear 1 Angular Movement of Driving Gear N2 Number of Teeth of Following Gear 2 Angular Movement of Following Gear 49 Mathematical Modelling of Gear Trains • In the system below, a torque, τa, is applied to gear 1 (with number of teeth N1, moment of inertia J1 and a rotational friction B1). • It, in turn, is connected to gear 2 (with number of teeth N2, moment of inertia J2 and a rotational friction B2). • The angle θ1 is defined positive clockwise, θ2 is defined positive clockwise. The torque acts in the direction of θ1. • Assume that TL is the load torque applied by the load connected to Gear-2. N1 N2 B1 B2 50 Mathematical Modelling of Gear Trains • For Gear-1 a J11 B11 T1 Eq (1) • For Gear-2 T2 J 22 B22 TL Eq (2) N1 N2 B1 • Since B2 N11 N 2 2 • therefore N1 2 1 N2 Eq (3) 51 Mathematical Modelling of Gear Trains • Gear Ratio is calculated as T2 N2 N1 T1 T2 T1 N1 N2 • Put this value in eq (1) N1 a J11 B11 T2 N2 N1 N2 B1 • Put T2 from eq (2) B2 N1 a J11 B11 ( J 22 B22 TL ) N2 • Substitute θ2 from eq (3) N1 N1 N1 N1 a J11 B11 (J2 1 B2 2 TL ) N2 N2 N2 N 2 52 Mathematical Modelling of Gear Trains N1 N1 N1 N1 a J11 B11 (J2 1 B2 2 TL ) N2 N2 N2 N2 • After simplification 2 N1 N1 J 21 B11 a J11 N2 N2 N1 a J1 N2 2 N1 J 2 1 B1 N2 2 J eq N J1 1 J 2 N2 2 N B21 1 TL N2 N1 B2 1 TL N2 2 2 Beq N B1 1 B2 N2 N1 a J eq1 Beq1 TL N2 53 Mathematical Modelling of Gear Trains • For three gears connected together J eq Beq 2 2 2 2 N1 N1 J 2 J1 N2 N2 N1 N1 B2 B1 N2 N2 2 N3 J 3 N4 2 N3 B3 N4 54 Example-15 • Drive Jeq and Beq and relation between applied torque τa and load torque TL for three gears connected together. 1 2 3 N1 N2 J1 τa J2 B2 B1 N3 J3 TL B3 55 To download this lecture visit http://imtiazhussainkalwar.weebly.com/ END OF LECTURES-3-4-5 56