MATHEMATICAL MODELING OF DYNAMIC SYSTEMS

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Sara Wade
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1 MTHEMTIL MODELIN OF DYNMI SYSTEMS Mechanical Translational System 1. Spring x(t) k F S (t) k x(t) x i (t) k x o (t) 2. Damper x(t) x i (t) x o (t) c c

2 3. Mass x(t) F(t) m EXMPLE I Produce the block diagram for the mass-spring system shown below by considering displacement x i as the input variable and displacement x o as the output variable. x i (t) x o (t) k m

3 EXMPLE II Produce the block diagram for the mass-spring-damper system shown below by taking force F a as the input variable and displacement x as the output variable. c k m x (t) F a (t) Mechanical Rotational System 4. Shaft stiffness θ o (t) Torque applied on the disc: θ i (t) k

4 5. Viscous damper θ i (t) B θ o (t) 6. Mass inertia τ(t) θ(t) J EXMPLE I Produce the block diagram for the mechanical rotational system shown below if angular displacement θ i is taken as the input variable and angular displacement θ o is taken as the output variable. θ i (t) k θ o (t) J B

Leverage and earing System Lever and gear are mechanical devices used to transfer energy from one section to another by transformation the force, torque, velocity and displacement.

5 Leverage and earing System Lever and gear are mechanical devices used to transfer energy from one section to another by transformation the force, torque, velocity and displacement. The inertia and friction effects are normally small and are neglected. Walking lever can be used for summing up mechanical displacement signals. 1. Fixed lever z l 1 l 2 y 2. Walking lever z x l 1 a b l 2 y

translational system shown below if displacement x i is

6 3. ear train ω,θ ear ratio: τ ω B,θ B τ B r r B EXMPLE I Produce the block diagram for the mechanical translational system shown below if displacement x i is the input and displacement x o is the output. k 2 x i a x o b m k 1

7 EXMPLE II Produce the block diagram for the rotational system shown below considering torque τ a as the input variable and angular displacement θ as the output variable. θ,ω τ a r 1 K r 2 B J Liquid Level System 1. Fluid Resistance, R The relation between liquid flow rate, q and system s hydrostatic head, h is not linear. For linear model approximation (if the change in head and flow rate are small), the flow rate is considered proportional to the system head multiply by a proportional constant, 1/R. R is known as fluid resistance in the liquid level system. h 1 R h 2 q

8 2. apacitance, hanges in the volume inside the tank are equals to the difference between the inflow rate and the outflow rate. apacitance is the cross-sectional area of the tank. q i Therefore: h q o onsider the liquid level system shown below. The variables to be monitored are the head, h and the flow rate, q. The system parameters are the fluid resistance, R and capacitance,. q i h R q o

9 EXMPLE I Produce the block diagram for the liquid level system shown below if inflow rate q i is taken as the input variable and head h 2 is taken as the output variable. q i 1 2 h 1 q o R 1 R 2 q h 2

10 Thermal System pemanas bendalir panas bendalir dingin Thermal Resistance, penebat Heat Balance Equation, Where, thermal capacitance (kcal/º) R thermal resistance (ºsec/kcal) hi heat input rate (kcal/sec) hi heat output rate (kcal/sec) θ - temperature of outflowing liquid (º) Electrical System 1. Resistance, R i R V R 2. Inductance, L i L V L

11 3. apacitance, i V EXMPLE I Produce the block diagram of the electrical system shown below by considering input voltage V i as the input variable and output voltage V o as the output variable. L V i R V o

12 DETERMINTION OF TRNSFER FUNTION USIN BLOK DIRM SIMPLIFITION Transfer Function Transfer function for a linear system is the ratio of the Laplace transform of the output variable (response function) to Laplace transform of the input variable (driving function) under the assumption that all the initial conditions are ZERO. The transfer function of a system represents the relationship which describes the dynamic behaviour of the system under study; describing the relationship between the system s input and output. However the transfer function does not provides information about internal structure and internal behaviour of the system. EXMPLE I onstruct the transfer function for a system represented by the following differential equation: 2 d x m 2 dt dx c dt kx = F (t) Block Diagram Simplification block diagram is regularly being used as pictorial representation of a control system. Block diagram depicts the interrelationships that exist among various components within the system. ll system variables are linked to each other through functional blocks. complex block diagram can be simplified using block diagram manipulation rules shown below. system block diagram normally shows the relationship of important variables only.

13 Block Diagram lgebra B -B -B -B B B -B B -B -B atau 2 1 B B

14 1 ± B ± B 1 ± H H 1 m

15 EXMPLE II onsider the mechanical translational system represented by the block diagram as shown below. Produce the system transfer function using the block diagram simplification method. k m x (t) F a (t) c F a 1 md 2 x c D k

16 EXMPLE III Produce the system transfer function using the block diagram simplification method. H 1 R 1 2 H 2

17 SINL FLOW RPH ND MSON S RULE Beside block diagram, signal flow graph is another way of illustrating a control system. For a complex system, signal flow graph has an advantage over block diagram as Mason s Rule can be used to determine the transfer function of the system. However, block diagram is more popular for uncomplicated system as it has close representation of the actual physical system. Signal flow graph is consisting of nodes and branches to represent linear relationships. Variables are represented by nodes and each transfer operator (internal relationship function) is a branch which connecting all the nodes. In general, branch is popularly known as gain which represents the relationship between two variables. Some characteristics of the nodes are: 1. Variable at a node is the summation of all the incoming signals into the node and the node variable is transferred to all outgoing branches. 2. The summing junction is just a node 3. Input node (source node) will only consist of outgoing branch. 4. Output node (sink node) will only consist of incoming branch

18 EXMPLE I Draw the signal flow graph for the control system represented by the block diagrams shown below. a) R 1 2 b) R H H 2 c) 4 R H 2 H 1

19 Mason s Rule Transfer function of a control system can be obtained using Mason s rule given below: Y(s) U(s) = PiΔ Δ i where, P i = i th forward path gain Δ = 1 (summation of all individual loops gain) (product of two non-touching loops) (product of three non-touching loops).. Δi = the value of Δ with all loop gains touching path P i are discarded EXMPLE II onstruct the signal flow graph and derive the system s transfer function by employing Mason s rule for the block diagram shown below. 2 R 1 H 1

20 EXMPLE III Using the Mason s rule, determine the transfer function of each signal flow graphs shown below. 6 x y -H 1 -H 2

21 EXMPLE IV onstruct the signal flow graph and derive the system s transfer function using Mason s rule for the block diagram shown below. 4 R H 1 H 2

Modeling and Simulation Revision III D R. T A R E K A. T U T U N J I P H I L A D E L P H I A U N I V E R S I T Y, J O R D A N

Modeling and Simulation Revision III D R. T A R E K A. T U T U N J I P H I L A D E L P H I A U N I V E R S I T Y, J O R D A N 0 1 4 Block Diagrams Block diagram models consist of two fundamental objects: