Transfer function of Mechanical translational system

ANNA UNIVERSITY,REGULATION 2017, EEE, IC8451-CONTROL SYSTEM

1. IC 8451&CONTROL SYSTEMS Department of Electrical and Electronics Engineering Approved by AICTE | Affiliated to Anna University | Accredited by NAAC | Accredited NBA | Recognized by UGC under 2(f) and 12(B) Chennai Main Road, Kumbakonam- 612 501. ARASU ENGINEERING COLLEGE 1

2. Prepared by Mrs .K. Kalpana .,M.E.,( Ph.D )., Department of Electrical and Electronics Engineering ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 2

3. TRANSFER FUNCTION Transfer function of a system is defined as the ratio of the Laplace Transform of the output variable to the Laplace Transform of the input variable assuming all the initial condition as zero TRANSFER FUNCTION = with zero initial conditions ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 3

4. Mechanical Translational systems The model of mechanical translational systems can be obtained by using three basic elements mass, spring and dashpot. When a force is applied to a translational mechanical system, it is opposed by opposing forces due to mass, friction and elasticity of the system. The force acting on a mechanical body is governed by Newton's second law of motion. For translational systems, it states that the sum of forces acting on a body is zero (or the sum of applied forces is equal to the sum of opposing forces on a body) ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 4

5. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 5 List of symbols used in mechanical translational system x = Displacement, m v = dx / dt = Velocity, m/s a = d 2 x /dt 2 = Acceleration, m/s 2 f = Applied force, N f m = Opposing force offered by mass of the body, N f k = Opposing force offered by elasticity of the body(spring),N f b = Opposing force offered by friction of the body(dash pot),N M = Mass, kg K = Stiffness of spring, N/m B = Viscous friction co efficient, N-s/m

6. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 6 Force balance equations of idealized elements: Consider an ideal mass element shown in fig. which has negligible friction and elasticity. Let a force be applied on it. The mass will offer an opposing force which is proportional to acceleration of a body. Let f = applied force f m =opposing force due to mass Here f m α (d 2 x / dt 2 ) or f m = M (d 2 x / dt 2 ) By Newton's second law, f = f m = M (d 2 x / dt 2 )

7. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 7 Consider an ideal frictional element dash-pot shown in fig. which has negligible mass and elasticity. Let a force be applied on it. The dashpot will be offer an opposing force which is proportional to velocity of the body.

8. When the dashpot has displacement at both the ends as shown in fig, the opposing force is proportional to differential velocity. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 8

9. Consider an ideal elastic element spring shown in fig ,which has negligible mass and friction. Let a force be applied on it. The spring will offer an opposing force which is proportional to displacement of the body. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 9

10. When the spring has displacement at both the ends as shown in figure, the opposing force is proportional to differential displacement. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 10

11. Guidelines to determine the transfer function of mechanical translational system In mechanical translational system the differential equations governing the system are obtained by writing force balance equations at nodes in the system. The nodes are meeting point of elements. Generally the nodes are mass elements in the system. In some cases the nodes may be without mass element. The linear displacement of the masses are assumed as x 1 ,x 2 ,x 3 ,....etc. and assign a displacement to each mass(node). The first derivative of the displacement is velocity and the second derivative of the displacement is acceleration. Draw the free body diagram of the system. The free body diagram is obtained by drawing each mass separately and then marking all the forces acting on that mass. Always the opposing forces acts in a direction opposite to applied force. The mass has to move in the direction of the applied force. Hence the displacement ,velocity and acceleration of the mass will be in the direction of applied force. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 11

12. For each free body diagram write one differential equation by equating the sum of applied forces to the sum of opposing forces. Take Laplace transform of differential equations to convert them to algebraic equations. Then rearrange the s domain equations to eliminate the unwanted variables and obtain the ratio between output variable and input variable. This ratio is the transfer function of the system. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 12

13. EXAMPLE PROBLEM -1 Write the differential equations governing the mechanical system shown in fig and determine the transfer function. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 13 The required transfer function is The system has two nodes and they are mass M 1 and M 2

14. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 14 Let the displacement of mass M 1 is x 1 .The opposing forces acting on mass M 1 are marked as f m1 ,f b1 ,f b ,f k1 ,f k 1

15. The free body diagram of mass M 2 is shown in fig. The opposing forces acting on M 2 are marked as f m2 ,f b2 ,f b and f k ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 15 2

16. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 16 Substituting x 1 (s) in equation 2

17. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 17 EXAMPLE PROBLEM -2 Determine the transfer function

18. ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 2 18 1

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21. HOME WORK PROBLEM ARASU ENGINEERING COLLEGE IC 8451&CONTROL SYSTEMS 21