Translational Mechanical Systems

1. Translational Mechanical Systems There are analogies between electrical and mechanical system: Mechanical force is analogous to electrical voltage Mechanical velocity is analogous to electrical current Mechanical displacement is analogous to electrical charge. Spring is analogous to capacitor Viscous damper is analogous to resistor Mass is analogous to inductor.

2. Electrical Components Impedance, Z(s)=V(s)/I(s) Admittance, Y(s)=I(s)/V(s) Voltage-current Current-voltage Voltage-charge Component

3. Mechanical Components

4. Mechanical Systems Find the transfer function, X(s)/F(s), for the system in (a):

5. Take Laplace Transform, Solving the transfer function,

6. Mechanical Systems-Two degrees of freedom Find the transfer function, X2(s)/F(s), for the system in (a):

7. Figure 2.18a. Forces on M1 due only to motion of M1b. forces on M1 due only to motion of M2c. all forces on M1 The forces on M1 are due to (1) its own motion and (2) the motion of M2 transmitted to M1 through the system. If we hold M2 still and move M1 to the right, we will see forces shown in Fig. 2.18(a). If we hold M1 still and move M2 to the right, we will see forces shown in Fig. 2.18(b). The total force on M1 is the superposition, or sum of the forces just discussed. It is shown in Fig. 2.18(c).

8. Figure 2.19a. Forces on M2 due only to motion of M2;b. forces on M2 due only to motionof M1;c. all forces on M2 (1) (2)

9. Solving the transfer function, (1) (2)

10. Mechanical Systems Write the equations of motion for the mechanical network of Figure below:

11. (1) (2) (3)

12. Equations of motion:

13. Electric Circuit Analogs Converting mechanical systems to electrical networks before writing the describing equations is an alternative problem solving approach. A n electric circuit that is analogous to a system from another discipline is called an electric circuit analog. When compared with mesh equations, the resulting electrical circuit is called series analog. When compared with nodal equations, the resulting electrical circuit is called parallel analog.

14. Series Analog Figure 2.41Development of series analog: a. mechanical system;b. desired electrical representation; c. series analog; d.parameters for series analog

15. Consider the mechanical system in Figure 2.41(a): Kirchhoff’s mesh equation for the simple RLC network in Figure 2.41(b): Displacement and current are not analogous. A direct analogy can be created by converting displacement to velocity by dividing and multiplying the left hand side by s, yielding Comparing the equations, the sum of impedance is recognized and a circuit shown in Figure 2.41 (c) can be drawn.

16. Parallel Analog Figure 2.43Development of parallel analog: a. mechanical system; b. desired electrical representation; c. parallel analog; d. parameters for parallel analog

17. Parallel Analog Consider mechanical system in Figure 2.43 (a), whose equation of motion is: Kirchhoff’s nodal equation for a simple RLC network in Figure 2.43 (b) is: Identify the sum of admittance and a circuit in Figure 2.43 (c) can be drawn.