![Laplace Transforms
• For M1:
• [M1 s^2 + (B1 + B)s + (K1 + K)] X1(s) = (B s + K)
X2(s)
• For M2:
• [M2 s^2 + (B + B2)s + K] X2(s) = (B s + K) X1(s)
+ F(s)](https://image.slidesharecdn.com/mechanicalsystemtransferfunctions-250614033912-0b2ca6dd/85/Mechanical_System_Transfer_Functions-pptx-6-320.jpg)
![Solve for X1(s)
• X1(s) = [(B s + K) X2(s)] / [M1 s^2 + (B1 + B)s +
(K1 + K)]](https://image.slidesharecdn.com/mechanicalsystemtransferfunctions-250614033912-0b2ca6dd/85/Mechanical_System_Transfer_Functions-pptx-7-320.jpg)
![Substitute into M2 Equation
• X2(s) * [M2 s^2 + (B + B2) s + K - ((B s + K)^2 /
[M1 s^2 + (B1 + B)s + (K1 + K)])] = F(s)](https://image.slidesharecdn.com/mechanicalsystemtransferfunctions-250614033912-0b2ca6dd/85/Mechanical_System_Transfer_Functions-pptx-8-320.jpg)
![Final Transfer Function
• X2(s)/F(s) = 1 / [M2 s^2 + (B + B2) s + K - (B s +
K)^2 / (M1 s^2 + (B1 + B)s + (K1 + K))]](https://image.slidesharecdn.com/mechanicalsystemtransferfunctions-250614033912-0b2ca6dd/85/Mechanical_System_Transfer_Functions-pptx-9-320.jpg)
Mechanical_System_Transfer_Functions.pptx
- 1. Transfer Function of a Mechanical Translational System • Two-mass Spring-Damper System with External Force • (Insert diagram of the system)
- 2. System Overview • - Two masses: M1, M2 • - Springs: K1 (wall to M1), K (M1 to M2) • - Dampers: B1 (M1 to ground), B (between M1 and M2), B2 (M2 to ground) • - Force f(t) applied on M2
- 3. Define Variables • - x1(t): Displacement of M1 • - x2(t): Displacement of M2 • - F(t): External force applied to M2 • - X1(s), X2(s), F(s): Laplace transforms
- 4. Equation of Motion for M1 • M1 * x1'' = -K1 * x1 - B1 * x1' + K(x2 - x1) + B(x2' - x1') • => M1 * x1'' + (B1 + B) * x1' + (K1 + K)x1 = B * x2' + K * x2
- 5. Equation of Motion for M2 • M2 * x2'' = -K(x2 - x1) - B(x2' - x1') - B2 * x2' + f(t) • => M2 * x2'' + (B + B2) * x2' + K * x2 = B * x1' + K * x1 + f(t)
- 6. Laplace Transforms • For M1: • [M1 s^2 + (B1 + B)s + (K1 + K)] X1(s) = (B s + K) X2(s) • For M2: • [M2 s^2 + (B + B2)s + K] X2(s) = (B s + K) X1(s) + F(s)
- 7. Solve for X1(s) • X1(s) = [(B s + K) X2(s)] / [M1 s^2 + (B1 + B)s + (K1 + K)]
- 8. Substitute into M2 Equation • X2(s) * [M2 s^2 + (B + B2) s + K - ((B s + K)^2 / [M1 s^2 + (B1 + B)s + (K1 + K)])] = F(s)
- 9. Final Transfer Function • X2(s)/F(s) = 1 / [M2 s^2 + (B + B2) s + K - (B s + K)^2 / (M1 s^2 + (B1 + B)s + (K1 + K))]
- 10. Summary and Applications • - Transfer function derived using Newton's laws and Laplace Transform • - System represents real-world systems: vehicles, robotics, etc. • - Useful for simulation and control design
- 11. Mechanical Rotational System - Free Body Diagram • System Components: • - Disk with J = 2 kg-m²/rad • - B = 4 N-m/(rad/s) • - K = 8 N-m/rad • - External torque T(t) • (Insert Free Body Diagram showing torque, damping opposing , and spring opposing θ) θ̇
- 12. Rotational System Equation • Using Newton’s Second Law: • J (t) + B (t) + K θ(t) = T(t) θ̈ θ̇ • Where: • J: Moment of inertia • B: Damping coefficient • K: Spring stiffness • T(t): Input torque • θ(t): Angular displacement
- 13. Summary • - Free-body diagram models rotational dynamics • - Equation from torque balance • - Analogous to translational systems