MESB 374 System Modeling and Analysis Translational Mechanical System

1. MESB 374 System Modeling and AnalysisTranslational Mechanical System

2. Translational Mechanical Systems • Basic (Idealized) Modeling Elements • Interconnection Relationships -Physical Laws • Derive Equation of Motion (EOM) - SDOF • Energy Transfer • Series and Parallel Connections • Derive Equation of Motion (EOM) - MDOF

3. Key Concepts to Remember • Three primary elements of interest • Mass ( inertia ) M • Stiffness ( spring ) K • Dissipation ( damper ) B • Usually we deal with “equivalent” M, K, B • Distributed mass  lumped mass • Lumped parameters • Mass maintains motion • Stiffness restores motion • Damping eliminates motion (Kinetic Energy) (Potential Energy) (Eliminate Energy ? ) (Absorb Energy )

4. Variables • x : displacement [m] • v: velocity [m/sec] • a: acceleration [m/sec2] • f: force [N] • p : power [Nm/sec] • w : work ( energy ) [Nm] 1 [Nm] = 1 [J] (Joule)

5. x2 K Basic (Idealized) Modeling Elements • Spring • Stiffness Element • Reality • 1/3 of the spring mass may be considered into the lumped model. • In large displacement operation springs are nonlinear. Linear spring  nonlinear spring  broken spring !! • Idealization • Massless • No Damping • Linear • Stores Energy Hard Spring Potential Energy Soft Spring

6. x1 x2 x fD fD f2 M f1 B f3 Basic (Idealized) Modeling Elements • Damper • Friction Element • Mass • Inertia Element • Dissipate Energy fD • Stores Kinetic Energy

7. Interconnection Laws • Newton’s Second Law Lumped Model of a Flexible Beam K,M • Newton’s Third Law • Action & Reaction Forces x x M K M K Massless spring E.O.M.

8. Modeling Steps • Understand System Function, Define Problem, and Identify Input/Output Variables • Draw Simplified Schematics Using Basic Elements • Develop Mathematical Model (Diff. Eq.) • Identify reference point and positive direction. • Draw Free-Body-Diagram (FBD) for each basic element. • Write Elemental Equations as well as Interconnecting Equations by applying physical laws. (Check: # eq = # unk) • Combine Equations by eliminating intermediate variables. • Validate Model by Comparing Simulation Results with Physical Measurements

9. Vertical Single Degree of Freedom (SDOF) System g • Define Problem The motion of the object • Input • Output xs • Develop Mathematical Model (Diff. Eq.) • Identify reference point and positive direction. x B,K,M f B K M g M • Draw Free-Body-Diagram (FBD) M x f • Write Elemental Equations From the undeformed position From the deformed (static equilibrium) position • Validate Model by Comparing Simulation Results with Physical Measurement

10. M f B x K Energy Distribution • EOM of a simple Mass-Spring-Damper System We want to look at the energy distribution of the system. How should we start ? • Multiply the above equation by the velocity term v : Ü What have we done ? • Integrate the second equation w.r.t. time: Ü What are we doing now ? Change of kinetic energy Change of potential energy Energy dissipated by damper

11. Example -- SDOF Suspension (Example) • Simplified Schematic (neglecting tire model) • Suspension System Minimize the effect of the surface roughness of the road on the drivers comfort. From the “absolute zero” From the path From nominal position g M x K B x x p

12. x2 x2 x2 x1 x1 xj x1 Û fS fS fS fS K1 K1 K2 K2 KEQ fS Series Connection • Springsin Series fS

13. x2 x1 x2 x1 fD fD Series Connection • Dampersin Series Û fD fD BEQ B1 B2

14. x2 x1 x2 x1 K1 fS fS KEQ K2 Parallel Connection • Springsin Parallel fS fS Û

15. x2 x1 x2 x1 B1 fD fD fD fD BEQ B2 Parallel Connection • Dampersin Parallel Û

16. Horizontal Two Degree of Freedom (TDOF) System • DOF = 2 • Absolute coordinates • FBD • Newton’s law

17. Horizontal Two Degree of Freedom (TDOF) System Static coupling • Absolute coordinates • Relative coordinates Dynamic coupling

18. Two DOF System – Matrix Form of EOM Input vector • Absolute coordinates Output vector Mass matrix Damping matrix SYMMETRIC • Relative coordinates Stiffness matrix NON-SYMMETRIC

19. TRY THIS x p MDOF Suspension • Simplified Schematic (with tire model) • Suspension System

20. Car body Suspension Wheel Tire Road Reference MDOF Suspension • Simplified Schematic (with tire model) • Suspension System Assume ref. is when springs are Deflected by weights

21. x2 M2 FS2 FD2 FS2 FD2 FD2 FS2 FD2 FS2 x1 M1 FD1 FS1 FD2 FS2 FD1 FS1 Example -- MDOF Suspension • Draw FBD • Apply Interconnection Laws

22. Example -- MDOF Suspension • Matrix Form Mass matrix Damping matrix Stiffness matrix Input Vector