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MESB 374 System Modeling and Analysis Translational Mechanical System

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## MESB 374 System Modeling and Analysis Translational Mechanical System

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**MESB 374 System Modeling and AnalysisTranslational**Mechanical System**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**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 )**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)**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**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**K,M**x x M K M K Interconnection Laws • Newton’s Second Law Lumped Model of a Flexible Beam • Newton’s Third Law • Action & Reaction Forces Massless spring E.O.M.**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**Vertical Single Degree of Freedom (SDOF) System**xs x B,K,M B K f g x f M M M g • Define Problem The motion of the object • Input • Output • Develop Mathematical Model (Diff. Eq.) • Identify reference point and positive direction. • Draw Free-Body-Diagram (FBD) • Write Elemental Equations From the undeformed position From the deformed (static equilibrium) position • Validate Model by Comparing Simulation Results with Physical Measurement**x**K M B 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 ? f Change of kinetic energy Change of potential energy Energy dissipated by damper**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**x2**x2 x2 x1 x1 xj x1 Û fS fS fS fS K1 K1 K2 K2 KEQ fS Series Connection • Springsin Series fS**x2**x1 x2 x1 fD fD Series Connection • Dampersin Series Û fD fD BEQ B1 B2**x2**x1 x2 x1 K1 fS fS KEQ K2 Parallel Connection • Springsin Parallel fS fS Û**x2**x1 x2 x1 B1 fD fD fD fD BEQ B2 Parallel Connection • Dampersin Parallel Û**K**K Horizontal Two Degree of Freedom (TDOF) System • DOF = 2 • Absolute coordinates • FBD K K • Newton’s law**K**K Horizontal Two Degree of Freedom (TDOF) System Static coupling • Absolute coordinates • Relative coordinates Dynamic coupling**Two DOF System – Matrix Form of EOM**K K Input vector • Absolute coordinates Output vector Mass matrix Damping matrix SYMMETRIC • Relative coordinates Stiffness matrix NON-SYMMETRIC**MDOF Suspension**• Simplified Schematic (with tire model) • Suspension System TRY THIS x p**MDOF Suspension**• Simplified Schematic (with tire model) • Suspension System Assume ref. is when springs are Deflected by weights Car body Suspension Wheel Tire Road Reference**FS2**FS1 FD1 FD2 x2 x1 FS2 FS2 FD2 FD2 FS2 FD1 FD2 FD2 FS2 M1 M2 FS1 Example -- MDOF Suspension • Draw FBD • Apply Interconnection Laws**Example -- MDOF Suspension**• Matrix Form Mass matrix Damping matrix Stiffness matrix Input Vector