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Translational_Mechanical System

# Translational_Mechanical System - ME 375 Handouts...

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Unformatted text preview: ME 375 Handouts Translational Mechanical Systems • • • • • • Basic (Idealized) Modeling Elements (Idealized) Modeling Elements Interconnection Interconnection Relationships -Physical Laws Derive Derive Equation of Motion (EOM) - SDOF Energy Transfer Series and Parallel Connections Derive Derive Equation of Motion (EOM) - MDOF School of Mechanical Engineering Purdue University ME375 Translation - 1 Variables • • • • • • x : displacement [m] displacement [m] v : velocity [m/sec] velocity [m/sec] a : acceleration [m/sec2] acceleration [m/sec f : force [N] p : power [Nm/sec] power [Nm/sec] w : work ( energy ) [Nm] work 1 [Nm] = 1 [J] (Joule) d ! x! x!v dt d d"d ! v!v! \$ dt dt & dt 2 #d x % ! 2 x ! !! ! a x ' dt d ! p ! f (v ! f ( x ! w dt t1 w(t1 ) ! w(t0 ) ) * p (t ) dt t0 t1 ! ! w(t0 ) ) * ( f ( x) dt t0 School of Mechanical Engineering Purdue University ME375 Translation - 2 1 ME 375 Handouts Idealized Modeling Elements • Inertia (mass) (mass) • Stiffness (spring) • Dissipation (damper) School of Mechanical Engineering Purdue University ME375 Translation - 3 Basic (Idealized) Modeling Elements • Spring – Reality – Stiffness Element x2 x1 fS fS K f S ! K + x2 - x1 , • 1/3 of the spring mass may be 1/3 of the spring mass may be considered into the lumped model. model. • In large displacement operation In springs are nonlinear. nonlinear fS – Idealization • Massless • No Damping • Linear – Stores Stores Energy School of Mechanical Engineering Purdue University (x2 - x1) ME375 Translation - 4 2 ME 375 Handouts Practical Nonlinear Spring Engine Mount: #T062 VERTICAL Experimental Analytical 4000 2 LOAD (N) 2000 1 0 -2000 -4000 -20 -15 -10 -5 0 5 DISP (mm) (mm) Restoring force ! +K ) /.x 2 ,.x 1 Small motions 0K for isolation 2 Large motions for static loads ! K ) /.x 2 School of Mechanical Engineering Purdue University ME375 Translation - 5 Series Connection • Springs in Series x2 x1 fS K1 K2 x2 x1 fS 1 fS School of Mechanical Engineering Purdue University KEQ fS ME375 Translation - 6 3 ME 375 Handouts Parallel Connection • Springs in Parallel x2 x1 K1 fS x2 x1 1 fS fS fS KEQ K2 School of Mechanical Engineering Purdue University ME375 Translation - 7 Basic (Idealized) Modeling Elements • Damper • Mass – Friction Element – Inertia Element x1 x2 fD x fD !! f D ! B + x2 - x1 , ! B + v2 - v1 , f2 M f3 f1 – Dissipate Energy M !! ! 2 fi ! f1 - f 2 - f3 x fD i – Stores Kinetic Energy !! + x2 - x1 , School of Mechanical Engineering Purdue University ME375 Translation - 8 4 ME 375 Handouts Series Connection • Dampers in Series fD 1 fD B1 x2 x1 x2 x1 B2 fD fD BEQ School of Mechanical Engineering Purdue University ME375 Translation - 9 Parallel Connection • Dampers in Parallel x2 x1 x2 x1 fD B1 fD 1 fD fD BEQ B2 School of Mechanical Engineering Purdue University ME375 Translation - 10 5 ME 375 Handouts Interconnection Laws • Newton’s Second Law d x + M v , ! M !! ! 2 f EXTi dt " i Linear Momentum • Newton’s Third Law x – Action & Reaction Forces K M M K • Displacement Law School of Mechanical Engineering Purdue University ME375 Translation - 11 Modeling Steps • Understand System Function, Define Problem, and Understand Identif 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. Free-Body– Write Elemental Equations as well as Interconnecting Write Elemental Equations as well as Interconnecting Equations Equations by applying physical laws. (Check: # eq = # unk) – Combine Equations by eliminating intermediate variables. Combine • Validate Model by Comparing Simulation Results Validate with Physical Measurements School of Mechanical Engineering Purdue University ME375 Translation - 12 6 ME 375 Handouts Energy Distribution • EOM of a simple Mass-Spring-Damper System Mass-SpringM !! ) x " Contribution of Inertia ! Bx " ) Contribution of the Damper ! Kx " Contribution of the Spring x K f (t ) " M B Total Applied Force f We want to look at the energy distribution of the system. How should we start ? • Multiply the above equation by the velocity term v : • Integrate the second equation w.r.t. time: Integrate t1 !! ( x dt x! * M%\$& #\$ t0 1 ! .KE ! M x2 2 6 ) t1 ! ! dt * Bx ( x& #\$ %\$ t0 t1 !2 *t0 Bx dt 5 0 6 ) 34What have we done ? 34What are we doing now ? t1 ! x(x * K%\$dt #\$ & t0 1 .PE ! K x2 2 6 ! t1 * f + t , ( v dt #\$%\$& t0 .E Total work done by the applied force f ( t ) from time t0 to t1 School of Mechanical Engineering Purdue University ME375 Translation - 13 School of Mechanical Engineering Purdue University ME375 Translation - 14 Examples 7 ME 375 Handouts Examples (Continued) School of Mechanical Engineering Purdue University ME375 Translation - 15 Examples (Continued) School of Mechanical Engineering Purdue University ME375 Translation - 16 8 ME 375 Handouts Examples (Continued) School of Mechanical Engineering Purdue University ME375 Translation - 17 Example Example -- SDOF Suspension • Suspension System – Simplified Schematic (neglecting tire model) Minimize Minimize the effect of the surface roughness of the road on the drivers’ comfort comfort. Define Define the reference position for the displacement of the car as the position when the spring does not have any deflection (i.e., the neutral position) School of Mechanical Engineering Purdue University ME375 Translation - 18 9 ME 375 Handouts SDOF Suspension – Apply Interconnection Laws – Draw FBD Q: Since gravity is always present, is there a Since way to represent the suspension system by subtracting the effect of gravity? School of Mechanical Engineering Purdue University ME375 Translation - 19 SDOF SDOF Suspension (II) • Relative Displacement Approach Define Define the reference position as the position of the car when the system is at rest in the gravity field, car when the system is at rest in the gravity field, i.e., i.e., the spring force balances the car’s weight. – FBD School of Mechanical Engineering Purdue University ME375 Translation - 20 10 ME 375 Handouts SDOF Suspension (II) – Interconnection Laws & Interconnection Simplification Simplification Q: What are the differences between the two What models? models? Q: Do the two models represent the same Do physical system? If they do, why are they different? different? School of Mechanical Engineering Purdue University ME375 Translation - 21 MDOF Suspension • Suspension System – Simplified Schematic (with tire model) School of Mechanical Engineering Purdue University ME375 Translation - 22 11 ME 375 Handouts MDOF Suspension – Draw FBD – Apply Interconnection Laws School of Mechanical Engineering Purdue University ME375 Translation - 23 MDOF Suspension – Matrix Form School of Mechanical Engineering Purdue University ME375 Translation - 24 12 ...
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