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translational_mechanical

# translational_mechanical - ME 375 Handouts Translational...

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Unformatted text preview: ME 375 Handouts Translational Mechanical Systems • • • • • • Basic (Idealized) Modeling Elements (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 School of Mechanical Engineering Purdue University ME375 Translation - 1 Variables • • • • • • x : displacement [m] displacement [m] v : velocity [m/sec] [m/sec] a : acceleration [m/sec2] [m/sec f : force [N] p : power [Nm/sec] [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 considered into the lumped model. • In large displacement operation springs are nonlinear. nonlinear fS – Idealization • Massless • No Damping • Linear – Stores (x2 − x1) Energy School of Mechanical Engineering Purdue University ME375 Translation - 4 2 ME 375 Handouts Basic (Idealized) Modeling Elements • Damper • Mass – Friction Element – Inertia Element x1 x2 x fD fD f2 && f D = B ( x2 − x1 ) = B ( v2 − v1 ) M f3 f1 – Dissipate Energy M && = ∑ fi = f1 − f 2 − f3 x fD i – Stores Kinetic Energy && ( x2 − x1 ) School of Mechanical Engineering Purdue University ME375 Translation - 5 Series Connection • Springs in Series x2 x1 fS fS K1 K2 x2 x1 ⇔ fS fS School of Mechanical Engineering Purdue University KEQ ME375 Translation - 6 3 ME 375 Handouts Parallel Connection • Springs in Parallel x2 x1 x2 x1 K1 fS ⇔ fS fS fS KEQ K2 School of Mechanical Engineering Purdue University ME375 Translation - 7 Series Connection • Dampers in Series fD fD B1 x2 x1 x2 x1 ⇔ fD fD BEQ B2 School of Mechanical Engineering Purdue University ME375 Translation - 8 4 ME 375 Handouts Parallel Connection • Dampers in Parallel x2 x1 x2 x1 fD B1 fD ⇔ fD fD BEQ B2 School of Mechanical Engineering Purdue University ME375 Translation - 9 Interconnection Laws • Newton’s Second Law d x ( M v ) = M && = ∑ 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 - 10 5 ME 375 Handouts 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. 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. • Validate Model by Comparing Simulation Results with Physical Measurements School of Mechanical Engineering Purdue University ME375 Translation - 11 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 Total Applied Force f B 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: w.r.t. t1 && ⋅ x dt x& ∫ M243 14 t0 1 & ΔKE = M x2 2 ⇓ + t1 && ∫ Bx ⋅ x dt 14 3 24 t0 t1 &2 ∫t0 Bx dt ≥ 0 ⇓ + ⇐ What have we done ? ⇐ What are we doing now ? t1 & x⋅x ∫ K24dt 14 3 t0 1 ΔPE = K x2 2 ⇓ School of Mechanical Engineering Purdue University = ∫ f ( t ) ⋅ v dt 14243 t1 t0 ΔE Total work done by the applied force f ( t ) from time t0 to t1 ME375 Translation - 12 6 ME 375 Handouts Examples School of Mechanical Engineering Purdue University ME375 Translation - 13 Examples (Continued) School of Mechanical Engineering Purdue University ME375 Translation - 14 7 ME 375 Handouts Example -- SDOF Suspension • Suspension System – Simplified Schematic (neglecting tire model) Minimize the effect of the surface roughness of the road on the drivers’ comfort comfort. Define the reference position for the displacement of the car 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 - 15 SDOF Suspension Q: Since gravity is always present, is there a way to represent the suspension system by subtracting the effect of gravity? School of Mechanical Engineering Purdue University ME375 Translation - 16 8 ME 375 Handouts SDOF Suspension (II) • Relative Displacement Approach 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. School of Mechanical Engineering Purdue University ME375 Translation - 17 SDOF Suspension (II) Q: What are the differences between the two models? Note: When gravity is involved, in order to simplify the analysis, always define the displacement from the static equilibrium of the system. system. Q: Q: Do the two models represent the same physical system? If they do, why are they different? School of Mechanical Engineering Purdue University ME375 Translation - 18 9 ME 375 Handouts MDOF Suspension • Suspension System – Simplified Schematic (with tire model) School of Mechanical Engineering Purdue University ME375 Translation - 19 MDOF Suspension School of Mechanical Engineering Purdue University ME375 Translation - 20 10 ...
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