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 Interconnection Relationships -Physical Laws Derive 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 1 ME 375 Handouts Variables • • • • • • x : displacement [m] [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] [Nm] 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 2 ME 375 Handouts Idealized Modeling Elements • Inertia (mass) • Stiffness (spring) • Dissipation (damper) School of Mechanical Engineering Purdue University ME375 Translation - 3 3 ME 375 Handouts 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 considered into the lumped model. • In large displacement operation springs are nonlinear springs are nonlinear. fS – Idealization • Massless • No Damping • Linear – Stores (x2 − x1) Energy School of Mechanical Engineering Purdue University ME375 Translation - 4 4 ME 375 Handouts Basic (Idealized) Modeling Elements • Damper • Mass – Friction Element – Inertia Element x1 x2 x fD fD && f D = B ( x2 − x1 ) = B ( v2 − v1 ) f2 f3 M f1 – Dissipate Energy M && = ∑ fi = f1 − f 2 − f 3 x fD i – Stores Kinetic Energy && ( x2 − x1 ) School of Mechanical Engineering Purdue University ME375 Translation - 5 5 ME 375 Handouts Series Connection • Springs in Series x2 x1 fS K1 fS K2 x2 x1 ⇔ fS fS School of Mechanical Engineering Purdue University KEQ ME375 Translation - 6 6 ME 375 Handouts Parallel Connection • Springs in Parallel x2 x1 x2 x1 fS K1 fS ⇔ fS fS KEQ K2 School of Mechanical Engineering Purdue University ME375 Translation - 7 7 ME 375 Handouts 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 8 ME 375 Handouts Parallel Connection • Dampers in Parallel Parallel x2 x1 x2 x1 fD B1 fD ⇔ fD fD BEQ B2 School of Mechanical Engineering Purdue University ME375 Translation - 9 9 ME 375 Handouts Interconnection Laws • Newton’s Second Law 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 10 ME 375 Handouts Modeling Steps • Identify reference point and positive direction. direction. • Draw Free-Body-Diagram (FBD) for each basic element. Free-Bodyelement. • Write Elemental Equations as well as Interconnecting El Equations Equations by applying Newton’s laws. • Obtain Equations of Motion (EOM): Combine Equations by eliminating intermediate variables. (Check: # eq = # unknown = #DOF) School of Mechanical Engineering Purdue University ME375 Translation - 11 11 ME 375 Handouts Energy Distribution • EOM of a simple Mass-Spring-Damper System of simple Mass System M && + 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 x 2 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 x 2 2 ⇓ School of Mechanical Engineering Purdue University = ∫ f ( t ) ⋅ v dt 14243 t1 t0 ΔE Total work done by the work done by the applied force f ( t ) from time t0 to t1 ME375 Translation - 12 12 ME 375 Handouts Examples School of Mechanical Engineering Purdue University ME375 Translation - 13 13 ME 375 Handouts Examples (Continued) School of Mechanical Engineering Purdue University ME375 Translation - 14 14 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. 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 - 15 15 ME 375 Handouts SDOF Suspension School of Mechanical Engineering Purdue University ME375 Translation - 16 16 ME 375 Handouts MDOF Suspension • Suspension System – Simplified Schematic (with tire model) School of Mechanical Engineering Purdue University ME375 Translation - 17 17 ME 375 Handouts MDOF Suspension School of Mechanical Engineering Purdue University ME375 Translation - 18 18 ...
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## This note was uploaded on 12/23/2011 for the course ME 375 taught by Professor Meckle during the Fall '10 term at Purdue.

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