Translational Mechanical Systems

Translational Mechanical Systems - 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 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] [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 = x = a ⎠ dt d p = f ⋅v = f ⋅ x = w dt t1 w(t1 ) = w(t0 ) + ∫ p(t ) dt t0 = w(t0 ) + ∫ ( f ⋅ x) dt t0 School of Mechanical Engineering Purdue University ME375 Translation - 2 t1 1 ME 375 Handouts Idealized Modeling Elements • Inertia (mass) • Stiffness (spring) • Dissipation (damper) School of Mechanical Engineering Purdue University ME375 Translation - 3 Basic (Idealized) Modeling Elements Basic • Spring – Stiffness Element x1 fS x2 fS – Reality • 1/3 of the spring mass may be considered into the lumped model. • In large displacement operation springs are nonlinear. nonlinear fS K f S = K ( x2 − x1 ) – Idealization • • • Massless No Damping Linear (x2 − x1) – Stores Energy School of Mechanical Engineering Purdue University ME375 Translation - 4 2 ME 375 Handouts Practical Nonlinear Spring Engine Mount: Experimental 4000 #T062 VERTICAL Analytical 2000 2 1 LOAD (N) 0 -2000 -4000 -20 -15 -10 DISP (mm) -5 0 5 Restoring force = (K + μΔx 2 )Δx 1 Small motions for isolation ≈K 2 Large motions for static loads = K + μΔx 2 School of Mechanical Engineering Purdue University ME375 Translation - 5 Basic (Idealized) Modeling Elements • Damper – Friction Element x1 fD x2 x fD f2 f3 M f1 • Mass – Inertia Element f D = B ( x2 − x1 ) = B ( v2 − v1 ) – Dissipate Energy fD M x = ∑ fi = f1 − f 2 − f3 i – Stores Kinetic Energy ( x2 − x1 ) School of Mechanical Engineering Purdue University ME375 Translation - 6 3 ME 375 Handouts Interconnection Laws • Newton’s Second Law d ( M v ) = M x = ∑ f EXTi dt i Linear Momentum • Newton’s Third Law – Action & Reaction Forces K M K M x • Displacement Law School of Mechanical Engineering Purdue University ME375 Translation - 7 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 Equations by applying physical laws. (Check: # eq = # unk) unk) – Combine Equations by eliminating intermediate variables. • Validate Model by Comparing Simulation Results with Physical Measurements School of Mechanical Engineering Purdue University ME375 Translation - 8 4 ME 375 Handouts In Class Exercise (Blood Sampler) Schematic: Schematic: Already done in Introduction; see those Filled Slides. School of Mechanical Engineering Purdue University ME375 Translation - 9 Energy Distribution • EOM of a simple Mass-Spring-Damper System Mass- SpringMx + Contribution of Inertia K M B x f Bx Contribution of the Damper + Kx Contribution of the Spring = f (t ) Total Applied Force We want to look at the energy distribution of the system. How should we start ? • Multiply the above equation by the velocity term v : ⇐ What have we done ? • Integrate the second equation w.r.t. time: ⇐ What are we doing now ? ∫ t1 t0 M x ⋅ x dt + ∫ t1 t0 Bx ⋅ x dt + ∫ t1 t0 K x ⋅ x dt = ∫ f ( t ) ⋅ v dt t1 t0 1 ΔKE = M x2 2 t1 2 ∫t0 Bx dt ≥ 0 1 ΔPE = K x2 2 ⇓ ⇓ ⇓ Total work done by the applied force f ( t ) from time t0 to t1 ΔE School of Mechanical Engineering Purdue University ME375 Translation - 10 5 ME 375 Handouts Example -- SDOF Suspension Example • Suspension System Minimize the effect of the surface roughness of the road on the drivers’ drivers’ comfort. – Simplified Schematic (neglecting tire model) 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 - 11 SDOF Suspension – Draw FBD – Apply Interconnection Laws 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 - 12 6 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, i.e., the spring force balances the car’s weight. car’ – FBD School of Mechanical Engineering Purdue University ME375 Translation - 13 SDOF Suspension (II) – Interconnection Laws & Simplification Q: What are the differences between the two models? 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 - 14 7 ME 375 Handouts Series Connection • Springs in Series x1 fS K1 K2 x2 fS x1 x2 fS KEQ ⇔ fS School of Mechanical Engineering Purdue University ME375 Translation - 15 Series Connection • Dampers in Series x1 fD B1 B2 x2 fD x1 x2 fD BEQ ⇔ fD School of Mechanical Engineering Purdue University ME375 Translation - 16 8 ME 375 Handouts Parallel Connection • Springs in Parallel x1 x2 x1 fS K1 K2 fS x2 fS fS KEQ ⇔ School of Mechanical Engineering Purdue University ME375 Translation - 17 Parallel Connection • Dampers in Parallel x1 x2 x1 fD B1 B2 fD x2 fD BEQ ⇔ fD 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 – Draw FBD – Apply Interconnection Laws School of Mechanical Engineering Purdue University ME375 Translation - 20 10 ME 375 Handouts MDOF Suspension – Matrix Form School of Mechanical Engineering Purdue University ME375 Translation - 21 11 ...
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