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 FreeBodyDiagram (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 MassSpringDamper 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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 Spring '08
 Leyla,O
 Mechanical Engineering, Force, Coil spring, School of Mechanical Engineering Purdue University

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