Unformatted text preview: ME 375 Handouts Translational Mechanical Systems
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• 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
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• 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 FreeBodyDiagram (FBD) for each basic element.
FreeBodyelement.
• 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 MassSpringDamper 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
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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.
 Fall '10
 Meckle

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