Unformatted text preview: ME 375 Handouts Translational Mechanical Systems
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• 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
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• 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 FreeBodyDiagram (FBD) for each basic element.
FreeBody– 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 MassSpringDamper System
MassSpringM && +
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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 Fall '10
 Meckle
 Mechanical Engineering, Force, Mass, Purdue University, School of Mechanical Engineering

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