Unformatted text preview: ME 375 Handouts Translational Mechanical Systems
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• Basic (Idealized) Modeling Elements
(Idealized) Modeling Elements
Interconnection
Interconnection Relationships Physical Laws
Derive
Derive Equation of Motion (EOM)  SDOF
Energy Transfer
Series and Parallel Connections
Derive
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]
velocity [m/sec]
a : acceleration [m/sec2]
acceleration [m/sec
f : force [N]
p : power [Nm/sec]
power [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 into the lumped
model.
model.
• In large displacement operation
In
springs are nonlinear.
nonlinear
fS – Idealization
• Massless
• No Damping
• Linear – Stores
Stores Energy School of Mechanical Engineering
Purdue University (x2  x1) ME375 Translation  4 2 ME 375 Handouts Practical Nonlinear Spring
Engine Mount:
#T062 VERTICAL Experimental Analytical 4000 2 LOAD (N) 2000 1 0 2000 4000
20 15 10 5 0 5 DISP (mm)
(mm) Restoring force ! +K ) /.x 2 ,.x
1 Small motions
0K
for isolation 2 Large motions
for static loads ! K ) /.x 2 School of Mechanical Engineering
Purdue University ME375 Translation  5 Series Connection
• Springs in Series
x2 x1
fS K1 K2 x2 x1
fS 1 fS School of Mechanical Engineering
Purdue University KEQ fS ME375 Translation  6 3 ME 375 Handouts Parallel Connection
• Springs in Parallel
x2 x1 K1 fS x2 x1 1 fS fS fS KEQ K2 School of Mechanical Engineering
Purdue University ME375 Translation  7 Basic (Idealized) Modeling Elements
• Damper • Mass – Friction Element – Inertia Element x1 x2 fD x
fD !!
f D ! B + x2  x1 , ! B + v2  v1 , f2 M f3 f1 – Dissipate Energy M !! ! 2 fi ! f1  f 2  f3
x fD i – Stores Kinetic Energy
!!
+ x2  x1 , School of Mechanical Engineering
Purdue University ME375 Translation  8 4 ME 375 Handouts Series Connection
• Dampers in Series
fD 1 fD
B1 x2 x1 x2 x1 B2 fD fD
BEQ School of Mechanical Engineering
Purdue University ME375 Translation  9 Parallel Connection
• Dampers in Parallel
x2 x1 x2 x1
fD B1 fD 1 fD fD
BEQ B2 School of Mechanical Engineering
Purdue University ME375 Translation  10 5 ME 375 Handouts Interconnection Laws
• Newton’s Second Law
d
x
+ M v , ! M !! ! 2 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  11 Modeling Steps
• Understand System Function, Define Problem, and
Understand
Identif
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.
Combine • Validate Model by Comparing Simulation Results
Validate
with Physical Measurements
School of Mechanical Engineering
Purdue University ME375 Translation  12 6 ME 375 Handouts 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
B Total
Applied Force f 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:
Integrate
t1 !! ( x dt
x!
* M%$&
#$
t0 1
!
.KE ! M x2
2 6 ) t1 ! ! dt
* Bx ( x&
#$
%$
t0 t1
!2
*t0 Bx dt 5 0 6 ) 34What have we done ? 34What are we doing now ? t1 !
x(x
* K%$dt
#$ &
t0 1
.PE ! K x2
2 6 ! t1 * f + t , ( v dt
#$%$&
t0 .E Total work done by the
applied force f ( t ) from
time t0 to t1 School of Mechanical Engineering
Purdue University ME375 Translation  13 School of Mechanical Engineering
Purdue University ME375 Translation  14 Examples 7 ME 375 Handouts Examples (Continued) School of Mechanical Engineering
Purdue University ME375 Translation  15 Examples (Continued) School of Mechanical Engineering
Purdue University ME375 Translation  16 8 ME 375 Handouts Examples (Continued) School of Mechanical Engineering
Purdue University ME375 Translation  17 Example
Example  SDOF Suspension
• Suspension System – Simplified Schematic (neglecting tire model) Minimize
Minimize the effect of the surface
roughness of the road on the drivers’
comfort
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  18 9 ME 375 Handouts SDOF Suspension
– Apply Interconnection Laws – Draw FBD Q: Since gravity is always present, is there a
Since
way to represent the suspension system by
subtracting the effect of gravity?
School of Mechanical Engineering
Purdue University ME375 Translation  19 SDOF
SDOF Suspension (II)
• Relative Displacement Approach
Define
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. – FBD School of Mechanical Engineering
Purdue University ME375 Translation  20 10 ME 375 Handouts SDOF Suspension (II)
– Interconnection Laws &
Interconnection
Simplification
Simplification Q: What are the differences between the two
What
models?
models? Q: Do the two models represent the same
Do
physical system? If they do, why are they
different?
different? School of Mechanical Engineering
Purdue University ME375 Translation  21 MDOF Suspension
• Suspension System – Simplified Schematic (with tire model) School of Mechanical Engineering
Purdue University ME375 Translation  22 11 ME 375 Handouts MDOF Suspension
– Draw FBD – Apply Interconnection Laws School of Mechanical Engineering
Purdue University ME375 Translation  23 MDOF Suspension
– Matrix Form School of Mechanical Engineering
Purdue University ME375 Translation  24 12 ...
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 Fall '10
 Meckle
 Mechanical Engineering, Force, Mass, Purdue University, School of Mechanical Engineering

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