Upload
rollo
View
36
Download
7
Embed Size (px)
DESCRIPTION
differential equations of spring pendulum system
Citation preview
Spring Pendulum Dynamic System Investigation K. Craig 1
Spring PendulumDynamic System Investigation
Dr. Kevin CraigProfessor of Mechanical Engineering
Rensselaer Polytechnic Institute
Spring Pendulum Dynamic System Investigation K. Craig 2
Spring-PendulumPhysical System
Spring Pendulum Dynamic System Investigation K. Craig 3
EngineeringSystem
InvestigationProcess
Spring-Pendulum
Dynamic SystemInvestigation
Engineering System Investigation Process
PhysicalSystem
SystemMeasurement
MeasurementAnalysis
PhysicalModel
MathematicalModel
ParameterIdentification
MathematicalAnalysis
Comparison:Predicted vs.
Measured
DesignChanges
Is The ComparisonAdequate ?
NO
YES
START HERE
Spring Pendulum Dynamic System Investigation K. Craig 4
Physical ModelSimplifying Assumptions
pure spring, i.e., negligible inertia and damping ideal (linear) spring frictionless pivot neglect all material damping and air damping point mass, i.e., neglect rotational inertia of mass two degrees of freedom, i.e., r and are the generalized
coordinates (this assumes no out-of-plane motion and no bending of the spring)
support structure is rigid
Spring Pendulum Dynamic System Investigation K. Craig 5
Physical Modelwith
Parameter Identification
m = pendulum mass = 1.815 kgmspring = spring mass = 0.1445 kg = unstretched spring length = 0.333 mk = spring constant = 172.8 N/mg = acceleration due to gravity = 9.81 m/s2Ft = 5.71 N = pre-tension of springrs = static spring stretch, i.e., rs = (mg-Ft)/k = 0.070 m rd = dynamic spring stretchr = total spring stretch = rs + rd
Spring Pendulum Dynamic System Investigation K. Craig 6
Spring Parameter Identification
spring
t
Spring Pendulum Dynamic System Investigation K. Craig 7
magnitude changedirection changemagnitude change
direction change2
r
r
r r
r
+
( ) ( )
r
r r r
2r
r r
r redr v re r e v e v edtdv a r r e r 2r edt
a e a e
== = + = +
= = + + = +
GGG GG
rv
v
r
r
de edde e
d
==
ree
Polar Coordinates:Position, Velocity, Acceleration
Spring Pendulum Dynamic System Investigation K. Craig 8
Rigid Body KinematicsXY: R reference frame (ground)xy: R1 reference frame (pendulum)
x cos sin 0 Xy sin cos 0 Yz 0 0 1 Z i Icos sin 0 j sin cos 0 J 0 0 1 Kk
= =
( )1 1 1 1 1 1R R R R R RR P R O R R OP R OP P R Pa a r r a 2 v = + + + + G G G G G G G G G G
m + r
k
X
Y
xy
P
O
Spring Pendulum Dynamic System Investigation K. Craig 9
Rigid Body Kinematics
After substitution and evaluation:
( ) ( )1
1
1
1
R O
RR
OP
RR
R P
R P
a 0 k K
r r j r sin I cos J
k K v rj r sin I cos J
a rj r sin I cos J
= = =
= + = + + = =
= = + = = +
GG G A AG G G
( ) ( )R P 2 a i r 2r j r r = + + + + + G A A
Spring Pendulum Dynamic System Investigation K. Craig 10
Mathematical Model
Free Body Diagram
Nonlinear Equationsof Motion
( )( )
2r rF ma m r r
F ma m r 2r
= = + = = + +
A A
( )( )
2tmr m r kr F mgcos 0
r 2r gsin 0
+ + + =+ + + =
A A
( )( )
2tkr F mg cos m r r
mgsin m r 2r
+ = + = + +
A A
t
Spring Pendulum Dynamic System Investigation K. Craig 11
Mathematical Model:Lagranges Equations Lagranges Equations
Generalized Coordinates
Kinetic Energy
Potential Energy
GeneralizedForces
Nonlinear Equationsof Motion
( )( )
2tmr m r kr F mgcos 0
r 2r gsin 0
+ + + =+ + + =
A A
( )21V kr mg r cos2
= + A A
( )22 21T m r r2
= + + A
1
2
q rq==
r tQ FQ 0
= =
ii i i
d T T V Qdt q q q + =
Spring Pendulum Dynamic System Investigation K. Craig 12
LabVIEW Simulation Diagram
Spring Pendulum Dynamic System Investigation K. Craig 13
Spring PendulumDynamic System
ttime
thetatheta position
u^2
square
0.333
spring lengthunstretched
(meters)
sin(u)
sin
rr position
95.21
k/mk=172.8 N/mm=1.815 kg
u (^-1)
inverse
9.81
gravity (m/s^2)
cos(u)
cos
Sum2
Sum
Sum
Product
Product
Product
Product
Product
1/s
Integrate r acc
1/s
Integratetheta vel
1/s
Integratetheta acc
1/s
Integrater vel
2
Gain
5.710/1.815
Ft=5.71 Nm=1.815 kg
Clock
MatLab Simulink Diagram
Spring Pendulum Dynamic System Investigation K. Craig 14
Simulation Results
0 10 20 30 40 50 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
time (sec)
r
a
d
i
a
l
a
n
d
a
n
g
u
l
a
r
p
o
s
i
t
i
o
n
(
r
a
d
o
r
m
)
Simulation Results with Initial Conditions: theta = -0.274 rad, r = 0.046 m
InitialConditions
0
0
0.274 radr 0.046 m =
=
Spring Pendulum Dynamic System Investigation K. Craig 15
0 10 20 30 40 50 60-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
time (sec)
r
a
d
i
a
l
a
n
d
a
n
g
u
l
a
r
p
o
s
i
t
i
o
n
(
r
a
d
o
r
m
)
Simulation Results with Initial Conditions: theta = 0.021 rad, r = 0.115 m
Simulation Results
0
0
0.021 radr 0.115 m =
=
InitialConditions
Spring Pendulum Dynamic System Investigation K. Craig 16
Actual Measured Dynamic Behavior
0 10 20 30 40 50 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
time (sec)
r
a
d
i
a
l
a
n
d
a
n
g
u
l
a
r
p
o
s
i
t
i
o
n
(
r
a
d
o
r
m
)
Experimental Results with Initial Conditions: theta = -0.274 rad, r = 0.046 m
InitialConditions
0
0
0.274 radr 0.046 m =
=
Spring Pendulum Dynamic System Investigation K. Craig 17
Actual Measured Dynamic Behavior
0 10 20 30 40 50 60-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
time (sec)
r
a
d
i
a
l
a
n
d
a
n
g
u
l
a
r
p
o
s
i
t
i
o
n
(
r
a
d
o
r
m
)
Experimental Results with Initial Conditions: theta = 0.021 rad, r = 0.115 m
InitialConditions
0
0
0.021 radr 0.115 m =
=
Spring Pendulum Dynamic System Investigation K. Craig 18
Comparison
0 10 20 30 40 50 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
time (sec)
r
a
d
i
a
l
a
n
d
a
n
g
u
l
a
r
p
o
s
i
t
i
o
n
(
r
a
d
o
r
m
)
Simulation Results with Initial Conditions: theta = -0.274 rad, r = 0.046 m
0 10 20 30 40 50 60-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
time (sec)r
a
d
i
a
l
a
n
d
a
n
g
u
l
a
r
p
o
s
i
t
i
o
n
(
r
a
d
o
r
m
)
Experimental Results with Initial Conditions: theta = -0.274 rad, r = 0.046 m
Initial Conditions: 00
0.274 radr 0.046 m =
=
Spring Pendulum Dynamic System Investigation K. Craig 19
Comparison
0 10 20 30 40 50 60-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
time (sec)
r
a
d
i
a
l
a
n
d
a
n
g
u
l
a
r
p
o
s
i
t
i
o
n
(
r
a
d
o
r
m
)
Simulation Results with Initial Conditions: theta = 0.021 rad, r = 0.115 m
0 10 20 30 40 50 60-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
time (sec)
r
a
d
i
a
l
a
n
d
a
n
g
u
l
a
r
p
o
s
i
t
i
o
n
(
r
a
d
o
r
m
)
Experimental Results with Initial Conditions: theta = 0.021 rad, r = 0.115 m
Initial Conditions: 00
0.021 radr 0.115 m =
=
Spring Pendulum Dynamic System Investigation K. Craig 20
Nonlinear Resonancemr m r kr F mg
r r gt cos
sin + + + =+ + + =
AAa f
a f
2 0
2 0Nonlinear Equations
of Motion
mr kr m r F mg
r g r
t !
!
+ = + + FHGIKJ
+ + FHGIKJ =
A
A
a f
a f
22
3
12
32
sin!
cos!
+
+
3
23
12
"
"
!
!
r km
r r Fm
g g
r g r g
t+ = + +
+ + = +
A
A
a f
a f
22
32
23
Spring Pendulum Dynamic System Investigation K. Craig 21
r r r
r mg Fk
t
= += +==
0
!
!
r km
r r r r Fm
g g
r r g r g
t+ + = + + +
+ + + = +
a f a f
a f
A
A
22
3
2
23
!
!
r km
r r r g
gr
rr
gr
rr
+ = + +
+ + =+ + + +
A
A A A A
a f
22
3
22
3
rkm
gr
2
2
=
= +A
!
!
r r r r g
rr
rr
r+ = + +
+ = + + +
2 22
2 23
22
3
A
A A
a f
Spring Pendulum Dynamic System Investigation K. Craig 22
A A A+ + = + + = +r r r x r r x a f a fa f a f1A A A+ + + = + +
+ = +
r x r x r x g
x x
ra f a f a fa f
!
!
2 22
2 23
12
23
Define:
!
!
x x x
x x
r+ = +
+ + = +
2 2 22
2 23
12
1 23
a f
a f
!
!
x x x
xxx x
r+ = +
+ + =+ + +
2 2 22
2
2 2 3
2
12
1 1 3
11
1 2 3+ + x x x x "
Spring Pendulum Dynamic System Investigation K. Craig 23
!
!
x x x
x x x x
r+ = +
+ = +
2 2 22
2
2 23
2
1 2 1 13
a f a f a f
!
!
x x x
x x x x
r+ = +
+ = + +
2 2 22
2
2 2 23
2
2 1 13
a f a f
x x
x x
r+ = ++ = + +
2 2 22
2 2
22
"
"Neglecting nonlinear terms
third order and higher
Spring PendulumDynamic System InvestigationSpring-Pendulum Physical SystemSlide Number 3Physical ModelSimplifying AssumptionsPhysical ModelwithParameter IdentificationSpring Parameter IdentificationPolar Coordinates:Position, Velocity, AccelerationRigid Body KinematicsRigid Body KinematicsMathematical ModelMathematical Model: Lagranges EquationsSlide Number 12Slide Number 13Simulation ResultsSimulation ResultsActual Measured Dynamic BehaviorActual Measured Dynamic BehaviorComparisonComparisonNonlinear ResonanceSlide Number 21Slide Number 22Slide Number 23