spring pendulum investigation

Spring Pendulum Dynamic System Investigation K. Craig 1

Spring PendulumDynamic System Investigation

Dr. Kevin CraigProfessor of Mechanical Engineering

Rensselaer Polytechnic Institute


Spring-PendulumPhysical System


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


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


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 Parameter Identification

spring

t


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


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


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


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


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 + =


LabVIEW Simulation Diagram


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


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 =

=


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


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 =

=


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 =

=


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 =

=


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 =

=


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


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


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 "


!

!

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

spring pendulum investigation

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