Spring-Pendulum Dynamic System Investigation · 2017. 4. 19. · Spring-Pendulum Dynamic System...

Spring-Pendulum Dynamic System Investigation K. Craig 1

Spring-Pendulum

Dynamic System Investigation


An Interesting Experiment

• Here’s an experiment to see the effects of nonlinearities.

– Take any spring and measure its unstretched length, ℓ.

– Attach a mass to the end of the spring. Choose a mass so that the

equilibrium length of the spring (i.e., the length of the spring when there

is no motion) is about (1.33)ℓ.

– Secure the upper end of the spring to a fixed pivot point.

– Pull the mass straight down a little bit from its equilibrium, release it, and

watch its motion.

– The mass moves up and down like a simple spring-mass system should.

But after a little while, the angle the spring makes with the vertical starts

to increase while the up-and-down (or in-and-out) motion decreases. It

decreases until the spring swings laterally only, like a simple pendulum,

essentially without in-and-out motion. Then the angle starts to

decrease, and the in-and-out motion starts to appear again.

– The sequence of events repeats itself. WHY?


Spring-Pendulum Physical System

There is a range of values for the mass

for which this described phenomenon

occurs. For values of mass outside that

range, the motion takes place essentially

along the vertical, where it started.

This is a nonlinear phenomenon called

Nonlinear Resonance!

It cannot be predicted by linearizing the

differential equations of motion for small

amplitudes of motion.

This phenomenon occurs in many

dynamical systems, e.g., satellites, ships,

airplanes, buildings, and machines.

Every engineer should be aware of it!


Topics

• Physical Modeling of the Spring-Pendulum System

– Simplifying Assumptions

– Parameter Identification

• Mathematical Modeling of the Physical Model

– Particle and Rigid Body Kinematics

– Particle and Rigid Body Kinetics

• Newton-Euler Equations

• Lagrange’s Equations

• Predicted Response vs. Experimental Measurements

– MatLab / Simulink and SimMechanics Simulations


Engineering

System

Investigation

Process

Spring-

Pendulum

Dynamic System

Investigation


Physical Model

Simplifying 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, so pivot point is fixed to ground


• These physical modeling assumptions lead to what we

call the design model, i.e., one that gives us insight and

understanding.

• There are a whole hierarchy of models possible.

• Modify assumptions that lead to the Truth Model

– Include pivot friction

– Treat spring as not pure, i.e., has inertia and energy dissipation

– Treat the attached mass as a rigid body and not a point mass –

this adds a degree of freedom


Physical Modelwith

Parameter Identification

m = pendulum mass = 1.815 kg

mspring = spring mass = 0.1445 kg

ℓ = unstretched spring length = 0.333 m

k = spring constant = 172.8 N/m

g = acceleration due to gravity = 9.81 m/s2

Ft = 5.71 N = pre-tension of spring

rs = static spring stretch, i.e., rs = (mg-Ft)/k = 0.070 m

rd = dynamic spring stretch

r = total spring stretch = rs + rd

m

k

θℓ + r


Spring Parameter Identification

Fspring (N)

Spring Pre-Tension

Ft = 5.71 N

K = 172.8 N/m

Spring Displacement

(meters)0.070 m

mg = 17.805 N


r

r r r

2

r

r r

ˆr re

drˆ ˆ ˆ ˆv re r e v e v e

dt

dvˆ ˆa r r e r 2r e

dt

ˆ ˆ a e a e

magnitude change

direction change

magnitude change

direction change2

r

r

r r

r

rv

v

r

r

ˆdee

d

ˆdee

d

O

PATH

r

re

e

Polar Coordinates:

Position, Velocity, Acceleration


Rigid Body Kinematics

XY: R reference frame (ground)

xy: R1 reference frame (pendulum)

x cos sin 0 X

y sin cos 0 Y

z 0 0 1 Z

ˆ ˆi Icos sin 0

ˆ ˆj sin cos 0 J

ˆ ˆ0 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

mℓ + r

k

X

Y

xy

P

O

θ

θ


Rigid Body Kinematics

After substitution and evaluation:

1

1

1

1

R O

RR

OP

RR

R P

R P

a 0

ˆ ˆk K

ˆ ˆ ˆr r j r sin I cos J

ˆ ˆk K

ˆ ˆ ˆv rj r sin I cos J

ˆ ˆ ˆa rj r sin I cos J

R P 2ˆ ˆa i r 2r j r r


Mathematical

Model

Free Body Diagram

Nonlinear Equations

of Motion

2

r rF ma m r r

F ma m r 2r

2

tmr m r kr F mgcos 0

r 2r gsin 0

2

tkr F mgcos m r r

mgsin m r 2r

kr + Ft

mg+r

+θ

+θℓ + r


Mathematical Model:

Lagrange’s EquationsLagrange’s Equations

Generalized Coordinates

Kinetic Energy

Potential Energy

Generalized

Forces

Nonlinear Equations

of Motion

2


r 2r gsin 0

21V kr mg r cos

2

22 21

T m r r2

1

2

q r

q

r tQ F

Q 0

i

i i i

d T T VQ

dt q q q


What is a Block Diagram?

• Block Diagram

– A block diagram of a system is a pictorial representation

of the functions performed by each component and of

the flow of signals. It describes a set of relationships

that hold simultaneously.

– A block diagram contains information concerning

dynamic behavior, but it does not include any

information on the physical construction of the system.

– Many dissimilar and unrelated systems can be

represented by the same block diagram.

– A block diagram of a given system is not unique.


What is Simulink?

• Simulink is an extension to MatLab that allows engineers to

rapidly and accurately build computer models of dynamic

physical systems using block diagram notation.

– linear and nonlinear systems

– continuous-time and discrete-time components

– graphical animations are possible

• Previously, a block diagram of the dynamic system

mathematical model was created and then the block diagram

was translated into a programming language.

• In Simulink, the computer program is the block diagram and

this eliminates the risk that the computer program may not

accurately implement the block diagram.


Why Is Simulink Important?

• The potential productivity improvement and cost savings

realized from the block diagram approach to programming is

dramatic.

• There are two principal strategies for Simulink employment.

– Rapid Prototyping

• This is the application of productivity tools to develop working

prototypes in the minimum amount of time. Here we optimize for

development speed, rather than execution speed or memory use. A

hierarchy of physical models is used in this phase. Physical system

design and control design are optimized simultaneously.

– Rapid Application Development

• Here the final computer program is the Simulink model or is derived

from the Simulink model through automatic C-code generation.


Simulink for the Spring Pendulum

• Nonlinear Equations of Motion

• Step 1: Solve for the highest derivative in each

equation.

2


r 2r gsin 0

2 tFkr r r gcos

m m

12r gsin

r


• Create two stems.

• Do what the equations say.

∫ ∫

∫ ∫

00

r r0r

r0r


MatLab Command Window

Select Simulink Library


Simulink Library Browser

Select New Model


Build Simulink Block Diagram here.

First Build the Two Stems.

Then Do What the Equations Say.

Save Model: Spring_Pendulum_Dynamic_System


File Name


Simulink Block Diagram Manipulations


These are the blocks we need for the Spring-Pendulum System. We know this

because we created the block diagram for the equations of motion on paper by

hand before we even turned the computer on. We know the block diagram is

correct because we can write the equations of motion from the block diagram.


Go to: Diagram, Format, Font Style to enlarge font.


Resize Blocks.


Lay out the Two Stems. Annotate the Stems.


Replicate blocks and lay out the block diagram.

Annotate Diagram.


MatLab M-File with Spring Pendulum Parameters

Never put numbers into the Simulink block diagram, unless the

number is part of the actual equation. Always use variables,

even for initial conditions. Define the variable values in the

workspace or create a m-file with the parameter values defined.

Save the m-file and then run it before you run the Simulink

program. See below.


Choose Simulation, Model Configuration Parameters


Make Selections as shown.


uncheck


scope parameters

uncheck

To Workspace block

Choose

Array

Scope

Block

NOTE


theta vs. timer vs. time

x-axis: theta

y-axis: -r

0.070 equilibrium


0

0

0.021 rad

r 0.115 m

InitialConditions

NotePlot was

created in the

MatLab

workspace

from the To

Workspace

block

outputs, r and

theta plotted

vs. tout.

0.070 equilibrium


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)

rad

ial a

nd

an

gu

lar

po

sitio

n (

rad

or

m)

Experimental Results with Initial Conditions: theta = 0.021 rad, r = 0.115 m

InitialConditions

0

0

0.021 rad

r 0.115 m

This plot was

created from two

sensor outputs: a

potentiometer

measuring theta

and an ultrasonic

sensor measuring r.


Comparison

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)

rad

ial a

nd

an

gu

lar

po

sitio

n (

rad

or

m)

Experimental Results with Initial Conditions: theta = 0.021 rad, r = 0.115 m

Initial Conditions:0

0

0.021 rad

r 0.115 m


Conclusion

Is the model adequate based on the comparison between predictions and measurements?

What attributes in the physical system are missing from your physical model?

How would you make the physical model more accurate, i.e., be a better representation of the

actual physical system?


What is SimMechanics?

• SimMechanics provides a multibody simulation environment

for 3D mechanical systems, such as robots, vehicle

suspensions, construction equipment, and aircraft landing

gear.

• You model the multibody system using blocks representing

bodies, joints, constraints, and force elements, and then

SimMechanics formulates and solves the equations of

motion for the complete mechanical system.

• Models from CAD systems, including mass, inertia, joint,

constraint, and 3D geometry, can be imported into

SimMechanics. An automatically generated 3D animation

lets you visualize the system dynamics.


• You can parameterize your models using MatLab

variables and expressions, and design control systems

for your multibody system in Simulink.

• You can add electrical, hydraulic, pneumatic, and other

components to your mechanical model using Simscape

and test them all in a single simulation environment.

• To deploy your models to other simulation

environments, including hardware-in-the-loop (HIL)

systems, SimMechanics supports C-code generation

(with Simulink Coder).


SimMechanics


body elements

constraints

forces

& torques

frames

& transforms

gears

utilities

joints


SimMechanics for the Spring Pendulum

Simulink Work Space

Saved File Name


Simulink

SimMechanics

Ft = 0

theta_0 = 0.021 rad

r_0 = 0.045 m

r = 0.1

ℓ+r = 0.433


Simulink

Ft = 5.71 N

theta_0 = 0.021 rad

r_0 = 0.045 m

SimMechanics

0.07

0.403

Spring-Pendulum Dynamic System Investigation · 2017. 4. 19. · Spring-Pendulum Dynamic System...

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