Close
Close window
  • PRODUCTS

    Maple

    Maple Professional

    Maple Academic

    Maple Student Edition

    Maple Personal Edition

    Maple Player

    Maple Player for iPad

    MapleSim

    MapleSim Professional

    MapleSim Academic

    Online Education

    Maple T.A. - Testing & Assessment

    Maple T.A. MAA Placement Test Suite

    Möbius - Online Courseware

    MapleNet

    Toolboxes & Connectors

    E-Books & Study Guides

    Professional Services

  • SOLUTIONS

    Engineering: Industry Solutions

    Machine Design / Industrial Automation

    Aerospace

    Vehicle Engineering

    Robotics

    Power Industries

    Engineering: Application Areas

    System Simulation and Analysis

    Model development for HIL

    Plant Modeling for Control Design

    Robotics/Motion Control/Mechatronics

    Other Application Areas

    Education

    Mathematics Education

    Engineering Education

    High Schools & Two-Year Colleges

    Testing & Assessment

    Students

    Applied Research

    Financial Modeling

    Operations Research

    High Performance Computing

    Physics

  • PURCHASE

    Purchase & Pricing Details

    Maplesoft Web Store

    Request a Price Quote

    Contact Maplesoft Sales

    Elite Maintenance Program

  • SUPPORT

    Tech Support & Customer Service

    Frequently Asked Questions

    Online Product Help

    Training

    Product Documentation

    Download Product Updates

  • RESOURCES

    Webinars & Events

    Live Webinars

    Recorded Webinars

    Upcoming Events

    Community

    MaplePrimes

    Maplesoft Blog

    Maplesoft Membership

    Maple Ambassador Program

    MapleCloud

    Publications

    Technical Whitepapers

    E-Mail Newsletters

    Maple Books

    Math Matters

    Examples & Applications

    Application Center

    MapleSim Model Gallery

    User Case Studies

    Exploring Engineering Fundamentals

    Teaching Concepts with Maple

    Resources

    Maplesoft Welcome Center

    Teacher Resource Center

    Student Help Center

  • COMPANY

    About Maplesoft

    Contact

    Careers

    Media Center

    Partners

    25th Anniversary Timeline

Application Center

  • Home
  • Editor's Choice Applications
  • MapleSim Model Gallery
  • New Applications
  • Tips & Techniques
  • Contribute your Work

Application Search


  • Advanced Search

Browse Categories

  • Mathematics
    • Precalculus
    • Abstract Algebra
    • Calculus I
    • Calculus II
    • Calculus III
    • Calculus of Variations
    • Chaos Theory
    • Combinatorics
    • Complex Analysis
    • Cryptography
    • Differential Equations
    • Differential Geometry
    • Engineering Mathematics
    • Game Theory
    • General
    • Geometry
    • Graph Theory
    • Group Theory
    • Linear Algebra
    • Logic
    • Number Theory
    • Numerical Analysis
    • Operations Research
    • Optimization
    • PDEs
    • Precalculus
    • Real Analysis
    • Tensors
    • Topology
    • Trigonometry
    • Vector Calculus
  • Computer Science
  • Education
  • Engineering
  • Finance
  • Maple Tools
  • Physics
  • Science
  • Statistics & Data Analysis

Subscribe

  • New Applications
  • Editor's Choice
Home : User Community : Application Center : Mathematics : Differential Equations : Pendulum with a moving pivot

App Preview:

Pendulum with a moving pivot

You can switch back to the summary page for this application by clicking here.

Learn about MapleDownload Application

 

Image 

Pendulum with a moving pivot 

Carl Madigan 

Nova Scotia Agricultural College 

Truro, N.S.  B2N 5E3 

Introduction 

 

The problem being considered is a non-linear pendulum where the point of suspension is moving.  Damping is ignored but can easily be included. 

                   Image 

If  X represents the horizontal and Y the vertical components of the motion of the pivot then by resloving the accelerations along the pendulum we have 

                                Typesetting:-mrow(Typesetting:-mi(   where g is the acceleration due to gravity. 

                         Image 

 

                                     Typesetting:-mrow(Typesetting:-mi( 

      the pendulum's Typesetting:-mrow(Typesetting:-mi( Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi( 

                             Typesetting:-mrow(Typesetting:-mi(  

                    Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

 

 

    ImageTypesetting:-mrow(Typesetting:-mi( 

Setting F = ma and equating  

                                     Typesetting:-mrow(Typesetting:-mo( Typesetting:-mrow(Typesetting:-mo(  

Typesetting:-mrow(Typesetting:-mi(   

           Typesetting:-mrow(Typesetting:-mo( 

elliminating T from these two equations we have 

                    Typesetting:-mrow(Typesetting:-mo(Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

simplifying       Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-mrow(Typesetting:-msup(Typesetting:-mo(                   

We will assume L =1 and consider three types of motion for the pivot      a)  horizontally    Typesetting:-mrow(Typesetting:-mi( 

                                                                                                             b)  vertically       X = 0     Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(   

                                                                                                             c)  circular         Typesetting:-mrow(Typesetting:-mi( Typesetting:-mrow(Typesetting:-mi(   

In the examples below XX(t) and YY(t) are used to define the pivot  

restart; -1; with(plots); -1; with(DEtools); -1 

 

Example 1    Horizontal motion of the pivot 

 

XX := proc (t) options operator, arrow; .5*cos(1.25*t-1/2*Pi) end proc 

 

deq1 := (diff(theta(t), t, t))+(diff(XX(t), t, t))*cos(theta(t))+9.81*sin(theta(t)) = 0 

slna := dsolve({theta(0) = .5, (D(theta))(0) = 1, deq1}, theta(t), numeric) 

 

Plot of the positon wrt time  

 

odeplot(slna, [t, theta(t)], t = 0 .. 60) 

 

animation of the motion for this pendulum 

 

Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
 

A1 := display([pll || (0 .. 300)], insequence = true); -1 

A2 := display([plla || (0 .. 300)], insequence = true); -1 

A3 := display([plbob || (0 .. 300)], insequence = true); -1 

A4 := display([plcv || (0 .. 300)], insequence = true); -1 

Typesetting:-mrow(Typesetting:-mi( 

Example 2     Vertical motion 

 

YY := proc (t) options operator, arrow; .75*sin(3.03*t) end proc 

deq2 := (diff(theta(t), t, t))+((diff(YY(t), t, t))+9.81)*sin(theta(t)) = 0 

slnb := dsolve({theta(0) = .5, (D(theta))(0) = -.5, deq2}, theta(t), numeric) 

 

Plots of the position with respect to time and also of the phase plane for this example 

 

odeplot(slnb, [t, theta(t)], t = 0 .. 100, numpoints = 5000) 

odeplot(slnb, [theta(t), (D(theta))(t)], t = 0 .. 100, numpoints = 5000) 

 

animation of the motion of this pendulum. 

 

Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
 

AY1 := display([pll || (0 .. 300)], insequence = true); -1 

AY2 := display([plla || (0 .. 300)], insequence = true); -1 

AY3 := display([plbob || (0 .. 300)], insequence = true); -1 

AY4 := display([plcv || (0 .. 300)], insequence = true); -1 

Typesetting:-mrow(Typesetting:-mi( 

 

Example 3    Circular motion 

 

XX := proc (t) options operator, arrow; cos(2*t) end proc 

YY := proc (t) options operator, arrow; sin(2*t) end proc 

 

deqc := (diff(theta(t), t, t))+(diff(XX(t), t, t))*cos(theta(t))+((diff(YY(t), t, t))+9.81)*sin(theta(t)) = 0
deqc := (diff(theta(t), t, t))+(diff(XX(t), t, t))*cos(theta(t))+((diff(YY(t), t, t))+9.81)*sin(theta(t)) = 0
 

slncirc := dsolve({theta(0) = .5, (D(theta))(0) = -.5, deqc}, theta(t), numeric) 

 

plots of the position with respect to time and of the phase plane  

 

odeplot(slncirc, [t, theta(t)], t = 0 .. 20, numpoints = 3000) 

odeplot(slncirc, [theta(t), (D(theta))(t)], t = 0 .. 100, numpoints = 3000) 

 

animation of the motion for this pendulum 

 

Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
Typesetting:-mrow(Typesetting:-mo(
 

 

AC1 := display([pll || (0 .. 300)], insequence = true); -1 

AC2 := display([plla || (0 .. 300)], insequence = true); -1 

AC3 := display([plbob || (0 .. 300)], insequence = true); -1 

AC4 := display([plcv || (0 .. 300)], insequence = true); -1 

display([AC1, AC2, AC3, AC4]) 

 

A Procedure for drawing the pendulums  

 

The procedure  is called drad and has the following  imputs   

                                 L = length ,  

                             angl = inital displacement angle,  

                              vel = initial velocity 

                                 a =  the x component of the path for the pivot 

                                 b =  the y component of the pivot's path 

                                 n = the number of iterations used to draw the annimations  

 

 

examples   for horizontal motion try  a = Typesetting:-mrow(Typesetting:-mn( 

                for vertical motion   try    a = 0    and b = Typesetting:-mrow(Typesetting:-mn( 

                for circular motion   try    a = Typesetting:-mrow(Typesetting:-mi(and b = sin(3t - π/2) 

                 

                experiment with other paths  such as  a = sin(2.25t - π/2)  b = cos(2t + π/4)  etc  try varying the lenght and the initial conditions.  Have Fun with it!!! 

 

 

Typesetting:-mrow(Typesetting:-mi( 

 

 

Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

 

 

Example 1.  pivot moving along a curve 

drad(2, 1, -.5, sin(2.25*t-1/2*Pi), cos(2*t+1/4*Pi), 400) 

Plot
 

 

Additional examples 

 

Example 1   horizontal motion of the pivot 

Typesetting:-mrow(Typesetting:-mi( 

 

Example 2 vertical motion of the pivot 

Typesetting:-mrow(Typesetting:-mi( 

 

Example 3 cicular motion of the pivot 

Typesetting:-mrow(Typesetting:-mi( 

 

Drawing the pendulums 

drad(2, .5, -.5, sin(3*t), sin(3*t-1/2*Pi), 400) 

Plot
 

 

 

plot([sin(3*t), sin(3*t-1/2*Pi), t = 0 .. 2]) 

Plot
 

 

 

Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.
 

Image 

 

Learn about Maple

Application Details

Author:
Prof. Carl Madigan
Application Type:
Maple Document
Publish Date: March 26, 2007
Created In:
Maple 10
Maple 11
Language: English
Category:
Mathematics: Differential Equations

Toolkit

Download Maple Document
View Details
Contact the Author
Evaluate Maple

Community Rating:

Your Rating:

 
Move the slider to rate

Tell others about this application!

Tweet



ExclamationPlease note that much of the Application Center contains content submitted directly from members of our user community. Although we do our best to monitor for objectionable content, it is possible that we occasionally miss something. If there is something objectionable on this page, please click here to report it.

Products
  • Maple
  • MapleSim
  • Maple T.A.
  • Professional Services and Consulting
  • MapleNet
  • Toolboxes & Connectors
  • E-books & Study Guides
Industry Solutions
  • Vehicle Engineering
  • Motion Control
  • Power Industries
  • Aerospace
Engineering Applications
  • Plant Modeling for Control Design
  • Virtual Prototyping
  • Real-Time Simulation
  • Optimization & Analysis
Education Solutions
  • Mathematics Education
  • Engineering Education
  • High Schools & 2-Year Colleges
  • Testing & Assessment
  • Students
Applied Research
  • Financial Modeling
  • Operations Research
  • High Performance Computing
  • Physics
Connect & Share
Facebook Twitter YouTube LinkedIn

Maplesoft E-Mail Lists
  • The Maple Reporter
  • Model-Driven Innovation Monthly
  • Other E-mail offerings
Maplesoft Membership
  • Login

Products  |  Solutions  |  Purchase  |  Support   |  Resources  |  Community  |  Company  |  Site-Map  | Login
Language:
English  |  Français  |  Deutsch  |  日本語

      © Maplesoft, a division of Waterloo Maple Inc. 2019. | Terms of Use | Privacy | Trademarks