Loading...

PPT – Initial Value Problems PowerPoint presentation | free to view - id: 1484cf-YWFlN

The Adobe Flash plugin is needed to view this content

Initial Value Problems

- MATH 224

General Linear Spring Model

- General Model for Spring/Mass system

Particular System

- k 90
- m 0.5
- Assume c 0 (no drag/friction)

Review of Solution

- Solve auxiliary equation
- Use m values to construct fundamental solution

set - Construct general solution

What does it mean?

- From general solution, we can tell that

Particular solutions

- Two unknown constants in general sol'n
- Determined by either
- initial conditions or
- boundary conditions
- Determine the predicted behaviour in one

particular scenario

Initial value problems (IVP)

- DE one initial condition per unknown constant
- e.g. initial position and velocity
- Let y(0) -2, and v(0) -0.5

Finding particular solution

- Use general solution, with given initial values
- Set up one equation for each condition
- Solve equations for c1, c2

Continued

Interpreting Particular Solution

- Solution is a prediction of behaviour
- We've got y , so we're predicting position of

mass (y) over time (t) based on DE rule, starting

at initial conditions - Particular solution, based on initial conditions,

is

Graphing analytic solution in MATLAB

- Graphing just like any other function

t linspace(0, 2, 1000) w sqrt(90/0.5) c1

-0.02 c2 -0.5/w y c1 cos(w t) c2

sin(w t) plot(t, y)

- Graph shows predicted position (y) over time (t)
- check satisfies given intial conditions?

Cantilevered beam

- Diving board
- Cantilevered structure
- Tip of beam behaves like spring/mass system
- to a first approximation!

Modelling

- Finding parameters can be done analytically or

experimentally - we'll use k 100, m 0.04, c 0.03
- DE is

General Solution

- Auxiliary equation is
- Gives roots of
- A fundamental solution set is
- General solution is

Comments on General Solution

- Basic structure
- Periodicity

Initial Conditions

- Take simple initial conditions
- y(0) - 0.02 m , v(0) 0 m/s
- Set up initial condition equations

Particular Solution

- Solve for c1, c2
- MATLAB, or by hand

MATLAB Assistance and Plot

Cantilevered beam example m roots(.04 .03

100) a real(m(1)) b imag(m(1))

initial conditions, using on-paper work c1

-0.02 c2 -c1 a / b t linspace(0, 10,

1000) y c1 exp(at) .cos(bt) ... c2

exp(at) . sin(bt) plot(t, y) xlabel('Time

(seconds)') ylabel('Height of beam tip

(m)') title('Motion of beam after release')

Analytic solutions for IVPs

- Find general solution to DE
- ignore initial conditions
- general solution applies to any starting point

for the system - Use initial values and general solution to find

values of undetermined constants - set up system of n equations for c1, cn

Just for fun

- There are other bending modes for cantilevered

beams, rather than simple sping model