Initial Value Problems

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Initial Value Problems

• MATH 224

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General Linear Spring Model

• General Model for Spring/Mass system

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Particular System

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

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Review of Solution

• Solve auxiliary equation
• Use m values to construct fundamental solution
  set
• Construct general solution

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What does it mean?

• From general solution, we can tell that

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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

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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

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Finding particular solution

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

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Continued
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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

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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?

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Cantilevered beam

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

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Modelling

• Finding parameters can be done analytically or
  experimentally
• we'll use k 100, m 0.04, c 0.03
• DE is

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General Solution

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

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Comments on General Solution

• Basic structure
• Periodicity

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Initial Conditions

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

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Particular Solution

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

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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')
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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

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Just for fun

• There are other bending modes for cantilevered
  beams, rather than simple sping model