Boundary Value Problems

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Boundary Value Problems

• MATH 224

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Beam Deformation
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Type of DE

• Order
• Linear or Non-Linear
• Homogeneous or Non-Homogeneous
• Supports define end conditions
• they don't affect DE

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Define load
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Solving

• yc

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Finding yp
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Boundary Conditions

• Previously, we studied motion of a mass
• Constants in yc determined by support type
• embedded
• free
• hinged

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IVP vs. BVP

• Note the difference boundary value problems have
  constants defined at two or more points IVPs all
  at t0
• An initial value problem for a linear DE always
  has a unique solution
• We always get n linear equations with n unknowns
• The same is not guaranteed for boundary value
  problems

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Column Buckling w/Axial Load

• Apply force P to fixed ends of thin column

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DE

• Note this is the same DE as a simple spring!
• We'll see how using boundary values instead of
  initial values changes the interpretation

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General Solution
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Boundary Conditions

• At the ends, the deformation is fixed at 0
• y(0) 0
• y(L) 0
• Solve for c1, c2

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Trivial Solution
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Any Non-Trivial Solutions?
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Modes
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Multiple Solutions to DE
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Reading

• Section 5.2
• Derivation of Buckling DE
• http//www.students.uwosh.edu/piehld88/ndcweb/ndc
  proj1.htm