SolSec 6.5 - 6-Problems and Solutions Section 6.5(6.40...

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Unformatted text preview: 6-Problems and Solutions Section 6.5 (6.40 through 6.47)6.40Calculate the natural frequencies and mode shapes of a clamped-free beam. Express your solution in terms of E, I, ρ, and l. This is called the cantilevered beam problem.Solution:Clamped-free boundary conditions arew(0,t)=ϖξ(0,τ29= 0ανδϖξξ(λ,τ29=ϖξξξ(λ,τ29= 0assume E, I, ρ, lconstant. The equation of motion is∂2ϖ∂τ2+ΕΙρΑ∂4ϖ∂ξ4= 0assume separation of variablesw(x,t)=φ(ξ29θ(τ29to getEIρΑ′′′′φφ= -&&θθ=ϖ2The spatial equation becomes′′′′φ-ρΑΕΙϖ2φ= 0define β4=ρΑϖ2ΕΙσο τηατ′′′′φ-β4φ= 0which has the solution:φ=Χ1σινβξ+Χ2χοσβξ+Χ3σινηβξ+Χ4χοσηβξApplying the boundary conditions w(0,t)=ϖξ(0,τ29= 0ανδϖξξ(λ,τ29=ϖξξξ(λ,τ29= 0 ⇒φ(029=′φ(029= 0ανδ′′φ(λ29=′′′φ(λ29= 0Substitution of the expression for φinto these yields:C2+C4= 0C1+ C3= 0-Χ1σινβλ-Χ2χοσβλ+Χ3σινηβλ+Χ4χοσηβλ= 0-Χ1χοσβλ+Χ2σινβλ+Χ3χοσηβλ+Χ4σινηβλ= 0Writing these four equations in four unknowns in matrix form yields:376-1111-σινβλ-χοσβλσινηβλχοσηβλ-χοσβλσινβλχοσηβλσινηβλχ1χ2χ3χ4= 0For a nonzero solution, the determinant must be zero to that (after expansion)-σινβλ- σινηβλ-χοσβλ- χοσηβ-χοσβλ- χοσηβσινβλ- σινηβ=(-σινβλ- σινηβλ29 (σινβλ- σινηβλ29 -(-χοσβλ- χοσηβλ29 (-χοσβλ- χοσηβλ29= 0Thus the frequency equation is cos βlcosh βl= -1 or cosβνλ= -1χοσηβνλand frequencies areϖν=βν4ΕΙρΑ. The mode shapes areφν=Χ1νσινβνξ+Χ2νχοσβνξ+Χ3νσινηβνξ+Χ4νχοσηβνξUsing the boundary condition information that C4= -Χ2ανδΧ3= -Χ1yields-Χ1σινβλ-Χ2χοσβλ-Χ1σινηβλ-Χ2χοσηβλ-Χ1(σινβλ+σινηβλ29=Χ2(χοσβλ+χοσηβλ29so thatC1= -Χ2χοσβλ+χοσηβλσινβλ+σινηβλand the mode shapes can be expressed as:φν= -Χ2ν-χοσβνλ+χοσηβνλσινβνλ+σινηβνλσινβνξ+χοσβνξ+χοσβνλ+χοσηβνλσινβνλ+σινηβνλσινηβνξ- χοσηβνξ386-6.41Plot the first three mode shapes calculated in Problem 6.40. Next calculate the strain mode shape [i.e.,′Ξ(ξ29], and plot these next to the displacement mode shapes X(x). Where is the strain the largest?...
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This note was uploaded on 11/28/2010 for the course ME 4440 taught by Professor Hill during the Winter '09 term at Detroit Mercy.

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SolSec 6.5 - 6-Problems and Solutions Section 6.5(6.40...

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