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SolSec 6.4 - 6-Problems and Solutions Section 6.4(6.30...

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Unformatted text preview: 6-Problems and Solutions Section 6.4 (6.30 through 6.39)6.30Calculate the first three natural frequencies of torsional vibration of a shaft of Figure 6.7 clamped at x= 0, if a disk of inertia J= 10 kg m2/rad is attached to the end of the shaft at x= l. Assume that l= 0.5 m, J= 5 m4, G= 2.5 ×109Pa, ρ= 2700 kg/m3.Solution:The equation of motion is&&θ=Γρ′′θ. Assume separation of variables: θ=φ(Ξ29θ(τ29to get φ&&θ=Γρ′′φ θορρΓ&&θθ=′′φφ= -σ2so that&&q+Γρσ2θ= 0 ανδ′′φ+σ2φ= 0where ϖ2=Γρσ2.The clamped-inertia boundary condition is θ(0,t) = 0, and -Γϑ′θ(λ,τ29=ϑ&&θ(λ,τ29 .This yields that φ(0) = 0 andGJ′φ(λ29θ(τ29=ϑφ(λ29 &&θ(τ29=ϑφ(λ29Γρσ2θ(τ29orJ′φ(λ29=ϑσ2ρφ(λ29The solution of the spatial equation is of the formφ(ξ29=Ασινσ ξ+Βχοσσ ξbut the clamped boundary condition yields B= 0. The inertia boundary condition yieldsJAσχοσσλ=ϑσ2ρΑσινσλτανσλ=ϑϑρλσλ=1σλ5 μ410κγ μ2(2700κγ/μ329 (0.5μ 29So the frequency equation istanσλ=675σλUsing the MATLAB function fsolve; this has the solutionsσ1λ= 1.5685σ2λ= 4.7054σ3λ= 7.8424ορσ1= 3.1369σ2= 9.4108σ3= 15.6847Thusϖ1= 3018.5 rad/s ⇒f1= 480.4 Hzϖ2=9055.6 rad/s ⇒f2= 1441.2 Hzϖ3= 15092.6 rad/s ⇒f3= 2402.1 Hz276-6.31Compare the frequencies calculated in the previous problem to the frequencies of the lumped-mass single-degree-of-freedom approximation of the same system.Solution:First calculate the equivalent torsional stiffness of the rod.k=Γϑλ=(2.5 × 10929 (5290.5= 2.5 × 1010ϑ&&θ= -κθϑ&&θ+κθ= 010&&θ+2.5 × 1010θ= 0ορ&&θ+2.5 × 109θ= 0so that ϖ2= 2.5 ×109, ϖ= 5 ×105rad/s or about 80,000 Hz, far from the 482 Hz of problem 6.30.286-6.32Calculate the natural frequencies and mode shapes of a shaft in torsion of shear modulus G, length l, polar inertia J, and density ρthat is free at x= 0 and connected to a disk of inertia Jat x= l.Solution:Assume zero initial conditions, i.e. θ(x,0) = &θ(ξ,029= 0. From equation 6.66∂2θ(ξ,τ29∂τ2=Γρ∂2θ(ξ,τ29∂ξ2(1)The boundary condition at x= land at x= 0 isGJ∂θ(λ,τ29∂ξ= -ϑ∂2θ(λ,τ29∂τ2∂θ(0,τ29∂ξ= 0Using separation of variable in (1) of form θ(x,t) = Θ(x)T(t) yields:′′...
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SolSec 6.4 - 6-Problems and Solutions Section 6.4(6.30...

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