{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

SolSec 6.4

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

This preview shows pages 1–4. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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:′′...
View Full Document

{[ snackBarMessage ]}

### Page1 / 11

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online