Concept explainers
Rotation of a Shaft Suppose the x-axis on the interval [0, L] is the geometric center of a long straight shaft, such as the propeller shaft of a ship. See Figure 5.2.12. When the shaft is rotating at a constant angular speed ω about this axis the deflection y(x) of the shaft satisfies the differential equation
where ρ is its density per unit length. If the shaft is simply supported, or hinged, at both ends the boundary conditions are then
y(0) = 0, y″(0) = 0, y(L) = 0, y″(L) = 0.
- (a) If λ = α4 = ρω2/EI, then find the eigenvalues and eigenfunctions for this boundary-value problem.
- (b) Use the eigenvalues λn in part (a) to find corresponding angular speeds ωn. The values ωn are called critical speeds. The value ω1 is called the fundamental critical speed and, analogous to Example 4, at this speed the shaft changes shape from y = 0 to a deflection given by y1(x).
Want to see the full answer?
Check out a sample textbook solutionChapter 5 Solutions
A First Course in Differential Equations with Modeling Applications (MindTap Course List)
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageElementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning