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- W2012309612 abstract "The study of buckling of a tapered rod leads to the nonlinear eigenvalue problem: (1){A(s)u′(s)}′+μ sinu(s)=0for alls∈(0,1),(2)u(1)=lims→0A(s)u′(s)=0 and (3)∫01A(s)u′(s)2ds<∞, where A∈C([0,1]) is such that A(s)>0 for all s>0 and lims→0A(s)/sp=L for some constants p⩾0 and L∈(0,∞). There is a number Λ(A)⩾0 such that, for μ⩽Λ(A), u≡0 is the only solution of the problem and it minimizes the energy in the space of all admissible configurations. For μ>Λ(A), the energy is minimized by a non-trivial solution. For p=0, this is a well understood classical problem studied by D. Bernoulli and Euler. For 0<p<2, the problem is singular but its bifurcation diagram remains similar to the case p=0. At p=2, striking changes occur. For 0⩽p<2, lims→0u(s)∈(−π,π) for all non-trivial solutions whereas lims→0u(s)=±π if p⩾2. For 0⩽p⩽2,Λ(A)>0 whereas Λ(A)=0 for p>2. For 0⩽p<2, bifurcation from the solution u≡0 occurs only at a discrete set of eigenvalues μi where μ1=Λ(A) and limi→∞μi=∞. For p=2, there is a number Λe(A)∈[Λ(A),∞) such that bifurcation occurs at every value μ∈[Λe(A),∞). The properties of the linearized problem, in which (1) is replaced by (4){A(s)u′(s)}′+μu(s)=0for alls∈(0,1), also change at p=2. For 0⩽p<2, its spectrum {μi} is discrete and all the eigenfunctions have only a finite number of zeros in [0,1]. For p=2,Λe(A) belongs to the essential spectrum and there may be no eigenfunctions. Furthermore, for p=2 and μ>Λe(A) all solutions of (4) have infinitely many zeros in [0,1], but solutions of the nonlinear problem have only a finite number of zeros." @default.
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- W2012309612 date "2001-04-01" @default.
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- W2012309612 title "Buckling of a heavy tapered rod" @default.
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- W2012309612 doi "https://doi.org/10.1016/s0021-7824(00)01196-x" @default.
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