On the stability of a flexible missile under an end thrust

Stability of a flexible missile described by asymptotics of the eigenvalues of fourth order boundary value problems

“…By definition of φ 1 in (4.8) and the estimate |φ 1 (µ)| < 1 for all µ ∈ S k , from Rouché's theorem we conclude that the functions φ 0 and φ have the same number of zeros in the square S k , for k ∈ N with k ≥ k1 . Since φ 0 has 4k + 4 zeros inside S k and thus 4k + 4 + 4 zeros inside S k+1 , it follows that φ has no large zeros other than the zeros ±μ k found above for |k| sufficiently large, and that there are μk for small |k| such that λk = μ2 k account for all eigenvalues of the problem (2.1), (3.4), where p 1 = 0, p 2 = 3, (p 3 , q 3 ) = (1, 0) and (p 4 , q 4 ) = (3,2). Since each of these eigenvalues gives rise to two zeros of φ, counted with multiplicity.…”

“…We recall that the quasi-derivatives associated to (2.1) are given by y [0] = y, y [1] = y ′ , y [2] = y ′′ , y [3] = y (3) − gy ′ , y [4] = y (4) − (gy ′ ) ′ , (2.4) see [7, Definition 10.2.1, page 256].…”

“…depends on the eigenvalue parameter. The fourth order problem with the differential equation (1.2) and the boundary conditions y ′′ (0) = y (3) (0) = 0 and y ′′ (a) = y (3) (a) = 0 describes the stability of a flexible missile, see [1,2,3]. This problem can be represented by the operator polynomial…”

Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problems

“…However, they are not quantitatively applicable to a specific missile. Guran and Ossia [6] have studied the dynamic stability of an uniform free-free rod, representing a flexible missile, subjected to an end thrust by applying finite difference techniques. They obtain eingenvalue curves and in graphical forms their convergence characteristics.…”

“…Studying a moving beam subjected to a tangential end force in space, Kirillov and Seyranian, see [7], have obtained its mass distribution with the critical flutter load and they have presented optimal distributions of nonstructural mass with a few switching points. The boundary conditions, of the problems investigated in [4,6,7], are y (3) (0) = y (0) = 0 and y (3) (a) = y (a) = 0 (1.3) and their operator polynomial representation is (1.4) in the Hilbert space L 2 (0, a). Recently, Xu, Rong, Xiang, Pan and Yin [24], using numerical calculations, have investigated the dynamic response and stability of a rotating and flexible missile under thrust.…”