Nonlinear vibrations of an infinitely long cylindrical shell.

Abstract

the transformed equation X"' + a.X" + b-X' + d-X + c = 0 (4) By making a further transformation of the type X + c /d = Y (5) we obtain 7'"+ aF" + 67'-+ d-7 = 0 (6) Equation (6) is a third-order linear differential equation and hence represents the equivalent linear system in the transformed plane (Y-T). Equation (6) can easily be solved to obtain F as a function of T. Using Eqs. (3) and (5) along with the solution to Eq. (6) gives, say, From Eq. (2),Substitution of Eq. (7) in (8) gives the relation between t and T as, say,Thus, the implicit relation between x and t is established by Eqs. (7) and (9) in terms of a common parameter T. In many cases, it may be possible to obtain the explicit relation, viz., a; as a function of t, if T is eliminated between Eqs.(7) and (9). The method just outlined is illustrated by means of the following suitable example. An equation of the type considered previously, viz., Eq. (1), arises in gyroscopic theory, while considering the inertial motion of a gyro rotating about its center of mass.Euler's equation of motion in body-fixed coordinates for a rigid body rotating about its center of mass under inertial motion (i.e., under no external moments) can be written as 4Equation (10) represents three coupled nonlinear first-order differential equations that on successive differentiation and mutual substitution can be reduced 5 to a set of uncoupled, third-order, nonlinear differential equations of the type x -(xx/x) + bx 2 x = 0 (H) Now Eq. (11) is a particular case of Eq. (1) with f(x) = x and a = c = d = 0. Hence, the necessary transformation function is i.e., dX/dx = dT/dt = x X = z 2 /2 X' = x Equation (13) in (11) gives X"' + bX' = 0 Letting X' = Y reduces Eq. (14) toY" + bY = 0 whose solution can at once be written as(13) (14) (15) (16) (17) Reverting back to the variable X gives X(T) = A 1 cos(bV*T + 6) + A, Using Eq. (13) in (18) gives x = {2[Ai cos(6 1 / 2 T + 0) + ^2]} 1/ and, from Eqs. (13) and (19), dT t J {214! 6) + A 2 ]} 1/2 (18) (19)Here Eqs. (19) and (20) together represent the response of the third-order system given by Eq. (11), and the constants Ai, 6, and A 2 can be chosen to suit the initial conditions.In case T is eliminated between Eqs. (19) and (20), the explicit relationship between x and t can be attained. Similar results can be obtained for the angular velocities along all of the three principal axes of the rigid body. Thus, the practical example just treated illustrates the value of the method outlined. ConclusionsFor any given nonlinear third-order system of the class represented by Eq. (1), the function f(x) and the constants a, b, c } and d are known, and hence the corresponding linear systems represented by Eq. (4) or (6) can be attained.By solving the resulting linear equations and working backwards, the solution to the original nonlinear equation can be arrived at, as can be seen from the example considered previously. Therefore, the method described here is expected to be of use in the study of many problems in various fields such as gyrodynamics, vi...

Nonlinear Vibrations of Simply Supported, Circular Cylindrical Shells, Coupled to Quiescent Fluid

“…However, the criticism raised by Evensen (1977) Some studies have addressed fluid-structure interaction in a light medium, like air, considering shell nonlinearity. For example, the nonlinear flutter of a circular cylindrical shell in supersonic flow was studied by Olson & Fung (1967) and Evensen & Olson (1967, 1968. In particular, Olson & Fung (1967) discussed the calculation of the aerodynamic pressure with the aid of both linear piston theory and potential flow theory; numerical results were obtained by using piston theory and a mode expansion involving two asymmetric modes having a different axial wavenumber plus two axisymmetric terms.…”

“…He studied the free vibrations and the response to a modal excitation without considering damping and discussed the stability of the response curves. Evensen (1968) extended his work to infinitely long shells vibrating in a mode in which the generators of the cylindrical surface remain straight and parallel, by using the method of harmonic balance, without considering the companion mode. He also wrote an interesting review paper (Evensen 1974).…”

“…In particular, Olson & Fung (1967) discussed the calculation of the aerodynamic pressure with the aid of both linear piston theory and potential flow theory; numerical results were obtained by using piston theory and a mode expansion involving two asymmetric modes having a different axial wavenumber plus two axisymmetric terms. Evensen & Olson (1967, 1968 used only piston theory, but discussed the mode expansion involving also companion modes. All the foregoing studies dealt with shells vibrating in vacuum or in a light fluid medium such as air.…”

“…Hence, the original idea for mode expansion was to add to asymmetric linear modes an axisymmetric term (mode) giving a contraction to the shell. The axisymmetric term added by Evensen (1967Evensen ( , 1968Evensen ( , 1974) is a square sine in the axial coordinate, the amplitude of which is related to amplitudes of linear terms in order to satisfy the continuity of the circumferential displacement.…”

Effects of in-plane inertia on natural frequency and stresses of long circular cylindrical shell

“…Readers interested by these works can turn to [1] and [2] or to the review paper presented recently by [3], and to a more recent review paper by [4], in which a set of papers are discussed and commented. It can be observed that although several works exist in literature on nonlinear vibrations of shells, only few papers are available which are concerned with the nonlinear free vibration of shells of infinite length.…”