Nonlinear Free Vibrations of Thin, Circular Cylindrical Shells

Mayers, J.; Wrenn, B. G.

doi:10.21236/ad0872813

Cited by 7 publications

(7 citation statements)

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“…[3][4][5][6][7][8][9][10][11]. Only some researchers used more refined nonlinear shell theories [12][13][14][15][16][17][18][19][20][21][22][23][24][25], as the Novozhilov, the Sanders-Koiter (also referred as Sanders) and the Flügge-Lur'e-Byrne nonlinear shell theory, or included shear deformation and rotary inertia. Numerical differences among the four most popular classical nonlinear shell theories have been numerically investigated in Ref.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear vibrations of clamped-free circular cylindrical shells

Kurylov

Amabili

2011

Journal of Sound and Vibration

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Section: Introductionmentioning

confidence: 99%

Nonlinear vibrations of clamped-free circular cylindrical shells

Kurylov

Amabili

2011

Journal of Sound and Vibration

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“…(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.…”

mentioning

confidence: 99%

Nonlinear vibrations of an infinitely long cylindrical shell.

Evensen¹

1968

AIAA Journal

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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...

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“…Hence, the modi"ed shape was used also for the non-linear vibration problem. However, he concluded that in contrast to the linear case in which the chessboard de#ection pattern is a natural choice, its selection in the non-linear problem must be more carefully assessed [75]. It may be worth noticing here that the question of the choice of the de#ection shape has been rediscussed recently in reference [92].…”

Section: Non-linear Free and Forced Vibrationsmentioning

confidence: 99%

“…An approximate solution was given for the symmetric response, i.e., when the companion mode is not ready to vibrate, and for the coupled response, i.e., when the companion mode participates in the vibration. Using a variational approach (Rayleigh}Ritz method), Mayers and Wrenn [75] used the more complicated shell theory of Sanders to surmount the restriction on the utilization of the Karman}Donnel formulation to shell problems when the number of circumferential waves is small. They arrived at the conclusion that free vibration is non-periodic and is of the hardening type.…”

Section: W(x Y T)"a(t) Sin(m X/¸) Cos(ny/r)!(na(t)/4r) Sin(m X/¸)mentioning

confidence: 99%