UDC 533.6.0113.42

AERODYNAMICS

A. I. SMIRNOV

NATURAL VIBRATIONS AND FLUTTER OF THREE-LAYER CYLINDRICAL SHELLS IN A SUPERSONIC GAS FLOW

(Presented by Academician L. I. Sedov on 25 X 1968)

1. The equations of elastic equilibrium of a three-layer cylindrical shell, when the assumptions of the semimembrane theory are used, can be represented in the form \((^1)\)

\[ D\frac{\partial^4}{\partial s^4} \left(\frac{\partial^2}{\partial s^2}+\frac{1}{R^2}\right)^2 \left(1-\frac{\vartheta h^2}{\beta}\frac{\partial^2}{\partial s^2}\right)\chi + \frac{Eh}{R^2}\frac{\partial^4}{\partial x^4} \left(1-\frac{h^2}{\beta}\frac{\partial^2}{\partial s^2}\right)\chi = \frac{1}{R}\frac{\partial^2 p}{\partial s^2}, \tag{1} \]

where \(D, E, R, h\) are the cylindrical stiffness, Young’s modulus, radius, and thickness of the shell; \(\vartheta, \beta\) are parameters characterizing, respectively, the bending stiffness of the load-bearing layers and the shear stiffness of the core; \(\chi(x,s,t)\) is a displacement function related to the deflection \(w\) by the relation

\[ w=\left(1-\frac{h^2}{\beta}\frac{\partial^2}{\partial s^2}\right)\chi . \tag{2} \]

The function \(p\) is determined through the normal component \(q\) and the tangential components \(p_1, p_2\) of the external surface load,

\[ p=R\,\partial^2 q/\partial s^2+\partial p_2/\partial s-\partial p_1/\partial x. \tag{3} \]

Let us assume that the shell is washed on the outside by a supersonic gas flow directed along the generator. Using the linear approximation of piston theory, we represent the normal load in the form

\[ q=N_x\frac{\partial^2 w}{\partial x^2} +N_s\left(\frac{\partial^2 w}{\partial s^2}+\frac{w}{R^2}\right) -\Omega\frac{\partial^2 w}{\partial t^2} -\frac{\varkappa p_0}{a_0} \left(\frac{\partial w}{\partial t}+V\frac{\partial w}{\partial x}\right), \tag{4} \]

where \(N_x, N_s\) are line forces in the longitudinal and circumferential directions; \(\Omega\) is the specific density of the three-layer package of shell material; \(p_0, a_0, V\) are the static pressure, speed of sound, and speed of the undisturbed flow; \(\varkappa=c_p/c_v\).

The tangential forces will be \((^1)\)

\[ p_1=0,\qquad p_2=N_x\frac{\partial^4}{\partial x^2\partial s^2} \left(\frac{\partial^2 w}{\partial s^2}-\frac{w}{R^2}\right). \tag{5} \]

We represent the displacement function in the form

\[ \chi(x,s,t)=\chi(x,s)e^{i\omega t}, \tag{6} \]

where \(\omega\) is the complex vibration frequency.

Substituting (2)—(6) into (1) and passing to dimensionless parameters and coordinates \(x/l,\ s/R\) (\(l\) is the length of the shell), while retaining the former notation for the coordinates, we obtain

\[ \frac{\partial^4}{\partial s^4} \left(\frac{\partial^2}{\partial s^2}+1\right)^2 \left(1-\vartheta k\frac{\partial^2}{\partial s^2}\right)\chi + \Theta_*\frac{\partial^4}{\partial x^4} \left(1-k\frac{\partial^2}{\partial s^2}\right)\chi - \]

\[ -\left[ n_x\frac{\partial^4}{\partial x^2\partial s^2} \left(\frac{\partial^2}{\partial s^2}-1\right) + n_s\frac{\partial^4}{\partial s^4} \left(\frac{\partial^2}{\partial s^2}+1\right) \right] \left(1-k\frac{\partial^2}{\partial s^2}\right)\chi - \]

\[ -\omega_*^2 \left(1-k\frac{\partial^2}{\partial s^2}\right)\chi + \left(i\varepsilon\omega_*+p_*\frac{\partial}{\partial x}\right) \left(1-k\frac{\partial^2}{\partial s^2}\right)\chi =0. \tag{7} \]

Here

\[ k=\frac{9h^{2}}{\beta R^{2}},\qquad \Theta_{*}=\frac{12}{\Theta}\left(\frac{R}{n}\right)^{2}\left(\frac{R}{l}\right)^{4},\qquad n_{x}=\frac{N_{x}R^{2}}{D}\left(\frac{R}{l}\right)^{2},\qquad n_{s}=\frac{N_{s}R^{2}}{D}, \]
\[ \omega_{*}^{2}=\frac{\Omega R^{4}}{D}\,\omega^{2},\qquad p_{*}=\frac{\chi p_{0}R^{3}}{D}\,\frac{R}{l}\,\mathrm{M},\qquad \varepsilon=\frac{\chi p_{0}R^{2}}{\sqrt{\Omega D}} \tag{8} \]

(\(\Theta\) is the dimensionless parameter (1), \(\mathrm{M}=V/a_{0}\) is the Mach number of the undisturbed flow).

We represent the solution of (7) in the form

\[ \chi(x,s)=\sin(ns)\chi(x), \tag{9} \]

where \(n\) is the prescribed number of waves in the circumferential direction. Substituting (9) into (7), we find

\[ \Theta_{*}\frac{d^{4}\chi}{dx^{4}} -n_{x}(1+n^{2})n^{2}\frac{d^{2}\chi}{dx^{2}} +p_{*}n^{4}\frac{d\chi}{dx} + \left[ \frac{1+\vartheta kn^{2}}{1+kn^{2}}(1-n^{2})n^{4} -n_{s}(1-n^{2})n^{4} -\omega_{*}^{2}n^{4} +i\varepsilon\omega_{*}n^{4} \right]\chi=0. \tag{10} \]

A particular solution of equation (10) will be

\[ \chi(x)=e^{i\alpha x}. \tag{11} \]

Substituting (11) into (10), we obtain the algebraic equation

\[ a_{4}\alpha^{4}+a_{2}\alpha^{2}+a_{1}\alpha+a_{0}=0, \tag{12} \]

where

\[ a_{4}=\Theta_{*},\qquad a_{2}=n_{x}(1+n^{2})n^{2},\qquad a_{1}=in^{4}p_{*}, \]

\[ a_{0}=\frac{1+\vartheta kn^{2}}{1+kn^{2}}(n^{2}-1)^{2}n^{4} -n_{s}(n^{2}-1)n^{4} -\omega_{*}^{2}n^{4} +i\varepsilon\omega_{*}n^{4}. \tag{13} \]

Determining the roots \(\alpha_i\) of equation (12), we construct the general solution (for a given \(n\)) of (10)

\[ \chi(x)=\sum_{i=1}^{4}c_i e^{i\alpha_i x}. \tag{14} \]

2. Let us consider several boundary-value problems.

A. Hingedly supported edges. In this case, for the function one can formulate two conditions at each edge \((x=0,\ x=1)\) of the form

\[ \chi=\chi''=0. \tag{15} \]

Substituting (14) into (15), we arrive at a system of linear algebraic equations with respect to the coefficients \(c_i\). The condition for a nontrivial solution will be

\[ \Delta\delta^{-1}=0, \tag{16} \]

where \(\Delta(\alpha_1,\alpha_2,\alpha_3,\alpha_4)\) is the determinant of the indicated system, and \(\delta\) is the Vandermonde determinant formed from the roots \(\alpha_i\). The determinant \(\Delta\) is a sum of 6 terms of the form

\[ (\alpha_{1}^{2}-\alpha_{2}^{2})(\alpha_{3}^{2}-\alpha_{4}^{2})e^{i(\alpha_{3}+\alpha_{4})}. \tag{17} \]

The remaining terms are obtained from (17) by cyclic permutation of the indices \(1,2,3,4\).

B. Clamped edges. The boundary conditions will be

\[ \chi=\chi'=0. \tag{18} \]

The determinant \(\Delta\) is composed of terms of the form

\[ (\alpha_{1}-\alpha_{2})(\alpha_{3}-\alpha_{4})e^{i(\alpha_{3}+\alpha_{4})}. \tag{19} \]

Fig. 1. Effect of aerodynamic damping on the critical flutter speed of a three-layer cylindrical shell with simply supported edges; \(\vartheta = 0.05;\ k = 1.00;\ n = 12\)

Fig. 2. Variation of the real part of the dimensionless vibration frequency \(\omega_*'\) of a three-layer cylindrical shell in a supersonic flow (simply supported edges); \(\vartheta = 0.05;\ k = 1.00;\ n = 12\)

Fig. 3. Effect of the shear stiffness of the core and the bending stiffness of the face layers on the critical flutter speed in a supersonic flow; the shell edges are simply supported; \(\Theta_* = 100;\ \varepsilon = 0.1;\ n = 12\)

Fig. 4. Effect of compressive forces in the circumferential direction on the vibration frequency of a three-layer shell in a supersonic flow (simply supported edges, frequencies \(1\)—\(2\)); \(\vartheta = 0.05;\ k = 1.00;\ n = 12\)

B. For a cantilever-clamped shell we have

\[ \chi = \chi' = 0 \quad (x = 0);\qquad \chi'' = \chi''' = 0 \quad (x = 1), \tag{20} \]

and the first term of the determinant \(\Delta\) is equal to

\[ (a_1 - a_2)a_3^2 a_4^2 (a_3 - a_4)e^{i(a_3+a_4)}. \tag{21} \]

For a clamped and simply supported shell, respectively, we obtain

\[ \chi = \chi' = 0 \quad (x = 0);\qquad \chi = \chi'' = 0 \quad (x = 1); \tag{22} \]

\[ (a_1 - a_2)a_3^2 a_4^2 (a_3 - a_4)e^{i(a_3+a_4)}. \tag{23} \]

Systematic numerical calculations were carried out on the “Strela” electronic digital computer over a wide range of dimensionless parameters. Some of the results are shown in Figs. 1–4. The calculations make it possible to clarify a number of interesting, previously unknown features of the behavior of a three-layer shell in a gas flow. The principal ones are as follows. The role of aerodynamic damping in determining the critical flutter speed depends to a considerable degree on the geometry of the shell itself (the parameter \(\Theta_*\), Fig. 1). Neglecting aerodynamic damping may lead to substantial errors in determining the flutter speed (Figs. 1 and 2). Other conditions being equal, the flutter speed of a three-layer shell decreases rapidly as the shear stiffness of the filler decreases (as the coefficient \(k\) increases, Fig. 3). Compression of the shell in the longitudinal and circumferential directions contributes to a reduction in the flutter speed (Fig. 4). The number of waves in the circumferential direction corresponding to the minimum flutter speed varies, depending on the boundary conditions, in the range \(n = 12\text{–}18\). At the same time, more rigid boundary conditions correspond to a larger value of \(n\).

Received
9 X 1968

REFERENCES

1. E. I. Grigolyuk, P. P. Chulkov, Critical Loads of Three-Layer Cylindrical and Conical Shells, Novosibirsk, 1966.