UDC 533.6.013.42

THEORY OF ELASTICITY

A. I. SMIRNOV

SUPERSONIC FLUTTER OF THREE-LAYER PLATES

(Presented by Academician L. I. Sedov on 3 IV 1968)

1°. The investigations presented below are based on a system of differential equations of elastic equilibrium for three-layer plates, obtained in (¹). In (², ³) a number of exact solutions of boundary-value problems for these equations were given for the case of free vibrations of a three-layer beam (an infinitely wide plate) in a vacuum and in a gas flow. In the present paper the investigations concern the behavior of three-layer plates of finite elongation in a supersonic gas flow.

The differential equation of elastic equilibrium of a three-layer plate in a gas flow, in the absence of shear forces, takes the form (¹)

\[ D(1-\vartheta h^2\beta^{-1}\nabla^2)\nabla^2\nabla^2\chi - N_x\frac{\partial^2}{\partial x^2}(1-h^2\beta^{-1}\nabla^2)\chi - \Omega\frac{\partial^2}{\partial t^2}(1-h^2\beta^{-1}\nabla^2)\chi - N_y\frac{\partial^2}{\partial y^2}(1-h^2\beta^{-1}\nabla^2)\chi - q=0. \tag{1,1} \]

Here \(\chi(x,y,t)\) is a displacement function related to the deflection \(w(x,y,t)\) by the relation

\[ w=(1-h^2\beta^{-1}\nabla^2)\chi, \tag{1,2} \]

where \(\nabla^2=\partial^2/\partial x^2+\partial^2/\partial y^2\).

The quantities \(D\), \(\vartheta\), \(\beta^{-1}\), \(\Omega\) characterize, respectively, the cylindrical stiffness of the three-layer package, the bending stiffness of the load-bearing layers, the shear stiffness of the filler, and the mass per unit area of the three-layer package (¹); \(h\) is the package thickness; \(N_x\), \(N_y\) are the compressive (tensile) forces in the longitudinal and transverse directions; \(q(x,y,t)\) is the transverse aerodynamic load.

2°. Suppose that the plate is flowed over on one side by a supersonic gas flow directed along the \(x\)-axis. We assume that two edges of the plate parallel to the velocity vector \(\vec V\) of the incident flow are hinged, while the other two edges, normal to the vector \(\vec V\), have arbitrary fixing.

We represent a particular solution of (1,1) in the form

\[ \chi(x,y,t)=\sin\frac{n\pi y}{b}\,\chi(x)e^{i\omega t}, \tag{2,1} \]

where \(n\) is the number of half-waves in the transverse direction; \(b\) is the width of the plate; \(\chi(x)\) is a complex function of the real variable \(x\); \(\omega\) is the complex frequency of vibrations; \(t\) is time.

We assume that the aerodynamic load can be calculated in the linear approximation of piston theory

\[ q=-\frac{\chi p_0}{a_0}\left(\frac{\partial w}{\partial t}+V\frac{\partial w}{\partial x}\right), \tag{2,2} \]

where \(p_0\), \(a_0\) are the static pressure and speed of sound in the undisturbed flow, and \(\chi=c_p/c_v\) is the ratio of the corresponding specific heats.

Substituting (2.1), (2.2) into (1.1) and passing to the dimensionless coordinates \(x/a,\ y/b\) (where \(a\) is the length of the plate) and to the dimensionless time \(\tau=t/\sqrt{\Omega D}\), we obtain

\[ a_6 \frac{d^6\chi}{dx^6}+a_4\frac{d^4\chi}{dx^4}+a_3\frac{d^3\chi}{dx^3}+a_2\frac{d^2\chi}{dx^2}+a_1\frac{d\chi}{dx}+a_0\chi=0, \tag{2.3} \]

where

\[ \begin{gathered} a_6=-\vartheta k;\quad a_4=1+3\vartheta k n^2\pi^2\bar a^2+k n_x;\\ a_3=-kp_*;\quad -a_2=2n^2\pi^2\bar a^2+3\vartheta k n^4\pi^4\bar a^4+(1+kn^2\pi^2\bar a^2)n_x+\\ +kn^2\pi^2\bar a^2 n_y-k\omega_*^2+ik\varepsilon\omega_*;\\ a_1=p_*(1+kn^2\pi^2\bar a^2);\quad a_0=(1+\vartheta k n^2\pi^2\bar a^2)n^4\pi^4\bar a^4+\\ +(1+kn^2\pi^2\bar a^2)n^2\pi^2\bar a^2 n_y-(1+kn^2\pi^2\bar a^2)\omega_*^2 +i\varepsilon\omega_*(1+kn^2\pi^2\bar a^2). \end{gathered} \tag{2.4} \]

Here

\[ k=h^2\beta^{-1}/a^2;\qquad n_x=N_xa^2/D;\qquad n_y=N_ya^2/D;\qquad \bar a=a/b; \]

\[ p_*=\chi p_0a^3M/D;\qquad M=V/a_0;\qquad \varepsilon=\chi p_0a^2/\sqrt{\Omega D};\qquad \omega_*=\omega a^2\sqrt{\Omega/D} \tag{2.5} \]

are dimensionless parameters.

Let a particular solution of (2.3) have the form

\[ \chi(x)=e^{i\alpha x}, \tag{2.6} \]

where \(\alpha\) is a root of the algebraic equation

\[ -a_6\alpha^6+a_4\alpha^4-ia_3\alpha^3-a_2\alpha^2+ia_1\alpha+a_0=0. \tag{2.7} \]

The general solution of (2.3) will be

\[ \chi(x)=\sum_{j=1}^{6} C_j e^{i\alpha_j x}, \tag{2.8} \]

where \(C_j\) are constants to be determined from the boundary conditions of the problem.

Fig. 1. Variation of the vibration frequency \(\omega_*\) and the critical flutter speed \(p_{*\mathrm{f}}\) of a three-layer plate as functions of the tension \(n_x\); \(\vartheta=0.05,\ k=1,\ \bar a=1,\ n=1,\ \varepsilon=0.1\) (simply supported edges, first two frequencies)

The solution of the original differential equation (1.1) for a certain fixed number \(n\) can be represented in the form

\[ \chi(x,y,t)=\sin\frac{n\pi y}{b}\cdot e^{i\omega t}\sum_{j=1}^{6} C_j e^{i\alpha_j x}. \tag{2.9} \]

\(3^\circ\). Consider a number of boundary-value problems.

a) Simply supported along all four edges. The boundary conditions for the displacement function \(\chi(x)\) at \(x=0,\ x=1\) and \(y=0,\ y=1\) have the form \((^1,\ ^2)\)

\[ \chi-\nabla^2\chi=\nabla^2\nabla^2\chi=0. \tag{3.1} \]

Substituting (2.8) into (3.1), we obtain a system of homogeneous linear algebraic equations for the coefficients \(C_j\). The condition for a nontrivial solution of the system will be

\[ \Delta/\delta=0, \tag{3.2} \]

where \(\Delta\) is the determinant of the system; \(\delta\) is the Vandermonde determinant formed from the roots \(\alpha_j\). The numerator of (3.2) is a sum of 10 terms \(\Delta_j\) of the form

\[ \Delta_j=A_jB_j \quad (j=1,2,\ldots,10), \tag{3.3} \]

where

\[ \begin{aligned} A_1/\left(e^{\alpha_1+\alpha_2+\alpha_3}-e^{-(\alpha_1+\alpha_2+\alpha_3)}\right) &=(123)+(231)+(312),\\ B_1&=(546)+(465)+(654); \tag{3.4} \end{aligned} \]

\[ \begin{aligned} (123)&=(\alpha_2^2-\alpha_1^2)(\alpha_1^2-n^2\pi^2\bar a^2)\times\\ &\quad\times(\alpha_2^2-n^2\pi^2\bar a^2), \tag{3.5}\\ (546)&=(\alpha_4^2-\alpha_5^2)(\alpha_5^2-n^2\pi^2\bar a^2)\times\\ &\quad\times(\alpha_4^2-n^2\pi^2\bar a^2). \end{aligned} \]

Fig. 2. Effect of tension \(n_y\) on the natural-frequency \(\omega_*\) of a three-layer plate with hinged supported edges, \(n_x=-3.25\), \(\vartheta=0.05\), \(k=1\), \(\bar a=1\), \(n=1\), \(\varepsilon=0.1\) (the first two frequencies)

The remaining terms of the sum, \(j=2,3,\ldots\), are obtained from (3.4) by the corresponding permutation of the indices of the roots \(\alpha_j\).

b) Hinged support along all four edges, but at the edges \(x=0\), \(x=1\) there are diaphragms rigid in the plane of the edge, preventing

Fig. 3. Variation of the critical divergence and flutter speed of a three-layer plate as a function of the tensions \(n_x, n_y\); \(\vartheta=0.05\), \(k=1.00\), \(\bar a=1\), \(n=1\), \(\varepsilon=0.1\) (hinged supported edges, first two harmonics)

shear of the load-bearing layers. The boundary conditions for \(x=0\), \(x=1\) will be

\[ \left(\frac{\partial^2}{\partial x^2}+\nu\frac{\partial^2}{\partial y^2}\right)(1-\vartheta k\nabla^2)\chi =(1-k\nabla^2)\chi =\frac{\partial}{\partial x}\nabla^2\chi=0, \tag{3.6} \]

where \(\nu\) is the generalized Poisson’s ratio \((^1)\). The components of the first term \(\Delta_j\,(j=1,2,\ldots,20)\) will take the form

\[ (123)=(\alpha_2-\alpha_1)(\alpha_3^2-\nu n^2\pi^2\bar a^2)[1-\vartheta k(\alpha_3^2-n^2\pi^2\bar a^2)]\times \]
\[ {}\times[\alpha_2^2+\alpha_2\alpha_1+\alpha_1^2-n^2\pi^2\bar a^2 -k(\alpha_1^2-n^2\pi^2\bar a^2)(\alpha_2^2-n^2\pi^2\bar a^2)]. \tag{3,7} \]

Similarly to (546), but with the factor \(\exp(\alpha_4+\alpha_5+\alpha_6)\).

c) Hinged support along the edges \(x=0,\ x=1\) and clamping along the edges \(y=0,\ y=1\). The boundary conditions for the clamped edge are

\[ (1-k\nabla^2)\chi=\partial\chi/\partial x=\nabla^2\partial\chi/\partial x=0. \tag{3,8} \]

For this case \((j=1,2,\ldots,10)\)

\[ (123)=\alpha_1\alpha_2[1-k(\alpha_3-n^2\pi^2\bar a^2)](\alpha_2-\alpha_1). \tag{3,9} \]

d) Two edges \(y=0,\ y=1\) are hinged, and the other two, \(x=0,\ x=1\), are free. The boundary conditions for the free edges are

\[ \left(\frac{\partial^2}{\partial x^2}+\nu\frac{\partial^2}{\partial y^2}\right)\chi = \left(\frac{\partial^2}{\partial x^2}+\nu\frac{\partial^2}{\partial y^2}\right)\nabla^2\chi = \]
\[ = \left[\frac{\partial^3}{\partial x^3}+(2-\nu)\frac{\partial^3}{\partial x\partial y^2}\right](1-\vartheta k\nabla^2)\chi=0. \tag{3,10} \]

In this case \((j=1,2,\ldots,20)\)

\[ (123)=\alpha_3[\alpha_3^2-(2-\nu)n^2\pi^2\bar a^2][1-\vartheta k(\alpha_3^2- \]
\[ {}-n^2\pi^2\bar a^2)](\alpha_1^2-\nu n^2\pi^2\bar a^2)(\alpha_2^2-\nu n^2\pi^2\bar a^2)(\alpha_2^2-\alpha_1^2). \tag{3,11} \]

Fig. 4. Effect of elongation on the critical flutter speed of a three-layer plate with hinged edges: \(\vartheta=0.05,\ k=1.00,\ n=1\).

From an analysis of the results of the calculation (Figs. 1–4) it follows that the dimensionless flutter speed of a three-layer plate \(p_*\) depends strongly on the magnitude of the longitudinal load \(n_x\) (Fig. 1) and on the elongation \(\bar a\) (Fig. 4), and depends practically not at all on the magnitude of the transverse load \(n_y\) (Fig. 2). However, the latter has a substantial effect on the divergence speed and, as a consequence, on the size of the stability region (Fig. 3). The effect of aerodynamic damping becomes noticeable only for sufficiently large values of the coefficient \(\varepsilon\) (Fig. 3).

The problem was suggested by E. I. Grigolyuk. I express my gratitude to him and to K. I. Babenko for their comments and advice.

All-Union Institute
of Scientific and Technical Information

Received
26 III 1968

REFERENCES

1. E. I. Grigolyuk, P. P. Chulkov, Izv. AN SSSR, Mekhanika i mashinostr., No. 1, 67 (1964).
2. A. I. Smirnov, DAN, 172, No. 3, 561 (1967).
3. A. I. Smirnov, DAN, 180, No. 5 (1968).