BRIEF COMMUNICATIONS

UDC 517.946

ON THE QUESTION OF AN APPROXIMATE SOLUTION OF THE PROBLEM OF PLANE BENDING OF A THIN ROD OF VARIABLE STIFFNESS

M. Yu. Karpova

The differential equation of equilibrium of a thin rod of variable stiffness has, as is known [1], the form

\[ \frac{d^{2}\zeta}{ds^{2}}+\frac{H'(s)}{H(s)}\frac{d\zeta}{ds} = \frac{H'(s)}{\rho_{0}H(s)}-\frac{P}{H(s)}\sin \zeta,\quad \zeta(0)=\zeta_{0},\quad \zeta'(0)=\frac{1}{\rho_{0}} \tag{1} \]

(the notation is as in the monograph [1]).

For an approximate solution of this equation we shall make use of the ideas of the works [2, 3, 4, 5, 6], repeatedly discussed at the Izhevsk mathematical seminar.

Equation (1) is equivalent to the integral equation

\[ \zeta(s)=\zeta_{0}+\int_{0}^{s} K(s,\gamma) \left\{ \frac{H'(\gamma)}{\rho_{0}H(\gamma)} - \frac{P}{H(\gamma)}\sin \zeta(\gamma) \right\}\,d\gamma, \tag{2} \]

where \(K(s,\gamma)\) is the Cauchy function [2, 3, 6] of the homogeneous equation

\[ \frac{d^{2}\zeta}{ds^{2}}+\frac{H'(s)}{H(s)}\frac{d\zeta}{ds}=0. \]

As the zero approximation we take \(\zeta^{0}(s)=\zeta_{0}+\dfrac{s}{\rho_{0}}\), the solution of the linear problem

\[ \frac{d^{2}\zeta}{ds^{2}}+\frac{H'(s)}{H(s)}\frac{d\zeta}{ds} = \frac{H'(s)}{\rho_{0}H(s)},\quad \zeta(0)=\zeta_{0},\quad \zeta'(0)=\frac{1}{\rho_{0}}. \]

Then the first approximation is

\[ \zeta^{1}(s)=\zeta_{0}+\frac{s}{\rho_{0}} - \int_{0}^{s} K(s,\gamma)\frac{P}{H(\gamma)}\sin \zeta^{0}(\gamma)\,d\gamma. \]

The error estimate \(\Delta\zeta=|\zeta-\zeta^{1}|\) of the approximate solution has the form \(\Delta\zeta\le \xi\), where \(\xi\) is the solution of the equation

\[ \xi(s)=\int_{0}^{s}\left|K(s,\gamma)\frac{P}{H(\gamma)}\right|\xi(\gamma)\,d\gamma+|\varphi(s)|, \]

\[ \varphi(s)=\zeta^{1}(s)-\zeta_{0}-\frac{s}{\rho_{0}} + \int_{0}^{s} K(s,\gamma)\frac{P}{H(\gamma)}\sin \zeta^{1}(\gamma)\,d\gamma. \]

Indeed, since

\[ \Delta\zeta=|\zeta-\zeta^{1}|= \left| -\int_{0}^{s}K(s,\gamma)\frac{P}{H(\gamma)} \left[\sin \zeta(\gamma)-\sin \zeta^{1}(\gamma)\right]\,d\gamma +\varphi(s) \right| \le \]

\[ \leqslant \int_0^s \left| K(s,\gamma)\,\frac{P}{H(\gamma)} \right|\,\Delta \zeta\,d\gamma + |\varphi|, \]

then, by virtue of the theorem on the integral inequality [5], we have the estimate \(\Delta \zeta \leqslant \xi\).

As an illustration of the effectiveness of the proposed procedure, let us consider the following problem.

A leaf spring of variable stiffness, whose curvature in the unloaded state is \(\dfrac{1}{\rho_0}\), is bent under the action of a force \(P\). Putting the stiffness of the spring \(H(s)=as^2+bs+c\), we have

\[ \zeta(0)=+\frac{\pi}{2},\qquad \zeta'(0)=\frac{1}{\rho_0}, \]

\[ K(s,\gamma)=\frac{2}{\sqrt{\Delta}}\,H(\gamma) \left( \operatorname{arctg}\frac{2as+b}{\sqrt{\Delta}} - \operatorname{arctg}\frac{2a\gamma+b}{\sqrt{\Delta}} \right), \]

where \(\Delta=4ac-b^2\). Then

\[ \zeta^1(s)=+\frac{\pi}{2}+\frac{s}{\rho_0} -\frac{2}{\sqrt{\Delta}}\,P \int_0^s \left( \operatorname{arctg}\frac{2as+b}{\sqrt{\Delta}} - \operatorname{arctg}\frac{2a\gamma+b}{\sqrt{\Delta}} \right) \cos \frac{\gamma}{\rho_0}\,d\gamma . \]

Putting \(a=10.95;\ b=-785;\ c=2.15\cdot10^4;\ E=2\cdot10^4\ \text{kg}/\text{mm}^2,\ P=10\ \text{kg},\ l=42\ \text{mm}\), we obtain \(\zeta^1(l)=-1.93\) (the integral was computed by Simpson’s formula).

Integrating (3) by the method of finite sums [7] for \(2n=20\) and denoting by \(\{\xi_i\}\) \((i=0,\ldots,20)\) the solution of the corresponding algebraic system, we have

Let \(z(s)=\xi_i\) for \(s\in[2.1i,\ 2.1(i+1)]\), \(i=0,\ldots,20\). It is not hard to verify that

\[ z>\int_0^s \left| K(s,\gamma)\,\frac{P}{H(\gamma)} \right| z\,d\gamma + |\varphi|. \]

Hence, from the theorem on the integral inequality [5], we have \(z(s)\geqslant \xi(s)\geqslant \Delta\zeta(s)\).

The relative error

\[ \frac{\Delta\zeta}{\zeta}\,100\% \leqslant \frac{\Delta z}{\zeta}\,100\% \]

of the approximate solution \(\zeta^1(s)\) proposed by us does not exceed \(6\%\) at the end of the spring \((s=l)\)—a result satisfying the requirements of engineering calculations.

Taking this opportunity, I thank the staff of the Izhevsk seminar for the discussion of the work and for valuable comments.

References

1. Popov E. P. Nonlinear Problems of the Statics of Thin Rods. Leningrad—Moscow, 1948.
2. Azbelev N. V., Tsalyuk Z. B. Doklady Akademii Nauk SSSR, 135, No. 3, 511—514, 1960.
3. Azbelev N. V., Tsalyuk Z. B. A note on iterative methods for solving differential equations. Izvestiya vuzov, Mathematics, 1, 1957.
4. Pak S. A. Siberian Mathematical Journal, 3, No. 4, 1962.
5. Azbelev N. V., Tsalyuk Z. B. On integral inequalities. Matematicheskii sbornik, 56 (93): 3, 1962, pp. 325—341.
6. Karpova M. Yu. Proceedings of the Izhevsk Mathematical Seminar, issue 1, 1963, pp. 49—51.
7. Demidovich B. P., Maron I. A., Shuvalova Z. Z. Numerical Methods of Analysis. Moscow, 1963.

Received by the editors July 13, 1965.

Izhevsk Mechanical Institute