A Course of Higher Mathematics. Volume II by V. I. Smirnov and A. J. Lohwater (Auth.)

By V. I. Smirnov and A. J. Lohwater (Auth.)

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19). We use the convention that forces acting on the beam are positive if directed down­ wards. We isolate section N of the beam with abscissa x. Let y denote the displacement of the point on the neutral axis, and R the radius of curvature of the deformed axis. I t is shown in the theory of strength of ma­ terials that, with certain assumption regard­ ing the character of the deformation and the position of the beam relative to axes OX, OY, the equation of equilibrium is to be obtained as follows: we neglect the part of F I G .

The problem therefore amounts to investigating the bending of a supported beam under the action of continuously distributed loading f(x) = — kx. We start by calculating P0 and Ph the reactions of the supports. The total loading is Jfc|d| = fc/2 2 The reactions at the supports O and L due to the elementary loading fc£d| are, in accordance with the usual law of levers: kHl- ■f) At A k & At —- d | and —=— d£. £_±,, , , - , - , . x-o = °; (34) i/Ix-r = °- The general solution is: -h Z2*3 ( x* , . n n \ Constants Gx and C2 are found from conditions (34): C2 = 0; C1 = 7 60 -^-^, whence finally: V = 3 " ^ (3* 5 - 10/2 ^3 + 7/%).

Which corresponds to a maximum for \y\. 54 ORDINARY DIFFERENTIAL EQUATIONS [17 Maximum deflection thus occurs towards the end L and not at the centre, its value being: ^ — l/lx-if. 348- 360j57/ I8QJM-• 17. Lowering the order of a differential equation. We notice a number of particular cases in which the order of an equation can be lowered. 1. Let the function y and a certain number of consecutive derivatives of y:y',y", . . , y**""1*, be excluded from the equation, which has t h e form: &(xyyW,y(k+1\..

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