The system experiences an increase in kinetic energy

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The system experiences an increase in kinetic energy for both y positive and y negative. The periodic path consists of four curves whose equations are listed below: C 1 : mcy 2 + (F 0 cx) 2 = (F 0 cx 0 ) 2 C 2 : mcy 2 + (F 0 cx) 2 = (F 0 cx 1 ) 2 C 3 : mcy 2 + (F 0 + cx) 2 = (F 0 + cx 1 ) 2 C 4 : mcy 2 + (F 0 + cx) 2 = (F 0 + cx 0 ) 2 The paths, the point (x 0 , 0 ) where the paths C 1 and C 4 meet, and the point (x 1 , 0 ) where the paths C 2 and C 3 meet are shown in Figure 1.24. At x = − α the energy is increased by E for both positive and negative y . The discontinuities at x = − α are shown in Figure 1.25. Therefore, at x = − α , E = 1 2 c [ (F 0 cx 1 ) 2 (F 0 + cα) 2 (F 0 cx 0 ) 2 + (F 0 + cα) 2 ] , E = 1 2 c [ (F 0 + cx 0 ) 2 (F 0 + cα) 2 (F 0 cx 1 ) 2 + (F 0 + cα) 2 ] .
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20 Nonlinear ordinary differential equations: problems and solutions x y C 1 C 2 C 3 C 4 x 0 x 1 a Figure 1.25 Problem 1.13. Simplifying these results E = 1 2 c [− 2 F 0 cx 1 + c 2 x 2 1 + 2 F 0 cx 0 c 2 x 2 0 ] , E = 1 2 c [ 2 F 0 cx 0 + c 2 x 2 0 2 F 0 cx 1 c 2 x 2 1 ] , Elimination of E gives x 1 = − x 0 , and x 0 = − x 1 = E 2 F 0 . 1.14 The ‘friction pendulum’ consists of a pendulum attached to a sleeve, which embraces a close-fitting cylinder (Figure 1.34 in NODE). The cylinder is turned at a constant rate > 0. The sleeve is subject to Coulomb dry friction through the couple G = − F 0 sgn ( ˙ θ ) . Write down the equation of motion, the equilibrium states, and sketch the phase diagram. 1.14. Taking moments about the spindle, the equation of motion is mga sin θ + F 0 sgn ( ˙ θ ) = − ma 2 ¨ θ . Equilibrium positions of the pendulum occur where ¨ θ = ˙ θ = 0, that is where mga sin θ F 0 sgn ( ) = mga sin θ + F 0 = 0, assuming that > 0. Assume also that F 0 > 0. The differential equation is invariant under the change of variable θ = θ + 2 so all phase diagrams are periodic with period 2 π in θ . If F 0 < mga , there are two equilibrium points; at θ = sin 1 F 0 mga and π sin 1 F 0 mga : note that in the second case the pendulum bob is above the sleeve.
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1 : Second-order differential equations in the phase plane 21 The phase diagram with the parameters = 1, g/a = 2 and F 0 /(ma 2 ) = 1 is shown in Figure 1.26. There is a centre at θ = sin 1 ( 1 2 ) and a saddle point at x = π sin 1 ( 1 2 ) . Dis- continuities in the slope occur on the line ˙ θ = = 1. On this line between θ = sin 1 ( 1 2 ) and θ = sin 1 ( 1 2 ) , phase paths meet from above and below in the positive direction of θ . Suppose that a representative point P arrives somewhere on the segment AB in Figure 1.26. The angular velocity at this point is given by ˙ θ(t) = (i.e. it is in time with the rotation of the spindle at this point). It therefore turns to move along AB , in the direction of increasing θ . It cannot leave AB into the regions ˙ θ > or ˙ θ < since it must not oppose the prevailing directions. Therefore the representative point continues along AB with constant velocity , apparently ‘sticking’ to the spindle, until is arrives at B where it is diverted on to the ellipse. Its subsequent motion is then periodic.
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