Download PDF by Derek F. Lawden: A Course in Applied Mathematics, Vol. 1 and 2

By Derek F. Lawden

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72) where D dfdt. 72) is a linear differential equation with constant co­ efficients, and we shall solve it by the standard procedure which is available for such equations. , (i) k > 2Cil, (ii) k 2Cil and (iii) k < 2Cil. 73) are both real and distinct. 73) greater than zero. Let the roots be -'A1 and - 'A2 ('A1 > 'A2). 76) and, hence, - 'Ale-A•t. 78) 44 A C O U R S E I N A P PLI E D M A T H E MAT I C S [CH. But the right-hand side of this equation is less than unity, whereas the left-hand side is greater than unity if t is positive, for then the exponent of e is also positive.

Provided the viscous friction is not too great, there will exist a frequency of oscillation of the point of suspension at which the particle will give a maximum response. At this frequency, the system is in the state of resonance. 72) . If, therefore, the particle is displaced from 0 and released, it will execute damped oscillations of the type described at the beginning of this section. A particle of mass m is hanging in equilibrium under gravity on the end of an elastic string having a Hooke's constant m.

R = -law2 cos lwt. w2(4 + 2y3)0N = -w2(4 + 2y3)MN. = fr is therefore representable, both in magnitude and direction, by the --+ vector (4 + 2y3)w2• PN. fo is similarly representable by the vector --+ (4 + 2y3)w2• NM. The total acceleration of P is now given by the vector sum --+ --+ (4 + 2y3)w2 • PN + (4 + 2y3)w2• NM = --+ --+ (4 + 2v3)w2 • (PN + NM) --+ = (4 + 2y3)w2. PM. EXERCISE 1 P and Q are moving around two coplanar circles in the same sense and with the same angular velocities w. If a is the distance I.

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