By S. A. Marshall (auth.)
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These results can be useful in establishing the approximate location of some or all of the roots of a polynomial. Polynomials of Odd Order When determining the roots of odd-order polynomials it is recognised, since all the coefficients are real, that the polynomial must have at least one real root, the remainder being real and/or complex, with the complex roots occurring as conjugate pairs. Thus, the real root is first determined using Newton's method and then factored out to leave a polynomial of even order.
16) a summation of the n solutions. 18) and the n unknown coefficients A I, A 2 , ... ,An are obtained from knowledge of the n initial conditions, that is,x(O), dx(O)/dt, d2 x(O)/dt 2 , . . ,dn-lx(O)/d~-I. 18 will be stable, that is, it will remain 27 The Need for Some Mathematics constant as time increases, if the exponential terms emit decay. Therefore, assuming that the roots mi are real, stability will be guaranteed if the mi are negative, that is mi < O. For a complex root, that is, mi = ai + jWi' the solution will be stable if the real part is negative, that is, ui < O.
Note that cos rf> =~, the damping factor. 5 for values of the damping factor ~ between oand 1; as is evident, it is this damping factor that dominates the response whilst Wn merely affects the time scale. When ~ == 0 (no damping), the response is a sinusoid of natural frequency Wn, the undamped natural frequency. / xU) ! /' 0:;'\ 1// .... 2' . ~ .... o 7' l,y - .. ~ : \ ". :""- --~r ........ / ;'/ / / -- -.... '-', The Need for Some Mathematics 45 as the damped frequency, and the percentage overshoot decreases.