By Henry G. Booker
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Extra resources for A Vector Approach to Oscillations
If we know the positions of the poles and zeros in Fig. 4, we may join them to the tip of any specified vector s and then use Eqs. 17) to evaluate the magnitude and direction of the vector F(s) defined by Eq. 12). 19) *l' = - 1 , s2' = - 2 . 20) Quadrature direction FIG. 5. Illustrating calculation of the vector F given by Eq. 18) under the circumstances described by Eqs. 21). 2. Algebraic Functions of a Vector 35 Let us evaluate the magnitude and direction of F(s) when s = 2j. 21) In these circumstances Fig.
Polynomial Functions of a Vector and Z'F = (0 + tan-M) - (tan" 1 ! 511 Z - 6 ? 9 . , a n are fixed vectors and s is a vector that may be varied in magnitude and direction. All products involved in Eq. 32) are planar products. The expression on the right-hand side of Eq. 4), if multiplied out, is a polynominal function of s of degree n\ in this case the coefficient aw is a unit vector in the reference direction. Thus the numerator and denominator of the expression on the right-hand side of Eq. 12) are polynominal functions of s, the numerator being of degree m and the denomi nator of degree w.
5). This means that Z P = «i + 0 a + - * » . 7) Thus the vector P defined by Eq. 4) is obtained by multiplying the magnitudes and adding the angles of the vectors given in Eqs. 5). The result described by Eqs. 7) is conveniently represented graphically as shown in Fig. 2. , sw and indicated by crosses. , s n are indicated in Fig. 2 only by the positions of their tips. In each case the vector is to be pictured as drawn from the origin to the point designated as the tip of the vector. , sn . In the same way there is shown in Fig.
A Vector Approach to Oscillations by Henry G. Booker