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1. APPENDIX 1 Introduction to Complex Numbers A complex number consists of a real (Re) and an imaginary (Im) part. For instance the complex number a ¼ b þ jc has the real part ‘‘Re ¼ b’’ and imaginary part ‘‘Im ¼ c’’, j ¼ ffiffiffiffiffiffiffi À1 p . The complex number ‘‘a’’ can be presented in the complex number plane (Re, Im) as a two-dimensional vector with the length ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 þ c2 p and angular orientation, measured from the axis ‘‘Re’’, as arctan (c/b): a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 þ c2 p e j arctan c=bð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 þ c2 p e j ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 þ c2 p cos þ j sin ð Þ ðA1:1Þ where arctanðc=bÞ (Figure A1.1). Note the important relationship, which ties complex number to trigonometric functions: e j ¼ cos þ j sin Using this equation, for the specific angles the following relationships hold true: e j90 ¼ j, e j180 ¼ À1, e j270 ¼ Àj, e j0 ¼ 1 The exponential function e j can also be denoted ‘‘exp ( j)’’. A complex conjugate of the number a ¼ b þ jc is a ¼ b À jc. Figure A1.1 Complex number as a vector in the complex number plane. 989 © 2005 by Taylor Francis Group, LLC

2. Note that in engineering use of complex numbers the imaginary part of the complex number is often called ‘‘quadrature’’, in order to avoid connotations with ‘‘unreality’’. The real part of the complex number is called ‘‘direct’’. Using Eq. (A1.1), the relationship between trigonometric and exponential functions can be derived as follows: cos ¼ e j þ eÀj 2 , sin ¼ e j À eÀj 2j ¼ Àj e j À eÀj 2 The hyperbolic functions look somewhat similar to the above functions, but they do not involve complex numbers in their exponential functions. The hyperbolic sine and cosine are as follows: sin h ¼ e À eÀ 2 , cos h ¼ e þ eÀ 2 The corresponding inverse functions are: ¼ arcsin hðAÞ ¼ ln A þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 þ 1 p , ¼ arccos hðAÞ ¼ ln A þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 À 1 p Similarly, the hyperbolic functions tangent and cotangent, and inverse functions arctan h, arccotan h, are introduced. The addition (subtraction) of two complex numbers, a1, a2 is as follows: a1 þ a2 ¼ b1 þ jc1 Æ ðb2 þ jc2Þ ¼ ðb1 Æ b2Þ þ jðc1 Æ c2Þ The addition (subtraction) of the complex numbers presented in the exponential format is as follows: A e j Æ B e j

3. ¼ C e j where C ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 þ B2 Æ 2AB cosð À

4. Þ p , ¼ arctan A sin Æ B sin

5. A cos Æ B cos

6. The multiplication of complex numbers is as follows: a1a2 ¼ ðb1 þ jc1Þðb2 þ jc2Þ ¼ b1b2 À c1c2 þ jðb1c2 þ c1b2Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 1 þ c2 1 q e j arctanðc1=b1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 2 þ c2 2 q e j arctanðc2=b2Þ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb2 1 þ c2 1Þðb2 2 þ c2 2Þ q e j arctanðc1=b1Þþarctanðc2=b2Þ½ Š ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb2 1 þ c2 1Þðb2 2 þ c2 2Þ q e j arctan ðb1c2þc1b2Þ=ðb1b2Àc1c2Þ½ Š 990 ROTORDYNAMICS © 2005 by Taylor Francis Group, LLC

7. The division of complex numbers is as follows: a1 a2 ¼ b1 þ jc1 b2 þ jc2 ¼ ðb1 þ jc1Þðb2 À jc2Þ ðb2 þ jc2Þðb2 À jc2Þ ¼ b1b2 þ c1c2 þ jðc1b2 À b1c2Þ b2 2 þ c2 2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 1 þ c2 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 2 þ c2 2 q e j arctanðc1=b1ÞÀarctanðc2=b2Þ½ Š ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 1 þ c2 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 2 þ c2 2 q e j arctan ðc1b2Àb1c2Þ=ðb1b2þc1c2Þ½ Š In the latter transformations the following trigonometric relationships were used (see also arctan A Æ arctan B ¼ arctan C, where A ¼ c1=b1, B ¼ c2=b2 Æ

8. ¼ , where A ¼ tan , B ¼ tan

9. , C ¼ tan tanð Æ

10. Þ ¼ tan tan ¼ tan Æ tan

11. 1 Ç tan tan

12. ¼ A Æ B 1 Ç AB C ¼ tan ¼ ðc1=b1Þ Æ ðc2=b2Þ 1 Ç ððc1c1Þ=ðb1b2ÞÞ ¼ c1b2 Æ b1c2 b1b2 Ç c1c2 A radical of a complex number is a complex number: ffiffiffiffiffiffiffiffiffiffiffiffi a þ jb p ¼ c þ jd a þ jb ¼ c2 þ 2jcd À d2 a ¼ c2 À d2 , b ¼ 2cd From these real and imaginary parts, the unknown c and d can be calculated: c2 À b 2c 2 ¼ a, c4 À ac2 À b2 4 ¼ 0 c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 a þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 p r , d ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Àa þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 p r Thus ffiffiffiffiffiffiffiffiffiffiffiffi a þ jb p ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 a þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 p r þ j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 Àa þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 p r The solution of a quadratic equation with complex coefficients, such as the characteristic equation of linear systems s2 þ ða þ jbÞs þ c þ jd ¼ 0 is as follows: s ¼ À a þ jb 2 Æ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ jb 2 2 Àc À jd s INTRODUCTION TO COMPLEX NUMBERS 991 © 2005 by Taylor Francis Group, LLC Appendix 6):

13. and further, using the formulas developed above: s ¼ À a 2 Æ 1 ffiffiffi 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 À b2 4 À c þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 À b2 4 À c 2 þ ab 2 À d 2 sv u u t 2 6 4 3 7 5þ À j b 2 Æ 1 ffiffiffi 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À a2 À b2 4 þ c þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 À b2 4 À c 2 þ ab 2 À d 2 sv u u t 2 6 4 3 7 5 8 : 9 = ; There are four solutions resulting from four combinations of signs ‘Æ’ of the quadratic equation with complex numbers. These four solutions are actually two pairs of complex conjugate numbers. In vibration measurements, the use of complex numbers in the representation of measured amplitudes and phases of the vibration waveforms, filtered to specific frequencies, makes data processing easier. The amplitude A and phase are presented in the form of a complex vector, Ae j . In industry, the formal representation of measured amplitudes and phases are usually presented in the notation Aff (amplitude A at angle ) and for algebraic transformations, the formulas are given following the above formal derivation. This is often applied in the routines of the one- or two-plane balancing. 992 ROTORDYNAMICS © 2005 by Taylor Francis Group, LLC

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