Newman, S.J.

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      Aerodynamics & Flight Mechanics Research Group            Coleman Transformation for N Rotor Blades  S. J. Newman  Technical Report AFM-11/20  January 2011     UNIVERSITY OF SOUTHAMPTON  SCHOOL OF ENGINEERING SCIENCES  AERODYNAMICS AND FLIGHT MECHANICS RESEARCH GROUP   Coleman Transformation for N Rotor Blades  by  S. J. Newman  AFM Report No. AFM 11/20   January 2011     © School of Engineering Sciences, Aerodynamics and Flight Mechanics Research Group     COPYRIGHT NOTICE (c) SES University of Southampton All rights reserved. SES authorises you to view and download this document for your personal, non-commercial use. 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This document, is protected by worldwide copyright laws and treaty provisions.     Definition of the Transformation The rotor consists of N blades, each of which is denoted by the suffix, k. Each individual blade then possesses a lag angle of ζk. The Coleman transformation converts the N lag angles (ζk) to N variables, j , via the following:    102 NkkNijk j e  ( 1 )  For brevity, define:   Nie2  ( 2 )  i.e.  is the primary Nth root of unity.  Thus;    10Nkk jk j    ( 3 )  for j=1,2,……,N‐1.     Therefore:      1 111 2211 0 11 214221 0 21122 1 0 11 2 1 0 0................                  N NNN NNNNNNN                          ( 4 )   The  terms represent the azimuthal spacing between the blades and are thus referenced to a rotating set of axes.  To cater for a non‐rotating set of axes, the above equations are multiplied by:   t i e ( 5 )  also, denoting the blade azimuth via:   kNt k    2  ( 6 )  from which:    k i k t i e e     ( 7 )     Since,  is an Nth root of unity we also have:   j jj jNj N       21( 8 ) Since:   1    ( 9 )  Rewriting the equations and premultiplying by powers of eit we find:       1 111 2211 0 11 2 212 2 222 21 02221 2 214 222 21 02221122 1 0 11 2 1 0 0....................                                                  N N t iNN t i N t i t iNt iN N t iNN t i N t i t iNt iN t iNt i t i t i t iN t iNt i t i t i t iNe e e e ee e e e ee e e e ee e e e e                                                  ( 10 )  Since  is an Nth root of unity the last two equations can be rewritten:    112211 0 11 2 214 222 21 0222........                             N t iNt i t i t iNt iN t iNt i t i t iNt ie e e e ee e e e e                        ( 11 )     Construction of the Coleman Variables The first equation  1 2 1 0 0 ....       N        ( 12 )  is left untouched. The equations are then paired:  333322221 1Nt i t iNt i t iNt i t ie ee ee e       ( 13 )  these are then added and subtracted to give real expressions in cosine and sine. For instance, let us examine the first pairing:                     1 11111 222211 10 01 1.... 212N N t iNN t iNN t i t iN t i t it i t iNt i t ie ee ee ee ee e                 ( 14 ) and since:     1 1 11 11    N N NNN   ( 15 )     we find:                     1111222211 10 01 1.... 212N t iNN t iNt i t it i t it i t iNt i t ie ee ee ee ee e                 ( 16 )  Recalling (7), we have finally:         1 12 21 10 01 1cos....coscoscos2N NNt i t i e e       ( 17 ) and by a similar process:        1 12 21 10 01 1sin....sinsinsin2N NNt i t i e e       ( 18 )   If the rotor has an odd number of blades (N) this process will complete the transformation. If the number of blades (N) is even then there will be one equation (M) left. In this case the multiplication by eMit does not take place and we have:     1122 1 0 ....     N MNM MM          ( 19 )      here we have:  2NM     as can be seen the multiplying factors for the N lag angles, k, are powers of M.  Since:   Nie2  ( 20 )  we find:   122   iNMiMNiMeee ( 21 )  In other words the coefficients of the k terms are powers of ‐1, in other words of alternating sign.  Reference R.P. Coleman, A.M. Feingold, Theory of self-excited mechanical oscillations of helicopter rotors with hinged blades, NACA Report TN 1351, 1958     Examples  We therefore have the following examples of the transformation:  N=2 1 0 11 0 0        -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.2-0.15-0.1-0.0500.050.10.150.2Coleman Mode - Symmetric     -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.16-0.14-0.12-0.1-0.08-0.06-0.04-0.0200.02Coleman Mode - Anti Symmetric   N=3  2 2 1 1 0 0 22 2 1 1 0 0 12 1 0 0sin sin sincos cos cos                          -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Symmetric -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Cosine 1    -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Sine 1    N=4 3 2 1 0 3213 2 1 0 0sincos                  kk kkk k     -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Symmetric  -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Cosine 1    -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Sine 1  -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Anti Symmetric   N=5          kk kkk kkk kkk k        2 sin2 cossincos43214 3 2 1 0 0  -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Symmetric     -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Cosine 1  -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Sine 1     -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Cosine 2  -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81Coleman Mode - Sine 2    Or in tabular form:   N  +  +‐  Cos  Cos 2  Cos 3  Cos 4  Sin  Sin 2  Sin 3  Sin 4 1                    2                  3                 4                 5                   6                   7                     8                     9                     10                       

Coleman transformation for N rotor blades

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