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3y 5 6 / 4 C( 37L NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1348 ON THE REPRESENTATION OF THE STABILITY REGION IN OSCILLATION PROBLEMS WITH THE AID OF THE HURWITZ DETERMINANTS* By E. Sponder For oscillation phenomena which may also have an unstable course, it is customary to represent the regions where stability or instability prevails in a plane as functions of two parameters x and y. In order to determine whether stability exists at any point of the plane represented (thus, whether a disturbance of the oscillation phe nomenon considered is damped in its course) it must be investigated whether all roots X of the frequency (characteristic) equation Xn + aln1 + + a = 0 have a negative real part at this arbitrary point. The ,mathematical condition for this is known to be that the n Hurwitz determinants D1 to Dn which are formed from the coefficients al to an and are functions of the two parameters x and y mentioned before are all positive for this point. Resulting from this criterion and completely equivalent to it is the fact that the coefficients al to an and only a few certain Hurwitz determinants must be positive. If, conversely, all roots X of the frequency equation have a nega tive real part, all values D1 to Dn are positive. If one now visu alizes the point considered before (for which we assume stability to have been established) as traveling in the representation plane of figure 1, thete vary with its parameters x and 'y also the coefficients al to an, the real parts of the roots X, and finally the n Hurwitz determinants. If one arrives at a point where for instance a real root *"Zur Darstellung des Stabilitatsgebietes bei Schwingungsaufgaben mit Hilfe der Hurwitz Determinanten." Schweizer Archiv, March 1950, PP. 9396. NACA TM 1348 disappears, an also disappears since lani is the product of the values of all roots; if one reaches, in contrast, a point where the real part of a complex root becomes zero, it can be shown that then Dn1 becomes zero. For every case, however, the product anDn1 disappears which is nothing else but the Hurwitz determinant of the nth degree anDn1 = Dn Thus the important theorem, the proof for which will be presented later, is valid: The limits of the (usually unique) stable region lie at Dn = anDn1 = 0. For graphical representation, it is therefore completely sufficient to plot Dn = 0 or more simply an = 0 and Dn1 = 0 as separate limiting curves of a region for which stability is known to prevail at an arbitrary point, as illustrated by figure 1. If the limit Dn_1 = 0 is exceeded, the course of the oscillation process is "dynamically" unstable because a damping becomes negative; beyond the limit an = 0, one usually calls the oscillation process "statically" unstable. In particular, the following is valid for oscillation phenomena which lead to frequency equations of the 4th degree (reference i). The Hurwitz determinant of the 4th degree formed from the coeffi cients A to E of the frequency equation AX4 + BX3 + CX2 + DX + E = 0 reads D4 = ED3 = E(BCD AD2 B2E) = ER with the expression in parentheses known as Routh's discriminant R; the latter is therefore nothing else but the Hurwitz determinant of the NACA TM 1348 3rd degree. Since the coefficient A is usually +1, it is generally valid as dynamic stability condition that the expression (BC D)D B2E turns out positive. The static stability is then guaranteed in addition by E > 0 so that the graphic representation of the region of figure 2 results. Therewith, every requirement has been met; for it is impossible that within the region denoted as stable a curve C = 0 or D = 0 could run and perhaps still further reduce this region. It is therefore completely superfluous to investigate further what sign the other coefficients of the frequency equation have once the boundaries for a region E = 0 and R = 0 have been fixed; however, and this is important, it must be known that, at an arbitrary point of this region, stability actually prevails. For this it is sufficient to check for a point conveniently situated (for instance on an axis) the signs of the remaining coefficients A to D of the frequency equation which is equivalent with the fact that there all Hurwitz deter minants D1 to D4 turn out positive. That this is important can, for instance, be recognized from the fact that, in the region indicated on the left in the plane of representation of figure 2, instability may prevail in spite of E > 0 and R > 0. The advantage of these recognized facts given above does not result in much simplification for frequency equations of the 4th, 5th; or even 6th degree; however, this advantage may save a great deal of unnecessary calculations. The expressions for the Hurwitz determinants become more and more cumbersome for higher degrees so that it will probably be a very welcome facilitation for the outlining of the stability region if the mere representation of the two curves an = 0 and Dn1 = 0 will be sufficient. There now follows the proof of the theorem given before that the limits of the (usually unique) stable region lie at Dn = anDn1 = 0. 1. For the present consideration, the existence of at least one stable region is presupposed. Not even one need necessarily always exist; on the other hand, sVeral stability regions, separated from one another, may occur. One can easily make sure of this by considering a frequency equation of the 2nd degree which pertains to an ordinary oscillation X2 + alk + a2 = 0 NACA TM 1348 The coefficients a, and a2 are dependent on two parameters x and y, corresponding to the two coordinate axes of the plane of repre sentation. For simplicity's sake, we shall assume that a2 everywhere has positive values only; then the sign of al alone is the criterion, for the stability, since al is a measure of the damping of the oscil lation considered and this oscillation is stable only if al is posi tive. The limit of stability now depends entirely on the sign of the analytical (or empirical) function of the coefficient al of the two parameters x and y, or, in other words, what form the curves al = 0 have in the plane of representation which separate the positive from the negative values. In this manner, one can easily understand that several stability regions may exist but that just as well not even one need exist anywhere in the entire plane of representation. For frequency equations of higher degree, the probability decreases that simultaneously in several regions all stability conditions found by Hurwitz will be satisfied. Usually, one will deal with only one single stability region which will then be considered more thoroughly. 2. Hurwitz' criterion (reference 2) signifies that a prescribed equation of the nth degree with real coefficients a0n + alkn1 + + an = 0 (ao >O) possesses roots exclusively of the ndeterminants Dk = with negative real parts only when the values al a3 a9 a0 a2 a4 0 al a3 are all positive. Therein, one generally subscript x is negative or larger than has to put ax = 0 when the n. a2k1 a2k2 S a2k3 S ak (k = 1,2,...,n) NACA TM 1348 5 It shall now be proved that the Hurwitz determinant of the (n l)th degree Dn_1 always disappears when the real part of a complex pair of roots is zero. Following the way Hurwitz pursued in deriving his criterion, one finds the remarkable presupposition that no purely imaginary roots must exist. This, however, is precisely the condition whose influence on a certain Hurwitzdeterminant is of special interest. However, for that reason, one need not abandon this presupposition; it is sufficient to interpret Dn1 simply as a prescribed calculation rule which one applies, without consideration of its derivation, to the coefficients of an equa tion of the nth degree. It will now be expedient to consider (corresponding to the assump tion that the real part of a complex pair of roots is to be zero) the equation of the nth degree as split into two factors: one is a quad ratic expression with the purely imaginary pair of roots and the other is an expression of the (n 2)th degree, thus (2 X aaon2 + alk3 + an2) = therein one may put the coefficient a0 = 1, without impairing the gen erality; the designation a0 will, however, be retained. For all remaining coefficients al to an2, no additional restriction will be prescribed other than that they are to be real and not all zero (which would be trivial for this statement of the problem). Multiplication of the two factors yields aoXn + an1Xn + (a5 + aao)n2 + (a3 + aa + n + AO Al A2 A3 aan2 = 0 An The combined coefficients A one now visualizes as substituted in the Hurwitz determinant Dn1, the general term of which has in the sth column and zth line the subscript x = 2s z. Thus, A2sz or, expressed by the coefficients a, the term NACA TM 1348 one has there a2sz + aa2sz2 The value of the determinant Dn1 is now extended by multiplying all terms of the first line by a8 and all terms of the second line by al. Furthermore, one makes use of the theorem that a determinant does not change its value if one adds to the elements of one line the elements of another line multiplied by an arbitrary number. This is done by adding to the'terms of the first line, which already have been multi plied by aO, the terms of the third, fifth, line multiplied by a2, a, .; thus, one obtains as the general term of the sth column in the first line z=1,3,5,... azl(a2sz + aa2sz2) Likewise, one adds to the terms of the second line already multiplied by a1 the terms of the fourth, sixth, line multiplied by a3, 85, .; one then finds as the general term of the sth column in the second line z=2,4,6,... azl(a2sz + aa2sz2) In order to calculate these sums, it is sufficient to note that one must generally put ax = 0 when the subscript x is negative; there then results for a term of the first line NACA TM 1348 > z=l, 3,5,... a,l(a2sz + aS2s_z2) = aOa2s1 + a232s3 + + a2s433 + a2s281 + (3a2s3 + 8a22s5 + + a2s4a1) and for a term of the second line z=2,, z=2,h,6,.... azl(a2s + 32sz2) = al2s2 + 2s4 + + a2s3a2 + a2sla0 + (l2s4 + s + ( + 2 3ao) One can see that both sums are of equal magnitude which, however, does not signify anything else but that the corresponding elements of the first two lines are equal and that therefore the Hurwitzdeterminant Dn1 extended by a0al identically disappears: aoalDn1 0 From the derivation follows that one could have extended initially, instead of the first and second line, two arbitrary other odd and even lines by as_1 and then have proceeded further in the same manner; one then would have obtained quite generally even odd n1 NACA TM 1348 Under the obvious assumption that there always exists such a pair of values of the coefficients a0 to an2, the product of which does not disappear, thus Dn_l 0, Q.E.D. is valid. 3. From the derivation of this proof, one may further conclude that part of the curve Dn_ = 0 may run entirely in the unstable region; thus it does not appear there as the required stability limit. Since such a possibility had been pointed out before, the motivation of this noteworthy indication should be mentioned. In splitting the prescribed equation of the nth degree into two factors, no more detailed data on the coefficients ao to an_2 of the expression of the (n 2)th degree have been given and accordingly none regarding the roots of the equation ( ,n2 ni ) ao n2 + alX n + + an_2 = 0 either. These roots may therefore have negative as well as positive real parts in the neighborhood of points of the plane of representation for which Dn1 disappears; in other words, they may signify stability and also instability. If there exists at least one root of the former expression of the (n 2)th degree with a positive real part, this fact signifies that the curve Dn1 = 0 lies in an unstable region. Figure 3 is to indicate how, for instance, three roots may vary their position in the complex number plane if their point of reference in the plane of representation shifts beyond the curve Dn1 = 0. Thus Dn_ = 0 is not necessarily always the boundary between two regions here stability and instability prevail. ileither does the sign of the coefficient a in the quadratic expres sion (X2 + a) play a role in the derivation of the proof. If a is NACA TM 1348 positive, a purely imaginary pair of roots satisfies the equation (X2 + a) = 0; this case has been considered in particular. However, a could just as well be negative without altering the result of the theorem. This signifies, however, that in the presence of two real oppositeequal roots of the equation of the nth degree, the Hurwitz determinant of the (n l)th degree also disappears. As figure 4 shows, here also an unstable region lies to both sides of the curve Dnl = 0 so that again it does not appear as the sta bility limit. Thus, these examples show that it is indispensable to make sure whether really all stability conditions on one side of the curve Dn_1 = 0 have been fulfilled; only then that curve forms together with the curve an = 0 the boundary against the unstable region. Translated by Mary L. Mahler National Advisory Committee for Aeronautics REFERENCES 1. Price, H. L.: The Lateral Stability of Aeroplanes. Aircr. Engng., Vol. 15 (1943), No. 173, 174. 2. Hurwitz, A.: Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negative reellen Teilen besitzt. Mathematische Annalen XLVI. NACA TM 1348 y Dynamically unstable D Dnio=O o Point at which stability prevails an =0 Statically unstable x Figure 1. Travel of the point after establishment of stability. NACA TM 1348 R=0 Dynamically, E>0,R> 0 Instability possible J E=O S\ tatically unstable Figure 2. Graphic representation of the region E > 0. R=O NACA TM 1348 S\\\\\\ s 1 \\\ \\% \ \\% ' ,Unstable i L DnlO " Unstable ~' \\\^\\ ^\l Figure 3. Change of position of the roots in the complex number plane. \\ \\\\\ Unstable\  Dn1, =0 i, ,Unstable \. S % Figure 4. Unstable region for the curve Dn1 = 0. C So 5 *' 3 g' .2g .0 ed n CD Id Cs z > c e ' Q a o oM'1^, ~ S E ^S^ a a iusSzA S g " c w a I 4 U I U0) u LzJ L. mN 11 .Jr j ^[ ilt* C.CCI ma o to W nl a I W A s ^ 1 Cgje .14 S  WLbD f i E < om14bfl : S : aS E V 0 >0 CQ c 1 I I 2 .1 I c: 5 i C~ ow0 ,E.,r 4 e w0. 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