Determinant of gyrostatic conservative system. The determinant (which for brevity we shall denote by D) in this case is what has been called by Cayley a skew determinant. What it would become if zero were substituted for λ2 +☎„ x2+ etc. in its principal diagonal is what is called a skew symmetric determinant. The known algebra of skew and skew symmetric determinants gives Square roots of skew symmetrics. + λ'°C (x2+ w ̧)(x2 + w ̧)... (\2 + w1) (12. 34 +31.24+23.14) (18), + d®°C (λ2 + π7) (λ2 + πg) ..... (λ2 + π¡) (12.34.56)2 + etc. + λ' (≥ 12.34.56...........i — 1, i)3 when i is even. the last term is For example see (30) below. When A is odd d'−1 Σ (λ2 + @,) (Σ12. 34. 56... i — 2, i − 1)3 .............(18'), and no other change in the formula is necessary. In each case the small denotes the sum of the products obtained by making every possible permutation of the numbers in the line of factors following it, with orders chosen acccording to a proper rule to render the sign of each product positive (Salmon's Higher Algebra, Lesson v. Art. 40). This sum is in each case the square root of a certain corresponding skew symmetric determinant. An easy rule to find other products from any one given to begin with is this:-Invert the order in any one factor, and make a simple interchange of any two numbers in different factors. Thus, in the last Σ of (18) alter i 1, i to i, i 1, and interchange i 1 with 3: so we find 12.i-1, 4.56...... i, 3 for a term similarly 12.64.53 ... i, i-1, and 62.14.53... i — 1, i, for two others. The same number must not occur more than once in any one product. Two products differing only in the orders of the two numbers in factors are not admitted. If n be the number of factors in each term, the whole number of factors is clearly 1.3. 5 ... (2n-1), and they may be found in regular roots of metrics. progression thus: Begin with a single factor and single term 12. Square Then apply to it the factor 34, and permute to suit 24 instead skew symof 34, and permute the result to suit 14 instead of 24. Thirdly, apply to the sum thus found the factor 56, and permute successively from 56 to 46, from 46 to 36, from 36 to 26, and from 26 to 16. Fourthly, introduce the factor 78; and so on. Thus we find 21, 0, 23, 24, 25, 26 +(12.53+13.52+23.51)46 31, 32, 0, 34, 35, 36 +(12.45+41.52+42.51)36 41, 42, 43, 0, 45, 46 51, 52, 53, 54, 0, 56 +(31.45+41.35+34.51)26 +(23.45+24.35+34.25)16 61, 62, 63, 64, 65, o The second member of the last of these equations is what is system with doms. contains λ as a factor. Hence, when all are expanded in powers Gyrostatic of X, the term independent of A is ... ☎j. If this be two freenegative there must be at least one real positive and one real negative root λ2. Hence for stability either must all of w1, W21 w be positive or an even number of them negative. Ex.:-Two modes of motion, x and y the co-ordinates. Let the equations of motion be Gyrostatic system with two freedoms. Gyrostatic influence dominant. The solution of this quadratic in λ may be put under the following forms,— − x2 = { (y2 + w + 5 ) ± § {[y2 + ( √w + √√5)3] [y°2 + ( √@ − √ ? )2] } ? ...(25). To make both values of - X' real and positive and must be of the same sign. If they are both positive no farther condition is necessary. If they are both negative we must have These are the conditions that the zero configuration may be stable. Remark that when (as practically in all the gyrostatic illustrations) y is very great in comparison with (w), the greater value of - λ2 is approximately equal to y3, and therefore (as the product of the two roots is exactly w), the less is approximately equal to /y. Remark also that 2/√ and 2/ are the periods of the two fundamental vibrations of a system otherwise the same as the given system, but with y=0. Hence, using the word irrotational to refer to the system with g = 0, and gyroscopic, or gyrostatic, or gyrostat, to refer to the actual system; From the preceding analysis we have the curious and interesting result that, in a system with two freedoms, two irrotational instabilities are converted into complete gyrostatic Gyrostatic stability (each freedom stable) by sufficiently rapid rotation; but that with one irrotational stability the gyrostat is essentially unstable, with one of its freedoms unstable and the other stable, if there be one irrotational instability. Various good illustrations of gyrostatic systems with two, three, and four freedoms (§§ 345*, and *) are afforded by the several different modes of mounting shown in the accompanying sketches, applied to the ordinary gyrostat* (a rapidly rotating fly-wheel pivoted as finely as possible within a rigid case, having a convex curvilinear polygonal border, in the plane perpendicular to the axis through the centre of gravity of the whole). stability. * xi Nature, No. 379, Vol. 15 (February 1, 1877), page 297. (Translational motions not considered) there are two freedoms, one azimuthal the other inclinational; the first neutral the other unstable when fly-wheel not rotating the first still neutral the second stable when fly-wheel in rapid rotation. Equations (23) with =0 express the problem, and (24) and (25) its solution. on universal- inverted (§ 319, Ex. D). Gyrostat on knife-edge gimbals with its axis vertical. Two freedoms; each unstable without rotation of the fly-wheel; each stable when it is rotating rapidly. Neglecting inertia of the knifeedges and gimbal-ring we have I-J in (20), and supposing the levels of the knife-edges to be the same, we have E = F. Thus its determinantal equation is (IA2+ E)2+ g2X2 = 0. A similar result, expressed by the same equations of motion, is obtained by supporting the gyrostat on a little elastic universal flexure-joint of, for example, thin steel pianoforte-wire one or two centimetres long between end clamps or solderings. A drawing is unnecessary. Two freedoms, one azimuthal the other inclinational, both unstable without, both stable with, rapid rotation of the fly-wheel. |