1、m ii E s NATIONAL ADVISORY COMMITTEE 4 z FOR AERONAUTICS TECHNICAL NOTE 3288 ON THE ANALYSIS OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS FROM TRJINSIENT-RESPONSE DATA By Marvin Shinbrot t J Ames Aeronautical Laboratory Moffett Field, Calif. . m Washington LOWBWV COPI December 1954 EEC 3 1954 Provided
2、by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR -AERONAlTICs TECRNICALNCTE 3288 ON THE ANALYSIS OF IINEAR AND NONLINE4R DYNAMICAL SYSTEMS FROMTRANSIENT-RESFONSEDATA By Marvin Shinbrot A general theory of the so-called “equation
3、s-of-motion“ methods for the analysis of linear dynamical systems is developed first. It is then shown that when viewed from this general point of vantage, all of these linear methods can be extended in a straightforward manner to apply.to the analysis of nonlinear systems. In addition, through use
4、of this theory, a new method is derived. It is essentially a variation of the well-known “Fourier transform“ method for the analysis of linear systems but possesses certain advantages over previous methods. Application and effectiveness of this method are demonstrated by three examples, two of which
5、 are nonlinear - one highly so - and the third being of the fourth order. INTRODUCTION It has often been suggested (e.g., in ref. 1) that nonlinearities which are ignored in the classical theory of the equations of motion of an aircraft may be responsible for certain unusual phenomena which have bee
6、n observed in flights of modern high-speed airplanes and missiles. Consequently, it seems desirable to develop methods for the analysis of such nonlinear systems - methods which allow the calculation from measured transient-response data of the nonlinear stability characteristics as well as the clas
7、sical linear stability derivatives of the aircraft. Several such methods are described in reference 2, the principal one consisting of a generalization of the so-called “derivative method“ which was orig- inally devised for use with linear systems (cf. ref. 3). However, the methods described in refe
8、rence 2 leave something to be desired from both points of view of accuracy of the results and lengthiness of the calcu- lations. In addition, application of these methods requires, in all but the simplest cases, the previous evaluation by some means of those sta- bility characteristics which are lin
9、ear. In view of these shortcomings, an attempt has been made in the present study to find simpler, more accu- rate, and more general procedures. The problem is attacked by first exam- ining several well-known methods for the analysis of simple linear systems and then modifying them as necessary to a
10、llow their application to more I general systems. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NACA TN 3288 Many methods for the analysis of linear systems have been proposed in the past (see, e.gD, refs. 3 and 4). In reference 5, these methods
11、have been classified under two main heads: “equations-of-motion“ method8 and “response-curve-fitting“ methods, the former title including the derivative method and what have been called the mace transform and the Fourier transform methods (ref. 31, and the latter consisting of such methods as Pronys
12、 (refs. 3 and 6) and the techniques of reference 4. SFnce the response-curve-fitting methods involve the explicit-soluMon of-the equations of motion in terms af the physical parameters of the system at hand, they do not seem suitable for use with nonlinear systems. Hence, we shall-be concerned solel
13、y with the equations-of-motion methods. Each of these methods has been consider-ed in the literature as an independent entity; apparently, no attempt has ever-been made to subsume all of them under a single general theory. For the purposes of the pre- sent study, such a theory would be desirabie sin
14、ce it seems reasonable to expect-first that when viewed from a more general point of view, a gen- eralization of the methods to nonlinear systems might appear; and second that once such a theory is known, it might be possible to develop new methods, superior in certain respects to the old ones from
15、which the theory sprang. In accordance with this plan, the paper beginstith a short-presen- tation of the three best known of the equations-of-motion methods. These methods are examined from a new point of view which138 then shown to lead to the general theory for linear systems; The further extensi
16、on to non- linear systems is considered next, B for a more detailed discussion of them, see reference 3. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 : NACA TN 3288 As a concrete example, let us consider an airplane operating under conditions wh
17、ere the stability characteristics are -effectively linear, so that, as in reference 3, the equations of its longftudinal motion can - be written 1 - aLcG -) + G(t) + h(t) = c k, Co, and C,. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3288
18、 5 The Laplace transform method.- Letting A(p) and A(p) denote the Laplace transforms of a(t) and 8(t),-respectively, so that A(P) = f me-pt a(t)dt 0 A(P) = f me-pt s(t)dt 0 it follows that if a(t) and 6(t) are related by equation (l), then (3) (P2 + bp + k) A(P) = (C,P + Co) A(P) (4) (ref. 7). In w
19、riting ity that a(0) = G(O) be removed later on. in b, k, Co, and C,. 6(t) for several such down equation (4), it has been aSSUEd for simplic- = 6(O) = 0; this restriction is inessential and will For any value of p, equation (4) is an equation After finding the Laplace transforms of a(t) and values
20、of p, say for p = pl, pp, . . ., PN, the .ing equations (4) can be set up and solved by least squares to correspond obtain b c Pi2A2(pi) + k .c PiA2(Pi) - CO c piA A - Cl c PisA(Pi) in practice, it would not be performed.- The Fourier transform method.- Finally, (1) If equation (1) is multiplied by
21、co8 wt and sin wt for several values of w, and (2) If the results h-e integrated from zero to infinity (as in the Iaplace transform method, integrating by parts to eliminate explicit dependence on the derivatives of a and S), one obtains a set of equations identical with those obtained from the Four
22、ier transform method. The general method for linear systems.- The general development of- equationa;of-motion methods is now manifest. One takes the equations of motion for the physical system under consideration - for definiteness, say equation (1) - and (1) Multiplies them by N arbitrary (but suff
23、iciently smooth) functions y,(t). (2) The resulting equations are then integrated between two definite limits, say, zero and T. In the three methods described above, T = 03, but this is not essential. In order to avoid some complications initially, we shall continue to integrate over this infinite i
24、nterval; this restriction will subsequently be removed, however, and T will be allowed to have finite values. In the case of equation (1), the proc-ess just described leads to N equa- tions of the form f co b y,(t) in such cases, equations (9) can be considered as N equations which are to be solved
25、by least squares for the desired parameters. Of course, $his process requires the calculation of the derivatives L(t), E(t), and 6(t). On the other hand, if the functions y,(t) are explicitly independent of a as is the case - Provided by IHSNot for ResaleNo reproduction or networking permitted witho
26、ut license from IHS-,-,-NACA TN 3288 9 In the Laplace and Fourier transform methods, the following formulas, obtained by integrating by parts, are used: J y,(ts(tdt = y,(O),(t then, we may write y,(t) = ain wVt, Y = 1, 2, . . ., N (20) This does not entirely eliminate the dependence on the initial c
27、onditione, however, as the term $v(Oa;(O) remains in equation (10). If this term can also be removed, the second weatiess in the Fourier transform method will have been entirely corrected. This will clearly be the case if gv(0) a8 well aa y,(O) ia zero for all V = 1, . . ., N. A possible choice of t
28、he method function8 for which this is 80, a choice which still retains the advantages of the favored Fourier transform method, is the following: yv(t) = sirP*t = 1 - CO8 2b+t 2 J Y=l,.,N ew With this choice of the method functions, equation (10) becomes -b s m s *I a.(t)jry(t)dt c k a(t)yv(t-)dt - C
29、o 0 0 s OD s(t)y+jdt + 0 s 00 s 03 (22) Cl G(t),(t)dt 0 0 The method functions (21a) would be used for systema satisfying dif- ferential equations, like (l), which involve derivatives of the second order. More generally, and for the same reasons, if the highest order derivative occurring in-any of t
30、he equations of motion of a system Is the nth, the following method functions are suggested: YVW = slnn WV-t, V=l,.,N lb) Thus, in the case of the two-degreee-of-freedom system described by equations (13), there is no.point in using formula (21a); the simpler method functLons (20) may aa well be use
31、d. As for the first of the weaknesses in the Fourier transform method, that which ia due to integration over an infinite Interval, it would . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NXA TN 3288 15 appear at first glance to be easily disposed
32、of by merely choosing some finite, positive number T and integrat5ng over the interval from zero This, however, introduces another difficulty. zvez*by equation (21a), so that (0) = however, the calculated pitching-moment curve M(u) = - Iy(koa + $a2 + Qa3) = - 5040 a + 3050 u2 - 8cdoo a“ can be plott
33、ed and compared with the true curve from which we started. This has been done in figure 1 from which it can be seen that the error at the least accurate point is less than 3 percent. It should be noted that the values of In(yv) given in table I can be used to solve any problem of the type considered
34、 here which depends on a second-order differential equation or on a system of-such equations. ILthe data run is 2 seconds long, it-ie only necessary to ineert the data in.table I in place of the data used in this example-and proceed aswe have just done-. As mentioned earlier, if the data run is more
35、 or less than 2 seconds long, it is only necessary to make a preliminary transformation of the time scale so that in the new time scale the-data run is 2 seconds long, a process illustrated-in example II. Example II The first example served to illustrate the application of the method to anequation o
36、fthe form (lb), corresponding in the missile pitch- response problem to the case where only u(t) and 6(t) are measured. A problem involving equations, like (L3), of-the first order, corresponding to the case where q(t) is available in addition to- u(t) and S(t), will be illustrated now. Provided by
37、IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3288 21 As in example I, the lift force will pitching moment nonlinear. The following assumed: m=2 v=750 * IY = 100 be assumed linear and the parametric values were r, = 1000 M since the lift ia linear, b,
38、= b2 = 0. and I there are no conceptual difficulties in the generalization to the nonlinear case. In addition, it will be assumed for simplicity that free oscillations are available for analysis. Thus, the system to be analyzed is a8SUned to be described by an equation of the form ak+ a3 d?x 2 dt4 d
39、t3 + a.2 s + a1 g + aox = 0 A solution of this equation was calculated over the interval from 0 to 2 seconds, for the following v8lues of the coefficients ai: a0 = 2544.9 a1 = 219.3 a2 = 132.87 a3 =2.000 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-
40、,-26 NACA TN 3288 The result, representing the free oscillations in r-esponse to some dis- turbance, is tabulated in table V and presented graphically a8 the solid curve in figure 5. The sums needed for the solution of the problem are displayed below table V and again in table VI. The least-squares
41、equa- tions for the parameters are 24.2035 a0 - 16.7225 a1 - 1098.l$-+ + 1603.19 a3 = - 81036.4 - 16.7225 a0 + 855.538 a1 + 664.495 a, - 82282.3 a3 = 85777.1 - 10g8.1gao + 664.495 a, + 77898.7 Q - 94741.8 a3 = 7533580 I 1603.19 a0 - 82282.3 a1 - 94741.8 a, + 8523670 a3 = - 10567800 J Solving these e
42、quations gives - a0 = 3098.7 “1 = 374-93 a, = 141.29 a3 = 3.367 It can be seen that these numbers are correct only to within order8 OP magnitude. On the other hand, it is not these coefficients which have direct physical significance; rather, it7Ls the damping and frequency of each of the components
43、 making up the oscillation which are important. In order to find these numbers, the following equation was set up h4 + 3.367 A” + 141.29 A2 + 374g3 h + 3098.7 = 0 and solved to find the roots: A= - I 0.046 t-10.7 i 1 - 1.64 f 4.96 i The true roots are obtained by solving the equation h4 + 2.000 A3 +
44、 132.87 A2 i- 219.32 A+ 2544x9 = 0 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-27 RACA TN 3288 This gives i 2 10.5 i h= - 1.00 + 4.71 1 Thus, the frequencies of the oscillation have been found quite accurately, as has the damping parameter of the
45、 undamped component. The only large effect of the errors in the coefficients 80, aI, a2, and a3 is in the damping of one of the cowonents. The apparent ill-conditioning of the problem with respect to this parameter is not too surprising, for after all, either the true or calculated value of this dam
46、Ring is large enough that the corresponding component of the motion is masked by the undamped component over a good part of the run. This may be seen best, perhaps, from figure 5 in which the solution of equation (32), using both the true and calculated values of the parameters, has-been plotted. It
47、 can be seen that the two curves do not differ by very much, indicating that the fit could not be much improved. CONCLUDING REMARKS A general theory of the so-called “equations-of-motion“ methoda for the analysis of dynamical systems has been presented. It ha8 been shown that, when looked at from a
48、new point of view, all such methods can be generalized so as to apply to linear and nonlinear systems alike. Using this theory, it has also been shown how new methods can be devel- oped in order to satisfy the requirements of particular problems. One new method has been described in detail. In certa
49、in cases, it reduces to one which is very similar to the well-known Fourier transform method (ref. 3) but in all cases has certain advantages over this latter method and over other method8 heretofore used. Its superiority is based on two facts. First, there is the heavy dependence on the initial con- ditions which
