THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING APPLICATION OF ULTRASPHERICAL POLYNOMIALS TO NON-LINEAR OSCILLATIONS - FREE OSCILLATIONS Harry H. Denman Y. King'~iu November, 1963 IP-641

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TABLE OF CONTENTS Page LIST OF FIGURES................ o.. o... o....... o...................... iii 1. Introduction.... o o o..... o o..... o................................. 1 2. The General Free Oscillation Problem......................... 1 3. The Ultraspherical Polynomials................................ 2 4. Application of the Linear Ultraspherical Approximation to Some Typical Problems..............oo.............oo 4 I. Cubic Non-linearityo.................................... 4 II. Sine and Hyperbolic Sine........o.....................o 7 IL. Hyperbolic Tangent Nonlinearity.......................... 8 5. Asymptotic Behavior of the Linear Ultraspherical Polynomial Approximation......o....... o...... o..o o.... 11 I. Lower Limit...................o o o o o o o o.o o..o o o e o......o 11 II. Large Asymptotic Limit.................................. 13 6. Cubic Ultraspherical Polynomial Approximation.............. 16 7. Connection with Method of Krylov and Bogoliuboffo o.........oo 21 8. Discussion......................................... 22 BIBLIOGRAPHY.....o.o........ *... e.. o...o o o o.... o.....o 24 ii

LIST OF FIGURES Figure Page 1 Period Ratio (T/T')-non-linearity Factor v Curve for Cubic Spring -- Free Oscillation........ 25 2 Period Ratio (T/T')-non-linearity Factor v Curve for Hard Cubic Spring -- Free Oscillation.... 26 3 Potential Curve for Softening-hardening Cubic Spring -- Free Oscillation........................ 27 4 Period Ratio (T/T ')-non-linearity Factor v Curve for Softening-hardening Cubic Spring -- Free Oscillation.................................. 28 5 Period Ratio (T/To)-amplitude Curve for sinh Spring -- Free Oscillation....................... 29 6 Period Ratio (T/To)-amplitude Curve for tanh Spring -- Free Oscillation........................ 30 7 Period Ratio (T/To)-amplitude Curve for sine Spring for Cubic Approximation -- Free Oscillation. 31 8 Period Ratio (T/To)-amplitude Curve for sinh Spring for Cubic Approximation -- Free Oscillation. 32 iii

1. Introduction. In a recent paper [1], Denman and Howard introduced a procedure for the linearization of the non-linear ordinary differential equation governing the free oscillation of the simple pendulum, by approximating the non-linear torque with an ultraspherical polynomialo This linearization yields an approximation for the period of the pendulum as a function of the amplitude of the motion~ The present paper applies this procedure to tne following non-linear functions: ax + bx3, sinh x and tanh x. Certain asymptotic results are obtainedo Further, it extends the procedure to the cubic ultraspherical polynomial approximation and applied it to the non-linear functions sin x and sinh Xo All results are either compared with the exact expressions for the period, if they are available, or to numerical results, if they are noto This extension yields a marked improvement over the linear approximation, witn little increase in complexityo 2. The general free oscillation problemo A general free oscillation problem is characterized by the differential equation d2x/dt2 + f(x) = 0o (1) -1 -

-2 - In this paper, f(x) is assumed to be an odd (non-linear) function representing a force or torque. The general initial conditions are: x(O) = xo and (dx/dt)o = v0o In the oscillatory case, these conditions may be replaced by xo = A, vo = 0, where A is the amplitude of motion. The first integral of (1) is (dx/dt)2/2 + V(x) = E, where V(x) is a potential function (dV/dx = f(x)), and E is proportional to the (constant) energy of the system. The turning points of the motion are (-A, A). The second integral of (1) can be written A -1/2 T/4 = f [2(E - V)] - dx, 0 where T is the period of oscillation, and is in general a function of A. Since E = V(A), one may write 3/2 A -1/2 T/23 = [V(A) - V(x)] dxo (2) 0 3. The ultraspherical polynomials. The ultraspherical polynomials on the interval [-1, 1] are the sets of polynomials orthogonal on this interval with respect to the weight factors (1 - x2) - 1/2, each set corresponding to a value of X > - 1/2. They may be obtained from Rodrigues' formula [2] Pn()(x) = An( (1 - x2)-X+l/2(d/dx)n (1-x2)n+X1 /2 where An) is a normalization factor. n

-3 - Some subsets of the ultraspherical polynominals are: X = 0, Tchebycheff polynomials of the first kind; X = 1/2, Legendre polynomials; X = 1, Tchebycheff polynomials of the second kind, and X -ao, the powers of x The ultraspherical polynomials on the interval [-A, A] are defined as the sets of polynomials orthogonal on this interval with respect to the weight factors [1 - (x/A)2]-1/2, X > - 1/2. For a function f(x) expandable in these polynomials one obtains f(x) = Z a() P(X) (x/A), (3) n=0 where the coefficients anX) may be written +1 f(Ax) PX) (x) (1 _x2)X - 1/2d a i -1 _ — * (4) +1 / [P(X) (x)]2 (1 - x2)X - 1/2 d -1 Since 4() P(X) (x/A) is unchanged if P(X) (x/A) is multiplied by a constant, one may choose any convenient normalization factor A(X) For all odd f(x), the approximate period T* resulting from the linear ultraspherical polynomial approximation in [-A, A], i.eo, f*(x) = a(X) P(X) (x/A), can be expressed as T* = 2i [i1/2 r(x + 1/2) A/4r(x + 2) S]l/2, (5) where S = 1 x f(Ax) (1 - x) -1/2 d, (6) 0

-4 - for X> - 1/2. (This expression for T* actually converges for values of X>- 2 for many f(x), but the accuracy of the approximate results for - 2 < X < - 1/2 has been found to be rather poor; therefore this range is not considered in this paper). 4. Application of the linear ultraspherical approximation to some typical problems. The restoring forces or torques encountered in non-linear oscillation problems may be roughly classified as: hardening, softening, flattening, and bottoming. Examples from the first three classes will be examined here, and include: (I) (odd) cubic, f(x) = ax + bx3; (II)sine, f(x) = sin x, and hyperbolic sine, f(x) = sinh x; and (III) hyperbolic tangent, f(x) = tanh x. Io Cubic Non-linearity. The equation of free oscillation is taken as d2x/dt2 + ax + bx3 = 0. (7) Equation (7) can be classified according to whether a and b are positive or negative. The four combinations are shown in Table I. Table I Classification Name Sign a Sign b Motion Case 1 Hardening + + Oscillatory Case 2 Softening + - Conditionally Oscillatory Case 3 Softening- -+ Oscillatory Hardening Case 4 Softening- - - NonSoftening oscillatory

-5 -Only the first three cases can yield bounded oscillatory motion; Case 4 is without any "restoring" force or torque regardless of the amplitude of motion. Case 3 is included both for completeness and because it will be used later (Section 6). A. Exact Solutionso Case lo Hardening Cubic. The expression for the period of (7) in terms of the complete elliptic integral of the first kind is well known [3]. The exact period is T = 4K(k1)/(a + bA2)1/2, (8) or, in dimensionless form, T/ T = 2K(kl)/A(l + v)1/2 (9) where T1 = 2i/a1/2 is the period of the system if b = O, K(kl) is the complete elliptic integral of the first kind, v = bA2/a is a dimensionless quantity, and kI = v/2(l+v) is the modulus of the elliptic integral. The quantity v is a measure of the non-linearity of the problem, since the maximum ratio of the non-linear to linear term in (7) is bA2/a = v. Case 2. Softening Cubic. If b is negative, i.e., the non-linear spring is soft, a solution similar to (9) is obtained by making the transformation k2 = -v/(2 + v), -1 < v < 0O After some simplification, the period ratio becomes T/T' =2(1 + k) 1/2K(k2)/ (10) K2k2)/ir

-6 - Note, however, that the above solution is no longer bounded, since v - -1 implies k2 - 1, and K - oo. This v (or amplitude) is called the critical v (or amplitude). Plots of T/T' against v for Cases 1 and 2 are shown in Figures 1 and 2. Case 3. Softening-Hardening Cubico This case possesses rather interesting properties: for x small, the linear term dominates the cubic, and vice-versa for large x; thus the force is away from the center for small x, but restoring for large x. The potential curve is shown in Fig. 3. For initial conditions represented by the energy El, the motion is periodic, but not symmetric about the origin. The conditions represented by E2; however, do give symmetric oscillations about the origin. The demarcation line is Eoo Only the case represented by E2 is considered here. Then one obtains T(-a)12 = 4(-v -1)-/2K(k3), (11) where v = bA2/a < 0, and k2 = v/2(1 + v). For E > Eo, v < - 2, so that 1/2 < k2 < 1. For T' = 2/|a| 1/ T/T' = 2K(k3)/)T(-v -I)1/2. (12) Equation (12) is plotted as curve E in Figure 4. B. Linear Ultraspherical Polynomial Approximation. In (7) if one expands f(x) = ax + bx3 in ultraspherical polynomials in [-A, A], one obtains the linear approximation (ax + bx3) = [a + 3bA /2(k + 2)] x, where * indicates a linear approximation. This linear approximation, when substituted into (7), results in the approximate linear differential equation

-7 - d2x/dt2 + [a + 3bA2/2( + 2) ]x = 0. (13) Therefore, for Cases 1 and 2, T*/T' = [1 + 3v/2( X + 2)]1/2, v > -1 (14a) and for Case 3, T/r= [-1 - 3v/2(x + 2)]-1/2, v < -2, (4b) where T' = 2n/ |a|1/2, and v = bA2/a. On Figs. 1, 2, and 4 are shown curves given by the above approximations for certain values of X, for these cubic springs. (For a = 0, To -, and v -, o, but by using (8) and (13) one obtains the usual results.) II. Sine and Hyperbolic Sine. A. Exact Solutions. The governing differential equations are written d2x/dt2 + o2 sin x = 0, (15) d2x/dt2 + W2 sinh x = 0 (16) Solutions of (15), both exact and for the linear ultraspherical polynomial approximation, are found in [l]. Equation (16) can also be solved in terms of the complete elliptic integral of the first kind. While the solution for (15) is oscillatory only if A < t, the solution to (16) has no such limitation; the exact period is given by T/T = (2/x)sech(A/2) K[tanh (A/2)]. (17)

-8 - B. Linear Ultraspherical Polynomial Approximation. If one expands sinh x in ultraspherical polynomials in [-A, A] and truncates after the linear term, one obtains (sinh x) [r( + 2) I+1 (A)/ (A/2) ] x (18) where In(x) = (-i)nJn(ix) is a modified Bessel function. The approximate period ratio T*/To is then T*/To = [(A/2)X+/r(X+2) I+l(A)]1/2, (19) The comparison between (19) and the exact solution (17) for various X values is shown in Fig. 5. III. Hyperbolic Tangent Nonlinearity. The governing differential equation of motion is dx/dt2 + 2 tanh x = 00 (20) No exact solution of (20) in simple functions is known to exist. To provide comparison for the approximate results, numerical quadrature is used to provide the "exact" results, From (2), the exact period in this case can be written T/TQ = (21/2/ ) f [ln(cosh A/cosh x)]1/2dx. (21) 0 For numerical quadrature, to remove the singularity at the upper limit, consider the expansion (x2 < x2/4) ln(cosh x) = x2/2 - x /12 +... (-l)n-1 22n-l(22n-l)B2nlx 2n(2n),

-9 - where B2n are the odd Bernoulli numbers [4]o Equation (21) becomes A -1/2 T/T = (21/2/t) f [(A 2-x2)/2-(A4 )/12 +..] dx. (22) 0 Setting x = A cos 0, one obtains (for A < t/2) it/2 2(2 2 -1/2 T/To = (2/it) f [1-A (l+coso )/6 + d] o/2 (23) 0 Equation (23) is now free of singularities and is susceptible to numerical integration. For small values of A, one can truncate after the second term and express T/To in terms of an elliptic integral T/To = (2/A) [6/(6-A2 )] K(k4), (24) where k2 = A2/(6-A2)o For intermediate and large A, expand ln(cosh x) in a Taylor series about A. Then In cosh x = in cosh A + tanh A (x-A) + (1/2) sech2A(x-A) - sech2A tanh A (x-A)3/3 +... Setting y2 = A-x, one obtains 3 1/2 A 2 2 T/To = (2/it)(6 cosh3A/sinh A) j [3 cosh2A-(5/2)y ctnh A 0 4 L-1/2 -y...] dy. If one truncates the series after the third term, the above integration can be carried out be means of an elliptic integral of the first kind [5] T/To ' (4//)g cosh A F(cp, k5), (25) 22 2 -:1/4 where k = (1/2)(1-g ), g = [+(16/3) sinh A] / and sin2p = 2/[ l+(2g2 inh 2A)/A-g2 ]

-10 - Retaining only two terms of the integrand yields the approximation T/To - (4/X) (cosh A)sin1 (A/sinh 2A)1/2 (26) For asymptotically large A, (26) approaches 1/2 1/2 T/T = (2/r)(2A) 0.9003 A o (27) Equations (24), (25), (26) and (27) span the entire range of A and overlap one another as far as graphical accuracy is concerned. The "exact" curve E in Fig. 6 was constructed using them. B. Linear Ultraspherical Polynomial Approximation. For f(x) = w tanh x, S in (6) becomes 1 2 X-l/2 S = f2 J x tanh Ax(l-x ) dx. (28a) 0 0 This integral does not seem to exist in closed form in terms of simple functions. For A < T/2, tanh Ax can be expanded in a Maclaurin series, and v~lr~ { ~1/2 Al[+ n-1 2n-1 22n 12 T*/T = { A/[4r(X+2) 2 (-1) (2A) (2 1) B2n lr (n + 1/2)/(2n)I}1/ (28b) Alternately, an integration of (28a) by parts yields -1 2 1 2X+l/2 2 S (2 + 1) W0A (1 - x sech Ax dx, (28c) 0 I which removes the singularity in the integrand of (28a) for x = 1, X < 1/2. Thus T*/To can be expressed as T*/T 8r(X+2) (-x 2)X1/2 dx -1 (29) I0 4n r(X+3/2) 0 e2AX(l+e-2Ax)2 i

-11 - For arbitrary X, numerical quadrature was used to evaluate (29). However, certain values of X yield somewhat simpler results. For example, if X = 1/2, 2 1 S = o I x tanh Ax dx. (30) 0 Since, for x > 0, A>0, x tanh Ax = x(1-2e2Ax + 2e4A - 2e 6Ax+ (31) the period ratio for X = 1/2 can be expressed as ) -I [5 + X (-l)n+l(2nA+l) I2 1 -1/2 1/2 A ( 1 (nA)2e2nA - (32) This expression is particularly convenient for large values of A, and for very large A, (T*/To)l/2 approaches 0.8165 A /2. For X — 1/2, (5) and (28c) yield T*/To)-1/2 -e [A/tanh A]l/'2 (53) Figure 6 compares the "exact" solution with the approximations corresponding to X values -1/2, 0, and 1/2. 5. Asymptotic behavior of the linear ultraspherical polynomial approximation. Two special regions of interest are those for which A approaches 0 and for A large. I. Lower Limit. For small oscillations in the neighborhood of the origin, one can expand V(x) in (2) in a Maclaurin series

-12 - V(x) = V + a2x2 + a4x4 +.., O0 or V(A) - V(x) = a2(A2-x2) [l+a4(A2+x2)/a2 +. ], where a2 must be positive for small oscillations about the origin to exist. Setting x/A = cos 0, one finds T/2 a2 [1+a4A (l+cos )/a2 +.. ]/2d r/23/2 a21/2 a /2 2~[lta dco +....G]-l/2 d. For sufficiently small values of A, one can expand the integrand by the Binomial Theorem, so that T/23/2 a21/2 [1-a4A2(1+cos2) /2a2.....]de The solution for small oscillation is, therefore, T = 2t(2a)-1/2 [-3a4A /4a2 +.] (34) In (6), setting x = cos 0, one obtains 0t/2 S = J f(A cos Q) sin2X 9 cos 9 doG (35) 0 Expanding f(A cos 9) in its Maclaurin series, 00 f(A cos 0) = Z cmAmcosme m=o Then (5) becomes T* = 2t [1/2 r(X+l/2)A/4r(X+2) Z cAml cos sin2 dx ] /(36) =) o ( For f(x) odd, (36) yields T. = 2c[cl+5c53A2/2(2+x) +. (37)

-13 - Since the ci's and ai's are the Maclaurin series coefficients for f(x) and V(x) respectively, their relationships are a2 = 1/2, a4 = c3.. Equation (37) becomes 1/2 T* = 2i(2a ) /[1-3a A /2a (x+2)...0]. (37') When one compares (37') with the expression for the exact period (34), one notes that the first two terms agree if and only if X = 0. Hence, for small free oscillations, X = 0, corresponding to Tchebycheff polynomials of the first kind, provides the most accurate period approximation of all the linear ultraspherical polynomials, for all odd analytic f(x). (This result can not be applied to the softening-hardening cubic; only the case E > Eo is considered here, for which arbitrarily small oscillations about 0 do not exist.) II. Large Asymptotic Limito For large non-linearities, each case was examined individually. A. Cubic Non-linearity. For the hardening and softening-hardening cases (Cases 1 and 3), the exact solutions are given by (9) and (12)o For Iv| > > 1, (9) and (12) yield = 1/2 lim (r/T') = 2(lo8541)/Iv 1/2i IVI —o while from (14a) and (14b), one finds 1/2 lim (T*/T') = [2(2+X)/31vH ] o Ivl- *o

-14 - Hence, the optimum value of X, for large non-linearity, is found from [2(x+2)/311/2 = 2(l.8541)/r, (58) [2(x+2)/5] (38) which yields X = 0.0899. For the softening case (Case 2), the exact solution (10) and the approximation (14a) behave as follows: T/T1 -oo as v -e -1, (39a) T*./T' -^o as 3v/2(2+X) -1. (39b) Hence, the value of X which yields the correct critical vc is X = -1/2. For the softening-hardening spring, the exact solution (12) becomes o for v = -2; the approximation (14b) gives the same result for X=l. B. Sine and Hyperbolic Sine Non-linearities. The exact solution for the free oscillation of a simple pendulum is T/T = (2/t)K(k), where k = sin (A/2). Thus, T/T0 approaches infinity as A-, t. The linear ultraspherical polynomial approximation yields [1] X+l 1/2 T*/o = [(A/2) /r(X+2)Jx+ (A)]/2 From a table of the first zero of Jp(x) as a function of p, such as in [6], one notes that J1/2(A) has its first zero at A=or Hence, the same critical amplitude occurs for the linear approximation when the index X is -1/2o

-15 - The exact solution by elliptic integrals for the sinh spring is given by (17). For A > > 1, k2 1, and T/T0o -C1A exp (-A/2), (40) where C1 is a constant. The linear ultraspherical polynomial approximation result is given by (19). For very large A [7], Ip(A) - eA/(2tA)1/2 and T*/To -e C2A exp (-A/2) (41) only if X = 1/2. C. The Hyberbolic Tangent Non-linearity. The asymptotic T/T is given by (27). tanh Ax -,1, x > 0, and (28a) and (5) yield For very large A, T*/TO e c A t/2 r (X+3/2)/2r(X+2 ) /2 Equating the coefficients of A1/2 in (27) and (42a), one obtains (42a) l /2r(X+3/2)/2r(X+2) = 8/I2, from which x = -0.075. (42b) These asymptotic results are summarized in Table II, and are verified by the computations leading to Figs. 1, 2, 4, 5, and 6.

-16 - Table II Type of Spring Optimum Value of X for Large Non-linearity or Correct Critical Value cubic (1) hard.089 (2) soft -.5 (3) softeninghardening.089 (large v) 1.0 (correct vc) sine -.5 sinh.5 tanh -o075 60 Cubic ultraspherical polynomial approximation. If, instead of truncating the ultraspherical polynomial expansion of f(x) in [-A, A] after the linear term, one does so after the cubic, much more accurate results are to be expected. Then f**(x) - a(X)P(X) (x/A) + (X)P(X) (x/A). 1 1 3 3 When this result is substituted in (1), the resulting approximate cubic differential equation can be either solved exactly by elliptic functions or, preferably, utilize the previously plotted "exact" results for the cubic springs to extract the approximate solution. The sin and sinh cases will be considered below (the hyperbolic tangent has no accurate cubic approximation for large amplitude, as is true for any flattening spring).

-17 - (x) The evaluation of the ultraspherical polynomial coefficient a5 will be shown for f(x) = sin x, that for sinh x being very similar. From (4), f sin (A cos a) sin2X@ P X)(cos Q) d~ (X) o a = f [P3 )(cos 0)] sin2X dO 0 [ The denominator is a standard integral, while the numerator is an integral found in [8]o Thus it can be shown (sin x)*= (x 1 [JX+(A) + (x+3) JX+(A)] x (sin x) = (A/2)X+l r(x+4) JX+(A) x, (43a) 6(A/2)X+3 A2 A = [a4+(A) -+3 A) - (A (A)x3,(43b) X+l 4(x+2) 7 6 x+3 where Ax(A) = r(X+l) JX(A)/(A/2)X. The function Ak is plotted in [6], while (43a) agrees with the result given in [9] for X = 0. The cubic ultraspherical polynomial approximation for sinh x is (sinh x)** - Ir(+2) I-+4)+) x3+ (44) X ~ (A/2)~X+l LX+l 6(A/2)X+3 2. Consider the case. = 0 and f(x) = no sin x. The differential equation (1) is replaced by d2x/dt2 + [2W2o(J1+3J3)/A]x- (&2oJ3/A )x3 = 0. (45) Since Jl+35J and J3 are positive in [0, A], (45) corresponds to the softening cubic of section 4, with

-18 - a = 2co2(J1+5J5)/A, b = Following solution (10), one finds 2 A2o (J +3J3 -&.2oJ3/A35 = 2/rt (l+k6) K(k6), (46) where T** is the approximation to the period resulting from the cubic ultraspherical approximation to f(x) for X = 0, and k2 = -v/2+v = 2J/J+J, v = -4j3/(J1+3J3), < k2 < 1, -1 < v < 0. (47) The critical amplitude Ac for this cubic approximation is found from k2 = 2J/ (J1+J) = 1, or J1(Ac) = J3(Ac). 6 33(1+3 The smallest solution of this equation is (A *) = 3.054, (48) compared with Ac = for the exact solution. Corresponding to a given value of A<Ac, one can compute v from (47). With this value of v known, one finds the corresponding numerical value of T/T' on the exact solution curve (E) in Fig. 1. This numerical value is set equal to the left-hand side of (46), whereupon one can obtain the new period ratio approximation T**/Too A numerical example will show the manipulations.

-19 - Example: Letting the amplitude A = 2, one finds v = -0o535, which corresponds to T/T' = 1.30 in Figo 1 (curve E). Hence, using (46) T**/T0 = 1.32. Instead of using Figo 1 to furnish the above result, one can, of course, evaluate (46) directly, or equivalently, use T**/T = (2/t) A21 (A) K(k6). (49) For comparison, using the value A = 2 of the above sample, one obtains T*'/T0 = 1.3296. The numerical difference between these results is due to graph-reading inaccuracy. For X = 1/2, the cubic Legendre polynomial approximation for sin x yields (T**/2,t)[3o (jl+ 2 3)/A]1 = 2(+k) 1/K(k)/ (50) where v = -3j /(6j+21j), k7 = 35j3/(12jl+7j3), and jl and j3 are spherical Bessel functions, i.e., jn(x) = (t/2x)l/2J n+(x) To establish the critical amplitude, set k7 = 1, which results in the equation 3jl(Ac) = 7j3(Ac). (51) The smallest solution of (51) is (A*) = 2.98. (52) c 1/2 Thus the critical amplitude Ac based on the cubic Legendre polynomial approximation is smaller than that from the Tchebycheff polynomials of

-20 - the first kind. This means that the Legendre approximation becomes poorer at a smaller amplitude A. This is substantiated by the curves of Figo 7, where these results and the exact solution are plotted. Included also in Fig. 7 is the linear approximation curve from [1] in order to demonstrate the improvement brought about by the cubic approximation. For sinh x and X = 0, one obtains from (44) (sinh x)** = (2/A)[I1-13(A)]x + [8I3/A3]x3 0 The cubic approximate differential equation of motion is d2x/dt2 + [2o2(I1-3I5)/A]x + [82I3/A3]x3 = 0, (53) When compared to (7) one notes a = 2wo(I1-3I)/A, b = 823/A It is interesting to observe that while b will always be positive, a becomes negative for certain values of the amplitude A, i.e., I < 3I3 for A > 3.91. When a becomes negative, one must treat (53) as a "softening-hardening" (Case 3) cubic spring. In the present case, v = bA2/a = 4i3/(I-3I3) (54) For a given A, one can compute Vo If v is positive, use Fig. 1 or 2; if 1/2 v is negative, use Fig. 4. Equate the period ratio found to T**(a) /2ir, and solve for the cubic-approximation period ratio T**/T0o An example is given to illustrate the procedure:

-21 - Example: Given A = 2. Then v = 0.8954, and Figo 1 gives T/To = 0.776. Hence, r**/To = 0.795, while a solution of (53) directly in terms of elliptic integrals yields 0.7954. Using the procedure shown for X 1/2 also, one obtains the results in Figo 8. It is worthwhile to note that the approximations for X = 0 and X = 1/2 are so close to the exact solution that the errors are hardly discernible. The results of the cubic Maclaurin series approximation has also been plotted for comparison. 7. Connection with method of Krylov and Bogoliuboff. The first approximation of Krylov and Bogoliuboff [10] yields the period approximation for (1) 1 2 -1/2 (Ti B = 2T[Ai f f(A cos e) cos 9 d9] / (55) On the other hand, the linear ultraspherical polynomial approximation gives, from (5) and (6), (4r (X+2_) t/2 2X -1/2 (T*) 2P [ =2T |r I/f f(A cos e)sin e cos e de] U.?P. 1/2,r(x+l/2)A 0 (56) We note that (55) and (56) are equivalent for odd f(x) when X = 0 in (56). Hence, a linear Tchebycheff polynomial approximation of f(x) in (1) gives results identical to those of the first approximation of Krylov and Bogoliuboff. Higher order approximations in the Krylov and Bogoliuboff method depend on an iteration procedure, whereas the cubic ultraspherical polynomial approximation yields a solution in elliptic integrals of the first kind. No general comparison was made,

-22 - 8. Discussion. The characteristic feature of any ultraspherical polynomial approximation (x < o) for arbitrary f(x) in (1) is that it yields an approximate period which will in general depend on the amplitude of the motion. The exact solution of a non-linear oscillation problem also has this property, while the linear Maclaurin series approximation does noto It is this feature which stimulated these investigations, while the use of particular ultraspherical polynomials permitted comparison of results for various values of Xo As was shown in section 5 and indicated in all the graphical results, for small oscillations and f(x) odd, the use of Tchebycheff polynomials (x = 0) not only gives the best results of all the ultraspherical polynomials, but also T* agrees with the exact result to terms 2 of order A o For large amplitudes or non-linearities, no such general results could be obtained. For the hardening cubics (Cases 1 and 3) and the hard sinh force, the X value corresponding to the correct asymptotic form for T as A or v become infinite was small and positive, while X = 0 yields smaller errors out to quite large amplitudes. For the only flattening spring considered here (tanh), for large amplitudes the choice X = -0.075 gives the correct asymptotic results. For the softening cubic (Case 2) and the soft sine force, oscillations at unlimited amplitudes cannot occur. Instead a critical amplitude exists at which the exact period becomes infinite. It can be shown that the linear ultraspherical polynomial approximation which produces the same critical amplitude corresponds to X = -1/2. Although the integrals for the coefficients do not exist for this value, the polynomials and the results for T1/2 can be obtained by a limiting process. The numerical results

-23 - were quite poor for the problems considered here, despite its yielding the correct v or A c c The extension to cubic ultraspherical approximations yields the expected increase in the accuracy of the results T*-o In comparing this extension with other higher order procedures, one notes that the cubic ultraspherical approximation method can be applied without a substantial increase in work, in contrast to higher order approximations in other methods, as the perturbation treatment. This is due to two factors. First, if the coefficient of the linear term a-() can be found, then aIX) can usually be found in a similar manner. If al() must be evaluated numerically, then a(X) will probably have to be also, but with no greater difficulty. Secondly, the procedure uses the fact that not only are the exact results for any cubic known in terms of elliptic integrals, but also T/T? depends only on one parameter v, so that one can obtain T/T' graphically or in tabular form once and use these reults for the cubic ultraspherical approximations o Finally, at no point in these approximation procedures was it necessary to restrict the non-linearity v or the amplitude A to a small value. In fact, all of ultraspherical polynomials (x < oo) gave reasonably good results as v (or A) became infinite. Given a (non-linear) f(x) other than those studied here, (and for which the amplitude does not become extremely large), then, if a single ultraspherical polynomial approximation is desired, the choice, X = 0 is indicated by our results.

BIBLIOGRAPHY lo Ho Ho Denman and Jo Eo Howard, Application of ultraspherical polynomials to non-linear oscillations Io Free oscillation of the pendulum, Quarterly of Applied Mathematics, in press. 2. Tables of the Chebyshev polynomials, National Bureau of Standards, Applied Mathematics Series 7 (Uo S. Government Printing Office, 1952), Introduction. 35. N W. McLachlan, Ordinary non-linear differential equations, 2nd Edition, Oxford University Press, London, 1956, p. 24-51, p. 39-40. 4. B. 0 Pierce and Ro Mo Foster, A short table of integrals, 4th Edition, Ginn and Co., Boston, 1956, po 99-102. 5. P. F. Byrd and MO Do Friedmann, Handbook of elliptic integrals for engineers and physicists, Springer, Berlin, 1954, p. 50. 60 Eo Jahnke, Fo Emde and Fo LBsch, Tables of higher functions, 6th Edition, McGraw-Hill, New York, 1960. 7. Po Franklin, Methods of advanced calculus, McGraw-Hill, New York, 1944, po 392. 80 Go No Watson, Theory of Bessel functions, 2nd Edition, Cambridge University Press, 1958, p. 369, Equation (8). 9. Ho Ho Denman, Computer generation of optimized subroutines, Jouro of the Assoc. for Comp. Macho (8), po 1 (1961)o 10. No Kryloff and No Bogolinboff, Introduction to non-linear mechanics, Princeton University Press, 1943o No Bogolinboff and Ao Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, 2nd Edition, Gordon and Breach, New York, 1961.

3.0!V- 2.01 Xs-1/4 I hO cn I Case I Cose 2 0.8.6.2 0.2.4.6.8 1.0 1.2 Figure 1. Period ratio (T/T')-non-linearity factor v curve for cubic spring -- free oscillation. 1.4 1.6 1.8 2.0

T/T' IIII I.1 X=-I/2' 1.0 10 102 Figure 2. Period ratio (T/T')-non-linearity factor v curve for hard cubic spring -- free oscillation. 0 0.1

Fgr\ 3./ P2 fo Eo IE, Figure 3. Potential curve for softening-hardening cubic spring -- free oscillation.

I# co \ -V 4 Or dsofte eninrg (~/~').uoolinerttYfactor v re -^^~~~~~,,,- — " t~~~~~~~~~~~Ia io^eTl.,~~~~~, osci.w i 4. period ra.tio fre e o iat~' Fi,'re cubic spring

io 1.0r e PT0 XCCO.901.80.70.60.50 I - I I I I I I I I I I II E.....X 1/2 I1 I I I I CD I.40.30.20.10 0 AI I....I..,I IA 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 c9 Figure 5. Period ratio (T/T )-amplitude curve for sinh spring -- free oscillation.

I CA) o I 3 4 5 Figure 6. Period ratio (T/To)-amplitude curve for tanh spring -- free oscillation.

-31 - 3.0 2.8 2.6 I I \O- <D X=I/2 2.; I }A I I X= I I (LINE I / I~ I I I -.5 1.0 1.5 2.0 2.5 3.0 Figure 7. Period ratio (-/T )-amplitude curve for sine spring for cubic approximation -- free oscillation. 1, 0 - A 3.5

-32 - 1.0.9.8.7.6.5 -E.4, X= 1/2.3.2 x=.1 0 I I I.....I I I I 0 2 3 4 5 6 Figure 8. Period ratio (T/TO)-amplitude curve for sinh spring for cubic approximation -- free oscillation.

- - (I — ~ ~ m ----i c-mm OD! ---- OD9D f (D 0~ a-mm~