Oscillation time and damping coefficients in a nonlinear pendulum
OOscillation time and damping coefficients in anonlinear pendulum
Jaime ArangoJanuary 26, 2021
Abstract
We establish a relationship between the normalized damping co-efficients and the time that takes a nonlinear pendulum to completeone oscillation starting from an initial position with vanishing velocity.We establish some conditions on the nonlinear restitution force so thatthis oscillation time does not depend monotonically on the viscositydamping coefficient.
ASC2020:
Keywords. oscillation time, damping, damped oscillationsThis paper is dedicated to the memory of Prof. AlanLazer (1938-2020), University of Miami. It was my pleasureto discuss with him some of the results presented here
The pendulum is perhaps the oldest and fruitful paradigm for the study ofan oscillating system. The apparent regularity of an oscillating mass going toand fro through the equilibrium position has fascinated the scientists well be-fore Galileo. There are plenty of mathematical models accounting for almostany observed behavior of the pendulum’s oscillation. From the sheer amountof the literature on the subject, one would expect that there is no reasonablequestion regarding a pendulum that has no been already answered. Andthat might be true. Yet, for whatever reason, it is not impossible to takeon a question whose answer does not seem to follow immediately from theclassical sources.In an ideal case with no damping, the oscillations are periodic and the pe-riod does exhibit a weakly dependence with respect to the initial conditions1 a r X i v : . [ m a t h . C A ] J a n rovided the amplitudes of the oscillation are small. Now, it is a differentmatter if the damping is noticeable. With damping there is no periodicity.However, we could still define the oscillation time as an analogy to the period(more later). But, how does depend this oscillation time on the character-istic of the medium, say on the viscosity of the surrounding atmosphere? Itseems that there is no much information on how the damping affects theoscillation time. There are plenty of new publications regarding dampingand oscillations, ranging from analytical solutions ([5], [3],[6]), to very cleverexperimental set ups (see for example [4]). The nature of the damping hasbeen also extensively consider ([8], [2]), but the dependence of the oscillationtime on the damping or on the non-linearity seems to be less investigated.For the sake of simplicity we analyze the oscillation time in the frame of amodel that appear in almost any text book of ordinary differential equations(see for example [1]): ¨ x + 2 α ˙ x + x (1 + f ( x )) = 0 , (1)where x = x ( t ) measures the pendulum’s deviation with respect to a verticalaxis of equilibrium and α ≥ x f ( x ) models the nonlinear part of the restoring force. We’ve rescaledthe time so that the period of the linear undamped oscillation is exactly 2 π .The math of the solutions x = x ( t ) is classical. If f is smooth and x and v are given real values, then there exists unique solution (defined for all t ≥
0) satisfying the given conditions x (0) = x and ˙ x (0) = v . Notice thatthe points of vanishing derivative of a solution x = x ( t ) to (1) are isolated andthose points correspond, either to local maxima or to local minima. Denoteby τ ( x , α ) the time taking by the mass to complete one oscillation startingfrom x with vanishing velocity ( v = 0). To be precise, if x = x ( t ) starts from x with vanishing velocity, then x reaches a local maximum at t = 0, and theoscillation is completed when x reaches the next local maximum. Certainly,the oscillation time generalizes the period of solutions for the undampedmodel ( α = 0). In this investigation we analyze the dependence of τ on x and on α under the following working hypothesis: Assumption 1.1.
On small (cid:15) − neighborhood of the function f is even andfor some constant a > we have f ( x ) = − a x + O (cid:0) | x | (cid:1) , We shall show that for x fixed, τ reaches a positive minimum at some0 < α < . It does not seem obvious that an increase in the dampingcoefficient α might cause a decrease in τ . It is also worth noticing that the2 . . . . . . Damping coefficient α π . . . τ ( x , α , ) x = 0 . x = 0 . x = 1 . x = 1 . Figure 1: Numerical simulation τ ( x , α ) depending on α for several values of x . The nonlinear term f was chosen so that x (1 + f ( x )) = sin x. existence of a minimum of τ is a consequence the sign of the constant a in theabove assumption. Indeed, τ does not reach a positive minimum neither inthe linear case ( f ≡ a < x (1 + f ( x )) = sin x. In that case, Figure 1 summarize our findingsby picturing the numerically simulated value for τ ( x , α ). Interestingly, ourqualitative analysis accurately reflects variations of τ that are not easy tospot numerically. For instance, the minimum of τ ( x , α ) for x = 0 . x = x ( t ) with respect to α . Definitions of underdamped oscillations in linear systems naturally carry overto solutions of (1). From now on, x ( · , x , α ) stands for the unique solutionto (1) satisfying the initial condition x (0) = x and ˙ x (0) = 0 . We also write τ ( x , α ) to highlight the dependence of the oscillation time on x and α . We3ill write simply τ or x when no confusion can arise. It is convenient torepresent (1) in the phase space ( x, v ) with ˙ x = v :˙ x = v ˙ v = − α v − x − x f ( x ) . (2)If we drop out the nonlinearity, equation (2) is explicitly solvable and in thatcase ( f ≡ x l ( t ) = e − α t ω ( ω cos ω t + α sin ω t ) x v l ( t ) = − e − α t ω sin ω t x (3)where ω = √ − α . Moreover, the oscillation time τ l is given by τ l = 2 πω = 2 π √ − α . Notice that τ l is an increasing function that solely depends on α .Although, a closed solution for (1) is either not known or impractical, wecould express the relevant solutions implicitly. To that end, we rewrite (2)so that the nonlinear term − x f ( x ) assumes the role of a non homogeneousforcing term. The expression for the solution ( x, v ) are implicitly given by x ( t ) = x l ( t ) − ω (cid:90) t e − α ( t − s ) sin ω ( t − s ) x ( s ) f ( x ( s )) dsv ( t ) = v l ( t ) − ω (cid:90) t e − α ( t − s ) ( ω cos ω ( t − s ) − α sin ω ( t − s )) x ( s ) f ( x ( s )) ds (4)Let us estimate some of the solution of (2) in the conservative case ( α = 0).By equations (3) and (4) and by Assumption 1.1, we see that x ( t ) = x ( t, x , v ( t ) = v ( t, x ,
0) are given by x ( t ) = x cos t + R ( t, x ) , x ( t ) = − x sin t + R ( t, x ) , (5)where | R i ( t, x ) | ≤ const | t | | x | , i = 1 , . At this point is convenient to define the half oscillation time ˆ τ = ˆ τ ( x , α )to be the time needed by the solution x ( t, x , α ) , t ≥ , to reach the nextlocal minimum. If α = 0 and f is even, the symmetry of the solution (1)yields. 2ˆ τ = τ. emma 2.1. If ˆ τ = ˆ τ ( x , α ) denote the half oscillation time and a is theconstant of Assumption 1.1, then ˆ τ ( x , α ) > π √ − α and lim x → + ˆ τ ( x ,
0) = π + a π x + o ( x ) . Proof.
We introduce introduce the polar coordinates r = √ x + v , tan θ = xv , to obtain ˙ θ = − (cid:0) α sin 2 θ + sin θ f ( x ) (cid:1) ˙ r = − vr (2 α v + x f ( x )) (6)As a consequence of equation (6) we obtain the following expression forthe half oscillation time ˆ τ = ˆ τ ( x , α )ˆ τ = (cid:90) π dθ α sin 2 θ + sin θ f ( x ( θ )) . (7)Now, the effect of the nonlinearity on the oscillation time is clear. By As-sumption 1.1 we obtainˆ τ ( x , α ) > (cid:90) π dθ α sin 2 θ = π √ − α . For α = 0 we use estimation (5) to obtainˆ τ ( x ,
0) = (cid:90) π dθ − a x sin θ cos t ( θ ) + o ( x ) . Now a straightforwards computation yieldslim x → + ˆ τ ( x ,
0) = π, lim x → + ∂ ˆ τ∂x ( x ,
0) = 0 . Now, taking the limit as x → τ with re-spect to x is almost straightforwards, except perhaps for one thing, namely:lim x → t ( θ ) = θ , so thatlim x → + ∂ ˆ τ∂x ( x ,
0) = (cid:90) π a sin θ cos θ dθ = 2 aπ , and the second claim of the lemma follows by the second order Taylor ex-pansion of ˆ τ ( x ,
0) around x τ ( x , α ) > π √ − α ≡ τ l . This inequality is illustrated in Figure 2 when a = 1. Had we considered inAssumption 1.1 negative values for a , then the inequality would reverse to τ ( x , α ) < τ l as it is depicted in Figure 2. It is not difficult at all to obtain a differential equation describing the themovement of the pendulum depending on the viscous damping coefficient.Indeed, writing X ( t, x , α ) = ∂x∂α ( t, x , α ) , V ( t, x , α ) = ∂v∂α ( t, x , α ) . Derivation of equation (2) with respect to α yields:˙ X = V ˙ V = − α V − X − v − ( x f (cid:48) ( x ) + f ( x )) X. (8)As for the initial conditions we have X (0 , x , α ) = 0 , V (0 , x , α ) = 0 , Let us write G ( x ) = − ddx ( x f ( x )) Again, as we did with equation (2), equa-tion (8) can be seeing as linear homogeneous part plus the forcing term − v + G ( x ) X. The solution
X, V is implicitly given by X ( t ) = 1 ω (cid:90) t e − α ( t − s ) sin ω ( t − s ) (cid:8) − v ( s ) + G ( x ( s )) X ( s ) } dsV ( t ) = 1 ω (cid:90) t e − α ( t − s ) ( ω cos ω ( t − s ) − α sin ω ( t − s )) (cid:8) − v ( s ) + G ( x ( s )) X ( s ) } ds In particular, for α = 0 the above expressions reduce to X ( t ) = (cid:90) t sin ( t − s ) (cid:8) − v ( s ) + G ( x ( s )) X ( s ) (cid:9) dsV ( t ) = (cid:90) t cos ( t − s ) (cid:8) − v ( s ) + G ( x ( s )) X ( s ) (cid:9) ds (9)6he following lemma does the heavy lifting to deliver the main result ofthe paper. Lemma 3.1.
Under Assumption 1.1, if ˆ τ = ˆ τ ( x , denotes the half oscil-lation time when α = 0 , then for < x << we have V (ˆ τ , x , > . Proof.
We start with an auxiliary estimate for X ( t ). By equation (9), byestimate (5) and by Assumption 1.1, for 0 < t ≤ π we have X ( t ) = x ( − t cos t + sin t ) + 3 a x (cid:90) t sin ( t − s ) cos s X ( s ) ds + O ( | x | )(10)Notice that X ( t ) ≡ x ( − t cos t + sin t ) does not vanish on (0 , π ) and that G ( x ( s )) > < x <<
1. Further, the initial conditions for X ( t, x , α ) at t = 0 and equation (8) yield that¨ X (0 , x , α ) = ˙ V (0 , x , α ) = 2 x (1 + f ( x ))meaning that X ( t ) is positive in an interval of the type (0 , (cid:15) ) with (cid:15) > X ( t ) > < t ≤ π , and by the equation (10) it follow immediatelythat X ( t ) = X ( t ) + O ( | x | ).Analogously, for V ( t ) we obtain V ( t ) = x t sin t + 3 a x (cid:90) t cos ( t − s ) cos s X ( s ) ds + O ( | x | ) ≡ V ( t ) + V ( t ) + O ( | x | )where V ( t ) ≡ x t sin t . Now, V ( t ) can be explicitly evaluated. However, forthe reader’s convenience, just in case she or he wants to check it, we writethe complete expression for V : V ( t ) =3 a x (cid:16) −
132 (6 t + 5) cos t − t sin 3 t − t sin t − t + 37128 cos t (cid:17) . Moreover, it is somewhat tedious but straightforward to show V is positiveand increasing in a small neighborhood of π . By Lemma 2.1 ˆ τ > π , therefore V (ˆ τ ) > V ( π ) = 9 a x π . V (ˆ τ ) = V ( π ) + (ˆ τ − π ) V (cid:48) ( π ) + O ( | x | )= − a x π O ( | x | ) , so that V (ˆ τ ) = V (ˆ τ ) + V (ˆ τ ) > . Now we are in a position to show the main result of the paper
Theorem 3.1.
Under Assumption 1.1, there exists a δ > such that for < x < δ fixed, the oscillation time τ ( x , α ) , for < α < , reaches apositive minimum at some < α < . Moreover, lim α → − τ ( x , α ) = ∞ . Proof.
We let 0 < x << x, v ) be the solutionof equation (2). By definition of ˆ τ we have v (ˆ τ , α ) = 0, so that the ImplicitFunction Theorem yields ∂ ˆ τ∂α ˙ v (ˆ τ , α ) + V (ˆ τ , α ) = 0 , therefore ∂ ˆ τ∂α = V (ˆ τ , α ) x (ˆ τ , α ) (1 + f ( x (ˆ τ , α ))) . Since x (ˆ τ , α ) is negative, it follows from Lemma 3.1 that and ∂ ˆ τ∂α | α =0 < τ .To do that, let us write ˆ x = − x (ˆ τ ( α, x ) , x ), and see that τ ( α, x ) = ˆ τ ( α, x ) + ˆ τ ( α, ˆ x ) . That is to say, the half oscillation time depend on | x | only. Notice thatˆ x ≤ x and the equality holds in the conservative case α = 0 only. Therefore ∂τ∂α ( α, x ) = ∂ ˆ τ∂α ( α, x ) + ∂ ˆ τ∂α ( α, ˆ x ) − ∂ ˆ x ∂α ( α, x ) ∂ ˆ τ∂α ( α, x ) = 0 . Moreover, since ∂ ˆ x ∂α ( α, x ) = v (ˆ τ ( α, x ) , x ) = 0 , we have that lim x → + ∂τ∂α ( α, x ) = 2 lim x → + ∂ ˆ x ∂α ( α, x )Finally, by the first claim of Lemma 2.1, τ ( α, x ) must attain a minimum atsome 0 < α <
1. 8igure 2: Numerical simulation τ ( x , α ) depending on α for x = 0 .
2. Thenon linearity is defined by f ( x ) = − a x , a = 1 , −
1. The curve with theround marker (blue in the online version) correspond to the oscillation time τ l of the linear case f ≡ The proof of existence of a positive minimum for the oscillation time dependson the sign of the nonlinear term f . Theorem 3.1 is underpinned in Assump-tion 1.1 that assumes a > . Notwithstanding Assumption 1.1, we carriedout some numerical simulations of τ ( x , α ) for several values of x assuming a = 1 and a = −
1. Those are especial case of an unforced Duffing oscillator[7]. The results can be seen in Figure 2. This pictures reflects very well thetheoretical results of the paper. If a = 1 we see that τ reach its minimumat a positive value for α . By contrast, if a = −
1, the minimum seems to beattained at α = 0. Just to compare, the curve with the round marker (bluein the online version) corresponds to the oscillation time of the linear case τ l = π √ − α .The numerical experimentation (not showed in this paper) assuming aquadratic damping exhibits the same behavior as the graphics of Figure 2.If the reader is curious about the numerical experiments, take a look at theauthor’s GitHub page ( h ttps://github.com/arangogithub/Oscillation-time)and download a Jupyter notebook Phython code with the details. References [1] V. I. Arnold.
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