Distinguished Limits and Vibrogenic Force revealed by Newton's Equation with Oscillating Force
aa r X i v : . [ phy s i c s . g e n - ph ] A p r Distinguished Limits and Vibrogenic Forcerevealed by Newton’s Equationwith Oscillating Force
By V. A. V l a d i m i r o v
York University, UK; Sultan Qaboos University, Oman; Leeds University, UK(Received 21 April 2020)
In this paper, we analyse the basic ideas of Vibrodynamics and the two-timing method.To make our analysis most instructive, we have chosen the Newton’s equation with ageneral oscillating force. We deal with its asymptotic solutions in the high frequencylimit. Our treatment is simple but general. The targets of our study are the distinguishedlimits and the universal vibrogenic force . The aim of the distinguished limit procedure isto identify how the small parameter can appear in an equation. The proper appearance ofa small parameter leads to valid successive approximations , and, in particular, to closedsystems of averaged equations. We show, that there are only two distinguished limits. Thismeans that Newton’s equation, with high-frequency forcing, has two types of interestingasymptotic solutions. The key item in the averaged equations for all distinguished limitsis the unique vibrogenic force . The current state-of-the-art in this area is: a large numberof particular examples are well-known, effective and advanced general methods (likethe Krylov-Bogolyubov approach) are well developed. However, the presented generaland simple analysis of distinguished limits and the vibrogenic force, formulated as acompact practical guide, is novel. An advantage of our treatment is the possibility of itsstraightforward use for various ODEs and PDEs with oscillating coefficients.
1. Introduction
This paper is devoted to the analysing of the basic ideas of Vibrodynamics and thetwo-timing method. To do this most instructive, we use Newton’s equation with a gen-eral high-frequency oscillating force. Under oscillations we understand any time-periodicprocess, including rotation. Our attention is focused at the distinguished limits (DLs) and the vibrogenic force . There are different definitions of DL , see e.g. Nayfeh (1981),Kevorkian & Cole (1996). All these definitions operate with ‘proper’ relations betweendifferent terms in a considered equation, such relations must lead to valid mathemati-cal results. For example, if we have two independent small parameters ε and ε in anequation, then we cannot say, what is greater ε or ε . As the result, the successive ap-proximations cannot be derived. For resolving this problem, we should study differentpathes in the ( ε , ε )-plane, such that ( ε , ε ) → (0 , ε ( ε ), ε ( ε ) with the only small parameter ε →
0. If such a limitbrings valid asymptotic results, then it is called a distinguished limit (DL) . A commonpoint of confusion is:
DLs are usually used as know-how , without any comments to theirappearance and to the presence or absence of any alternatives. Often, DL is implicitlyused without naming it. The existence of DLs demonstrates that equations are ‘smart’by themselves: the structure of an equation dictates the placing a small parameter within
V. A. Vladimirov it. It is important to emphasise that the
DL procedure represents a purely mathematicaltool . The fitting of a particular physical problem into the
DLs represents the next task.The term vibrogenic force , for the averaged force, generated by oscillations/vibrations wasintroduced by Yudovich (2006). Similarly, any averaged product of oscillating functionscan be called a vibrogenic term . A number of references could be very significant, thedirectly related treatments are Vladimirov (2005), Vladimirov (2017), Yudovich (2006).In this paper we show that there are two
DLs for Newton’s equation. Our results areexposed briefly, but with a natural generality and simplicity. In Section 2, we present thegeneral setting of the two-timing method and Vibrodynamics. All the required notationsand suggestions are introduced in Section 3.1. The next Section 3.2 is devoted to the gen-eral properties of considered distinguished limits. The list of averaged equations is givenand analysed in Section 3.3. In Section 4, we illustrate our approach by the paradigmexample of the Stephenson-Kapitza pendulum (a pendulum with an oscillating pivot),where our attention is focused at fitting the pendulum’s equation into the described
DLs .Section 5 contains the formulation of the practical algorithm for using our results. Finally,the detailed calculations of successive approximations are presented in the Appendix.
2. Two-Timing Setting
We consider a system of ODEs d x ∗ /dt ∗ = f ∗ ( x ∗ , τ ) , τ ≡ ω ∗ t ∗ (2.1)where x ∗ = ( x ∗ , x ∗ , x ∗ ) and t ∗ are cartesian coordinates and time, x ∗ ( t ∗ ) is the unknownfunction, f ∗ ( x ∗ , τ ) represents a given oscillating force, a constant mass of a particleis included in the definition of force, ω ∗ is a given frequency, and asterisks mark thedimensional variables. We assume that there are mutually independent characteristicscales of length L , time T , and force F . Then, the above equation in the dimensionlessform is: d x dt = ( F T /L ) f ( x , τ ) , (2.2)where x ∗ = L x , t ∗ = T t, f ∗ = F f , ω ∗ = ω/T, τ = ωt = ω ∗ t ∗ The first step of the two-timing method is the introduction of two ( originally mutuallydependent ) time-variables s and τ ; we call s slow time and τ fast time : τ ≡ ωt, s ≡ t/ω α ; α = const > − , ω ≫ α > − τ and s as the fast time variable and theslow time variable for ω → ∞ . The use of the chain rule brings (2.2) to the form (cid:18) ω ∂∂τ + 1 ω α ∂∂s (cid:19) x = ( F T /L ) f ( x , τ ) (2.4)where we have two independent scaling parameters ω and ( F T /L ). Next, we reduce theproblem to a single parameter by introducing a functional dependence F T /L ≡ ω β ,where β = const. Then (2.4) is transformed to the equation x ττ + 2 ε x τs + ε x ss = ε κ f ( x , τ ) , ε ≡ ω − − α , κ ≡ − β α (2.5) istinguished Limits and Vibrogenic Force ε is the only small parameter and the subscripts τ and s stand for the partialderivatives. The key suggestion of the two-timing method is τ and s are treated as mutually independent variables (2.6)which converts (2.5) from an ODE with the only independent variable t into a PDEwith two independent variables τ, s . As a result, any solutions of (2.5) must have thefunctional form: x = x ( s, τ ) (2.7)A regular perturbation procedure starts from substituting the series x ( s, τ ) = ∞ X n =0 ε n x n ( s, τ ) , n = 0 , , , . . . (2.8)into (2.5). After x ( s, τ ) is found, one can use (2.3) to return to the original variable t : X ( t ) ≡ x ( s ( t ) , τ ( t )). Mathematical justification of the obtained solution X ( t ) consists ofproving that the difference between X ( t ) and the exact solution to (2.2) is small. In thispaper, we accept (2.6) and (2.7) and do not consider any mathematical justifications.
3. Asymptotic Procedure and Distinguished Limits
Required Notations, Operations, and Suggestions
To make our treatment self-consistent and efficient, we introduce some notations andagreements. Let g = g ( s, τ ) be any dimensionless function, which could be scalar, vecto-rial, or tensorial . We accept that: (A) The order agreement: g ∼ O (1), and its s -, and τ -derivatives (required for ourconsideration) are also O (1). All small parameters are represented by various degrees of ε only, they appear as explicit multipliers. If we do not accept this, then the use of thesuccessive approximations is impossible. In particular, it makes impossible the change of κ by dividing (2.5) by any degree of ε . (B) τ -periodicity: g is 2 π -periodic in τ , i.e. g ( s, τ ) = g ( s, τ + 2 π ), which represents atechnical simplification; (C) Average, bar-functions and tilde-functions: g has an average given by g ≡ h g i ≡ π Z τ +2 πτ g ( s, τ ) dτ ∀ τ = const , (3.1)hence any g ( s, τ ) can be split into the sum of its average part and purely oscillatingpart g ( s, τ ) = g ( s ) + e g ( s, τ ), where the tilde-function (or purely oscillating function) issuch that h e g i = 0 and the bar-function g ( s ) is τ -independent. The introduced functions x ( s, τ ), f ( x , t ), and x n ( s, t ) represent the combinations of bar- and tilde- parts, e.g. x n ( s, τ ) = x n ( s ) + e x n ( s, τ ); (D) Tilde-integration: e g τ (with a superscript τ ) stands for the operation: e g τ ≡ G − G, G ( s, τ ) ≡ Z τ e g ( s, τ ′ ) dτ ′ We call it the tilde-integration , since it keeps the result within the tilde-class. The tilde-integration is inverse to the τ -differentiation ( e g τ ) τ = ( g τ ) τ = e g ; (E) Taylor series:
The related to (2.8) expansion for f = ( f , f , f ) appears as f ( x , τ ) = f + ε ( x · ∇ ) f ,k + ε ( x · ∇ ) f + 12 x k x l f ,kl + O ( ε ) , (3.2) V. A. Vladimirov
We use the summation convention and shorthands f ≡ f ( x , τ ), f ,k ≡ ∂ f /∂x k , f ,kl ≡ ∂ f /∂x k ∂x l , x = ( x , x , x ), ∇ ≡ ( ∂/∂x , ∂/∂x , ∂/∂x ), f =( f , f , f ), etc. For brevity, we simulaneously use both the vectorial notations andthe component/subscript notations. (F)
The class of motion:
We accept that x ( s, t ) = x ( s ) or e x ( s, t ) ≡ The constraint (3.3) means that the amplitude of oscillations/vibrations is small comparedwith the amplitude of the average motion.
This constraint is both mathematical andphysical.
Mathematically: If e x = 0, then the series (3.2) has the coefficients like f ≡ f ( x + e x , τ ), etc. Such an expression, for a general function f , makes impossible thesplit of (3.2) into their bar parts and tilde parts, and hence the procedure of successiveapproximations fails. Physically: If e x = 0, then the main term of velocity grows toinfinity | ω e x τ | → ∞ as ω → ∞ . Then, e x τ = 0 represents a natural physical constraint,while e x = 0 follows after its integration, due to the zero average of a tilde-function. It isimportant, that we can consider only the versions of the equation (2.5) not contradictingto (3.3). 3.2. General Properties of Distinguished Limits
The consistency between equations (2.5), (2.8) and the agreement (A) above, requirethat parameter κ in (2.5) represents a non-negative integer: κ = 0 , , , , . . . . Then eachDL appears as the value of κ which, after the substitution of (2.8) into (2.5) , leads to avalid procedure of successive approximations . To obtain a full set of asymptotic equations,one should consider all available values of κ . Our calculations for (2.5)-(3.2) representthe standard procedure of successive approximations, complimented by splitting solutionsinto the averaged part and the purely oscillating part. This procedure allows to calculatethe full solution, including its average and oscillating parts. However, for practical use,we will present only the calculations until the approximation, in which the vibrogenicforce F vg appears. Below we present the final results for DLs , while all the calculationsof successive approximations are moved to the Appendix. The general results for different κ are: • κ = 0: The amplitudes of the oscillatory part and the averaged part in x ( s, τ ) = x ( s ) + e x ( s, τ ) are of the same order. It violates our main constraint (3.3), and thereforeit is out of our consideration. • κ = 1: A valid/closed system of successive approximations does exist, we refer toit as Distinguished Limit-1 or DL-1 . F vg appears in the zeroth approximation of theaveraged motion. • κ = 2: A valid/closed system does exist, we refer to it as DL-2 . The same as in
DL-1 vibrogenic force F vg appears in the second approximation of the averaged motion. • κ = 3: A closed system of successive approximations appears, however it degenerates.It appears since the forcing is so small that x ( s ) represents a free rectilinear motion witha constant velocity and F vg appears only in the fourth approximation of the averagedmotion. • κ = N >
3, with an integer N : Closed systems also appears, however the degreesof their degeneration are even higher than for κ = 3. The orders of approximation,describing the rectilinear motion and the the orders of averaged equations where F vg appears, are increasing with the grow of N . istinguished Limits and Vibrogenic Force The List of Averaged Equations for DL-1,2,3
In this subsection, we present only the averaged equations and discuss their properties.The related calculations of successive approximations are given in the Appendix.The averaged
DL-1 equations have been obtained by considering three successive ap-proximations n = 0 , , κ = 1: x ss = F vg , the required constraint f ≡ DL-2 equations have been obtained by considering five successive approx-imations n = 0 , , , , κ = 2: x ss = f , (3.5) x ss = ( x · ∇ ) f (3.6) x ss = ( x · ∇ ) f + 12 x i x k f ,ik + F vg (3.7)The averaged κ = 3 equations have been obtained by considering seven successive ap-proximations n = 0 , , , , , , x ss = 0 or x = a s + b , where a and b are arbitrary constants (3.8) x ss = f (3.9) x ss = ( x · ∇ ) f (3.10) x ss = ( x · ∇ ) f + x i x k f ,ik / x ss = ( x · ∇ ) f + x i x k f ,ik + x i x k x l f ,ikl / F vg (3.12)In all cases κ = 1 , ,
3, the expression for vibrogenic force is the same: F vg ≡ −h ( e f τ · ∇ ) e f τ i (3.13)The important properties of the above averaged equations with κ = 1 , , (i) κ = 1 , ,
3: All three systems are presented only until the approximation, where F vg appears; (ii) κ = 1 , ,
3: For F vg ≡
0, each averaged system coincides with the successive approx-imations of the original equation (2.2), taken for the τ -independent function f ; (iii) DL-1 : Equation (3.4) provides the promising option of entering F vg in the zero/mainapproximation. The constraint f ≡ f = e f = 0. Such a force often appears in applications,see Sect.4 below. (iv) DL-1,2:
Equations (3.5), and (3.9) coincide with the original equation (2.2), takenfor the τ -independent function f ; (v) DL-2:
Equation (3.6) represents the linearized version of (3.5). Equation (3.7) de-scribes the second-order perturbations for (3.5), complemented by F vg . For a betterdemonstration of the effects of the vibrogenic force, one can choose x ( s ) ≡ V. A. Vladimirov (3.7), which yields: x ss = f , x ss = ( x · ∇ ) f + F vg (3.14)It reveals the appearance of F vg for linear perturbations, which is promising for variousapplications. In particular, it can be important for the studies of long-time evolution,for deviations from symmetry, etc. Relevant applications can be found, for example, inastronomy. (vi)
DL-1, link to physics:
The
DL-1 corresponds to κ = 1 which gives α + β = 1, see(2.5). Two physically promising cases are ( α, β ) = (0 ,
1) (when s = t and ε = 1 /ω ) and( α, β ) = (1 ,
0) (when s = t/ω and ε = 1 /ω ). In both cases, the same F vg (3.13) appearsin the average equations of zeroth approximation, however the slow time variables aredifferent. (vii) How to compare DL-1 and DL-2?
Let us consider solutions with β = 0 in (2.5).They could be interesting physically, since they correspond to the dimensionless force oforder one. Then ( α, β ) = (1 , , s = εt in DL-1 , and ( α, β ) = (0 , , s = t in DL-2 . The
DL-1 averaged equation (3.13) (where always f ≡ x ss = F vg , where s = εt (3.15)The DL-2 averaged equations are (6.10), (6.13), (6.16). Considering a special case f ≡ x ss = 0 , x ss = 0 , x ss = F vg , where s = t (3.16)Then a special DL-2 solution can be chosen as x = 0 , x = 0 , x ss = F vg (3.17)One can see that in terms of the original physical time t , the equations can be (informally)rewritten as x tt = F vg for DL-2 , and x tt = ε F vg for DL-1 (3.18)which shows that the main (zeroth)
DL-1 approximation is the same as the second ap-proximation ε x of the special DL-2 solution (3.17). Making these asymptotic solutionsidentical to each other violates our strict setting, in which all our functions are of orderone. However, an informal renormalization of the involved function allows us to concludethat, in some cases, the solutions
DL-1 represents a special case of
DL-2 . (viii) κ ≥
3: For κ = 3 system (3.8)-(3.12) degenerates. It appears since all the coefficientsin Taylor’s series (3.2) are calculated at the arbitrary rectilinear motion x = x = a s + b .In this situation all the averaged equations have a structure when ‘an acceleration isequal to a given force’, hence the secular behaviour of their solutions is common. Forexample, one can consider the state of rest x = x = 0. Then (3.9) simplifies to x ss = f (0) = const, which gives the quadratic growth of x ( s ), etc. Similar degeneration takesplace for the averaged equations with any κ ≥
3. Therefore, one can consider all thesedegenerated cases as useless for applications . The question of their mathematical validityremains open. (ix)
Potential forces:
For a potential vibrational force e f = ∇ e ϕ the vibrogenic force alsohas a potential: F vg = −∇ Π , where Π ≡ h ( ∇ e ϕ τ ) i / istinguished Limits and Vibrogenic Force
4. The Stephenson-Kapitza Pendulum fitted to Distinguished Limits
The mathematical part of the job is over after all the
DLs are determined and asymp-totic equations/solutions are calculated. The next step is to choose the values of α and β in (2.5), which are suitable for particular physical applications. Let us consider amathematical pendulum with a vertically vibrating pivot. Such a pendulum represents aparadigm of Vibrodynamics and averaging methods, therefore it is instructive to demon-strate how it can be seen in the DLs framework. The dimensional equation is lθ t ∗ t ∗ = − ( g + Y ∗ t ∗ t ∗ ) sin θ (4.1)where θ is the angle of pendulum’s deviation from the vertical, g is the homogeneousvertical gravity field, l is the length of the pendulum, Y ∗ ( τ ) is the periodically changingvertical coordinate of the pivot, τ ≡ ω ∗ t ∗ , ω ∗ is the given frequency of the pivot’soscillations. The use of characteristic scales of length L = l and time T = p l/g leads tothe dimensionless equation: θ tt = − (1 + ω e Y ττ ) sin θ, e Y = e Y ( τ ) , τ = ωt (4.2)This equation represents a special one-dimensional (scalar) form of (2.2); in (4.2) theunknown function θ ( t ) plays a part of x ( t ). If e Y ∼ O (1) then (4.2) is out of our consid-eration due to the main constraint (3.3). Hence, we abolish the requirement e Y ∼ O (1)by introducing either of the two constraints for the amplitude of pivot’s vibrations(a) e Y = e ξ ( τ ) /ω, (b) e Y = e ξ ( τ ) /ω , (4.3)where e ξ ( τ ) ∼ O (1) is a given function. In the limit ω → ∞ , the case (a) corresponds tothe finite velocity and infinite acceleration, while (b) gives the limit of zero velocity andfinite acceleration. Then the particular forms of (2.2) are(a) x tt = − (1 + ω e ξ ττ ) sin x, (b) x tt = − (1 + e ξ ττ ) sin x, (4.4)where, to keep similarity with (2.2), the unknown function θ is replaced with x . Thenthe only component of force f ( x, τ ) appears as(a) x tt = ω ( f ( x, τ ) + g ( x, τ ) /ω ) = − ω ( e ξ ττ ( τ ) + 1 /ω ) sin x (4.5)(b) x tt = f ( x, τ ) = − ( e ξ ττ ( τ ) + 1) sin x (4.6)The minor difference between (2.2) and (4.5) is: instead of the function f in (2.2), we usea sightly more general function f + g/ω , where both functions f and g are of order one.As one can see, (a) belongs to DL-1 with ( α, β ) = (0 , DL-2 with( α, β ) = (1 , DL-1 equation (3.4) follows automatically(we return notations back from x to θ ): θ ss = − sin θ + F vg , F vg = − h e ξ τ i sin(2 θ ) (4.7)The averaged equations for (b) are also straightforward, we do not write them here. The DL-1 equation (4.7) leads to the conservation of ‘energy’ E = K + Π = const , K ≡ ( θ s ) / , Π ≡ − cos θ − ( h e ξ τ i /
4) cos(2 θ ) (4.8)which is always used for the demonstration of the upside down pendulum or invertedpendulum . The upside down equilibrium at θ = π appears due to the vibrogenic termin the potential energy Π ( θ ), this term agrees with (3.19). A physically minded readermay be surprised here that Nature has chosen the limit
DL-1 , which corresponds toinfinite acceleration.
V. A. Vladimirov
5. Conclusions and Discussion
A Practical Algorithm:
Our consideration leads to a simple and highly practical al-gorithm of building averaged equations for (2.1)-(2.4). The only two productive optionsare
DL-1 with the averaged equation (3.4) and
DL-2 with the averaged equations (3.5)-(3.7). The latter, for a better physical exposition, can be replaced by (3.14). The choicebetween these two options is determined by the structure of the main approximation ofthe force f ≡ f ( x , τ ). Under the constraint (3.3), it takes a form f = f ( x ( s ) , τ ). Ifthe average f ≡ h f ( x ( s ) , τ ) i = 0, then we have the case of DL-1 . If f = 0, then wedeal with DL-2 . The value of f is easy to calculate, since x ( s ) is treated as a constant.After the choice between DL-1 and
DL-2 has been done, the only remaining job is tocalculate F vg (3.13), and then go for a particular physical application. Second Order ODEs versus
First Order ODEs and PDEs:
Mathematically, a systemof the first order ODEs is more general, than a second order equation. However, theobtaining of the second-order equation from a system of first-order equations requires awell known degeneration of the latter. Therefore the second-order ODEs are worth toconsider separately. Very similar ideas can be used for PDEs, see e.g.
Vladimirov (2005),Vladimirov (2017).
6. Appendix: Calculations for
DL-1 and
DL-2
Calculations for the
DL-1 ( κ = 1): Equation (2.5) takes the form: x ττ + 2 ε x τs + ε x ss = ε f ( x , τ ) (6.1)The substitution of (2.8), (3.2) into (6.1) produces the successive approximations: The zero-order equation (6.1) is x ττ = 0. The substitution of x = x ( s ) + e x ( s, τ )gives e x ττ = 0. Integrating it twice yields e x = h ( s ) τ + h ( s ), where the arbitraryfunctions h ( s ) and h ( s ) vanish due to the τ -periodicity and zero average of e x . Hence,the unique solution is e x ≡ e x ( s, τ ) ≡ x ( s ) is not defined (6.2) The first-order equation (6.1) is x ττ + 2 x τs = f . Its averaging leads to the constraint f ≡
0, whilst the tilde-part is e x ττ = e f . Hence, we obtain: e x ( s, τ ) = e f ττ ; x ( s ) is not defined (6.3) The second-order equation (6.1) is x ττ + 2 x τs + x ss = ( x · ∇ ) f (6.4)Its bar-part appears as: x ss = h ( e x ·∇ ) e f i ; the substitution of (6.3) into this expressionand integration by parts lead to (3.4) and (3.13): x ss = F vg , F vg ≡ −h ( e f τ · ∇ ) e f τ i (6.5)Since the vibrogenic force F vg appears in the zeroth approximation, we do not presentthe calculations for any further steps. We claim that any further approximation can becalculated, including its average and oscillatory parts.Calculations for the DL-2 ( κ = 2): Equation (2.5) takes the form: x ττ + 2 ε x τs + ε x ss = ε f ( x , τ ) (6.6)The substitution of (2.8), (3.2) into (6.6) produces the successive approximations: istinguished Limits and Vibrogenic Force The zero-order equation (6.6) is again x ττ = 0. Hence, the results are the same as (6.2): e x ( s, τ ) ≡ x ( s ) is not defined . (6.7) The first-order equation (6.6) is x ττ + 2 x τs = 0. It leads to e x ( s, τ ) ≡ x ( s ) is not defined . (6.8) The second-order equation (6.6) is x ττ + 2 x τs + x ss = f + e f , (6.9)which produces the bar-part (3.8) x ss = f , (6.10)whilst the tilde-part leads to e x ττ = e f , e x = e f ττ . (6.11) The third-order equation (6.6) x ττ + 2 x τs + x ss = ( x · ∇ ) f (6.12)Its averaging with the use of (6.8) produces (3.6): x ss = ( x · ∇ ) f (6.13) The forth-order equation (6.6) x ττ + 2 x τs + x ss = ( x · ∇ ) f + 12 x i x k f ,ik (6.14)produces the bar-part x ss = ( x · ∇ ) f + h ( e x · ∇ ) e f i + 12 x i x k f ,ik (6.15)The substitution of e x from (6.11) and integration by parts produce (3.7) x ss = ( x · ∇ ) f + 12 x i x k f ,ik + F vg (6.16)where the vibrogenic force F vg is again given by (3.13). Since its appearance, we donot present the calculations of further approximations; we claim here that DL-2 alsoproduces a valid asymptotic procedure.
Acknowledgements:
The paper is devoted to the memory of Prof. A.D.D. Craik, FRSE,who read this manuscript and made useful critical comments. A special thank to Prof.M.R.E. Proctor, FRS, for a productive discussion, and to Mr. A. A. Aldrick for help withthe manuscript. This research is partially supported by the grant IG/SCI/DOMS/ 18/16from SQU, Oman.
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