ISSN 10526188, Journal of Machinery Manufacture and Reliability, 2013, Vol. 42, No. 4, pp. 261–268. © Allerton Press, Inc., 2013. Original Russian Text © I.I. Vulfson, 2013, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2013, No. 4, pp. 3–11.

MECHANICS OF MACHINES

Energy Transfer in Vibratory Systems of Drives with Cyclic Mechanisms I. I. Vulfson St. Petersburg, Russia Received March 4, 2013

Abstract—The exchange of vibrational energy between an external source and subsystems of the cyclic mechanism schematized as a dynamic model with slowly varying parameters is studied. The condi tions are obtained whose violation yields the formation of regions where the energy of vibrations appear in the time interval of the kinematic cycle, even in the absence of external perturbations. This effect is shown to be caused by the operation of the external source upon implementation of nonsta tionary links and the dynamic instability of the system in a finite time. The dynamic effects and key conclusions are illustrated by results of computer simulation. Engineering recommendations for decreasing the vibrational activity of cyclic mechanisms are presented. DOI: 10.3103/S1052618813040171

When solving problems of machine dynamics, researchers often encounter different effects related to the transfer of energy from one subsystem to another, or the energy exchange between different forms of vibrations. Sometimes these effects are positive and facilitate vibrational protection of machines and mechanisms. One of the most widely known and vivid examples of this effect is dynamic damping. When it is properly set, the reaction from the side of the damper onto the key mass in the steadystate mode is equal to the driving force, but with opposite sign. In this case, the energy of the external source is purposely transferred from the object under vibrational protection to the dynamic damper. The analogous effect is observed upon dynamic unloading of the drive of cyclic mechanisms where the energy exchange between the effector and the dynamic unloader takes place [1, 2]. In other cases, these effects yield an undesirable redistribution of vibrations and their localization in certain units, links, cross sections, etc. Here, a large role is often played by the variability of the parameters of the vibratory system which are inherent in the drives of machines with cyclic operation. In particular, it is quite possible that local disturbances of the dynamic stability conditions, when the energy “replenish ment” of the system yields an intense increase in the vibration amplitudes, take place at limited time inter vals [3–5]. Spatial localization in vibratory chains [6–8], when strict dynamic regularity of the system is violated in certain sections due to socalled “inclusions” that manifest themselves as pronounced extrema in the forms of vibration, belongs to this class of problems. Based on the classical model, this problem is defined concretely in [6] by the example of vibrations of a string and Bernoulli–Euler and Timoshenko beams with concentrated inclusions. Similar effects are also observed upon the analysis of the dynamics of machines and automated lines with repetitive sections (modules). In particular, elimination of spatial localization is necessary in design ing machines with an increased extension of the region of technological treatment of items when vibra tions of long actuators should be similar to inphase ones [1, 3–5]. The violation of this requirement yields (apart from undesirable dynamic effects) different flaws in the products, e.g., nonuniformity of the yarn and selvage defects in the manufacture of fabrics, thread breaks, defects of printed output in printing machines, violation of the specified accuracy and purity of treated surfaces in metalcutting machines, etc. Preservation of the inphase vibration forms over a relatively large number of cyclic drive mechanisms is a rather complicated problem that requires further investigation. The analysis showed that, as applied to the problems of machine dynamics, this problem is beyond the framework of classical theory. This is related to the fact that repetitive modules which form complicated systems with variable parameters and nonlinear elements appear instead of the point masses. In addition, the “inclusions,” which are attributed in the vibratory systems of machines with deviations from regularity due to design and other factors that break the strict dynamic identity of the repetitive modules, are dras tically different. 261

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Note that the problem analyzed here is quite gen eral and it can be encountered in many physical problems. For example, the socalled Landau–Zener tunneling, in which the energy exchange takes place between two levels, is known in quantum mechanics [9, 10]. A mechanical analogue of tunneling, which represents a system of two weakly coupled pendulums 0 t where the partial frequency of one slowly varies in time and covers the region of internal resonance, was proposed in [10]. In this case, if the transfer of energy from one vibrator to another one turns out to be irre versible, there appears a trap for “capturing” the vibratory frequency. This effect, illustrated in Fig. 1 by the plots showing the vibration transfer ui from one Fig. 1. subsystem (solid line) to another (dashed line), is presented in [9]. Let us consider the transfer of vibrational energy by ϕ0 ϕ1 ϕ2 c1 c2 the example of the dynamic model of a cyclic mecha J0 J1 Π J2 nism with an elastic drive in the series connection of ψ1 ψ2 the elements J0 – c1 – J1 – Π – c2 – J2 (Fig. 2). Let us use the following notation: Ji is the moment of inertia; Fig. 2. ci are the coefficients of rigidity; ψ i are the scattering coefficients; ϕ1 = ϕ0 + q1; ϕ2 = Π(ϕ0) + q2, where ϕi are the absolute angular coordinates of the corre sponding inertial elements; ϕ 0 = ω0t is the ideal coordinate of the element J0 at ω0 = const; qi are the gen eralized coordinates which are equal to the absolute dynamic errors, i.e., deviation from the specified pro grammed motion; and Qi are the generalized forces. Assuming the function Π(ϕ1) to be continuous and differentiable, let us linearize this function and its first two derivatives in the vicinity of the programmed motion [1, 2]: ui

Π ( ϕ 0 + q 1 ) ≈ Π * + Π '* q 1 ,

Π' ( ϕ 0 + q 1 ) ≈ Π '* + Π ''* q 1 ,

(1)

where the asterisk corresponds to the argument ϕ0 = ω0t; ( )' = d/dϕ. According to Eq. (1) the following set of differential equations corresponds to this model after linear ization in the vicinity of the programmed motion: q·· 1 + k 1 ( 2δ 1 q· 1 + k 1 q 1 ) + μk 2 Π '* [ 2δ 2 ( Π ' q· 1 – q· 2 ) + k 2 ( μΠ ' q 1 – q 2 ) ] = W 1 ( t ), *

*

(2)

q·· 2 + k 2 [ 2δ 2 ( q· 2 – Π ' q· 1 ) + k 2 ( q 2 – Π ' q 1 ) ] = W 2 ( t ), *

*

where ki = c i /J i , μ = J 2 /J 1 , δi = ψi/(4π) at (i = 1, 2); and Wi(t) is the external excitation. To understand the observed dynamic effects, let us consider the particular case when the problem is reduced to the analysis of the vibratory system with variable parameters and one degree of freedom. If we schematize the output unit of the cyclic mechanism as an absolutely rigid disc (c2 → ∞), the system is described with the following nonlinear differential equation: 2

J 1 q·· 1 + c 1 q 1 = – Π' ( ϕ 1 ) { J 2 [ Π'' ( ϕ 1 ) ( q· 1 + ω 0 ) + Π' ( ϕ 1 )q·· 1 ] + Q 1*},

(3)

where, in addition to the introduced denomination, we assume that q1 = ϕ1 = ϕ0; Q* is the external non conservative generalized force. According to Eqs. (1) and (3) the linearized differential equation in this case takes on the form 2 q·· 1 + 2n ( t )q· 1 + p ( t )q 1 = W ( t ), 2

2

where p = k1/ 1 + μ Π ' ; k1 = *

(4)

c 1 /J 1 ; the function n(t) consists of the dissipative and gyroscopic com

ponents n(t) = n0(t) + n1(t). JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY

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It can be shown that n0(t) = δ1p(t), where δ = ψ1/(4π), ψ1 is the coefficient of scattering at given the positional dissipative force. Distinguishing in Eq. (3) the terms which are proportional to q· 1 we have 2

n 1 = ω 0 μΠ ' Π '' / ( 1 + μΠ ' ). * *

(5)

*

Note that Eq. (4) is valid for any vibratory system with one degree of freedom which describes a drive with a reduced moment of inertia J(ϕ) = J1(1 + μ2Π'2), where μ2 = (Jmax – J1)/J1 or with the variable reduced coefficient of rigidity [2]. Let us make use of the method of conditional vibrator which is a slowly varying function of position yields the solution at the level of the WKB approximation of the first order [1, 2]. Then t

t

∫

q 1 = A 0 exp – n ( t ) dt

k 1 /p ( t ) cos

0

∫ k ( t ) dt + α

+ Y ( t ),

(6)

0

where A0, α are determined by the initial conditions, and the partial solution Y(t) is defined as t

t

∫

∫

1   W ( u ) exp – n ( ξ ) dξ Y ( t ) =  p(t) p(u) 0

t

∫

sin p ( ξ ) dξ du.

u

(7)

u

At p = const, dependence (7) coincides with the Duhamel formula. In the vibratory system analyzed, the energy exchange with the external energy source of unlimited power occurs apart from the energy losses from overcoming the dissipative forces. Let us restrict ourselves to the analysis of free vibrations (Y(t) ≡ 0). In the practice of dynamic calcula tion of machines, free vibrations are often of interest from the standpoint of frequency analysis which is an important stage in the estimation of forced vibrations. In this case, it is taken into account that free vibrations formed due to the energy introduced into the system at the initial instant are damped fairly quickly and, consequently, do not actually affect the steadystate vibratory modes. Meanwhile, the vibra tional activity of cyclic machines mainly depends on the level of the socalled free accompanying vibra tions [1–5, 11] which, with complex laws of motion, gaps, and other perturbations of a pulsed nature, do not only damp, but also excite. The key sources of excitation of such vibrations in cyclic mechanisms are discontinuous or sharp variations in the derivatives of the function of position. Strictly speaking, these vibrations should be referred to the class of forced vibrations; however, from the standpoint of the fre quency spectrum, remoteness from resonances, and methodical considerations it is more convenient to consider them as free vibrations appearing at t = ti > 0. In this case, the corresponding “initial” conditions are determined when partial solution (7) is represented as a rapidly convergent series over the derivatives of the perturbation function W(t) [1, 2]. Let us single out the variable component of the vibrational amplitude which, according to Eq. (6), is described by the function t

∫

S = exp – ( n 0 + n 1 ) dt

k 1 /p ( t ).

(8)

0

According to Eq. (7), at dS/dt < 0 the free vibrations of the system at any t will be decreasing. In this case, according to Eq. (8) the following condition should be met [1, 2] n + p· / ( 2p ) > 0. (9) It is interesting that condition (9) can be obtained at arbitrary variations in p(t) based on the direct Lyapunov method that sets sufficient conditions of asymptotic stability. Actually, if we take the square of the amplitude of free vibrations as the Lyapunov function, we have U = q2 + ( q· 1 /p)2. Omitting the com putations, we can write as follows 2 d –2 2 U· = q· 1  ( p ) – 4n/p . dt

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VULFSON δ* 0.05

According to the second Lyapunov theorem, the sufficient condition for the asymptotic stability has the form U· < 0. One can easily ensure that when

μ2 = 0.2 5 1

0 −0.05

Eq. (10) is taken into account this condition coin cides with condition (9). This means that for esti mating the amplitudes at slow variation in the parameters at the level of the WKB approximation of the first order, condition (9) is not only sufficient, but also necessary.

0.5 1.0 1.5 2.0 2.5 3.0 ϕ0

Fig. 3.

Based on Eq. (10), for the analyzed model we obtain the following 2

μ Π ' Π '' ω 0 * * . δ 1 > δ 1* =  2 2 2k 1 1 + μ Π '

(11)

*

Bearing in mind that the parameter δ1 is proportional to the scattering coefficient ψ1, this condition points to the fact that the variability of the system parameters does not only change the intensity of damp ing of free vibrations, but can also lead to “negative” damping, i.e., to substantial qualitative changes in the vibratory process. It follows from Eq. (11) that the amplitude increase should be expected at the brak ing sections when the kinetic power proportional to the Π ' Π '' product is negative. At the constant value * *

of the first transfer function Π ' = const, we have Π'' = 0 and δ 1* > 0 that, as expected, corresponds to * damped free vibrations during the entire kinematic cycle. To estimate the energy variation intensity let us make use of the function 2

2

E = p ( t )S ( t ),

(12)

to which the amplitude values of the energy are proportional. Based on Eqs. (8) and (12), the condition dE/dt < 0 yields the following result: n – p· / ( 2p ) > 0.

(13)

According to Eqs. (5) and (13), the condition which is analogous to relation (11) takes on the form 2

3μ Π ' Π '' ω 0 * * . δ > δ * =  2 2 2k 1 1 + μ Π '

(14)

*

Figure 3 shows the family of critical values of the δ∗ (ϕ 0, μ2) parameter. The energy of vibratory systems increases in the process of acceleration ( Π ' Π '' > 0) and decreases during deceleration ( Π ' Π '' < 0). As μ * * * * approaches unity, the intensity of vibrational energy variations decreases. Let us now return to the analysis of the initial model with two degrees of freedom (Eqs. (2)). As a rule, the parameter n in problems of the dynamics of mechanisms negligibly affects the “natural” frequency p and, at the same time, substantially influences the vibration frequencies. The analogous situation is iden tified when analyzing the systems with variable parameters which are reflected in models with many degrees of freedom. Then, with discretely specified parameters after linearization in the vicinity of the programmed motion, the frequency and modal analysis is based on the set of homogeneous differential equations a ( t )q·· + c ( t )q = 0,

(15)

where a(t), c(t) are the square matrices of inertial and quasielastic coefficients, and q is the vector function of the generalized coordinates. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY

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ENERGY TRANSFER IN VIBRATORY SYSTEMS OF DRIVES p1, p2

(a)

(b)

40

40

20

20

π

0

265

π

2π 0

2π ϕ0

Fig. 4.

It was shown in [12] that the kinetic and potential energies are determined (accurate within the mag nitudes of the first order) by dependences H

T = 0.5

∑

H

2 a *r η· r ,

V = 0.5

r=1

∑ c* η , r

2 r

r=1

where ηr are quasinormal coordinates; H is the number of degrees of freedom of the oscillatory system. The variable “natural” frequencies in the first approximation can be defined on the basis of formal fre quency equation, in which time plays a parameter role: 2

det ( c ij ( t ) – a ij p ( t ) ) = 0.

(16)

In quasinormal coordinates, the set of equations takes on the form a *r ( t )η·· r + [ b r ( t ) + a· *r ( t ) ]η· r + c *r ( t )η r = M r ( t ),

r = 1, H ,

(17)

H

where Mr(t) =

∑α

ir Q * ir ,

Q *ir is the nonconservative generalized force from external loads and kinematic

i=1

excitation; br(t) is the coefficient of the equivalent linear resistance reduced to the form r; H

a *r =

H

∑∑α

H

ir α jr a ij ,

c *r =

i = 1j = 1

H

∑∑α

ir α jr c ij ;

i = 1j = 1

where αij are the nonstationary formfactors. In matrix form we have T

α aα = diag { a *1 , …, a *H },

T

α cα = diag { c *1 , …, c *H }.

In the given method for determining quasinormal coordinates, the forms of vibrations (based upon physical prerequisites) are assumed to be slowly varying functions. As for the rest, their determination does not differ from the analogous procedure with constant parameters. Note that, as distinct from the tradi tional frequency indexing when a higher frequency corresponds to a larger index, the indexing at which kr = lim p r is more preferable in this case. Then, at the dwell of the output unit when the system degen Π' → 0

erates into two uncoupled vibratory circuits the frequency pr is equal to the partial frequency with the same index. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY

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VULFSON q1, q2

(a)

(b)

ϕ0

Fig. 5.

E

(b)

E

(a)

1

1

2

2 0

1

2

E'

3 (c)

4

5

6

π/2

0

1

2

3

4

5

6 ϕ0

π ϕ0

Fig. 6.

According to Eqs. (16), (17), the slowly varying “natural” frequencies for the analyzed model are the roots of the following biquadratic equation: 4

2

2

2

2

2

2 2

p – { k 1 + k 2 [ 1 + μ Π' ( ϕ 0 ) ] }p + k 1 k 2 = 0.

(18)

The free term in Eq. (18) does not depend on time and, thus, based on one of the Viet theorems p1(t)p2(t) = const. This means that the minimum of one frequency corresponds to the maximum of the other. Figure 4 shows typical plots of frequency variations p1(ϕ0), p2(ϕ0) for two combinations of partial fre quencies: k1 = 20, k2 = 30 (mode 1, Fig. 4a), and k1 = k2 = 30 (mode 2, Fig. 4b). Here and hereinafter, dimensionless (normalized) frequencies that correspond to the dimensionless time ϕ0 = ω0t are used. In this case, a threeperiod structure of the motion law was used in the forward trace (speedingup, the sec tion of uniform motion, runningout) when accelerations varied following the “modified trapezium” law; with the return trace the harmonic law of acceleration variations was used [2]. Note that at similar partial frequencies there appear regions with an increased density of the frequency spectrum where the intensity of the energy exchange between vibration forms drastically increases. For the fixed value f = μΠ' < 1, the frequencies are the closest at κ = k2/k1 =

2

1 – f /(1 + f 2). At this value of f it follows that p2/p1 =

( 1 – f )/ ( 1 + f ) . JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY

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Suppose when ϕ0 = 0 the subsystem of the output unit receives the pulsed perturbation ( q· 2 (0) ≠ 0). Fig ure 5 (a is mode 1, b is mode 2) shows the dependencies for q1(t) (solid line) and q2(t) (dashed line) plotted on the basis of the computer simulation of the set of equations (2). For clarity, the plots the dissipative terms in set (2) are omitted. At Π'(0) = 0, the subsystems are not interrelated and, thus, the initial energy store completely “belongs” to subsystem 2 (the q2 coordinate). At Π'(ϕ) ≠ 0, the energy is redistributed between both subsystems. When the partial frequencies are equal (k1 = k2) the initial level of amplitudes is restored almost completely at the end of the kinematic cycle. Figure 6a plots the variations of the normalized total energy E in both subsystems (curve 1) and the ini tial energy E(0) (line 2) for mode 2. On the whole, the operation of gyroscopic forces arising with variable parameters is almost zero when the vibrations are close to harmonic, and a nearly complete exchange of the vibratory energy between the subsystems is observed. As distinct from the case considered in [9] (Fig. 1), the cyclic system plays the function of a “trap” only partially, namely within the kinematic cycle. This is observed most clearly at the sections of uniform motion and partially at intermediate dwell. The plots of variations in the total energy in Figs. 6b and 6c correspond to the harmonic law of motion that is widely used in engineering in which Π'(ϕ0) = r0 sin ϕ 0. At similar partial frequencies (Fig. 6b, mode 3: p1 = 23, p2 = 27, μ2 = J2/J1 = 1) the initial energy level is not restored within one kinematic cycle. In this case, the energy accumulates over a relatively long time and yields a substantial increase in the vibrational activity of the entire system. The analysis showed that this is related to formation of the beat mode when the work of gyroscopic forces vanishes only after the beat period (which often exceeds the period of the kinematic cycle τ = 2π/ω0 by one order of magnitude) is over. Figure 6c plots the E ' = dE/dϕ0 function which characterizes the intensity of energy variations for mode 4 (p1 = 10, p2 = 30, μ2 = J2/J1 = 0.2). This mode is interesting since, in fact, the situation considered above for the model with one degree of freedom (Fig. 3) which corresponds to the limiting case when 2 2 p 2 /p 1 Ⰷ 1 is repeated. The comparison of the E' plot with that shown in Fig. 3 shows that the plot of δ∗(ϕ0, μ) obtained at the limiting transition serves as an analogue for the envelope for highfrequency vibrations that were identified when the elasticity of the output unit was taken into account. This study shows that the energy accumulated in the vibratory system in cyclic machines is mainly determined by the work performed by the external source upon implementation of nonstationary links that manifests itself most clearly in local violations of stability conditions at certain sections of the kine matic cycle. According to Eq. (14), these conditions can be restored by increasing the dissipative forces. The analysis showed that when dissipation is taken into account one of the subsystems (at relatively high scattering coefficients) can serve as an effective means for decreasing the level of vibrations in the second subsystem. The distribution of the energy that was transferred between the subsystems mainly depends on the ratio of partial frequencies. In cyclic systems with a circular structure, the role of the distribution of the energy between subsystems is even greater since the energy factors substantially affect the transformation of nonstationary forms and spatial localization of vibrations [3, 4]. However, this problem requires separate consideration. REFERENCES 1. Vul’fson, I.I., Kolebaniya mashin s mekhanizmami tsiklovogo deistviya (Oscillations for Machines with Cycle Mechanisms), Leningrad: Mashinostroenie, 1990. 2. Vul’fson, I.I., Dinamicheskie raschety tsiklovykh mekhanizmov (Dynamic Calculations for Cycle Mechanisms), Leningrad: Mashinostroenie, 1976. 3. Vul’fson, I.I., Phase synchronism and space localization of vibrations of cyclic machine tips with symmetrical dynamic structure, J. Mach. Manuf. Reliab., 2011, vol. 40, no. 1, p. 9. 4. Vul’fson, I.I. and Preobrazhenskaya, M.V., Investigation of vibration modes excited upon reversal in gaps of cyclic mechanisms connected with a common actuator, J. Mach. Manuf. Reliab., 2008, vol. 37, no. 1, p. 28. 5. Vul’fson, I.I., Kolebaniya v mashinakh (Oscillations in Machines), St. Petersburg: St. Petersburg Gos. Univ. Technol. Design, 2008. 6. Indeitsev, D.A., Kuznetsov, N.G., Motygin, O.V., et al., Lokalizatsiya lineinykh voln (Linear Waves Localiza tion), St. Petersburg: Izd. St. Petersburg. Univ., 2007. JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY

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7. Manevich, L.I., Mikhlin, D.V., and Pilipchuk, V.N., Metod normal’nykh kolebanii dlya sushchestvenno neli neinykh sistem (Normal Oscillations Method for Strongly Nonlinear Systems), Moscow: Nauka, 1989. 8. Brillouin, L. and Parodi, M., Wave Propagation in Periodic Structures, New York: Dover Publ., 1953. 9. Kovaleva, A., Manevitch, L., and Kosevich, Yu., Fresnel integrals and irreversible energy transfer in an oscilla tory system with timedependent parameters, Phys. Rev. E, 2011, vol. 83, pp. 0266021–12. 10. Kosevich, Yu.A., Manevich, L.I., and Manevich, E.L., Oscillation analog of nonadiabatic LandauZener tun neling and possibility for creating the new type of power traps, Usp. Fiz. Nauk, 2010, vol. 180, no. 12, pp. 1331– 1334. 11. Babakov, I.M., Teoriya kolebanii (Oscillation Theory), Moscow: Nauka, 1965. 12. Mitropol’skii, Yu.A., Problemy asimptoticheskoi teorii nestatsionarnykh kolebanii (Problems in Asymptotic The ory of Nonstationary Oscillations), Moscow: Nauka, 1964.

Translated by L. Borodina

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