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Slow motions in systems with fast modulated excitation
Journal of Sound and Vibration 383 (2016) 295–308
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Slow motions in systems with fast modulated excitation E. Kremer a,b,n a b
Institute of Problems of Mechanical Engineering, V.O. Bolshoj pr. 61, St. Petersburg 199178, Russia LuK GmbH & Co.KG, Industriestrasse 3, 77815 Buehl/Baden, Germany
a r t i c l e i n f o
abstract
Article history: Received 23 October 2015 Received in revised form 26 June 2016 Accepted 4 July 2016 Handling Editor: M.P. Cartmell Available online 3 August 2016
It is well known that high-frequency excitation can modify the behavior of systems with respect to slow motions. The goal of this study is consideration of these effects in a rather general case of analytical systems with modulated sinusoidal excitation. The method of direct separation of motions proposed by I.I. Blekhman was applied in a modified form with the explicit introduction of a small parameter. Equations for the slow motions are obtained and an analysis of how they depend on the structure of the original equations is performed. Five basic effects corresponding to different possible dependencies of the modulation amplitude on position, velocity, and slow time are selected (some of them for the first time). These effects offer a possibility for designing a high-frequency control of the slow motions with specified properties. For example, high-frequency excitation in a system with a nonlinear friction can essentially increase the effective damping. The results are also of significance for system identification and diagnostics. Analysis of a hydraulic valve is given as an example of application. & 2016 Elsevier Ltd. All rights reserved.
Keywords: High-frequency excitation Vibrational mechanics Strobodynamics Vibration-transformed forces Direct separation of motions Modulation Vibrational control Hydraulic valves
1. Introduction 1.1. Vibrational mechanics The concept of vibrational mechanics and the method of direct separation of motions were proposed by I.I. Blekhman [1] and systematically presented in books written by him [2,3]. This approach was further developed and applied to many practical problems by Fidlin [4–6], Malakhova [7], Thomsen [8–10], Sorokin [11], Sperling [12], and others. The main idea consists of the following steps. Let a dynamical system be described as 2 d x dx ; t; ωt (1) ¼ Θ x; dt dt 2 The system must not be a one-mass mechanical oscillator because Eq. (1) is valid for many physical processes. If that is the case, the parameters are assumed to have been normalized so that the mass is equal to 1 unit. In any case, the function Θ is interpreted as a force and can be presented as a sum of slow and fast components F and Φ dx dx (2) Θ ¼ F x; ; t þΦ x; ; t; ωt dt dt
n
Corresponding author at: LuK GmbH & Co.KG, Industriestrasse 3, 77815 Buehl/Baden, Germany. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.jsv.2016.07.006 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
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Fig. 1. Character of the solution (solid line – full system, dashed line – averaged system). The units of the axes x and t are here meters and seconds.
A solution can be obtained as a superposition of the time-dependent mean value X and the fast oscillation ψ: x ¼ XðtÞ þ ψðt; ωtÞ; ⟨ψ⟩ ¼ 0:
(3)
The function ψ is 2π-periodical and has a mean value equal to 0 with respect to the fast time θ¼ωt. The following is an averaging operation: Z 2π 1 f ðt; θÞdθ: (4) ⟨f ⟩ ¼ 2π 0 Fig. 1 illustrates the character of the solution. The method of direct separation of motions [2] introduces two integral-differential equations for X and ψ as follows: 2 d X dX dψ þ ; t; ωt (5) ¼ Θ X þ ψ; dt dt dt 2 2 d ψ dX dψ dX dψ þ ; t; ωt Θ X þ ψ; þ ; t; ωt ¼ Θ X þ ψ; dt dt dt dt dt 2
(6)
Eq. (5) is the result of the direct averaging and Eq. (6) is the complement of Eq. (5) to the original Eq. (1). The aim of further consideration is to obtain an equation for X, which does not include ψ and, therefore, enables the calculation the slow motion without solution of the full system. In order to achieve it, one should obtain ψ through X from Eq. (6) and substitute it into Eq. (5). This leads to an equation for the averaged motion, which is called the main equation of vibrational mechanics (also the equation of strobodynamics [13] or the vibration-transformed equation [2]): 2 d X dX ; t þ V: (7) ¼ F X; dt dt 2 The additional slow term V in this equation for the averaged motion is the so-called vibrational force, which takes into account the influence of now hidden fast motions. The vibrational force V can depend on the slow variable X, their derivatives with respect to the slow time t, and the slow time t explicit. There are many known nontrivial and unexpected physical effects caused by vibrational forces. Well-known examples are Chelomei's pendulum, the Stephenson–Kapitza pendulum, the Indian rope, vibrational transportation, and many other phenomena presented in the books written by Blekhman [2,3]. Thomsen [8,9] selected three effects, which are characteristics of many different systems with fast excitation – stiffening, biasing, and smoothening. These approaches to systematic effects were developed by him, later together with Fidlin [5] for strong excitation, especially in systems with friction.
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The objective of this study is to analyze vibrational effects with a focus not on the properties of vibrational forces that comprise the vibrations within the structures or machines in which they appear, but on the causes of them. The term effect is used here to define a possible combination of variables, which lead to the appearance of vibrational forces and, therefore, to nontrivial strobodynamical phenomena. It will be shown that there are five basic effects, corresponding to different possible dependencies of the modulation amplitude on position, velocity, and slow time. The subject of this investigation is an analytical system with relative weak modulated sinusoidal excitation in a smooth system. This case is used as a minimal generic model, which provides all the basic effects already in the first approach. The class of systems with modulated excitation is not only of theoretical interest. A well known parametrically excited system is the pendulum with the vibrating pivot (Stephenson–Kapitza pendulum [3,14,15]) where the modulation depends on position x. The case of velocity-depending modulation was considered in Ref. [4] for the first time as a pure mathematical example for a demonstration of the developed technique. Interesting physical and technical applications for this system were suggested later in Ref. [11]. An example of a technical application for a more general case of modulated excitation – a model of a hydraulic valve – is considered in Section 8. 1.2. Investigation technique The method of direct separation of motions is modified in comparison with its original version. [2,3] This modification concerns solving the equation of the fast motion (Eq. (6)). Of particular concern is that the original variant of the method with the frozen x and x_ in Eq. (6) provides correct results in the great majority of practical cases, but sometimes it needs more accurate consideration similar to the one made in Ref. [11] for a system with high-frequency modulation of the dissipation coefficient. In this case, an effect of the slow time on the fast motion was taken into account, but it was not clear in advance whether a more accurate consideration of this form would have been needed. The modification of the method proposed in this study consists of the explicit introduction of a small parameter in Eqs. (5) and (6) that enables automatic analysis of comparable importance of different terms. It makes the method of direct separation of motion more strict and robust. The corresponding analytical results are approximate in nature, because they correspond to the first approach to the solution. Therefore, it was important to verify them through a comparison with numerical calculations for the full system. This verification was fulfilled and is presented here.
2. The problem of modulated excitation The problem of modulated excitation is described by the following equation: 2 d x dx dx ¼ F x; ; τ þB x; ; τ sin ðθÞ; 2 dt dt dt
(8)
where τ ¼ t, θ ¼ ωt, ωc 1. This equation corresponds to Eq. (1) where the fast excitation is a sinusoid with an amplitude depending on the position x, velocity dx , and slow time τ. Let x be presented as a sum of slow and fast motions in accordance dt with Eq. (3). The problem consists in obtaining V in Eq. (7) for slow motion. The questions are: how does the vibrational force V depend on position, velocity, and slow time in general and how does the structure of the functions F and Φ affect it?
3. Equation for the slow motion Frequency ω is considered as being large and the corresponding value 1/ω as a small parameter. In this context, the order of magnitude of different terms can be estimated. If both F and B are of the order 1, the solution for the fast motion has the order 1/ω2. For this reason, it is convenient to introduce a new variable ξðτ; θÞ ¼ ψðτ; θÞω2 , where ξðτ; θÞ has the order 1. The full derivatives of every function f, which is to be differentiated with respect to t, can be calculated as 2
df _ 0 d f 0 00 ¼ f þωf ; 2 ¼ f€ þ2ωf_ þ ω2 f ; dt dt
(9)
∂f where f_ ¼ ∂f ∂τ and f ¼ ∂θ are partial derivatives with respect to slow and fast times. These derivatives are introduced here in a manner similar to that of the method of multiple scales [16] to provide here an explicit estimation of the orders for different terms. The equations of the method of direct separation of motions (Eqs. (5) and (6)) can be rewritten in terms of the order 1/ω2 in the following form: * + 2 ⟨ξ02 ⟩ ∂2 F 1 ∂Φ 1 _ ∂Φ 1 0 ∂Φ 1 02 ∂ Φ € þ 2 ξ ξ þ ξ X ¼Fþ 2 (10) þ þ 2 ξ 2 ω 2ω ∂X_ 2 ω2 ∂X ω 2ω ∂X_ ∂X_ ∂X_ 0
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0 ∂2 F 1 2 1 ξ ∂F ξ 1 ∂Φ ∂Φ 1 _ ∂Φ ξ_ ∂F _ ∂Φ ξ þ ξ þ 2 ξ00 þ ξ_0 þ 2 ξ€ ¼ Φ þ 2 þ þ ξ þ 2 ξ02 ⟨ξ02 ⟩ ξ 2 ω ∂X ω ω ∂X_ 2ω ω ω ∂X ω2 ∂X ω2 ∂X_ ∂X_ ∂X_ * +! 1 0 ∂Φ ∂Φ 1 ∂2 Φ ∂2 Φ ξ ξ0 þ 2 ξ02 2 ξ02 2 þ _ _ ω 2ω _ ∂X ∂X ∂X ∂X_
(11)
where F¼F(X, X_ ,τ) and Φ ¼B(X, X_ ,τ)sin(θ). In accordance with the method of direct separation of motions, the averaged motion X and its derivatives are considered initially as given and Eq. (11) is solved with respect to ξ. This solution is obtained asymptotically as ξ ¼ ξ0 þξ1/ω. The procedure consists of balancing the terms of the same order and integrating the corresponding equations related to θ. The integration constants can be chosen such that ξ0 and ξ1 are equal to 0. In this case, the appropriate particular solution is 1 ∂F 1 ∂B ξ ¼ B sin ðθÞ þ 2B_ cos ðθÞ þ B cos ðθÞ B sin ð2θÞ (12) ω 8 ∂X_ ∂X_ with ∂B ∂B ∂B þ : B_ ¼ X_ þ X€ ∂X ∂X_ ∂τ
(13)
The obtained expression for ξ is substituted in Eq. (10). It leads to the following result: X€ ¼ F þ V 2
_
B ∂ :F B ∂B ∂F ∂B B ∂B V ¼ 4ω 2ωB 2 ∂X þ 2ω 2 2 _ 2 2ω2 _ ∂X ∂X_ ∂X 2
(14)
∂X
_ This phenomenon The vibrational force V here depends on “slow” acceleration X€ if the modulation depends on velocity x. of the added mass is known in the context of vibrational mechanics from Ref. [11], where a linear system with velocity proportional modulation was considered. This problem was previously solved in Ref. [4] with another asymptotic technique. Solving Eq. (14) with respect to X€ correct to order 1/ω2 leads to the following final equation similar to Eq. (7): X€ ¼ F þ V ef f V ef f ¼
F ∂B 2 B2 ∂2 F B ∂B ∂F B ∂B þ 2 2 2 2 2ω ∂X_ 2ω ∂X_ ∂X_ 2ω2 ∂X 4ω ∂X_ þ
1 ∂B ∂B X_ ∂B ∂B þ : 2ω2 ∂X_ ∂X 2ω2 ∂X_ ∂τ
(15)
This equation allows the description of the averaged motion X without solving the full original Eq. (8) and without calculating the fast component ξ from Eq. (11). The vibrational force Veff as a function of X, X_ and τ is determined here explicitly from the expressions for the original forces F and B, and it can be directly calculated in every concrete applied problem. In the next section, the mechanism of the structure of the forces F and B affecting the structure of the vibrational force Veff is analyzed.
4. Basic strobodynamical effects 4.1. General approach The solution for the vibrational force (Eq. (15)) can be refined by taking into account further terms of the Taylor series. However, it is more important to note that this expression enables us to select the main effects of the excitation on the slow motion. The effects occurring already in this first approach can be called basic effects. There are five different basic effects, which correspond to the different terms of the vibrational force V eff . These effects can be classified depending upon the modulation structure as follows:
Arbitrary excitation (even with constant amplitude) in a system with nonlinearity relative to velocity Excitation dependent on position only (x -modulation) _ Excitation dependent on velocity only (x-modulation) Excitation dependent on velocity and position ðx_ x modulationÞ Excitation dependent on velocity and slow time ðx_ τ modulationÞ These effects are now considered in more detail.
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Fig. 2. Cubic dissipative force and excitation with constant amplitude generate an effective linear friction: (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), B¼ 50, F¼ 25x 3x_ 3, ω ¼60. The units of the axes x and t are here meters and seconds.
Fig. 3. Cubic dissipative force and excitation with constant amplitude stabilize a system with additional negative dissipation (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), B¼ 50, F ¼ 25x þ 0.5x_ 3x_ 3, ω¼ 60. The units of the axes x and t are here meters and seconds.
4.2. A system with nonlinearity relative to velocity under arbitrary excitation The first effect is a combination of the arbitrary excitation (even with constant amplitude) with nonlinearity of the unexcited system relative to velocity. In this case, the equation for the vibrational force (Eq. (15)) gives V eff ¼ V 1 ¼
B2 ∂2 F : 4ω2 ∂X_ 2
(16)
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Let the force in the unexcited system be a sum of two forces. The first of them depends on position and slow time and the second force is a nonlinear friction with velocity to the power of m: F ¼ F p ðX; τÞ βX_ m signðX_ Þ: (17) Then, the vibrational force in accordance with Eq. (16) is a pure dissipative force with velocity to the power of m 2: V1 ¼
βB2 mðm 1Þ _ m 2 X signðX_ Þ: 4ω2
(18)
For example, a cubic dissipative force generates an additional linear friction. This is illustrated in the numerical examples presented in Fig. 2 for F ¼ 25x 3x_ 3 and in Fig. 3 for F ¼ 25x þ0:5x_ 3x_ 3 . In both examples, B¼ 50 and ω¼60. The units of the variables x and t are assumed for definiteness being meters and seconds, but they could be also scaled or normalized in some way. The solid, dotted, and dashed lines are used here and further, respectively, for the original system (Eq. (8)) with excitation, for the same system without excitation (B¼0) and the vibration-transformed (averaged) system (Eq. (15)). The difference between the data of Figs. 2 and 3 is the additional term 0.5x_ in F for Fig. 3. It corresponds to some positive linear damping causing low-frequency self-exited oscillations in the system without high-frequency excitation. A significant increase of damping due to excitation is observed in both examples. In the second of them, the linear friction induced by the excitation is capable of stabilizing the system, which without any excitation executes low-frequency self-exited oscillation. The plots show a good agreement between the analytical and numerical results. It is valid as well for all presented examples although B is not of the order 1, as assumed above, but typically larger. The explanation of this is given in Section 5. 4.3. Excitation depends on position only (x-modulation) The next effect is the well-known transformation of the position-dependent modulation in an additional conservative vibrational force: V eff ¼ V 2 ¼ V 1
B ∂B 2ω2 ∂X
(19)
with additional potential energy D ¼ B2 =ð4ω2 Þ
(20)
This effect was described firstly for the pendulum with a vibrating pivot by Stephenson [14] and Kapitza [15] and later generalized by Blekhman. [2] For this pendulum (Fig. 4), additional potential energy is calculated in accordance with Eq. (20) as D ¼(Aω sin(X))2/(4ω2). It leads to the known results for this system [2], for example, to the condition of the upper position pffiffiffiffiffiffiffi stability: Aω 4 2gl. The method of direct separation of motions in its original form [1] with a frozen x in Eq. (6) can be used in the case of x-modulation directly.
Fig. 4. Pendulum of Stephenson–Kapitza.
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_ Fig. 5. Example of x-modulation: stiffening is ever-positive (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), _ F ¼ x 0.2x, _ ω¼ 40. The units of the axes x and t are here meters and seconds. B¼ 35x,
_ Fig. 6. Example of x x-modulation: positive stiffening (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), _ ω¼ 40. The units of the axes x and t are here meters and seconds. B¼ 35x_ 10x, F¼ x 0.2x,
_ 4.4. Excitation depends on velocity only (x-modulation) _ The next case is x-modulation. The corresponding expression for the vibrational force in accordance with Eq. (15) is as follows: F ∂B 2 B ∂B ∂F V ef f ¼ V 3 ¼ V 1 þ 2 2 : 2ω ∂X_ 2ω ∂X_ ∂X_
(21)
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Let the force in the unexcited system be described by Eq. (17) and B is a linear function of X_ : B ¼ αX_ . Then, Eq. (21) is transformed to V3 ¼
α2 βmðm 1Þ _ m jX j signðX_ ÞÞ: ðF p ðX; τÞ 2 2ω2
(22)
It shows that the parts of the unexcited system, depending on position and time, as well as its dissipative part are involved in vibrational force linear with different coefficients. In the case of m¼ 1, the vibrational force is proportional only to the positional part of the original slow force Fp with some coefficient b: V3 ¼bFp(X,τ). It can be characterized as the _ phenomenon of stiffening [8], and it coincides with the results of Refs. [4,11] for a full linear system with x-modulation. The coefficient b is equal to α2/(2ω2) and therefore always positive. A numerical verification of this result for the system with F ¼ _ B ¼ 35x, _ and ω¼40 is presented in Fig. 5. It is evident from the figure that the vibration-transformed system x 0.2x, (dashed line) is really an average of the excited system (solid line) and has a larger eigenfrequency as the unexcited system. For the case of m ¼2, a phenomenon of mass reduction takes place – the vibrational force is proportional to the full slow force: V3 ¼ bF(X, X_ , τ). _ 4.5. Excitation depends on velocity and position (x x-modulation) _ If the excitation amplitude depends on velocity and position (x x-modulation), the vibrational force has the form V eff ¼ V 4 ¼ V 1 þ V 2 þ V 3 þ
X_ ∂B ∂B : 2ω2 ∂X_ ∂X
(23)
Let us consider Eq. (17) with m ¼1 and B ¼ α1X þ α2X_ . The vibrational force has the form V4 ¼bFp(X, τ) as in the case of
_x-modulation, but with coefficient calculated as b ¼ α21 α22 þβα1 α2 ð2ω2 Þ. It indicates that b40 if βoβcr and b o 0 if 2 β4βcr, where βcr ¼ α1 þα22 =ðα1 α2 Þ. The cases of b40 and bo0 can be characterized as positive and negative stiffening, respectively. Corresponding numerical verifications are presented in Figs. 6 and 7 for F¼ x 0.2x_ and F¼ x 5x_ correspondingly. In both examples B ¼ 35x_ 10x and ω¼40. 4.6. Excitation depends on velocity and slow time (x_ τ-modulation) The last effect is the x_ τ-modulation. In this case, the general expression for the vibrational force has the following form: V eff ¼ V 5 ¼ V 3 þ
1 ∂B ∂B : 2ω2 ∂X_ ∂τ
(24)
_ Fig. 7. Example of x x-modulation: negative stiffening (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), _ ω¼ 40. The units of the axes x and t are here meters and seconds. B ¼ 35x_ 10x, F¼ x 5x,
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Fig. 8. Phenomenon of low-frequency oscillations as example of x_ τ -modulation (solid line – without excitation, dotted line – with excitation, dashed line _ ω¼ 50. The units of the axes x and t are here meters and seconds. – averaged system), B¼ 50x_ 50 sin(2t), F¼ 4x 1.2x,
Fig. 9. Phenomenon of low-frequency oscillation disappears without dependence on velocity in the excitation (solid line – without excitation, dotted line – _ ω¼ 50. The units of the axes x and t are here meters and seconds. with excitation, dashed line – averaged system), B¼ 50sin(2t), F¼ 4x 1.2x,
Two phenomena within the scope of this effect will be considered as examples. The first of them occurs if the excitation amplitude is a sinusoidal function of slow time plus a linear function of velocity: B ¼as sin(τΩ) þ av X_ , and the function F has the form Eq. (17) with m ¼1. Then, the vibrational force V5 is calculated as V5 ¼
a2v av as F p þ 2 ðβ sin ðτΩÞ þΩ cos ðτΩÞÞ 2ω2 2ω
(25)
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Fig. 10. Phenomenon of drifting as an example of x_ τ-modulation (solid line – without excitation, dotted line – with excitation, dashed line – averaged _ ω¼ 50. The units of the axes x and t are here meters and seconds. system), B¼ 15x_ 2t, F¼ x 1.2x,
Fig. 11. Phenomenon of drifting disappears without dependence on velocity in the excitation (solid line – without excitation, dotted line – with excitation, _ ω¼ 50. The units of the axes x and t are here meters and seconds. dashed line – averaged system), B¼ 2t, F¼ x 1.2x,
It includes low-frequency excitation. This phenomenon of low-frequency oscillation disappears if dependence on velocity in modulation vanishes. This is illustrated by two numerical examples in Figs. 8 and 9 for B ¼ 50x_ 50 sin(2t) and B ¼ 50 sin(2t). In both examples, F ¼ 4x 1.2x_ and ω¼50. Another example of the x_ τ-modulation is a linear function of
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time and velocity: B ¼ at τ þ av X_ . If the function F has again the form Eq. (17) with m ¼1, the vibrational force is calculated as V5 ¼
a2v av at F p þ 2 ðβτ þ 1Þ 2ω2 2ω
(26)
It is apparent from Eq. (26) that vibrational force increases linearly with slow time τ. The corresponding increase of the averaged variable X under acting of excitation with an increasing amplitude can be called vibration drifting. It should be emphasized that this phenomenon exists only in the case of combined dependencies of modulation on x_ and τ. In this case, a phenomenon of vibration drifting takes place. It is illustrated in Figs. 10 and 11 for B ¼ 15x_ 2t and B ¼ 2t. In both examples, F ¼ x 1.2x_ and ω¼50. In the first case, the averaged x increases. In the second case, it is equal to 0.
5. Generalizations The presented theory enables some generalizations. 5.1. Stronger excitation The first generalization is a modest stronger excitation. Let the excitation amplitude have now the magnitude of order ωλ: 2
d x ¼ F þ ωλ B sin ðθÞ dt 2
(27)
If λ o 1, then the following expression for the vibrational force is valid in the first approach V eff ðλÞ ¼ ω2λ V eff ð0Þ
(28)
It explains the results in the numerical examples above, which are valid for a stronger excitation as it can be expected from the initial assumption that F is of the same magnitude order as B. 5.2. Multiharmonic excitation Another generalization is a multiharmonic excitation: 2 N X d x ¼Fþ BðksÞ sin ðkθÞ þBðkcÞ cos ðkθÞ 2 dt k¼0
(29)
In this case, the effects of different harmonics are additive due to their orthogonality, and the vibrational force is calculated as V eff ¼
N . X 2 k V ðsÞ þ V kðsÞ k
(30)
k¼0
The parameters V kðsÞ and V ðcÞ represent the vibrational forces corresponding to the excitations with amplitudes BkðsÞ and BðcÞ , k k respectively, in accordance with Eq. (15). 5.3. N-dimensional case The presented results can be directly generalized to the case of vectorial x, if a system consists of equations of second order.
Fig. 12. Magnet valve (1 – domain of the adjusted pressure p, 2 – piston, 3 – changing gap, 4 – pump, 5 – sump, 6 – solenoid, 7 – the area of the pressure action on the piston).
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6. Example of technical application An example of technical application of the developed theory in the dynamics of hydraulic valves is presented in this section [17,18]. A typical magnet proportional valve is depicted in Fig. 12. The purpose of this valve is to adjust pressure in domain 1. It is achieved due to a motion of piston 2, changing gap 3, and thus the hydraulic resistance between pump 4 to sump 5. The following forces act on the piston:
Control magnet force Fc of solenoid 6. _ , Viscous friction having two components – with linear and square dependence on velocity: Fc ¼ Dx_ with D ¼ β þ kxj where β and k are coefficients.
Pressure force pA, where p is the adjusted pressure in domain 1 and A is the area 7 of the pressure action on the piston. The area A depends generally on the piston position x: A ¼A(x).
Force cx of the spring 8.
A rather typical case of a very light piston is considered here. Therefore, inertia forces are neglected in comparison to viscous friction even for relative high excitation frequency. In this case, the equation of motion has the form Dx_ þ cx ¼ Ap F c :
(31)
Another equation expresses the volume balance in domain 1 and has the following form: _ ch p_ ¼ Q 0 Q Ax;
(32)
where ch is the hydraulic capacity of the domain 1, Q is the flow rate through gap 3, and Q0 is the pump flow. The flow rate Q depends on position x and pressure p according to the following known hydraulic formula [17]: pffiffiffi Q ¼ γAB ðxÞ p (33) where γ is a constant and AB(x) is the cross-sectional area of the gap, depending on the piston position x. The last term in Eq. (32) represents the pump effect of the piston – the flow rate of the fluid due to the piston motion. Differentiating Eq. (31) with respect to t and following substitution of x_ from Eq. (31) leads to an equation of second order for the piston position x: _ x€ þcx_ ¼ ðβ þkjxjÞ
pffiffiffi A dA _ F_ c ; _ þ xp ðQ γAB p AxÞ ch 0 dx
(34)
where pressure p is calculated through x and x_ in accordance with Eq. (31): 1 p ¼ ðF c þ βx_ þcxÞ: A
(35)
A typical magnetic force Fc in such valves is a sum of a slow component and a fast-oscillating component F0(t) with frequency ω and slowly changing amplitude F1(t): F c ¼ F 0 ðtÞ þ F 1 ðtÞ cos ðωtÞ:
(36)
It is usually larger as the term βx_ þ cx in the expression for pressure (Eq. (35)), and one can expect that averaged pressure is approximately equal to p ¼ F 0 ðtÞ=A. This is really the case, but the oscillations of the magnet force lead to some departure from this nominal value. A practically important problem is to describe the effect of magnet force oscillations on averaged controlled pressure. The following are some assumptions:
The frequency of excitation is large relative to a characteristic frequency of the main control force F0 as well as of the excitation amplitude F1.
The excitation amplitude F1 is small relative to main control force F0. Then, Eq. (35) can be presented in the form of Eq. (8) with ! ! pffiffiffiffiffi dA dx A2 A 1 dx 1 _ _ _ _ F 0 þ βx_ þ cxÞBðx; ; t ¼ U 1 F 1 ωU ¼ ðβ þ kjxjÞ F x; ; t ¼ U cþ x þ ðQ 0 γAB p1 Þ þ xp1 F 0 p1 ¼ dt ch dx A dt ch (37) The system in the presented formulation is a type of x_ τ-modulation with a nonlinear dependence of the main slow force on velocity. The slow motion can be described by Eq. (7), where V ¼Veff is calculated directly with the help of Eq. (24). This expression takes the following form: 1 1 00 _ 2ksign X_ F 1 @@ 2 F 1 ∂2 F 2ksign X ∂F A F1 (38) V eff ¼ F_ 1 A 4k 2 þ þ 2 ∂X_ 2 U U ∂X_ 2U 2 U
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It is beyond the scope of this study to analyze in detail the technical consequences of the phenomenon. The following points should be still noted: 1. The developed theory enables analysis of the causes of the departure of the controlled variables from their nominal characteristics to improve valve design. 2. Here are two causes of vibrational forces. The first of them is a nonlinear dependence of the main slow force F on velocity due to a dependence of the acting area A on position x and the square component of the friction (k a 0). Another cause is the dependence of modulation B in Eq. (37) on time and velocity if the amplitude of the magnet force changes and the friction has a square component in its dependence on velocity (k a0). Increasing the amplitude of the magnet force can lead to an increase in the averaged pressure (vibration drift). 3. In the case of pure linear friction (k ¼0), the expression for vibrational force V ¼Veff has a particularly simple form: V eff ¼
F 21 ∂2 F : 4U 2 ∂X_ 2
(39)
7. Conclusions
A problem of vibrational mechanics (strobodynamics) for an analytical system with modulated high-frequency excitation is considered and for this case a general expression for vibrational force is obtained.
It is shown to be possible to analyze in general the mechanism of the structure of the original equation affecting the equation for slow motions. It is shown particularly that only five possible effects cause vibrational forces. These effects can be classified as follows: – – – – –
Arbitrary excitation (even with constant amplitude) in a system with nonlinearity relative to velocity Excitation dependent on position only (x-modulation) _ Excitation dependent on velocity only (x-modulation) _ Excitation dependent on velocity and position (x x-modulation) Excitation dependent on velocity and slow time (x_ τ-modulation)
With each of these five effects, some specific vibration force is connected, which causes, in turn, different physical phenomena. In particular the following are shown: – A combination of a nonlinear friction with an excitation of a constant amplitude can lead to essential increasing of damping in slow motion. It can be used to stabilize a system and to avoid self-oscillations. _ – The case of combined dependence of modulation on position and velocity (x x-modulation) causes a vibration force depending only on position. They can either increase or decrease an existing stiffness depending on damping, which can find application in different control systems. – Dependence of modulation simultaneously on slow time and velocity (x_ τ-modulation) can cause low-frequency oscillations and vibrational drifting of averaged variables.
The results are generalized for a stronger excitation (of the order ωλ with λ o 1) and multiharmonic excitation. The method of direct separation of motions is applied in a modified form with the explicit introduction of a small
parameter. This enables one to take into account the slow time in the equation of fast motion automatically wherever it is needed. The analytical results are verified by comparison with numerical simulation. The investigated effects are of interest not only from the theoretical point of view but at least for two practical fields:
– development of systems for high-frequency control of slow motions with predefined properties; – identification of systems and experimental modal analysis by taking into account effects of high-frequency excitations on low-frequency motions. Analysis of a hydraulic valve is given as an application example.
Acknowledgment The author is grateful to I.I. Blekhman for interesting discussion and useful advice. This study was supported by the Russian Scientific Foundation (Grant No. 14-19-01190).
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
I. Blekhman, Method of direct motion separation in problems of vibration acting on nonlinear mechanical systems, Mechanics of Solids 11 (1976) 7–19. I. Blekhman, Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications, World Scientific, Singapore, 2000. I. Blekhman, Selected Topics in Vibrational Mechanics, World Scientific, Singapore, 2004. A. Fidlin, On the separation of motions in systems with a large fast excitation of general form, European Journal of Mechanics A/Solids 18 (1999) 527–539. A. Fidlin, J. Thomsen, Non-trivial effects of high-frequency excitation for strongly damped mechanical systems, International Journal of Non-Linear Mechanics 43 (2008) 569–578. A. Fidlin, Nonlinear Oscillations in Mechanical Engineering, Springer, Berlin, Heidelberg, New York, 2006. I. Blekhman, O. Malakhova, Quasi-equilibrium positions of the chelomei pendulum, Soviet Physics Doklady 31 (1986) 229–231. J. Thomsen, Some general effects of strong high-frequency excitation: stiffening, biasing and smoothening, Journal of Sound and Vibration 253 (2002) 807–831. J. Thomsen, Slow high-frequency effects in mechanics: problems, solutions, potentials, International Journal of Bifurcation and Chaos 15 (2005) 2799– 2818. J. Thomsen, Vibrations and Stability – Advanced Theory, Analysis, and Tools, Springer, Berlin, Heidelberg, New York, 2013. I. Blekhman, V. Sorokin, On the separation of fast and slow motions in mechanical systems with high-frequency modulation of the dissipation coefficient, Journal of Sound and Vibration 329 (2010) 4936–4949. L. Sperling, f Merten, H. Duckstein, Rotation und vibration in beispielen zur methode der direkten bewegungsteilung, Technische Mechanik 17 (3) (1997) 231–243. (in German). I. Blekhman, Oscillatory strobodynamics a new area in nonlinear oscillations theory, nonlinear dynamics and cybernatical physics, Cybernatics and Physics 1 (1) (2012) 5–10. A. Stephenson, On a new type of dynamic stability, Memoirs and Proceedings of the Manchester Literary and Philosophical Society 52 (1908) 1–10. P. Kapitza, Dynamic stability of a pendulum with an oscillating point of suspension, Zurnal Eksperimentalnoj i Teoreticeskoj Fiziki 21 (1951) 588–597. (in Russian). A.H. Nayfeh, Perturbation Methods, John Wiley and Sons, New York, 2008. A.K. Mohieddine Jelali, Hydraulic Servo-systems: Modelling, Identification and Control, Springer, Berlin, Heidelberg, New York, 2003. E. Kremer, About absolute stability of control valves, Proceedings of the 6th EUROMECH Nonlinear Dynamics Conference (ENOC 2008), 2008, 〈http://lib. physcon.ru/doc?id¼ d354de706a04〉.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Slow motions in systems with fast modulated excitation E. Kremer a,b,n a b
Institute of Problems of Mechanical Engineering, V.O. Bolshoj pr. 61, St. Petersburg 199178, Russia LuK GmbH & Co.KG, Industriestrasse 3, 77815 Buehl/Baden, Germany
a r t i c l e i n f o
abstract
Article history: Received 23 October 2015 Received in revised form 26 June 2016 Accepted 4 July 2016 Handling Editor: M.P. Cartmell Available online 3 August 2016
It is well known that high-frequency excitation can modify the behavior of systems with respect to slow motions. The goal of this study is consideration of these effects in a rather general case of analytical systems with modulated sinusoidal excitation. The method of direct separation of motions proposed by I.I. Blekhman was applied in a modified form with the explicit introduction of a small parameter. Equations for the slow motions are obtained and an analysis of how they depend on the structure of the original equations is performed. Five basic effects corresponding to different possible dependencies of the modulation amplitude on position, velocity, and slow time are selected (some of them for the first time). These effects offer a possibility for designing a high-frequency control of the slow motions with specified properties. For example, high-frequency excitation in a system with a nonlinear friction can essentially increase the effective damping. The results are also of significance for system identification and diagnostics. Analysis of a hydraulic valve is given as an example of application. & 2016 Elsevier Ltd. All rights reserved.
Keywords: High-frequency excitation Vibrational mechanics Strobodynamics Vibration-transformed forces Direct separation of motions Modulation Vibrational control Hydraulic valves
1. Introduction 1.1. Vibrational mechanics The concept of vibrational mechanics and the method of direct separation of motions were proposed by I.I. Blekhman [1] and systematically presented in books written by him [2,3]. This approach was further developed and applied to many practical problems by Fidlin [4–6], Malakhova [7], Thomsen [8–10], Sorokin [11], Sperling [12], and others. The main idea consists of the following steps. Let a dynamical system be described as 2 d x dx ; t; ωt (1) ¼ Θ x; dt dt 2 The system must not be a one-mass mechanical oscillator because Eq. (1) is valid for many physical processes. If that is the case, the parameters are assumed to have been normalized so that the mass is equal to 1 unit. In any case, the function Θ is interpreted as a force and can be presented as a sum of slow and fast components F and Φ dx dx (2) Θ ¼ F x; ; t þΦ x; ; t; ωt dt dt
n
Corresponding author at: LuK GmbH & Co.KG, Industriestrasse 3, 77815 Buehl/Baden, Germany. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.jsv.2016.07.006 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
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Fig. 1. Character of the solution (solid line – full system, dashed line – averaged system). The units of the axes x and t are here meters and seconds.
A solution can be obtained as a superposition of the time-dependent mean value X and the fast oscillation ψ: x ¼ XðtÞ þ ψðt; ωtÞ; ⟨ψ⟩ ¼ 0:
(3)
The function ψ is 2π-periodical and has a mean value equal to 0 with respect to the fast time θ¼ωt. The following is an averaging operation: Z 2π 1 f ðt; θÞdθ: (4) ⟨f ⟩ ¼ 2π 0 Fig. 1 illustrates the character of the solution. The method of direct separation of motions [2] introduces two integral-differential equations for X and ψ as follows: 2 d X dX dψ þ ; t; ωt (5) ¼ Θ X þ ψ; dt dt dt 2 2 d ψ dX dψ dX dψ þ ; t; ωt Θ X þ ψ; þ ; t; ωt ¼ Θ X þ ψ; dt dt dt dt dt 2
(6)
Eq. (5) is the result of the direct averaging and Eq. (6) is the complement of Eq. (5) to the original Eq. (1). The aim of further consideration is to obtain an equation for X, which does not include ψ and, therefore, enables the calculation the slow motion without solution of the full system. In order to achieve it, one should obtain ψ through X from Eq. (6) and substitute it into Eq. (5). This leads to an equation for the averaged motion, which is called the main equation of vibrational mechanics (also the equation of strobodynamics [13] or the vibration-transformed equation [2]): 2 d X dX ; t þ V: (7) ¼ F X; dt dt 2 The additional slow term V in this equation for the averaged motion is the so-called vibrational force, which takes into account the influence of now hidden fast motions. The vibrational force V can depend on the slow variable X, their derivatives with respect to the slow time t, and the slow time t explicit. There are many known nontrivial and unexpected physical effects caused by vibrational forces. Well-known examples are Chelomei's pendulum, the Stephenson–Kapitza pendulum, the Indian rope, vibrational transportation, and many other phenomena presented in the books written by Blekhman [2,3]. Thomsen [8,9] selected three effects, which are characteristics of many different systems with fast excitation – stiffening, biasing, and smoothening. These approaches to systematic effects were developed by him, later together with Fidlin [5] for strong excitation, especially in systems with friction.
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The objective of this study is to analyze vibrational effects with a focus not on the properties of vibrational forces that comprise the vibrations within the structures or machines in which they appear, but on the causes of them. The term effect is used here to define a possible combination of variables, which lead to the appearance of vibrational forces and, therefore, to nontrivial strobodynamical phenomena. It will be shown that there are five basic effects, corresponding to different possible dependencies of the modulation amplitude on position, velocity, and slow time. The subject of this investigation is an analytical system with relative weak modulated sinusoidal excitation in a smooth system. This case is used as a minimal generic model, which provides all the basic effects already in the first approach. The class of systems with modulated excitation is not only of theoretical interest. A well known parametrically excited system is the pendulum with the vibrating pivot (Stephenson–Kapitza pendulum [3,14,15]) where the modulation depends on position x. The case of velocity-depending modulation was considered in Ref. [4] for the first time as a pure mathematical example for a demonstration of the developed technique. Interesting physical and technical applications for this system were suggested later in Ref. [11]. An example of a technical application for a more general case of modulated excitation – a model of a hydraulic valve – is considered in Section 8. 1.2. Investigation technique The method of direct separation of motions is modified in comparison with its original version. [2,3] This modification concerns solving the equation of the fast motion (Eq. (6)). Of particular concern is that the original variant of the method with the frozen x and x_ in Eq. (6) provides correct results in the great majority of practical cases, but sometimes it needs more accurate consideration similar to the one made in Ref. [11] for a system with high-frequency modulation of the dissipation coefficient. In this case, an effect of the slow time on the fast motion was taken into account, but it was not clear in advance whether a more accurate consideration of this form would have been needed. The modification of the method proposed in this study consists of the explicit introduction of a small parameter in Eqs. (5) and (6) that enables automatic analysis of comparable importance of different terms. It makes the method of direct separation of motion more strict and robust. The corresponding analytical results are approximate in nature, because they correspond to the first approach to the solution. Therefore, it was important to verify them through a comparison with numerical calculations for the full system. This verification was fulfilled and is presented here.
2. The problem of modulated excitation The problem of modulated excitation is described by the following equation: 2 d x dx dx ¼ F x; ; τ þB x; ; τ sin ðθÞ; 2 dt dt dt
(8)
where τ ¼ t, θ ¼ ωt, ωc 1. This equation corresponds to Eq. (1) where the fast excitation is a sinusoid with an amplitude depending on the position x, velocity dx , and slow time τ. Let x be presented as a sum of slow and fast motions in accordance dt with Eq. (3). The problem consists in obtaining V in Eq. (7) for slow motion. The questions are: how does the vibrational force V depend on position, velocity, and slow time in general and how does the structure of the functions F and Φ affect it?
3. Equation for the slow motion Frequency ω is considered as being large and the corresponding value 1/ω as a small parameter. In this context, the order of magnitude of different terms can be estimated. If both F and B are of the order 1, the solution for the fast motion has the order 1/ω2. For this reason, it is convenient to introduce a new variable ξðτ; θÞ ¼ ψðτ; θÞω2 , where ξðτ; θÞ has the order 1. The full derivatives of every function f, which is to be differentiated with respect to t, can be calculated as 2
df _ 0 d f 0 00 ¼ f þωf ; 2 ¼ f€ þ2ωf_ þ ω2 f ; dt dt
(9)
∂f where f_ ¼ ∂f ∂τ and f ¼ ∂θ are partial derivatives with respect to slow and fast times. These derivatives are introduced here in a manner similar to that of the method of multiple scales [16] to provide here an explicit estimation of the orders for different terms. The equations of the method of direct separation of motions (Eqs. (5) and (6)) can be rewritten in terms of the order 1/ω2 in the following form: * + 2 ⟨ξ02 ⟩ ∂2 F 1 ∂Φ 1 _ ∂Φ 1 0 ∂Φ 1 02 ∂ Φ € þ 2 ξ ξ þ ξ X ¼Fþ 2 (10) þ þ 2 ξ 2 ω 2ω ∂X_ 2 ω2 ∂X ω 2ω ∂X_ ∂X_ ∂X_ 0
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0 ∂2 F 1 2 1 ξ ∂F ξ 1 ∂Φ ∂Φ 1 _ ∂Φ ξ_ ∂F _ ∂Φ ξ þ ξ þ 2 ξ00 þ ξ_0 þ 2 ξ€ ¼ Φ þ 2 þ þ ξ þ 2 ξ02 ⟨ξ02 ⟩ ξ 2 ω ∂X ω ω ∂X_ 2ω ω ω ∂X ω2 ∂X ω2 ∂X_ ∂X_ ∂X_ * +! 1 0 ∂Φ ∂Φ 1 ∂2 Φ ∂2 Φ ξ ξ0 þ 2 ξ02 2 ξ02 2 þ _ _ ω 2ω _ ∂X ∂X ∂X ∂X_
(11)
where F¼F(X, X_ ,τ) and Φ ¼B(X, X_ ,τ)sin(θ). In accordance with the method of direct separation of motions, the averaged motion X and its derivatives are considered initially as given and Eq. (11) is solved with respect to ξ. This solution is obtained asymptotically as ξ ¼ ξ0 þξ1/ω. The procedure consists of balancing the terms of the same order and integrating the corresponding equations related to θ. The integration constants can be chosen such that ξ0 and ξ1 are equal to 0. In this case, the appropriate particular solution is 1 ∂F 1 ∂B ξ ¼ B sin ðθÞ þ 2B_ cos ðθÞ þ B cos ðθÞ B sin ð2θÞ (12) ω 8 ∂X_ ∂X_ with ∂B ∂B ∂B þ : B_ ¼ X_ þ X€ ∂X ∂X_ ∂τ
(13)
The obtained expression for ξ is substituted in Eq. (10). It leads to the following result: X€ ¼ F þ V 2
_
B ∂ :F B ∂B ∂F ∂B B ∂B V ¼ 4ω 2ωB 2 ∂X þ 2ω 2 2 _ 2 2ω2 _ ∂X ∂X_ ∂X 2
(14)
∂X
_ This phenomenon The vibrational force V here depends on “slow” acceleration X€ if the modulation depends on velocity x. of the added mass is known in the context of vibrational mechanics from Ref. [11], where a linear system with velocity proportional modulation was considered. This problem was previously solved in Ref. [4] with another asymptotic technique. Solving Eq. (14) with respect to X€ correct to order 1/ω2 leads to the following final equation similar to Eq. (7): X€ ¼ F þ V ef f V ef f ¼
F ∂B 2 B2 ∂2 F B ∂B ∂F B ∂B þ 2 2 2 2 2ω ∂X_ 2ω ∂X_ ∂X_ 2ω2 ∂X 4ω ∂X_ þ
1 ∂B ∂B X_ ∂B ∂B þ : 2ω2 ∂X_ ∂X 2ω2 ∂X_ ∂τ
(15)
This equation allows the description of the averaged motion X without solving the full original Eq. (8) and without calculating the fast component ξ from Eq. (11). The vibrational force Veff as a function of X, X_ and τ is determined here explicitly from the expressions for the original forces F and B, and it can be directly calculated in every concrete applied problem. In the next section, the mechanism of the structure of the forces F and B affecting the structure of the vibrational force Veff is analyzed.
4. Basic strobodynamical effects 4.1. General approach The solution for the vibrational force (Eq. (15)) can be refined by taking into account further terms of the Taylor series. However, it is more important to note that this expression enables us to select the main effects of the excitation on the slow motion. The effects occurring already in this first approach can be called basic effects. There are five different basic effects, which correspond to the different terms of the vibrational force V eff . These effects can be classified depending upon the modulation structure as follows:
Arbitrary excitation (even with constant amplitude) in a system with nonlinearity relative to velocity Excitation dependent on position only (x -modulation) _ Excitation dependent on velocity only (x-modulation) Excitation dependent on velocity and position ðx_ x modulationÞ Excitation dependent on velocity and slow time ðx_ τ modulationÞ These effects are now considered in more detail.
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Fig. 2. Cubic dissipative force and excitation with constant amplitude generate an effective linear friction: (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), B¼ 50, F¼ 25x 3x_ 3, ω ¼60. The units of the axes x and t are here meters and seconds.
Fig. 3. Cubic dissipative force and excitation with constant amplitude stabilize a system with additional negative dissipation (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), B¼ 50, F ¼ 25x þ 0.5x_ 3x_ 3, ω¼ 60. The units of the axes x and t are here meters and seconds.
4.2. A system with nonlinearity relative to velocity under arbitrary excitation The first effect is a combination of the arbitrary excitation (even with constant amplitude) with nonlinearity of the unexcited system relative to velocity. In this case, the equation for the vibrational force (Eq. (15)) gives V eff ¼ V 1 ¼
B2 ∂2 F : 4ω2 ∂X_ 2
(16)
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Let the force in the unexcited system be a sum of two forces. The first of them depends on position and slow time and the second force is a nonlinear friction with velocity to the power of m: F ¼ F p ðX; τÞ βX_ m signðX_ Þ: (17) Then, the vibrational force in accordance with Eq. (16) is a pure dissipative force with velocity to the power of m 2: V1 ¼
βB2 mðm 1Þ _ m 2 X signðX_ Þ: 4ω2
(18)
For example, a cubic dissipative force generates an additional linear friction. This is illustrated in the numerical examples presented in Fig. 2 for F ¼ 25x 3x_ 3 and in Fig. 3 for F ¼ 25x þ0:5x_ 3x_ 3 . In both examples, B¼ 50 and ω¼60. The units of the variables x and t are assumed for definiteness being meters and seconds, but they could be also scaled or normalized in some way. The solid, dotted, and dashed lines are used here and further, respectively, for the original system (Eq. (8)) with excitation, for the same system without excitation (B¼0) and the vibration-transformed (averaged) system (Eq. (15)). The difference between the data of Figs. 2 and 3 is the additional term 0.5x_ in F for Fig. 3. It corresponds to some positive linear damping causing low-frequency self-exited oscillations in the system without high-frequency excitation. A significant increase of damping due to excitation is observed in both examples. In the second of them, the linear friction induced by the excitation is capable of stabilizing the system, which without any excitation executes low-frequency self-exited oscillation. The plots show a good agreement between the analytical and numerical results. It is valid as well for all presented examples although B is not of the order 1, as assumed above, but typically larger. The explanation of this is given in Section 5. 4.3. Excitation depends on position only (x-modulation) The next effect is the well-known transformation of the position-dependent modulation in an additional conservative vibrational force: V eff ¼ V 2 ¼ V 1
B ∂B 2ω2 ∂X
(19)
with additional potential energy D ¼ B2 =ð4ω2 Þ
(20)
This effect was described firstly for the pendulum with a vibrating pivot by Stephenson [14] and Kapitza [15] and later generalized by Blekhman. [2] For this pendulum (Fig. 4), additional potential energy is calculated in accordance with Eq. (20) as D ¼(Aω sin(X))2/(4ω2). It leads to the known results for this system [2], for example, to the condition of the upper position pffiffiffiffiffiffiffi stability: Aω 4 2gl. The method of direct separation of motions in its original form [1] with a frozen x in Eq. (6) can be used in the case of x-modulation directly.
Fig. 4. Pendulum of Stephenson–Kapitza.
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_ Fig. 5. Example of x-modulation: stiffening is ever-positive (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), _ F ¼ x 0.2x, _ ω¼ 40. The units of the axes x and t are here meters and seconds. B¼ 35x,
_ Fig. 6. Example of x x-modulation: positive stiffening (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), _ ω¼ 40. The units of the axes x and t are here meters and seconds. B¼ 35x_ 10x, F¼ x 0.2x,
_ 4.4. Excitation depends on velocity only (x-modulation) _ The next case is x-modulation. The corresponding expression for the vibrational force in accordance with Eq. (15) is as follows: F ∂B 2 B ∂B ∂F V ef f ¼ V 3 ¼ V 1 þ 2 2 : 2ω ∂X_ 2ω ∂X_ ∂X_
(21)
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Let the force in the unexcited system be described by Eq. (17) and B is a linear function of X_ : B ¼ αX_ . Then, Eq. (21) is transformed to V3 ¼
α2 βmðm 1Þ _ m jX j signðX_ ÞÞ: ðF p ðX; τÞ 2 2ω2
(22)
It shows that the parts of the unexcited system, depending on position and time, as well as its dissipative part are involved in vibrational force linear with different coefficients. In the case of m¼ 1, the vibrational force is proportional only to the positional part of the original slow force Fp with some coefficient b: V3 ¼bFp(X,τ). It can be characterized as the _ phenomenon of stiffening [8], and it coincides with the results of Refs. [4,11] for a full linear system with x-modulation. The coefficient b is equal to α2/(2ω2) and therefore always positive. A numerical verification of this result for the system with F ¼ _ B ¼ 35x, _ and ω¼40 is presented in Fig. 5. It is evident from the figure that the vibration-transformed system x 0.2x, (dashed line) is really an average of the excited system (solid line) and has a larger eigenfrequency as the unexcited system. For the case of m ¼2, a phenomenon of mass reduction takes place – the vibrational force is proportional to the full slow force: V3 ¼ bF(X, X_ , τ). _ 4.5. Excitation depends on velocity and position (x x-modulation) _ If the excitation amplitude depends on velocity and position (x x-modulation), the vibrational force has the form V eff ¼ V 4 ¼ V 1 þ V 2 þ V 3 þ
X_ ∂B ∂B : 2ω2 ∂X_ ∂X
(23)
Let us consider Eq. (17) with m ¼1 and B ¼ α1X þ α2X_ . The vibrational force has the form V4 ¼bFp(X, τ) as in the case of
_x-modulation, but with coefficient calculated as b ¼ α21 α22 þβα1 α2 ð2ω2 Þ. It indicates that b40 if βoβcr and b o 0 if 2 β4βcr, where βcr ¼ α1 þα22 =ðα1 α2 Þ. The cases of b40 and bo0 can be characterized as positive and negative stiffening, respectively. Corresponding numerical verifications are presented in Figs. 6 and 7 for F¼ x 0.2x_ and F¼ x 5x_ correspondingly. In both examples B ¼ 35x_ 10x and ω¼40. 4.6. Excitation depends on velocity and slow time (x_ τ-modulation) The last effect is the x_ τ-modulation. In this case, the general expression for the vibrational force has the following form: V eff ¼ V 5 ¼ V 3 þ
1 ∂B ∂B : 2ω2 ∂X_ ∂τ
(24)
_ Fig. 7. Example of x x-modulation: negative stiffening (solid line – without excitation, dotted line – with excitation, dashed line – averaged system), _ ω¼ 40. The units of the axes x and t are here meters and seconds. B ¼ 35x_ 10x, F¼ x 5x,
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Fig. 8. Phenomenon of low-frequency oscillations as example of x_ τ -modulation (solid line – without excitation, dotted line – with excitation, dashed line _ ω¼ 50. The units of the axes x and t are here meters and seconds. – averaged system), B¼ 50x_ 50 sin(2t), F¼ 4x 1.2x,
Fig. 9. Phenomenon of low-frequency oscillation disappears without dependence on velocity in the excitation (solid line – without excitation, dotted line – _ ω¼ 50. The units of the axes x and t are here meters and seconds. with excitation, dashed line – averaged system), B¼ 50sin(2t), F¼ 4x 1.2x,
Two phenomena within the scope of this effect will be considered as examples. The first of them occurs if the excitation amplitude is a sinusoidal function of slow time plus a linear function of velocity: B ¼as sin(τΩ) þ av X_ , and the function F has the form Eq. (17) with m ¼1. Then, the vibrational force V5 is calculated as V5 ¼
a2v av as F p þ 2 ðβ sin ðτΩÞ þΩ cos ðτΩÞÞ 2ω2 2ω
(25)
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Fig. 10. Phenomenon of drifting as an example of x_ τ-modulation (solid line – without excitation, dotted line – with excitation, dashed line – averaged _ ω¼ 50. The units of the axes x and t are here meters and seconds. system), B¼ 15x_ 2t, F¼ x 1.2x,
Fig. 11. Phenomenon of drifting disappears without dependence on velocity in the excitation (solid line – without excitation, dotted line – with excitation, _ ω¼ 50. The units of the axes x and t are here meters and seconds. dashed line – averaged system), B¼ 2t, F¼ x 1.2x,
It includes low-frequency excitation. This phenomenon of low-frequency oscillation disappears if dependence on velocity in modulation vanishes. This is illustrated by two numerical examples in Figs. 8 and 9 for B ¼ 50x_ 50 sin(2t) and B ¼ 50 sin(2t). In both examples, F ¼ 4x 1.2x_ and ω¼50. Another example of the x_ τ-modulation is a linear function of
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time and velocity: B ¼ at τ þ av X_ . If the function F has again the form Eq. (17) with m ¼1, the vibrational force is calculated as V5 ¼
a2v av at F p þ 2 ðβτ þ 1Þ 2ω2 2ω
(26)
It is apparent from Eq. (26) that vibrational force increases linearly with slow time τ. The corresponding increase of the averaged variable X under acting of excitation with an increasing amplitude can be called vibration drifting. It should be emphasized that this phenomenon exists only in the case of combined dependencies of modulation on x_ and τ. In this case, a phenomenon of vibration drifting takes place. It is illustrated in Figs. 10 and 11 for B ¼ 15x_ 2t and B ¼ 2t. In both examples, F ¼ x 1.2x_ and ω¼50. In the first case, the averaged x increases. In the second case, it is equal to 0.
5. Generalizations The presented theory enables some generalizations. 5.1. Stronger excitation The first generalization is a modest stronger excitation. Let the excitation amplitude have now the magnitude of order ωλ: 2
d x ¼ F þ ωλ B sin ðθÞ dt 2
(27)
If λ o 1, then the following expression for the vibrational force is valid in the first approach V eff ðλÞ ¼ ω2λ V eff ð0Þ
(28)
It explains the results in the numerical examples above, which are valid for a stronger excitation as it can be expected from the initial assumption that F is of the same magnitude order as B. 5.2. Multiharmonic excitation Another generalization is a multiharmonic excitation: 2 N X d x ¼Fþ BðksÞ sin ðkθÞ þBðkcÞ cos ðkθÞ 2 dt k¼0
(29)
In this case, the effects of different harmonics are additive due to their orthogonality, and the vibrational force is calculated as V eff ¼
N . X 2 k V ðsÞ þ V kðsÞ k
(30)
k¼0
The parameters V kðsÞ and V ðcÞ represent the vibrational forces corresponding to the excitations with amplitudes BkðsÞ and BðcÞ , k k respectively, in accordance with Eq. (15). 5.3. N-dimensional case The presented results can be directly generalized to the case of vectorial x, if a system consists of equations of second order.
Fig. 12. Magnet valve (1 – domain of the adjusted pressure p, 2 – piston, 3 – changing gap, 4 – pump, 5 – sump, 6 – solenoid, 7 – the area of the pressure action on the piston).
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6. Example of technical application An example of technical application of the developed theory in the dynamics of hydraulic valves is presented in this section [17,18]. A typical magnet proportional valve is depicted in Fig. 12. The purpose of this valve is to adjust pressure in domain 1. It is achieved due to a motion of piston 2, changing gap 3, and thus the hydraulic resistance between pump 4 to sump 5. The following forces act on the piston:
Control magnet force Fc of solenoid 6. _ , Viscous friction having two components – with linear and square dependence on velocity: Fc ¼ Dx_ with D ¼ β þ kxj where β and k are coefficients.
Pressure force pA, where p is the adjusted pressure in domain 1 and A is the area 7 of the pressure action on the piston. The area A depends generally on the piston position x: A ¼A(x).
Force cx of the spring 8.
A rather typical case of a very light piston is considered here. Therefore, inertia forces are neglected in comparison to viscous friction even for relative high excitation frequency. In this case, the equation of motion has the form Dx_ þ cx ¼ Ap F c :
(31)
Another equation expresses the volume balance in domain 1 and has the following form: _ ch p_ ¼ Q 0 Q Ax;
(32)
where ch is the hydraulic capacity of the domain 1, Q is the flow rate through gap 3, and Q0 is the pump flow. The flow rate Q depends on position x and pressure p according to the following known hydraulic formula [17]: pffiffiffi Q ¼ γAB ðxÞ p (33) where γ is a constant and AB(x) is the cross-sectional area of the gap, depending on the piston position x. The last term in Eq. (32) represents the pump effect of the piston – the flow rate of the fluid due to the piston motion. Differentiating Eq. (31) with respect to t and following substitution of x_ from Eq. (31) leads to an equation of second order for the piston position x: _ x€ þcx_ ¼ ðβ þkjxjÞ
pffiffiffi A dA _ F_ c ; _ þ xp ðQ γAB p AxÞ ch 0 dx
(34)
where pressure p is calculated through x and x_ in accordance with Eq. (31): 1 p ¼ ðF c þ βx_ þcxÞ: A
(35)
A typical magnetic force Fc in such valves is a sum of a slow component and a fast-oscillating component F0(t) with frequency ω and slowly changing amplitude F1(t): F c ¼ F 0 ðtÞ þ F 1 ðtÞ cos ðωtÞ:
(36)
It is usually larger as the term βx_ þ cx in the expression for pressure (Eq. (35)), and one can expect that averaged pressure is approximately equal to p ¼ F 0 ðtÞ=A. This is really the case, but the oscillations of the magnet force lead to some departure from this nominal value. A practically important problem is to describe the effect of magnet force oscillations on averaged controlled pressure. The following are some assumptions:
The frequency of excitation is large relative to a characteristic frequency of the main control force F0 as well as of the excitation amplitude F1.
The excitation amplitude F1 is small relative to main control force F0. Then, Eq. (35) can be presented in the form of Eq. (8) with ! ! pffiffiffiffiffi dA dx A2 A 1 dx 1 _ _ _ _ F 0 þ βx_ þ cxÞBðx; ; t ¼ U 1 F 1 ωU ¼ ðβ þ kjxjÞ F x; ; t ¼ U cþ x þ ðQ 0 γAB p1 Þ þ xp1 F 0 p1 ¼ dt ch dx A dt ch (37) The system in the presented formulation is a type of x_ τ-modulation with a nonlinear dependence of the main slow force on velocity. The slow motion can be described by Eq. (7), where V ¼Veff is calculated directly with the help of Eq. (24). This expression takes the following form: 1 1 00 _ 2ksign X_ F 1 @@ 2 F 1 ∂2 F 2ksign X ∂F A F1 (38) V eff ¼ F_ 1 A 4k 2 þ þ 2 ∂X_ 2 U U ∂X_ 2U 2 U
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It is beyond the scope of this study to analyze in detail the technical consequences of the phenomenon. The following points should be still noted: 1. The developed theory enables analysis of the causes of the departure of the controlled variables from their nominal characteristics to improve valve design. 2. Here are two causes of vibrational forces. The first of them is a nonlinear dependence of the main slow force F on velocity due to a dependence of the acting area A on position x and the square component of the friction (k a 0). Another cause is the dependence of modulation B in Eq. (37) on time and velocity if the amplitude of the magnet force changes and the friction has a square component in its dependence on velocity (k a0). Increasing the amplitude of the magnet force can lead to an increase in the averaged pressure (vibration drift). 3. In the case of pure linear friction (k ¼0), the expression for vibrational force V ¼Veff has a particularly simple form: V eff ¼
F 21 ∂2 F : 4U 2 ∂X_ 2
(39)
7. Conclusions
A problem of vibrational mechanics (strobodynamics) for an analytical system with modulated high-frequency excitation is considered and for this case a general expression for vibrational force is obtained.
It is shown to be possible to analyze in general the mechanism of the structure of the original equation affecting the equation for slow motions. It is shown particularly that only five possible effects cause vibrational forces. These effects can be classified as follows: – – – – –
Arbitrary excitation (even with constant amplitude) in a system with nonlinearity relative to velocity Excitation dependent on position only (x-modulation) _ Excitation dependent on velocity only (x-modulation) _ Excitation dependent on velocity and position (x x-modulation) Excitation dependent on velocity and slow time (x_ τ-modulation)
With each of these five effects, some specific vibration force is connected, which causes, in turn, different physical phenomena. In particular the following are shown: – A combination of a nonlinear friction with an excitation of a constant amplitude can lead to essential increasing of damping in slow motion. It can be used to stabilize a system and to avoid self-oscillations. _ – The case of combined dependence of modulation on position and velocity (x x-modulation) causes a vibration force depending only on position. They can either increase or decrease an existing stiffness depending on damping, which can find application in different control systems. – Dependence of modulation simultaneously on slow time and velocity (x_ τ-modulation) can cause low-frequency oscillations and vibrational drifting of averaged variables.
The results are generalized for a stronger excitation (of the order ωλ with λ o 1) and multiharmonic excitation. The method of direct separation of motions is applied in a modified form with the explicit introduction of a small
parameter. This enables one to take into account the slow time in the equation of fast motion automatically wherever it is needed. The analytical results are verified by comparison with numerical simulation. The investigated effects are of interest not only from the theoretical point of view but at least for two practical fields:
– development of systems for high-frequency control of slow motions with predefined properties; – identification of systems and experimental modal analysis by taking into account effects of high-frequency excitations on low-frequency motions. Analysis of a hydraulic valve is given as an application example.
Acknowledgment The author is grateful to I.I. Blekhman for interesting discussion and useful advice. This study was supported by the Russian Scientific Foundation (Grant No. 14-19-01190).
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