Control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator

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DOI 10.1007/s11071-010-9784-5 O R I G I N A L PA P E R

Control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator

Amine Bichri·Mohamed Belhaq· Joël Perret-Liaudet

Received: 11 March 2010 / Accepted: 11 July 2010

Abstract The control of vibroimpact dynamics of a single-sided Hertzian contact forced oscillator is investigated analytically and numerically in this paper.

The control strategy is introduced via a fast excitation and attention is focused on the response near the primary resonance. The fast excitation is added to the basic harmonic force, either through a harmonic force applied from above, or via a harmonic base displacement added from bellow, or by considering the stiffness of the oscillator as a periodically and rapidly varying in time. The results reveal that the threshold of vibroimpact response initiated by jump phenom- enon near the primary resonance can be shifted toward lower or higher frequencies of the slow dynamic system depending on the fast excitation taken into consid- eration. It was also shown that the most realistic and practical way for controlling the vibroimpact dynamics is the introduction of a fast harmonic base displacement.

Keywords Hertzian contact·Vibroimpact· High-frequency excitation·Active control· Perturbation analysis

A. Bichri·M. Belhaq (⁾

Laboratory of Mechanics, University Hassan II-Casablanca, Casablanca, Morocco e-mail:mbelhaq@hotmail.com J. Perret-Liaudet

Ecole Centrale de Lyon, Laboratoire de Tribologie et Dynamique des Systèmes, UMR 5513, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France

1 Introduction

The study of the dynamics of mechanical systems evolving in a Hertzian contact regime constitutes an important branch in various scientific and engineering applications. For instance, rolling contact mechanism, bearing, gear drive, and, in general, mechanisms trans- forming rotation or translations, operate in Hertzian contact regimes. Vibration analysis of such contacts has been carried out in several papers [1–5]. The approach adopted in these studies uses a single-degree- of-freedom system for modeling the behavior of a moving surface under a Hertzian contact force. Mann et al. [6] carried out experimental study of an impact parametric oscillator with viscoelastic and Hertzian contact with a motion-dependent discontinuity. Rong et al. [7] investigated the resonant response of a single- degree-of-freedom nonlinear vibroimpact oscillator, with cubic nonlinearities, to combined deterministic harmonic and random excitations. Recently, the dynamics of an idealized preloaded and non-sliding dry Hertzian contact was studied near the primary resonance [8] and near the subharmonic [9] and the su- perharmonic [10] resonances of order 2. The analysis was based on numerical simulations, analytical approximation and experimental testing. The vibration of an impacting Hertzian contact under random excitation was also investigated numerically and experi- mentally [11,12]. Results of these works [8–10] con- cluded that vibroimpact responses resulting from the loss of contact are principally initiated by jump phe- nomena near the resonances. The results indicated that

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the nonlinearity in the Hertzian contact law that causes hysteresis and jumps has to be taken into considera- tion for predicting the birth of vibroimpact responses.

Thus, to keep the system operating in desired values of the frequency excitation without the loss of contact, it is necessary to develop control methods. This paper attempts to propose and analyze some possibilities to control such vibroimpact dynamics by adding fast excitations. In this respect, we refer to [8], where an idealized preloaded and non-sliding dry Hertzian contact is considered.

Effect of fast excitations on nonlinear dynamics of mechanical systems has been considered previously by a number of authors [13–15]. Recent works focused attention on the effect on stability, stiffness, frequencies, resonance, symmetry breaking, limit cycle, delay, hysteresis and pull-in instability in MEMS [16–27]. To control the vibroimpact occurrence, we propose three strategies based on adding a fast harmonic excitation to the basic harmonic force. The first strategy uses a fast harmonic force added from above, the second one introduces a fast harmonic base motion, whereas in the third strategy we consider the stiffness of the nonlinear restoring force as a rapidly and periodically varying in time (fast parametric stiffness).

The rest of the paper is organized as follows. In Sect.2, we consider the case of a rapid forcing added from above and we use an averaging technique to split up the fast and the slow dynamics. Then, a multiple scale technique is applied on the slow dynamic to derive the modulation equations and the frequency response curves. These curves are used to analyze the effect of the fast excitation on the vibroimpact near the primary resonance. In Sect.3, we follow a similar approach as used in Sect. 2 to analyze the effect of a fast harmonic base displacement on the frequency response. Section 4 deals with the effect of a fast parametric stiffness on the frequency response, while Sect.5concludes the work.

2 Effect of a fast forcing from above

In this section, we analyze the effect of a fast forcing applied from above on the frequency response curve and on the vibroimpact threshold.

2.1 Equation of motion and slow dynamic

Consider a single-degree-of-freedom system consist- ing in a vertically and periodically forced impact os-

Fig. 1 Dynamic model of the single-degree-of-freedom impact oscillator. The moving rigid massmis subjected to a purely harmonic normal force superimposed on the static load and excited by a fast forcing

cillator, as shown in Fig.1. Assume that the moving massmsupported by a nonlinear restoring force [28]

is under a static load and is excited by a fast forcing added to the basic harmonic normal force. The equation of motion can be written in the form [9]

m¨z+c˙z+kz³² =N (1+σcosωt )+acos(Ωt ) (1) where z is the normal displacement of the rigid mass m, c the damping coefficient, k the constant given by the Hertzian theory, N the static normal force, σ andωare the level of the excitation and its frequency, respectively, while a and Ω are the amplitude and the frequency of the fast forcing, respectively. The static contact compressionzs is given by zs=(^N_k)²³.

Introducing the dimensionless variable change:

¯

ω=(^ω_ν),ν²=(_2m^3k)z

1

s2,ξ=_2mν^c ,q=^3(z_2z⁻_s^z^s⁾,τ=νt, Ω¯ =(^Ω_ν)anda=(_N^a), the dimensionless equation of motion takes the form

¨

q+2ξq˙+

1+2 3q

³

2

=1+σcosωτ¯ + ¯acosΩτ¯ (2) Expanding the nonlinear restoring force in Taylor series around the static load and neglecting terms of order greater than three inq, (2) reads

¨

q+q+αq˙+βq²−γ q³=σcosωτ¯ + ¯acosΩτ¯ (3) whereα=2ξ,β=¹₆andγ=₅₄¹.

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To derive the slow dynamic of system (3), we use the method of separation of motion [29,30] by introducing two different time scales, a fast timeT0= ¯Ωτ and a slow timeT1=τ, and we split upq(τ ) into a slow partx(T₁)and a fast partψ (T₀, T₁)as follows:

q(τ )=x(T₁)+ψ (T₀, T₁) (4) Here,x describes the slow main motions at time-scale of oscillations andψstands for an overlay of the fast motions. The frequencyΩ¯ is considered as a large parameter, for convenience. The fast partψand its derivatives are assumed to be 2π-periodic functions of fast timeT0with zero mean value with respect to this time, so thatq(τ ) =x(T1)where ≡ _2π¹ 2π

0 ( ) dT0de- fines time-averaging operator over one period of the fast excitation with the slow timeT₁fixed.

Introducing D_i^j ≡ _∂^∂j^jT_i yields _dτ^d = ¯ΩD₀+D₁,

d²

dτ² = ¯Ω²D₀²+2ΩD¯ ₀D₁+D₁² and substituting (4) into (3) gives

¨

x+ ¨ψ+x+ψ+αx˙+αψ˙ +βx²+βψ²+2βxψ

−γ x³−3γ ψ x²−3γ xψ²−γ ψ³

=σcosωτ¯ + ¯acosΩτ¯ (5) Averaging (5) leads to

¨

x+x+αx˙+βx²+β ψ²

−γ x³−3γ x ψ²

−γ ψ³

=σcosωτ¯ (6)

Subtracting (6) from (5) yields ψ¨ +ψ+αψ˙+βψ²−β

ψ²

+2βxψ

−3γ x²ψ−3γ xψ²+3γ x ψ²

−γ ψ³+γ ψ³

= ¯acosΩτ¯ (7)

The equation for the fast motions (7) may be solved in rough approximation without introducing a significant error into the solution of the slow motions. Here, we use the so-called inertial approximation, i.e. all terms in the left-hand side of (7), except the first, are ig- nored [29]. Thus, the fast dynamicψis written as ψ= − a¯

Ω¯²cosΩτ¯ (8)

Inserting (8) into (6), we find the approximate equation for the slow motion

Fig. 2 The full motion, (3), and the slow dynamic, (9), for ξ=0.01,σ=0.04,Ω¯=8,ω¯=0.98 anda¯=60

¨

x+ω²₀x+αx˙+βx²−γ x³+G=σcosωτ¯ (9) whereω²₀=1−³₂γ ( _¯^a^¯

Ω²)²andG=^β₂(_¯^a^¯

Ω²)².

In Fig. 2we show a comparison between the full motion q(τ ), (3), and the slow dynamic x(T1), (9), for the given values of parametersξ=0.01,σ =0.04, Ω¯ =8, ω¯=0.98 and a¯ =60. A good agreement between the full motion and the slow main dynamic is illustrated.

2.2 Frequency response near the principal resonance To study the frequency response near the principal resonance, we express the resonance condition by introducing a detuning parameterλaccording to

ω²₀= ¯ω²+λ (10)

Introducing a small bookkeeping parameter μ and scaling as α = μα, β = μβ,σ = μσ, λ = μλ, G= μGandγ=μ²γ, (9) reads

¨

x+ ¯ω²x=μ

−αx˙−βx²+σcosωτ¯ −λx−G

+μ²γ x³ (11)

Using the multiple scales method [31], we seek a solution to (11) in the form

x(t )=x0(T0, T1, T2)+μx1(T0, T1, T2) +μ²x2(T0, T1, T2)+0

μ³

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Fig. 3 Amplitude–frequency response. Analytical approximation (solid lines for stable and dashed line for unstable) and numerical simulation (circles) forξ=0.01,σ=0.04 andΩ¯=8

whereT0=τ, T1=μτ and T2=μ²τ. In terms of the variables Ti, the time derivatives become _dt^d = D0+μD1+μ²D2+O(μ³)and ^d²

dt² =D₀²+2μD01+ μ²D₁²+2μ²D02+O(μ³), whereDi=_∂T^∂_i andDij=

∂²

∂TiTj. Substituting (12) into (11) and equating terms of same power ofμ, we obtain the following hierarchy of problems

D²₀x₀+ ¯ω²x₀=0 (13) D²₀x1+ ¯ω²x1= −(2D01+αD0+λ)x0

−βx₀²−G+σcosωτ¯ (14) D²₀x₂+ ¯ω²x₂= −(2D01+αD₀+λ)x₁

−(2D02+D₁₁+αD₁)x₀

−2βx0x1+γ x₀³ (15) The solution to the first order is given by

x₀(T₀, T₁, T₂)=r(T₁, T₂)cos

¯

ωτ+θ (T₁, T₂) (16) Substituting (16) into (14) and (15) and removing secular terms, we obtain the following slow flow modulation equation of the amplitude and the phase:

⎧⎪

⎪⎨

⎪⎪

⎩ dr

dt =Ar+H₁sinθ+H₂cosθ rdθ

dt =Br+Cr³+H1cosθ−H2sinθ

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with A= −^α₂, B = ₂^λ_ω_¯ − ₈^α_ω²_¯ − ^βG_ω_¯3 − ₈^λ_ω_¯²3, C =

−(^5β²

12ω¯³ +^3γ₈_ω_¯),H1=₈^λσ_ω_¯3 −₂^σ_ω_¯ andH2=₈^ασ_ω_¯2.

Equilibria of the slow flow (17), corresponding to periodic solutions of (9), are determined by settingr˙= θ˙=0. Using the relation cos²θ+sin²θ=1, we obtain the amplitude–frequency response equation

C²r⁶+2BCr⁴+

A²+B² r²

−

H₁²+H₂²

=0 (18)

Figure 3a shows the frequency response curve, as given by (18), in the absence of the fast excitation force (a¯ =0), while Figs. 3b and c display the effect of the amplitude of the fast excitation on the frequency response curve fora¯=60 anda¯=70, respectively. The solid lines correspond to the stable solutions, while the dashed line corresponds to the unstable branch. Numerical simulations (circles), obtained by using the fourth-order Runge–Kutta method, are compared to analytical approximations for comparisons. The numerical amplitudes are obtained using the steady-states from the time series. The plots in these figures show that as the amplitude of the fast forcing increases, the resonance curve shifts left indi- cating that the birth of the vibroimpact response can be shifted toward lower frequencies. The figures also show that the control of the vibroimpact dynamics by adding a superimposed fast harmonic force from above requires a large amplitude which is difficult to achieve in practice.

3 Effect of a fast harmonic base motion

In this section, we analyze the effect of a fast harmonic base motion on the response curve and on the locus of

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Fig. 4 Dynamic model with a fast harmonic base motion

the vibroimpacting setting. The schematic model with harmonic base motion is shown in Fig.4. In this case, the equation of motion can be written in the form mz¨+c(z˙− ˙ze)+k(z−ze)³² =N (1+σcosωt ) (19) wherez_e=acosΩtis the fast harmonic base motion.

By introducing the dimensionless variable changes as before, withq_e=³₂^z_z^e_s, (19) can be written in the following dimensionless form:

¨

q+2ξ(q˙− ˙qe)+

1+2

3(q−qe) ³₂

=1+σcosωτ¯ (20) Using a truncated Taylor series of the term (1 +

2

3(q−qe))³² around the origin andqeand substituting into (20), one obtains the following equation:

¨

q+q+αq˙+βq²−γ q³

=σcosωτ¯ +αq˙_e +

1+2βq−β

3q²+γ q³

q_e (21)

where α=2ξ, β = ¹₆, γ = ₅₄¹, q_e = ¯acosΩτ¯ and

¯ a=³₂_z^a_s.

Averaging (21) as in the previous section, we obtain the following equation for the slow dynamic:

¨

x+ω₁²x+αx˙+β1x²−γ1x³+G1=σcosωτ¯ (22) whereω²₁,β1,γ1andG1are given inAppendix.

Figure5illustrates the comparison between the full motionq(τ ), (21), and the slow dynamicx(T1), (22), validating the averaging procedure. To approximate

periodic solutions of (22) near the principal resonance, we use the multiple scales technique as before to obtain the modulation equations of amplitude and phase

⎧⎪

⎪⎨

⎪⎪

⎩ dr

dt =A1r+H3sinθ+H4cosθ rdθ

dt =B1r+C1r³+H3cosθ−H4sinθ

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whereA1,B1,C1,H3andH4are given inAppendix.

Eliminating the phaseθ from the system (23), we obtain the following amplitude–frequency response equation

C₁²r⁶+2B1C1r⁴+

A²₁+B₁² r²

−

H₃²+H₄²

=0 (24)

In Fig.6, we show the frequency response curve as given by (24). Figure6a shows the frequency response in the absence of the fast base displacement (a¯=0), while Figs.6b and c display the effect of the amplitude of the base motion on the frequency response for

¯

a=5 anda¯=8, respectively. The solid lines correspond to the stable branches, while the dashed line corresponds to the unstable one. Numerical simulations (circles) are reported for comparisons. As in the previous case, the plots in these figures show that as the amplitude of the base displacement increases, the

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threshold of vibroimpact dynamics shifts left. Note that the plots in Fig. 6 indicate that it is possible to control the vibroimpact response because the required amplitude of the base displacement remains relatively small.

4 Effect of a fast parametric stiffness

Here, we investigate the effect of a fast parametric stiffness on the response curve. The dynamic model with parametric stiffness is shown in Fig. 7 and the equation of motion can be written in the form

mz¨+cz˙+(k₀+k₁cosΩt )z³²

=N

1+σcos(ωt )

(25) where k₀ and k₁ are constant. Introducing the variable changeω¯=(^ω_ν),ν²=(^3k_2m⁰)z

1

s2,ξ=_2mν^c ,τ=νt, Ω¯ =(^Ω_ν)anda=(^k_k¹

0)withzs =(_k^N

0)²³, scaling (25) and definingq =^3(z_2z⁻_s^z^s⁾, the dimensionless equation of motion reads

¨

q+2ξq˙+

1+2 3q

³₂ + ¯a

1+2

3q ³₂

cosΩτ¯

=1+σcosωτ¯ (26) We focus the analysis on small vibrations around the static load by expanding in Taylor series the term (1+²₃q)³² keeping only the terms up to the third

Fig. 7 Dynamical model with parametric stiffness

order in q. Then, the equation of motion takes the form

¨

q+q+αq˙+βq²−γ q³ + ¯a

1+q+βq²−γ q³ cosΩτ¯

=σcosωτ¯ (27)

whereα=2ξ,β=¹₆andγ=₅₄¹. Following a similar analysis as in Sect.1, we split up the motion into fast and slow parts as

q(τ )=x(T1)+ψ (T0, T1) (28) Substituting (28) into (27) and averaging, one obtains the stationary solution to the first order for ψ as

ψ= a¯ Ω¯²

1+x+βx²−γ x³

cosΩτ¯ (29)

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and the approximate equation for the slow motion

¨

x+ω₂²x+αx˙+β₂x²−γ₂x³+G₂=σcosωτ¯ (30) whereω²₂,β₂,γ₂andG₂are given inAppendix.

Figure 8 shows the comparison between the full motionq(τ ), (27), and the slow dynamicx(T₁), (30).

A good approximation of the slow dynamic is obtained.

The 1:1 resonance condition is considered by introducing a detuning parameterλaccording to

ω²₁= ¯ω²+λ (31)

Introducing a bookkeeping parameterμsuch that (30) is rewritten as

¨

x+ ¯ω²x=μ

−αx˙−β2x²+σcosωτ¯ −λx−G2

+μ²γ₂x³ (32) and using the multiple scales technique, we seek a solution to (32) in the form

x(t )=x0(T0, T1, T2)+μx1(T0, T1, T2) +μ²x₂(T₀, T₁, T₂)+0

μ³

(33) Substituting, equating terms of same power ofμand removing secular terms as usual, we obtain the slow flow modulation equation of amplitude and phase

⎧⎪

⎪⎨

⎪⎪

⎩ dr

dt =A2r+H5sinθ+H6cosθ rdθ

dt =B2r+C2r³+H5cosθ−H6sinθ

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whereA2,B2,C2,H5andH6are given inAppendix.

Equilibria of the slow flow (34), corresponding to periodic solutions of (30), are determined by setting

˙

r= ˙θ=0. Using the relation cos²θ+sin²θ=1, we obtain the amplitude–frequency response equation C₂²r⁶+2B2C₂r⁴+

A²₂+B₂² r²

−

H₅²+H₆²

=0 (35)

In Fig.9, we show the frequency response curve, as given by (35). Figure9a depicts the frequency response curve in the absence of the fast parametric

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stiffness (a¯ =0), while Figs. 9b and c display the effect of the amplitude of the fast parametric stiffness on the frequency response curve for a¯=4 and

¯

a =5, respectively. The solid lines correspond to the stable branches, whereas the dashed line corresponds to the unstable one. In contrast to the two previous cases, Fig. 9 shows clearly that as the amplitude of the parametric stiffness is increased, the resonance curve shifts right. This is confirmed by per- forming numerical simulation (circles) of (30). Thus, a time-varying stiffness can also be used to control the vibroimpact dynamics since the required amplitude level for the control purpose still relatively small.

5 Conclusions

In this paper, we have proposed three strategies for controlling the vibroimpact dynamics of a single-sided Hertzian contact excited by an external harmonic force near the primary resonance. The first strategy is based on adding a fast harmonic force superimposed to the basic harmonic one. The second method uses a fast harmonic base displacement added to the basic harmonic force, while the third strategy considers the stiffness of the nonlinear restoring force as a rapidly and periodically time-varying stiffness. The separation of motion procedure and the multiple scales method are performed, in each case, to drive a slow dynamic and its slow flow. Analysis of equilibria of the slow flows provides analytical expressions of the frequency response allowing the analysis of the effect of different control strategies on the occurrence of the vibroimpact responses triggered near the primary resonance. Results show that adding a fast harmonic force from above or a fast base displacement causes the resonance curve to shift left, whereas the introduction of a rapidly parametric stiffness in the nonlinear restoring force shifts the response curve right.

From a practical point of view, the first strategy based on adding a fast harmonic force appears to be difficult to achieve. Actually, the necessary amplitude of the superimposed force to ensure a practical effect is too high. The third strategy using a time-varying stiffness is more suitable in term of the amplitude level. However, from a mechanical view point, this strategy is very difficult to realize. Indeed, this requires, for instance, time-varying radii of curvature,

which can be induced by involving rolling motion of a special cam. Therefore, the best practical way for controlling the system seems to be the second strategy because the needed amplitude level of the base displacement to produce a shift remains relatively small (typically a factor ten compared to the static displacement which is generally around some mi- crons).

The results of the present work open some per- spectives in active control of the vibroimpact occurrence in mechanisms using single-sided Hertzian contacts.

Appendix

ω²₁=

1+ βa¯²

3Ω¯⁴−3γa¯² 2Ω¯²

α²− 1 Ω¯²

+ γa¯⁴ 8Ω¯⁴

α²+ 3

Ω¯²

ω²₂=1+

β−3 2γ

a¯² Ω¯⁴+

1 2+β

a¯² Ω¯²

− 9γa¯⁴

8Ω¯⁶

β1=

β−γa¯² Ω¯⁴ + a¯²

2Ω¯²

3γ−β 3

− γa¯⁴ 8Ω¯⁴

α²β− 1 Ω¯²

γ₁=

γ−βa¯² Ω¯⁴

γ−β 9

− a¯² 3Ω¯²

2γ+β² 3

−γa¯⁴ 8Ω¯⁴

3γ α²+ 1 9Ω¯²

G1=

α²+ 1 Ω¯²

βa¯²

2Ω¯²+3γa¯⁴ 8Ω¯⁴

+ a¯² 6Ω¯² A1= −α

2 B₁= λ

2ω¯ − α²

8ω¯ −β1G1

¯

ω³ − λ² 8ω¯³ C1= −

5β₁² 12ω¯³+3γ1

8ω¯

H3= λσ 8ω¯³− σ

2ω¯