Effect of fast parametric viscous damping excitation on vibration isolation in sdof systems

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Short communication

Effect of fast parametric viscous damping excitation on vibration isolation in sdof systems

Lahcen Mokni, Mohamed Belhaq^⇑, Faouzi Lakrad

Laboratory of Mechanics, University Hassan II-Casablanca, Morocco

a r t i c l e i n f o

Article history:

Received 19 April 2010

Received in revised form 2 August 2010 Accepted 24 August 2010

Keywords:

Parametric excitation Nonlinear damping Vibration isolation Fast excitation

Direct partition of motion

a b s t r a c t

In this work, we investigate analytically the effect of cubic nonlinear parametric viscous damping on vibration isolation in sdof systems. Attention is focused on the case of a fast parametric damping excitation. The method of direct partition of motion is used to derive the slow dynamic and steady-state solutions of this slow dynamic are analyzed to study the inﬂuence of the fast nonlinear parametric damping on the vibration isolation. This study shows that adding periodic nonlinear damping variation to the vibration isolation device can reduce transmissibility over the whole frequency range. The results also reveals that this nonlinear parametric viscous damping enhances vibration isolation comparing to the case where the cubic nonlinear damping is time-independent.

1. Introduction

Reducing transmitted vibrations to a support structure is an active topic of research in many engineering applications; see for instance[1,2]and references therein. Among the methods used to reduce transmitted vibrations, there is one that uses viscous damping in the vibration isolation device. However, it is known that when the viscous damping is linear, the transmissibility is reduced in the resonant region and increased elsewhere. To solve this problem and enhance vibration isolation in the whole frequency range, cubic nonlinear viscous damping has been introduced[3]. In a recent study[4], a theoretical analysis on vibration isolation of a single degree of freedom (sdof) spring damper system revealed that cubic nonlinear damping can produce an ideal vibration isolation such that the non-resonant regions remain unaffected.

In the present paper, we study the effect of cubic nonlinear parametric damping on vibration isolation of a sdof system with nonlinear stiffness. We focus attention on the case where the frequency of the parametric excitation, due to periodic damping variation, is large comparing to that of the harmonic external excitation. To perform the analysis, we use an analytical treatment based on averaging method[5]to derive the main equation governing the slow dynamic. Then, we apply the multiple scales method[13]on the slow dynamic to obtain its slow ﬂow. Analysis of steady-state solutions of this slow ﬂow provides an analytical relationship of transmissibility versus the system parameters allowing us to study the effect of the fast parametric viscous damping on vibration isolation.

2. Equation of motion and slow dynamic

Consider a sdof model of a suspension system with nonlinear stiffness and nonlinear parametric viscous damping in the form X€þx²XþB1X³þ ðB2X_ þB3X_³Þð1þbm²cosmtÞ ¼ gþYX²cosXt; ð1Þ

doi:10.1016/j.cnsns.2010.08.031

⇑ Corresponding author.

E-mail address:mbelhaq@yahoo.fr(M. Belhaq).

Contents lists available atScienceDirect

Commun Nonlinear Sci Numer Simulat

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c n s n s

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wherex²¼^k_m¹,B1¼^k_m²,B2¼^c_m¹andB3¼^c_m². Heremis the body mass,k1andk2are the linear and nonlinear stiffness coefﬁ- cients,c1andc2are the linear and nonlinear damping,gis the acceleration gravity andXis the relative vertical displacement of the mass. The parametersYandXare, respectively, the amplitude and the frequency of the external excitation, whilebm²

andmare the acceleration amplitude and the frequency of the parametric excitation, respectively. The parametric frequency

mis supposed to be large comparing to the external frequencyX.

Note that the particular case of linear stiffness (B₁= 0) and basic nonlinear damping (b= 0) has been studied in[3]and it was shown that the nonlinear viscous damping component,B3, has a beneﬁcial effect on vibration isolation. Our purpose here is to study the inﬂuence of the nonlinear parametric damping amplitude,b, on vibration isolation of system(1)taking into account the nonlinear stiffness (B1–0).

Equation(1)contains a slow dynamic due to the external excitation and a fast dynamic produced by the frequencym. To investigate the effect of the rapid parametric excitation on the slow dynamic, we use the method of direct partition of motion [5–12]. This method consists in introducing two different time scales, a fast timeT0=mtand a slow timeT1=t, and splitting upX(t) into a slow partz(T1) and a fast part/(T0,T1) as

XðtÞ ¼zðT1Þ þ/ðT0;T1Þ; ð2Þ

wherezdescribes the slow main motions at time-scale of oscillations and/stands for an overlay of the fast motions. The fast part/and its derivatives are assumed to be 2p-periodic functions of fast timeT0with zero mean value with respect to this time, so thathX(t)i=z(T1) whereh i ₂¹_pR2p

0 ðÞdT0deﬁnes time-averaging operator over one period of the fast excitation with the slow timeT1ﬁxed.

IntroducingD^j_i_@^@jT^jiyields_dt^d¼mD0þD1,^d²

dt²¼m²D²₀þ2mD0D1þD²₁and substituting Eq.(2)into Eq.(1)gives

€zþ/€þx²zþx²/þB1ðzþ/Þ³þB2z_þB2/_ þB2bm²cosðmtÞz_þB2bm²cosðmtÞ/_ þB3ðz_þ/Þ_ ³þB3bm²cosðmtÞðz_þ/Þ_ ³

¼ gþYX²cosðXtÞ: ð3Þ

Averaging(3)leads to

€zþx²zþB1z³þ3B1zh/²i þB1h/³i þB2z_þB2bm²hcosðmtÞ/i þ_ B3z_³þ3B3zh_ /_²i þB3h/_³i þ3B3bm²z^_²hcosðmtÞ/i_ þ3B3bm²zhcosð^_ mtÞ/_²i þB3bm²hcosðmtÞ/_³i

¼ gþYX²cosðXtÞ: ð4Þ

Subtracting(4)from(3)yields

/€þx²/þ3B1z²/þ3B1z/²3B1zh/²i þB1/³B1h/³i þB2/_ þB2bm²cosðmtÞ/_B2bm²hcosðmtÞ/i þ_ 3B3z_²/_ þ3B3z_/_²3B3zh_ /_²i þB3/_³B3h/_³i þ3B3bm²z^_²/_cosðmtÞ 3B3bm²z^_²h/_cosðmtÞi þ3B3bm²z^_/_²cosðmtÞ 3B3bm²zh^_ /_²cosðmtÞi þB3bm²/^_³cosðmtÞ B3bm²h/_³cosðmtÞi

¼ bm²ðB2z_þB3z_³ÞcosðmtÞ: ð5Þ

Using the so-called inertial approximation[5], i.e. all terms in the left-hand side of Eq.(5), except the ﬁrst, are ignored, one obtains

/¼bðB2z_þB3z_³ÞcosðmtÞ: ð6Þ

Inserting/from Eq.(6)into Eq.(4), using that hcos²T0i= 1/2, and neglecting terms of orders greater than three inz, give the equation governing the slow dynamic of the motion

€zþx²zþB1z³þB2z_þBz_³þHzz_²¼ gþYX²cosXt; ð7Þ

whereB¼B3ð1þ³₂B²₂b²m²ÞandH¼³₂B1B²₂b². 3. Frequency response and transmissibility

In this section we examine analytically the effect of nonlinear parametric damping on the slow dynamic(7)by perform- ing a perturbation method. Introducing a bookkeeping parameterand scalingY=Y,B=B,B₁=B₁,B₂=B₂andH=H, Eq.(7)reads

€zþx²z¼ gþ½YX²cosXtB1z³B2z_Bz_³Hzz_²: ð8Þ

We seek a two-scale expansion of the solution in the form

zðtÞ ¼z0ðT0;T1Þ þz1ðT0;T1Þ þOð²Þ; ð9Þ

whereTi=ⁱt.

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In the case of the principal resonance, i.e.X=x+r, whereris a detuning parameter, standard calculations yield the ﬁrst-order solution

zðtÞ ¼ g

x²^þ^a^cosð^X^tcÞ þOðÞ ð10Þ

and the modulation equations of the amplitudeaand the phasec

a_ ¼^YX₂_x²sinðcÞ ^B₂²a³₈Bx²a³; ac_ ¼ r³₂^B_x¹^g5²

h i

a³₈^B_x¹þ^H₈^x

a³þ^YX₂_x²cosðcÞ:

8<

: ð11Þ

Periodic solutions of Eq.(8)correspond to stationary solutions of the modulation equation(11), i.e.a_ ¼c_¼0. These stationary solutions are given by the following algebraic equation

s²₂þs²₃

a⁶þ ð2s1s2þ2s3s4Þa⁴þs²₁þs²₄

a² YX² 2x

!2

¼0 ð12Þ

wheres1¼^B₂²,s2¼^3B₈^x²,s3¼^3B₈_x¹þ^H₈^xands4¼^3B₂¹_x^g²5r. On the other hand, the relationship between displacement transmissibility (TR) and the system parameters is deﬁned by

TR¼X Y¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þa

Y cosðcÞ

²

þ a Y

2

sin²ðcÞ r

: ð13Þ

We consider the case of a quarter-car model such that the supported massm= 240 kg and we focus attention on the case of softening stiffness (k1= 160000 N/m,k2=30000 N/m³) and hardening damper (c1= 250 N s/m,c2= 25 N s³/m³). InFig. 1, we illustrate the relative amplitude of motion a versus the frequency X, as given by Eq. (12), for x= 25.82, m= 35, Y= 0.11 and b= 0.0001. For validation, the analytical prediction (solid line) is compared to the numerical integration (crosses) obtained by using a Runge–Kutta method.

Fig. 2shows the effect of the parametric excitation amplitudebon the TR, as given by Eq.(13). This ﬁgure reveals that increasingb, the transmissibility is reduced over the whole frequency range.

InFig. 3, we illustrate, for comparisons, the TR versusrfor different cases of viscous damping. The standard caseB3= 0, b= 0 corresponds to the linear damping, the caseB3= 0.1,b= 0 is related to the basic nonlinear damping in the absence of the nonlinear parametric damping, as considered in[4], while the case under study,B3= 0.1,b= 0.01, is the general case con- cerned with the nonlinear basic and parametric damping.

Fig. 1.Amplitudeaversusr. Analytical prediction: solid line, numerical simulation: crosses.

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4. Conclusions

This work proposes a technique to minimize transmitted vibrations to a support structure. The strategy is based on adding a nonlinear parametric viscous damping to the basic cubic nonlinear damper. The analytical predictions, based on the direct partition of motion and on the multiple scale technique, shown clearly that increasing the amplitude of the parametric damping can enhance signiﬁcantly the vibration isolation over the whole frequency range. This strategy of adding parametric damping to the vibration isolation device improves vibration isolation with respect to the case where the nonlinear damping is time-independent.

Fig. 2.Transmissibility versusrforY= 0.11,m= 35 and for different values ofb.

Fig. 3.Transmissibility versusrfor different cases of viscous damping. Continuous (or squares) line for classical linear damping, dashed line for basic nonlinear damping[4]and dotted line for nonlinear fast parametric damping.

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