Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 1, pp. 61-72, Warsaw 2007 REGULAR AND CHAOTIC VIBRATIONS OF A VIBRATION-ISOLATED HAND GRINDER Jan Łuczko Piotr Cupiał Urszula Ferdek Institute of Applied Mechanics, Cracow University of Technology e-mail: jluczko@mech.pk.edu.pl The paper is concerned with qualitative analysis of a non-linear model describing vibration of a vibration-isolated hand grinder. A discontinu- ous description of grinding forces is introduced, which accounts for the possible separation of the grindingwheel from the object during the pro- cess. Eight non-linear ordinary differential equations are obtainedwhich describe dynamics of the system. Numerical analysis is done using me- thods of numerical integration and the Fast Fourier Transform. The in- fluence of selected parameters on the character of vibration is studied and somemeasures are calculated which characterize the quality of the vibration isolation system. Key words: vibrations, chaos, grinding, vibration isolation, non-linear 1. Introduction Harmful vibrations duringmechanical processing (such as grinding ormilling) have to be avoided since they deteriorate the quality of the product as well as have a negative effect on the human-operator. The main sources of the- se vibrations are kinematic and inertial excitations (Alfares and Elsharkawy, 2002; Gradǐsek et al., 2001; Karube et al., 2002; Łuczko andMarkiewicz, 1986; Suh et al., 2002). In order to reduce vibration levels transmitted to the ope- rator, vibration isolation systems aremounted between the tool body and the handle. In the case of passive vibration isolation systems their parameters are usually selected using linear models. However, for large vibration amplitudes it is necessary to account for non- linear phenomena, such as brought about e.g. by the loss of contact of the grinding wheel with the object being worked (Łuczko et al., 2003). Moreover, 62 J. Łuczko et al. it is desirable to determine the influence of parameters that undergo changes during the grinding process, such as rotational speed or pressure on the tool handle, on dynamic characteristics of the system. 2. Model of the system Figure 1 shows a schematic view of a hand grinder equipped with a vibration isolation system. The following elements have been taken into account in the model: tool body (1) (with the subassembly motor-spindle-grinding wheel), handle (2) and object being processed (base) (3). The resilient connections between the tool body and the handle represent the passive vibration isolation system, whereas the flexible elements which link the body to its surroundings are a simplified model of the operator interaction. The flexible connections consist of extension-compression springs of stiffnesses cj (j =1,2) and torsion springs of stiffnesses kj, which can be considered as a result of the reduction of any resilient connections to the point at a distance lnj from the respective centre ofmass Sn (n=1,2). In a similarway, one candefineparameterswhich describe energydissipation.Assuming that thedampingmatrix is proportional to the stiffness matrix, the damping of a given connector is given by a single coefficient εj. In order to limit the displacements of the handle relative to the body, a motion limiter (4) is introduced into the model. The properties and geometry of the motion limiter are determined by the parameters c4, ε4, δ, l14, l24. Fig. 1. Model of the system: 1 – body, 2 – handle, 3 – workpiece, 4 – limiter It has been assumed that the tool body and the handle can undergo a ge- neralmotion in space, with the exclusion of longitudinal and torsional degrees of freedom.Therefore, longitudinal and torsional vibrations are excluded from Regular and chaotic vibrations... 63 the analysis. The kinematic excitation is accounted for by defining parame- ters m0 (the unbalanced mass) and e0 (eccentricity), and assuming that the rotational speed remains constant. Introducing moving co-ordinate systems with origins at the centres of mass Sn of subassemblies: the body with the rotating elements (n=1) and the handle (n=2), motion of the system can be described by specifying the co-ordinates xn and yn of points Sn and the angles αn and βn, which for small vibrations describe rotations of the local axes with respect to the fixed ones. Fig. 2. Kinematic relations: (a) grinding forces, (b) reaction of the limiter The case of processing of a flat surface is being considered. It has been assumed that when the grinding wheel remains in contact with the base (for xA = x1 + lα1 > 0), a linear relationship T = fN holds between the tan- gential and normal components of the grinding force (Fig.2a). Additionally, it has been assumed that the normal reaction force N is described by the Voigt-Kelvin model, but it cannot take on negative values. By introducing the notation s= d/dt, this reaction force is given by the following formula N(xA,sxA)= c3(1+ε3s)xAH(xA)H[(1+ε3s)xA] (2.1) where H is the Heaviside step function. In a similar way, the normal reaction force of the limiter (Fig.2b) is written as R(rB,srB)= c4(1+ε4s)rBH(rB − δ)H[(1+ε4s)rB] (2.2) Here, it hasbeenassumedthat the impact takes placewhen rB = √ x2B +y 2 B > >δ, where xB =(x1+l14α1)−(x2+l24α2) yB =(y1−l14β1)−(y2−l24β2) (2.3) The Cartesian components of the limiter reactions are calculated as Rx =R xB rB Ry =R yB rB (2.4) 64 J. Łuczko et al. Using the laws of change of momentum and angular momentum about the centres of mass Sn, motion of the system can be described by the following set of eight second-order ordinary differential equations m1s 2x1+ c1(1+ε1s)(x1−x2− l11α1+ l21α2)+Rx+N =m0e0Ω 2cosΩt m1s 2y1+ c1(1+ε1s)(y1−y2+ l11β1− l21β2)+Ry +T =m0e0Ω 2 sinΩt I1s 2α1− I0Ωsβ1+(1+ε1s)[k11α1−k21α2− c1l11(x1−x2)]+ +Rxl14+Nl=m0e0lΩ 2cosΩt I1s 2β1+ I0Ωsα1+(1+ε1s)[k11β1−k21β2+c1l11(y1−y2)]+ −Ryl14−Tl=−m0e0lΩ 2 sinΩt m2s 2x2− c1(1+ε1s)(x1−x2− l11α1+ l21α2)+ c2(1+ε2s)(x2− l22α2)+ −Rx =Qx (2.5) m2s 2y2− c1(1+ε1s)(y1−y2+ l11β1− l21β2)+ c2(1+ε2s)(y2+ l22β2)+ −Ry =Qy I2s 2α2− (1+ε1s)[k21α1− (k1+ c1l 2 21)α2− c1l21(x1−x2)]+ +(1+ε2s)(k22α2− c2l22x2)−Rxl24 =Mα I2s 2β2− (1+ε1s)[k21β1− (k1+ c1l 2 21)β2+ c1l21(y1−y2)]+ +(1+ε2s)(k22β2+ c2l22y2)+Ryl24 =−Mβ Here k11 = k1+ c1l 2 11 k21 = k1+ c1l11l21 k22 = k2+ c2l 2 22 (2.6) In equations (2.5), m1 and m2 are respectively themasses of subassemblies 1 and 2, I1 and I2 are moments of inertia of the subassemblies with respect to the x-axis (or the y-axis, thanks to axial symmetry) passing through points S1 and S2, and I0 is themoment of inertia of the rotating elements about the axis of symmetry. The generalised forces Qx, Qy, Mα and Mβ represent the operator action on the system. When the grinding wheel remains in contact with the object, one can assume that these forces remain constant. The case when the grinding wheel loses contact with the object is more complex, espe- cially as far as the generalized forces Qy and Mβ are concerned. The human operator is an active system since he reacts to changes in the working condi- tions. During grinding, the operator tries to adjust themagnitudes of forces so as to equilibrate the corresponding components of the grinding forces. When the loss of contact occurs, the operator tries on onehand tobring the tool back into contact with the object (without changing the values of Qx, Mα), and, on the other hand, counteracts suddenmovements of the tool in the direction tangent to the surface being processed by a sudden change (in the model an Regular and chaotic vibrations... 65 instant change) of the force Qy and the moment Mβ. To account for this behaviour, the following simplified relations are used in the model Qy = fQxH(N) Mα =Qxd Mβ =Qyd (2.7) where d is the distance of the resultant operator’s action from the centre of mass S2. The analysis is done in a dimensionless form, and non-dimensional quan- tities are used where the amplitudes are calculated relative to the effective amplitude of inertial excitation e = m0e0/(m1 +m2), angles are taken rela- tive to e/l (l = l13) and the non-dimensional time τ = ω0t is referred to the circular frequency ω0 = √ c2/(m1+m2) of the simplified linear model. The equations, when written in the dimensionless form, depend on the following parameters ω= Ω ω0 ∆= δ e q= Qx c2e µn = mn m1+m2 ρn = In mnl2 χ= I0 I1 γj = cj c2 κj = kj c2l2 ζj = εjω0 2 λnj = lnj l λ0 = l0 l η= d l (2.8) and on the coefficient of dry friction f. In equations (2.8), the index n is the number of the subsystem (n = 1,2), and the index j corresponds to the number of the flexible element (j = 1,2,3,4). Moreover, the following conditions hold: γ2 =1, λ21 =λ0−λ11, λ24 =λ0+λ14 and µ1+µ2 =1. By introducing the state vector u= [x1 e , y1 e , lα1 e , lβ1 e , x2 e , y2 e , lα2 e , lβ2 e ]⊤ (2.9) the system of equations (2.5) can be written in a compact matrix form Mu”+(G+2ζ1C1+2ζ2C2)u ′+(C1+C2)u=p(τ)+q+r+s (2.10) Here, the matrices M, G, C1 and C2 are respectively the mass-, gyroscopic- and stiffness matrices, and p(τ) is the vector of inertial excitation p(t)=ω2[cosωτ,sinωτ,cosωτ,−sinωτ,0,0,0,0]⊤ (2.11) The vectors q, r and s describe non-linear terms, respectively related to the model of the operator q= [0,0,0,0,q,fqH(n),ηq,−ηfqH(n)]⊤ (2.12) 66 J. Łuczko et al. the model of the limiter r= [−rx,−ry,−λ14rx,λ14ry,rx,ry,λ24rx,−λ24ry] ⊤ (2.13) and themodel of the grinding forces s= [−n,−fn,−n,fn,0,0,0,0]⊤ (2.14) In order to calculate n, rx and ry one makes use of formulae (2.1)-(2.4), where dimensionless components of the state vector (2.9) and the respective non-dimensional parameters (e.g. parameters γ3, γ4, 2ζ3, 2ζ4 and ∆ in lieu of c3, c4, ε3, ε4 and δ) are introduced. The matrix M is diagonal and has the following form M= diag[µ1,µ1,µ1ρ1,µ1ρ1,µ2,µ2,µ2ρ2,µ2ρ2] (2.15) The only non-zero terms of the gyroscopic matrix G are given by G43 = =−G34 =χµ1ρ1ω. In order to define the stiffness matrix, we introduce an auxiliary matrix Γ j(λnj,λmj)=      γj 0 −γjλmj 0 0 γj 0 γjλmj −γjλnj 0 κj +γjλnjλmj 0 0 γjλnj 0 κj +γjλnjλmj      (2.16) The matrices C1 and C2, which represent respective flexible links can be written as the following block matrices C1 = [ Γ1(λ11,λ11) −Γ1(λ11,λ21) −Γ1(λ21,λ11) Γ1(λ21,λ21) ] C2 = [ 0 0 0 Γ2(λ22,λ22) ] (2.17) 3. Results Results of the qualitative analysis of the vibration-isolated hand grinder will be described below, with emphasis put on the selection of some of the para- meters of the vibration isolation system and on the explanation of physical phenomena brought about by the percussive nature of the grinding forces. The results have been obtained using methods of numerical integration and theFast FourierTransform,which have been used in studyingnon-linear oscil- lations, e.g. byAwrejcewicz and Lamarque (2003).More details about the use of spectrum analysis to determine the character of vibrations have been di- scussed inFerdek andŁuczko (2003). In discussion of the results, the criterion Regular and chaotic vibrations... 67 index J1 (or J2) of the efficiency of the vibration isolation system will be used. It is defined as the ratio of the rms values of accelerations (respectively velocities) at the point B on the handle (front grip – see Fig.1), calculated for the tool with- and without the vibration isolation system. By analysing the influence of the parameters γ1, κ1 and λ11 on the value of the criterion indices Jk, estimates of the optimumparameter values from the point of view ofminimising vibration levels have been found.The character of vibration has also been studied, depending on the values of these parameters and the values of parameters (ω,q)which characterize the grindingprocess.The following set of values of parameters have beenused in thenumerical calculations: µ1 =0.8, µ2 =0.2, ρ1 = 1.5, ρ2 =0.5, χ=0.1, f =0.5, γ1 =1.5, γ2 =1, γ3 =400, γ4 = 100, κ1 = 1.5, κ2 = 1, ζ1 = 0.1, ζ2 = 0.5, ζ3 = 0.05, ζ4 = 0.05, λ0 = 0.5, λ11 = 0.25, λ21 = 0.25, λ22 = 0, λ13 = 0.5, λ23 = 1.5, λ14 = 0.5, λ24 =1, ∆=10, η=0.75, q=10, ω=5. Figure 3a shows thedependenceof the criterion index J1 on theparameters γ1, κ1 (for ω = 5 and q = 10), and Fig.3b illustrates the zones of different vibration types in the (γ1,κ1) plane. The minimum of J1 (and also of J2) is achieved in the neighbourhood of the point γ1 = 1.5, κ1 = 1.5. As seen in Fig.3b, in the neighbourhood of this point, sub-harmonic vibrations of type 1:2 take place. Fig. 3. Influence of parameters c1 and κ1 (ω=5, q=10, λ11 =0.25) on: (a) criterion index J1, (b) zones of different vibration types In a similar way, Figs. 4a and 4b illustrate the influence of the parameters ω and λ11. In a relatively wide neighbourhood of λ11 = 0.25, the index J2 (Fig.4a) assumes a value close to the minimum, regardless of the rotational speed ω. This holds true, even though for high values of ω the vibrations are chaotic, as is seen in Fig.4b. The type of vibrations and the value of the criterion index depend also on the remaining parameters, characterizing both the model of the tool and the 68 J. Łuczko et al. Fig. 4. Influence of ω and λ11 (c1=1.5, λ11 =1.5, q=10) on: (a) criterion index J2, (b) zones of vibration types grinding process. A proper choice of the vibration isolation system requires evaluation of the sensitivity of the solution to changes in these parameters. Below, the discussion is limited to the influence of the rotational speed and the operator pressure on the tool handle, since these two parameters have been found to have the biggest effect on the dynamic behaviour of the sys- tem operator-tool-base. The operator controls the grinding processmainly by changing the force q. The rotational speed ω also undergoes changes as a result of the limited power of the motor. Fig. 5. Influence of ω and q (c1 =1.5, κ11 =1.5, λ11 =0.25) on: (a) efficiency index J2, (b) vibration zones Figures 5a and 5b illustrate, in the same format as in the previous figures, the influence of the parameters ω and q on the criterion index J2 and on the character of vibrations. By studying the zones in the (ω,q) plane in Fig.5b, one can note that only for small values of the non-dimensional rotational spe- ed ω the grindingprocess takes placewithout separation of the grindingwheel Regular and chaotic vibrations... 69 from themotion limiter. This zone becomes a little wider as the dimensionless pressure q increases. For higher values of ω, which is the case of most prac- tical applications, there appear zones of sub-harmonic vibrations of an order increasing with ω separated by narrow zones of sub-harmonic (mostly of ty- pe 1:4) or chaotic oscillations. The criterion index (Fig.5a) decreases with the rotational speed, which signifies that the selected vibration isolation system is efficient also in the case of irregular vibrations. Fig. 6. Bifurcation diagram (c1 =1.5, κ11 =1.5, λ11 =0.25, q=5) Figure 6 shows a bifurcation diagram obtained using the stroboscopic me- thod by taking the displacement u1 at selected time instants (every excitation period). This diagram corresponds to the section of the (ω,q) plane shown in Fig.5b taken for q = 5. In the bifurcation diagram, one can distinguish alternate regions of sub-harmonic and chaotic oscillations, where the order of the sub-harmonic vibrations increases with the rotational speed. Fig. 7. Time history, spectrum, phase portrait and trajectory of the point B on the handle (c1 =1.5, κ11 =1.5, λ11 =0.25, q=5): (a) 4T-periodic vibrations, ω=4.6, (b) chaotic vibrations, ω=5.0 Figure 7 shows time histories of the displacement xB(t), phase portraits (xB,x ′ B) and frequency spectra S(ν) of the signal xB for two values of the excitation frequency ω. For the first value of ω, the period of oscillations is four times of that of the excitation (sub-harmonic oscillations of type 1:4), 70 J. Łuczko et al. the spectrum has pronounced maxima at points ν/ω = k/4, where k is a natural number, and the corresponding curve in the phase plane is closed. For the second value of ω, the oscillations are chaotic, which is manifested in the irregular time history by the continuous spectrum and irregular phase portraits. For chaotic vibrations, the phase portraits are difficult to interpret. Much more information is gainedbymaking stroboscopicportraits orPoincarémaps. For the subsequent regions of chaotic vibrations shown in the bifurcation dia- gram in Fig.6, one obtains different shapes of fractals shown in Fig.8. Fig. 8. Stroboscopic portraits of chaotic oscillations (c1 =1.5, κ11 =1.5, λ11 =0.25, q=5): (a) ω=5.0, (b) ω=6.5, (c) ω=7.5, (d) ω=9.0 4. Conclusions In the paper an approach to the modelling and to the qualitative analysis of vibrations of a non-linear system in the presence of unilateral constraints resulting both from the grinding process and the presence of the motion li- miter is discussed. The adopted model of the grinding process can predict sub-harmonic and chaotic vibrations brought about by the percussive charac- ter of grinding forces. The numerical analysis has shown that the parameters ω and q have a pronounced qualitative effect onmotion of the system.A little less important, but not negligible, is the effect of the parameters f, γ3, ζ3. The regions of different vibration types shown in Fig.3-Fig.5 as well as the bifurcation diagram (Fig.6) do not undergo qualitative changes for different values of the parameters γ3, ζ3, f that describe the model of the grinding forces. By increasing the values of the parameters f and γ3 or by reducing the value of ζ3, the zones of chaotic vibrations tend to get bigger. The inclusion of the motion limiter in the model prevents excessive static displacements and allows an efficient choice of the parameters of the passive vibration isolation system without a need of imposing additional constraints. Withabadlydesignedvibration isolation system, theparameters of themotion limiter can have an important effect on the type of vibration in the system. Regular and chaotic vibrations... 71 The proposed approach has proved to be very efficient for qualitative ana- lysis of vibrations of non-linear systems. A similar approach can be used in the study of models of other mechanical machining processes. References 1. AlfaresM., ElsharkawyA., 2002,Effect of grinding forces on the vibration of grindingmachine spindle system, International Journal ofMachineTools and Manufacture, 40, 2003-2030 2. Awrejcewicz J., Lamarque C.-H., 2003, Bifurcation and Chaos in Non- Smooth Mechanical Systems, World Scientific Series of Nonlinear Science, Se- ries A, 45, Singapore 3. Ferdek U., Łuczko J., 2003, Qualitative analysis of vibro-impact systems [in Polish],Czasopismo Techniczne, 6-M, 1-16 4. Gradǐsek J., Govekar E., Grabec I., 2001, Chatter onset in non- regenerative cutting: A numerical study, Journal of Sound and Vibration, 242, 829-838 5. Karube S., Hoshino W., Soutome T., Sato K., 2002, The non-linear phenomena in vibration cutting system. The establishment of dynamic model, International Journal of Non-Linear Mechanics, 37, 541-564 6. Łuczko J., Markiewicz M., 1986, Dynamical analysis of the system: tool- base-operator in hand grinding with a pneumatic tool [in Polish], Zeszyty Na- ukowe Politechniki Rzeszowskiej, 31,Mechanika, 12, 51-55 7. Łuczko J., Cupiał P., Ferdek U., 2003, Regular and chaotic vibration in themanual grindingprocess,Proc. of the 7thConference onDynamical Systems – Theory and Applications, 1, 207-214 8. SuhC.S., KhurjekarP.P., YangB., 2002,Characterisationand identifica- tion of dynamic instability inmilling operation,Mechanical Systems and Signal Processing, 16, 853-872 Drgania regularne i chaotyczne w procesach obróbki wibroizolowaną ręczną szlifierką Streszczenie Praca dotyczy analizy jakościowej nieliniowegomodelu, opisujący drgania wibro- izolowanej ręcznej szlifierki. Model opisano układem ośmiu równań różniczkowych zwyczajnych drugiego rzędu. Wprowadzono nieciągły opis sił skrawania, uwzględ- niający możliwość chwilowego oderwania się ściernicy od obrabianego przedmiotu. 72 J. Łuczko et al. Do analizy wykorzystano procedury matematycznego całkowania skojarzone z algo- rytmami szybkiej transformaty Fouriera. Zbadano wpływ parametrów na charakter drgań oraz wyznaczono pewne wskaźniki jakości działania zastosowanego układu wi- broizolacji. Manuscript received August 3, 2006; accepted for print August 18, 2006