Role of parameters in the stochastic dynamics of a stick-slip oscillator

Transcription

1 Proceeing Series of the Brazilian Society of Applie an Computational Mathematics, v. 6, n. 1, 218. Trabalho apresentao no XXXVII CNMAC, S.J. os Campos - SP, 217. Proceeing Series of the Brazilian Society of Computational an Applie Mathematics Role of parameters in the stochastic ynamics of a stick-slip oscillator Roberta Lima 1 Mechanical Engineering Department, PUC-Rio, Rio e Janeiro, RJ Rubens Sampaio 2 Mechanical Engineering Department, PUC-Rio, Rio e Janeiro, RJ Abstract. In this paper a parametric analysis of a sample of responses of a ry-friction oscillator is performe in orer to construct a statistical moel. The system consists of a simple oscillator moving on a base with a rough surface. Due to this roughness, the mass is subject to a ry-frictional force moele as a Coulomb friction. It is consiere that the base has an impose stochastic bang-bang motion which excites the system in a stochastic way an, inuces stochastic stick-slip oscillations. The base velocity is moele by a Poisson process for which a probabilistic moel is fully specifie. The system response is compose by a ranom sequence alternating stick an slip-moes. With realizations of the system, a statistical moel is constructe for this sequence. Statistics an histograms of the ranom variables which characterize the stick-slip process are estimate. The objective of the paper is to analyze how these estimate statistics an histograms vary with the system parameters, i.e., to make a parametric analysis of the statistical moel of the stick-slip process. Keywors. Stick-slip, Dry-friction oscillator, Stick uration, Statistical moel, Parametric analysis 1 Introuction Dry-friction appears in several situations, as in rilling process an in mechanical gear systems. Despite the great number of papers in the area, few of them aress the problem with a stochastic approach. The majority of the references only uses a eterministic approach. They o not iscuss or quantify the uncertainties that are involve in the ynamics, although the ry friction force itself presents an inherent ranom behavior [1]. Therefore, a stochastic approach is the ieal way to aress problems with ry-friction. In this paper, we analyze the ynamics of a ry-friction oscillator which moves over a base with a rough surface. The base has an impose stochastic bang-bang motion which excites the system in a stochastic way. The non-smooth behavior of the ry-friction force [2] associate with the non-smooth stochastic base motion inuces in the system stochastic stick-slip oscillations. The system response is compose by a ranom sequence alternating stick an slip-moes. To characterize it, we construct a statistical moel, in which the 1 robertalima@puc-rio.br 2 rsampaio@puc-rio.br DOI: 1.4/ SBMAC

2 Proceeing Series of the Brazilian Society of Applie an Computational Mathematics, v. 6, n. 1, variables of interest of the ranom sequence are moele as ranom variables [3, 4]. To estimate statistics an histograms of the system responses, samples of the ranom sequence of stick an slip-moes are compute by the integration of the ynamic equation of the system using inepenent samples of the base motion. The objective of the paper is to analyze how the estimate statistics an histograms vary with the system parameters, i.e., to make a parametric analysis of the statistical moel of the stick-slip process. 2 Dynamics of a stick-slip oscillator The system analyze is compose by a simple oscillator (mass-spring) moving on a rough surface, as shown in Fig. 1. The roughness inuces a ry-frictional force between Figure 1: Stick-slip oscillator. the mass an the base which is moele as a Coulomb friction. Due to this friction moel, the resulting motion of the mass can be characterize in two qualitatively ifferent moes: the stick-moe (in which the mass an base have the same velocity uring an open time interval) an the slip-moe, in which mass an base have ifferent velocities. The position of the mass over the base is represente by x an its equation of motion is m ẍ(t)+k x(t) = f(t), (1) where m is the mass, k is the spring stiffness an f is the frictional force between mass an base. During the slip-moe, it is assume that f(t) = mgµsgn(v(t) ẋ(t)), where v is the base spee, g is the acceleration of gravity an µ is the friction coefficient (assume to be constant). Thus, uring the slip-moe, the value of the frictional force, f, is known. Its absolutevalueisequaltothemaximumfrictionforce, f max = µmg. Duringthestick-moe, ẋ(t) = v(t), an thus equation of motion Eq. (1) can be rewritten as m v +k x(t) = f(t). The value of the frictional force uring the stick-moe varies an it is confine to the interval f max f f max. When the base spee is constant in time, uring the stickmoe we have f max k x(t) f max. Then, once in a stick-moe, the mass stays moving with the base until x(t) = mgµ k in case of positive base velocity, or until x(t) = mgµ k in case of negative base velocity. Observe that uring the stick-moe, the moulus of the elastic force increases up to the limit value f max. Remark that the uration of the stick-moe is boune an its maximum value is max = 2 mgµ kv. For the slip one can make no preiction, in principle it can last forever. 3 Construction of a probabilistic moel for the base motion Consiering that the ry-friction oscillator has an impose stochastic bang-bang motion, we propose to moel its velocity as a Poisson process, with constant rate λ, rep- DOI: 1.4/ SBMAC

3 Proceeing Series of the Brazilian Society of Applie an Computational Mathematics, v. 6, n. 1, resente by V. We consier that V is constant by parts an assumes only two values: 1, m/s an 1, m/s. A realization of such stochastic process consists of point events in time which represents the instants in which occur changes of the velocity sign of the base motion. The parameter λ represents the expecte value of number of changes per unit of time. As V is moele as a Poisson process, the instants of change are given by ranom variables which can be orere as < Y 1 < Y 2 < Y 3 <... From this sequence, it is possible to efine the inepenent ranom variables W 1 = Y 1 an W j = Y j Y j 1, with j 2. Each of them has exponential probability ensity function p Wj (t) = 1 [, ) (t)λe λt, with mean 1/λ. The ranom variable W j, with j 2, inicates the waiting time between two consecutive change of the velocity sign of the base motion. Observe that a higher λ correspons to a smaller average waiting time. The first change is at W 1, the secon at W 1 +W 2, et cetera. Due to the bang-bang base motion, if the mass is in the stick-moe in the instant just before the iscontinuity on the base velocity, it must be in the slip-moe in the instant just after the iscontinuity. Thus, the stick is interrupte by the iscontinuities on the base velocity, as if the ynamics were reinitialize; all previous information lost. 4 Construction of a statistical moel of the stick-slip process As it was assume that the base motion is uncertain, the response of the stochastic stick-slip oscillator is a ranom process which presents a sequence alternating stick an slip-moes. Define a time interval for analysis, the variables of interest in the ynamics are moele as stochastic objects. We have two iscrete ranom variables, which are the number of time intervals in which stick occurs (S T ) an the number of time intervals in which slip occurs (S L ). We have also iscrete ranom processes, which are the instants at which the sticks begin (T 1,,T ST, where the subscripts 1,,S T inicate the orer that they occur), the uration of the sticks (D 1,,D ST ), the instants at which the slips begin (L 1,,L SL ) an, the uration of the sticks (H 1,,H SL ). Figure 2 shows a sketch of the sequence of sticks an slips in the system response. Observe that we count the first slip just after the first stick. Figure 2: Sketch of the sequence of sticks an slips in the system response with S T = S L. Strategy for parametric analysis Two system parameters are consiere in the parametric analysis of the statistical moel of the stick-slip process. One of them is relate to the probabilistic moel of the DOI: 1.4/ SBMAC

4 Proceeing Series of the Brazilian Society of Applie an Computational Mathematics, v. 6, n. 1, base motion λ an the other is the friction coefficient µ of the friction force. We performe numerical simulations combining ifferent values of these parameters. To λ, 4 values were selecte nonuniformly in the interval [.1, 3, ]. To µ, 8 values were selecte nonuniformly in the interval [., 7,]. For each combination of λ an µ, the ynamics equations were integrate 2, times using inepenent realizations of the base movement. A previous convergence stuy was evelope to etermine the acceptable number of realizations. In total, 64, integrations were performe. In orer to compute all these integrations, we aopte the strategy of parallelization of the simulations. Using a cluster compose of sixteen computers, as shown in Fig. 3, the computation time necessary to perform the 64, integrations was approximately, ays. Without the parallelization, it woul be nee approximately 2, years to compute the integrations, which is infeasible. For computation, uration t a was chosen as secons. For the integration, it was use the function oe4 of the Matlab software, which applies the Runge-Kutta 4th/th-orer metho as time-integration scheme with a varying time-step algorithm. The maximal step size is equal to 1 4 secons, an the relative an absolute tolerance are equal to 1 9. The values of the parameters are m = 1. kg an k = 4. N/m. The initial conitions of the system were moele as inepenent ranom variables, uniform istribute over [ 1, 1]. Figure 3: Parallelization of the simulations in the parametric analysis. 6 Influence of λ in the statistical moel First, we analyze the influence of λ in the statistical moel of the stick-slip process for a fixe value of µ. We took µ =.. In [3], it is possible to verify that for a fixe value of λ the uration of sticks are ientically istribute. Then, we take D 1,,D ST as ientically istribute ranom variables an, we call them as D. To unerstan the influence of λ in D, we investigate the variation of the normalize histograms for ifferent values λ between.1 1/s an 1. 1/s. The results are shown in Fig. 4. For λ =.1 1/s, we verify that the normalize histogram of D has one pick near to the maximum stick uration, which is max = 2 mgµ kv = 2. s. However as λ grows, this peak isappears, an the support of the normalize histogram is reuce. We conclue that the estimate mean of the stick DOI: 1.4/ SBMAC

5 Proceeing Series of the Brazilian Society of Applie an Computational Mathematics, v. 6, n. 1, 218. uration ecreases as λ grows. To quantify this ecay, we plotte the estimate mean of the stick uration ˆµ D as a function of λ, shown in Fig.. 1 λ=.1 1 λ=.2 1 λ= λ= λ= λ= λ= λ= λ= Figure 4: Normalize histograms constructe with 2, samples of the uration of the first stick for six ifferent values of λ % CI ˆµ D s Figure : Estimate mean of the stick uration, ˆµ D, an 9 % confience interval as a function of λ. The graph of the estimate mean of the number of sticks ivie by the uration t a, as a function of λ is shown in Fig. 6(a). Observing it, we verify that the mean of the total number of sticks increases as λ grows. As we known from Fig. that the stick uration ecreases as λ grows, we conclue that as λ grows, the system response presents on average a higher number of sticks, but these sticks have on average a lower uration. Given that, we may ask what happens with the total time of stick as λ grows. Computing the sum of the uration of all sticks, an iviing it by the uration t a, we get the total time of stick in relation to the time interval analyze. We call this ranom variable R. DOI: 1.4/ SBMAC

6 Proceeing Series of the Brazilian Society of Applie an Computational Mathematics, v. 6, n. 1, The question is what happens with the mean of R when λ grows. Is it possible that, on average, a higher number of sticks with lower uration give us a higher total time of stick than a lower number of sticks with a higher uration? To answer this question, we plotte the estimate mean of R as function of λ. The obtaine graph is shown in Fig. 6(b). Observing it, we verify that the mean of total time of stick reaches a maximum value, ˆµ R, which is almost 8% an occurs at λ = 3.8 1/s. The conclusion is that for λ [., 3.8] 1/s, the increase of the number of sticks causes the increase of the total time of stick, even though the sticks have a lower uration. The larger number of sticks compensates its shorter uration. However, when λ excees 3.8 1/s, the increase of the number of sticks compensates no more the reuction of its uration, so ˆµ R ecreases. 1 9% CI ˆµS T [%] ta 1 8 9% CI ˆµ R [%] (a) (b) Figure 6: (a) Estimate mean of the number of sticks, an 9 % confience interval as a function of λ. (b) Estimate mean of the total stick uration, ˆµ R, an 9 % confience interval as a function of λ. 7 Influence of the friction coefficient on the statistical moel To quantify the influence of µ on the statistical moel of the stick-slip process, we analyze the graph of the estimate mean of the total time of stick ˆµ R as a function of λ, Fig. 7. Observing it, we verify that the maximum of the estimate mean of total time of stick, ˆµ R, grows as µ increases. For µ =., the maximum is 1.%, an for µ = 7., the maximum is 81.89%. Besies, we verify also that the maximum is always reache for λ in the short interval [3., 3.8] 1/s. From these results, we conclue that the friction coefficient has a lot of influence on ˆµ R. However, very little influence on the position of the maximum. 8 Conclusions The obtaine results showe that the normalize histograms of the stick uration are sensitive to variations on λ an µ. As λ grows, the estimate mean of the stick uration ecreases. Besies of this, the system response presents on average a higher number of DOI: 1.4/ SBMAC

7 Proceeing Series of the Brazilian Society of Applie an Computational Mathematics, v. 6, n. 1, ˆµ R % Figure 7: Estimate mean of the total stick uration ˆµ R as a function of λ for ifferent values of µ. The ˆµ R is highlighte for each µ with markers. sticks, however these sticks have on average a lower uration. The relative total time of stick, R, showe that for λ lower than approximately 3.8 1/s, the increase of the number of sticks causes the increase of the total time of stick, even though the sticks have a lower uration, in a way that the large number of sticks compensates its shorter uration. However, when λ excees approximately 3.8 1/s, the increase of the number of sticks oes not compensate the reuction of its uration anymore, so the total time of stick ecreases. From the analysis of the influence of µ, we conclue that maximum of the estimate mean of total time of stick, ˆµ R, grows as µ increases. However, the value of µ oes not change the behavior of R in relation to the position of its maximum. Acknowlegments The authors acknowlege the support given by FAPERJ, CNPq an CAPES. References [1] M. Bengisu an A. Akay. Stick-slip oscillations: ynamics of friction an surface roughness, Journal of the Acoustical Society of America, 1(1):194 2, [2] R. Lima an R. Sampaio. Stick-moe uration of a ry-friction oscillator with an uncertain moel, Journal of Soun an Vibration, 33:29 271, 21. [3] R. Lima an R. Sampaio. Construction of a statistical moel for the ynamics of a base-riven stick-slip oscillator, Mechanical Systems an Signal Processing, 91:17 166, 217. [4] R. Lima an R. Sampaio. Parametric analysis of the statistical moel of the stick-slip process, Journal of Soun an Vibration, 397:141 11, 217. DOI: 1.4/ SBMAC