This paper deals with nonlinear mechanics of an elevator brake system subjected to uncertainties. A deterministic model that relates the braking force with uncertain parameters is deduced from mechan- ical equilibrium conditions. In order to take into ac- count parameters variabilities, a parametric probabilis- tic approach is employed. In this stochastic formalism, the uncertain parameters are modeled as random vari- ables, with distributions speci ed by the maximum en- tropy principle. The uncertainties are propagated by the Monte Carlo method, which provides a detailed sta- tistical characterization of the response. This work still considers the optimum design of the brake system, for- mulating and solving nonlinear optimization problems, with and without the uncertainties e ects. Keywords elevator brake system nonlinear mechan- icsnonlinear optimization uncertainty quanti ca- tionparametric probabilistic approach P. Wolszczak G. Litak Lublin University of Technology, Faculty of Mechanical En- gineering, Nadbystrzycka 36, PL-20-618 Lublin, Poland E-mail: p.wolszczak@pollub.pl E-mail: g.litak@pollub.pl P. Lonkwic State School of Higher Education, Mechanical Engineering Faculty, Pocztowa 54, PL-22-100 Chelm, Poland E-mail: plonkwic@gmail.com A. Cunha Jr Universidade do Estado do Rio de Janeiro, Nucleus of Mod- eling and Experimentation with Computers { NUMERICO, Rua S~ ao Francisco Xavier, 524, Rio de Janeiro, 20550-900, RJ, Brazil E-mail: americo@ime.uerj.br S. Molski AGH University of Science and Technology, Department of Rope Transport, Mickiewicza 30, PL-30-059 Krakow, Poland E-mail: molski@agh.edu.pl1 Introduction Considerations regarding the construction of lifting de- vices (design of cranes), and in particular brake sys- tems, are not often discussed in the scienti c literature, as the corresponding dynamical conditions are dicult to determine. The rst person who addressed the is- sue of the impact of safety gears construction on the braking distance was Elisha Graves Otis, who in 1853 built the rst safety gears and subjected them to exper- imental studies [1, 2]. Subsequent works on this subject, studying several aspects of cranes mechanics, appeared in the twentieth century and were published in journals and conference proceedings [3, 4, 5, 6, 7, 8, 9]. For instance, Yost and Rothen uth [3] describe how to con gure a lifting device and how to select the cor- rect components. These issues constituted a signi cant contribution to the development of the con guration of lifts, ensuring a trouble-free operation. Lonkwic [4] presents a comparative analysis of the operation of slip safety gear of his own design study with the models by leading European manufacturers. Deceleration (braking time) values obtained in the phys- ical experiment are analyzed. In [5], the same author and collaborators address, by means of wavelet analysis, how certain variables in uence on the operating condi- tions of deceleration. A similar analysis is presented in [6], which concerns the selected braking parameters of CHP2000 and PP16 type chaters using the analysis of recursive patterns. Regarding the study of elevator systems with un- certain operating conditions, the literature is not very comprehensive. The only works in this line known by the authors are [7, 8], developed by Kaczmarczyk et al., who attempt to analyze the behavior of balance ropes due to harmonic and stochastic excitations, and 2 P. Wolszczak et al. Col on et al. [9], who calculate the propagation of the rail pro le uncertainties and study the e ectiveness of a closed-loop control law. Even with the scienti c literature being rich in stud- ies regarding the behavior of vehicles brake systems under changing operating conditions [10, 11, 12], it is surprising that, to the best of authors' knowledge, no similar research description on lift brakes has been re- ported up to the present date. Only general provisions contained in the British Standard Document BS EN 81 [13, 14] are to be found. Thus, seeking to ll this gap, the present work aims to study the in uence of some operating conditions on the eciency of an elevator brake device, by analyzing how the operating parameters underlying uncertainties propagate through the mechanical system. In particu- lar, the cam brake angle and the spring reaction force are of interest. In addition to quantifying the e ects of uncertainties in operating conditions, this study also aims to achieve a robust design of a brake system by solving a nonlinear optimization problem, considering (or not) the uncertainty e ects. The remaining part of this paper is organized as fol- lows: the deterministic modeling of the lift brake sys- tems under study is presented in section 2. In section 3, the construction of a consistent stochastic model of uncertainties to deal with variabilities in the uncertain parameters, is presented. Two optimization problems, one classical and one robust, which seek to nd an opti- mal design for the brake system are formulated in sec- tion 4. In section 5, numerical experiments are reported and discussed. Finally, in section 6, concluding remarks are presented. 2 Deterministic modeling 2.1 Elevator brake system A schematic representation of the CHP 2000 safety gear, used by a typical friction crane brake system, is pre- sented in Figure 1. It consists of a monolithic steel body (1), in which a braking cam (5) is mounted on a bolt. The braking cam moves the brake roller (2), which has a knurled surface. This irregular surface is responsible for the braking process and for the cooperation with the guide roller surface. The brake roller moves over the braking cam surface until it contacts the lift guide (6). It is a free movement that does not cause any brak- ing e ect. The second part of the braking process is in constant contact with the lift guide surface. An illustration of a typical friction crane brake sys- tem used by lifting devices is shown in Figure 2, which indicates the di erent components of the mechanism Figure 1 Schematic representation of the CHP 2000 safety gear used by a friction crane brake system, which consists of the following parts: 1 - steel body; 2 - brake roller; 3 - thrust plate; 4 - spring package; 5 - braking cam; 6 - lift guide. (see the caption). It consists of two safety gears, mov- ing on the lift guides, connected to each other to ensure simultaneous operation when the brakes are activated by means of a trigger lever. A lift safety gear is placed in the frame, under a safety gear cabin. Its trigger is attached to the trigger lever, in which the end is con- nected to the rope speed limiter. In the upper part of the elevator shaft there is a speed limiter supervising the work of the safety gear, and in its lower part load responsible for causing the proper tension of the speed limiter rope is located. The speed limiter triggers the braking process when the nominal speed of the elevator car is increased by 0.3 m/s. After exceeding the nomi- nal speed, the speed limiter is blocked, and the rope is also immobilized. During the movement of an elevator car with locked components, the lever is moved in the opposite direction to the cabin, triggering the brake safety gear roll. In its turn, the roll is pressed against the guide causing elastic deformation towards the thrust plate located on the other side of the disk spring package, which induces the loss of energy in the accelerating mass. Therefore, the disc spring package is responsible for a variable force that presses the roller to the guide during the braking process. Robust optimization and uncertainty quanti cation in the nonlinear mechanics of an elevator brake system 3 Figure 2 Illustration of a typical friction crane brake system used by lifting devices, which consists of the following parts: 1 - safety gear cabin; 2 - trigger lever; 3 - safety gears connector; 4 - rope speed limiter. 2.2 Mathematical model The design assumptions and safety gear structure shown in Figure 1 are taken into account to construct a math- ematical model that relates the braking force with geo- metric parameters and other characteristics of the me- chanical system. In this sense, equilibrium conditions for the system are deduced below. A free-body diagram can be seen in Figure 3, which shows a schematic representation of the forces (in red) acting on the safety gear steel body, and the underlying geometric dimensions (in blue). A balance of forces and the moments acting on the steel body gives rise to the equations N4+Fs+Rx= 0; (1) T4Ry(Fg+Fb)=2 = 0; (2) FsaRyl+N4mT4n= 0; (3) whereFsis the spring reaction force; Fbis the inertial force from the cabin and lifting capacity; Fgis the cabin and lifting capacity weight; T4is the friction force be- tween the guide and brake retaining block, and N4is the corresponding normal force; RxandRyare the reaction forces in the braking cam rotation point; while a,l,m andnare geometric dimensions depicted in Figure 3. The forces (in red) acting on the wedge during brak- ing and immediately after stopping the cabin, until the safety gears are unlocked by technical maintenance of the lift, are shown in Figure 4, along with the relevant geometric dimensions (in blue). A new balance of forces and moments provides T2+N1cos +T1sin FsRx= 0; (4) Figure 3 Illustration of the forces (in red) acting on the safety gear steel body and the underlying geometric dimen- sions (in blue). Figure 4 Illustration of the forces (in red) acting on the wedge and the underlying geometric dimensions (in blue) and Fsa+T1bN1cN2dT2e= 0; (5) whereT1andT2are friction forces between brake ele- ments (roller and cam), N1andN2are the correspond- ing normal forces; is the braking cam angle; b,c,d andeare other geometric dimensions of the problem, shown in Figure 4. In Figure 5 the reader can see characteristic dimen- sions (in blue) and forces (in red) acting on the brake roller inside the safety gear. Now the balance of forces gives N3T2N1cos T1sin = 0; (6) T3N2+T1cos N1sin = 0; (7) 4 P. Wolszczak et al. Figure 5 Illustration of the forces (in red) acting on the brake roller inside the safety gear and and the underlying geometric dimensions (in blue). whereT3andN3respectively denotes the frictional and the normal forces between brake roller and the guide. The frictional forces T1,T2andT4are, respectively, related to the normals N1,N2andN4through a Coulomb friction model, so that T1=1N1; (8) T2=2N2; (9) T4=4N4; (10) where1,2and4are friction coecients. On the other hand, the relationship between the fric- tional force T3and the normal N3takes into account the plastic deformation occurring in the contact between brake roller and the guide, so that T3=f RN3; (11) wherefandRare geometric dimensions de ned in Fig- ure 5. The vertical reaction force Rycan be obtained from Eqs.(2) and (3), Ry=T4+(Fg+Fb) 2; (12) Ry=FsaN4m+T4n l; (13) which, when combined together with Eq.(10), allows one to express N4as N4=(Fg+Fb)l=2Fsa 4(n+l)m: (14) Similarly, from suitable manipulations of Eqs.(6), (8) and (9), it can be concluded that N3=2N2+ (1sin + cos )N1; (15)as well as, from Eqs.(5), (8) and (9), it is possible to obtain N2=aFs+ (b1c)N1 d+e2; (16) which, in combination with Eqs.(1), (4), (8) and (9), gives rise to N1=N4a2Fs d+e2 1sin + cos +2(b1c) d+e2: (17) The braking force, resulting from the joint superpo- sition of all frictional forces, is given by Fh=T1+T2+T3+T4; (18) which, with aid of Eqs.(8) to (11), can be rewritten as Fh=1N1+2N2+f RN3+4N4: (19) Note that, once normal forces N1,N2,N3andN4 present explicit dependence on geometric dimensions, frictional coecients, and non-frictional forces, the brak- ing forceFhis also a function of these parameters, i.e., Fh=Fh( ;Fs;Fg;Fb;1;2;4; a;b;c;d;e;f;l;m;n;R ):(20) 3 Stochastic modeling The angle and the spring reaction force Fsare sub- jected to variabilities during the operation conditions of the brake system, so that their actual values may be very di erent from the nominal project values. Since they are the critical parameters for the brake system eciency, studying the e ect of such variabilities on the braking force is essential for a good design. In this way, a parametric probabilistic approach [15, 16] is em- ployed here to construct a consistent stochastic model for uncertain parameters andFs. 3.1 Probabilistic framework Let (;;P) be the probability space used to describe the model parameters uncertainties [16, 17], where is the sample space, a- eld over, andP:![0;1] a probability measure. In this probabilistic setting, the parameters and Fsare respectively described by the random variables X1:!RandX2:!R, which are lumped into the random vector X:!R2, which associates to each elementary event 2a vector X= (X1;X2). Robust optimization and uncertainty quanti cation in the nonlinear mechanics of an elevator brake system 5 The probability distribution of Xis characterized by the mappX:R2![0;+1), dubbed the joint proba- bility density function (PDF). The mean value of Xis de ned in terms of the ex- pected value operator EfXg=Z Z R2xpX(x)dx; (21) in which x= (x1;x2) anddx=dx1dx2. 3.2 Maximum entropy principle To perform a judicious process of uncertainty quanti - cation, it is essential to construct a consistent stochastic model for the random vector X, that represents the un- certainties in andFsin a rational way, trying to be unbiased as possible. In this sense, in order to avoid pos- sible physical inconsistencies in the probabilistic model, only available information must be used in its construc- tion [16, 17]. When this information materializes in the form of a large set of experimental data, the standard procedure is to use a nonparametric statistical estima- tor to infer the joint distribution of X[16, 17]. However, if little (or even no) experimental data for X1andX2is available, as is the case of this paper, such construction can be done based only on known theoretical informa- tion, with the aid of the maximum entropy principle [16, 17]. The available theoretical information about the ran- dom parameters X1andX2encompasses a range of possible values for each of then, i.e., (X1;X2)2[ 1; 2][ 1; 2][0;2](0;+1);(22) as well as their nominal values X1andX2, that are assumed to be equal to their mean values, i.e., EfX1g=X12[ 1; 2]; (23) EfX2g=X22[ 1; 2]: (24) This information is translated into the statistical language through the normalization condition Z 2 1Z 2 1pX(x)dx= 1; (25) and the rst order moment equation Z 2 1Z 2 1xpX(x)dx= (X1;X2): (26) From the information theory point of view, the most rational approach to specify the distribution of Xinthis scenario of reduced information is through the max- imum entropy principle (MaxEnt) [16, 17, 18], which seeks the PDF that maximizes the entropy functional S(pX) =Z 2 1Z 2 1pX(x) lnpX(x)dx; (27) respecting the restrictions (information) de ned by (25) and (26). Using the Lagrange multipliers method it is possible to show that such joint PDF is given by pX(x) =pX1(x1)pX2(x2); (28) with marginal densities pX1(x1) = exp (1011x1)1[ 1; 2](x1); (29) pX2(x2) = exp (2021x2)1[ 1; 2](x2); (30) where10,11,20and21are parameters of the dis- tribution of X, and 1I(x) =( 1; x2I; 0; x62I;(31) denotes the indicator function of the interval I. Note that, since no information relative to the cross statis- tical moments between X1andX2has been provided, MaxEnt provides independent distributions. The parameters 10,11,20and21depend on 1, 2, 1, 2,X1andX2. They are computed through the nonlinear system of equations obtained by replacing (28) in (26) and in the normalization conditions of the marginal PDFs (29) and (30). In a scenario with little information, it is practically impossible not to be biased in choosing a probability distribution. The MaxEnt formalism provides the least biased distribution that is consistent with the known in- formation, therefore constituting the most rational ap- proach [18]. 3.3 Uncertainty propagation The mathematical model relating the braking force Fh with braking cam angle and the spring reaction force Fs, Eq.(20), can be thought abstractly as a nonlinear deterministic functional Mthat maps a vector of input parameters x= ( ;Fs) into a scalar quantity of interest y=Fh, i.e., x7!y=M(x): (32) Thus, if the uncertain parameters andFsare rep- resented by the known random vector X, the braking force becomes the random variable Y=M(X), for which the distribution must be estimated. 6 P. Wolszczak et al. The process of determining the distribution of Y, once the probabilistic law of Xis known, is dubbed un- certainty propagation problem [15, 16], being addressed in this paper via the Monte Carlo simulation [19, 20]. In this stochastic calculation technique, indepen- dent samples of Xare drawn according to the density (28), giving rise to statistical realizations X(1);X(2);;X(): (33) Each of these scenarios for Xis given as input to the nonlinear deterministic map x7!y=M(x), result- ing in a set of possible realizations for the quantity of interest Y(1);Y(2);;Y(); (34) whereY(j)=M(X(j)); j= 1;;. These samples are used to estimate statistics of Ynon-parametrically, i.e., without assuming the PDF shape known [21]. 4 Optimization framework Regarding the improvement of brake system eciency, an optimal design of its components is required. This work addresses this question by solving nonlinear opti- mization problems that seeks to maximize the braking force, using geometric dimensions of the system as de- sign variables. Two optimization approaches are employed. The rst one, named classical , is based on deterministic formal- ism of nonlinear programming [22], while the latter, dubbed robust , takes into account the model parame- ters uncertainties, in order to reduce the optimum point sensitivity to small disturbances [23, 24]. In this framework, a set of two design variables (ge- ometric dimensions) is denoted generically by the vec- tors. The other parameters of the model are denoted generically by x, and the model response is given by the nonlinear map ( s;x)7!y=M(s;x) The quantity of interest to be optimized (objective function) is denoted generically byJ. 4.1 Classical optimization In this classical optimization approach the scompo- nents are employed as design variables, while the brak- ing force is adopted as objective function, i.e., JC(s) =y: (35) The admissible set for this optimization problem is de ned byAC= [smin 1;smax 1][smin 2;smax 2], so that it can be formally stated as nd an optimal design vector sopt C= arg max s2ACJC(s): (36)4.2 Robust optimization In this robust optimization framework, which is based on those shown in [25, 26], the uncertainties are de- scribed according to the formalism of the section 3, where xbecomes the random vector X, and, as a con- sequence,y=M(s;x) becomes the random variable Y=M(s;X). Thus, the robust objective function is not constructed directly from the model response, but with the aid of statistical measures of Y, which aims to guarantee greater stability to small disturbances (robustness) to an opti- mum point. Speci cally, the robust objective function is given by a convex combination between minimum, maximum, mean and standard deviation inverse, so that JR(s) = 1min 2fYg+ 2max 2fYg+ 3EfYg+ 41q EfY2gEfYg2;(37) where 1+ 2+ 3+ 4= 1. Note that by maximizing this robust objective func- tion, it is sought to raise both the lowest and the high- est possible value, the mean, in addition to reducing the dispersion, by reducing the standard deviation. Additionally, in order to avoid excessively small brak- ing forces, the following probabilistic constraint is im- posed P jYj>y 1Pr; (38) whereyis a lower bound for the magnitude of Y, and Pris reference probability. Therefore, the admissible set for the robust opti- mization problem, denoted by AR, is de ned as the sub- set ofACfor which the probabilistic constraint (38) is respected. In this way, the robust optimization problem is formally de ned as nd an optimal design vector sopt R= arg max s2ARJR(s): (39) 5 Results and discussion The simulations reported below, conducted in Matlab, use the following numerical values for the deterministic parameters of the mechanical model: Fg= 50 kN;Fb= 30 kN;1= 0:10;2= 0:10;4= 0:15;a= 55:0 mm; b= 16:6 mm;c= 52:7 mm;d= 34:5 mm;e= 60:7 mm; f= 0:005 mm;l= 49:0 mm;m= 40:0 mm;n= 17:5 mm;R= 29:0 mm. Regarding the two random parameters, the follow- ing information is assumed: [ 1; 2] = [0;18]o; [ 1; 2] = [0;56] kN;X1= 6o; andX2= 42 kN. Robust optimization and uncertainty quanti cation in the nonlinear mechanics of an elevator brake system 7 0 1000 2000 3000 4000 sample index51015 cam angle (degree) sample mean mean std 95% prob. 0 1000 2000 3000 4000 sample index12345 spring force (N)104 sample mean mean std 95% prob. Figure 6 Generated samples and statistical measures for the cam angle (top) and the spring force reaction (bottom). 5.1 Uncertainty quati cation The calculation of the propagation of uncertainties of X= (X1;X2) through the mechanical-mathematical model (32) initially involves the generation of random samples according to the probabilistic model de ned by Eq.(28). A set of 4096 random samples for X1(top) andX2(bottom) can be seen in Figure 6, which also shows some statistics (mean, standard deviation and 95% con dence interval) for this set of values. In Figure 7 the reader can see the statistics shown in Figure 6 compared to the analytical curves for the PDFs ofX1andX2, and histograms constructed with the underlying random samples. It can be observed that the sampling process is well conducted, since the his- tograms and analytical curves present great similarity. Note that the brake cam angle X1is modeled ac- cording to a probability density with a descending expo- nential behavior, which decays slowly between the ends of the support [ 1; 2] = [0;18]o, whereas the spring 0 5 10 15 cam angle (degree)02468 probability density functionPDF mean mean std 95% prob. 1 2 3 4 5 spring force (N) 10401234567 probability density function10-5 PDF mean mean std 95% prob.Figure 7 Probability density functions and statistical mea- sures for the cam angle (top) and the spring force reaction (bottom). reaction force X2is described by probabilistic law with an increasing exponential density, which grows rapidly from the left to the right extreme of [ 1; 2] = [0;56] kN. It is worth noting that, of course, the real system pa- rameters do not follow these probability distributions. These are only approximations of the real distributions, constructed with the aid of the maximum entropy prin- ciple and the available information about these param- eters. However, as in this paper the authors do not have experimental data to infer the real form of these distri- butions, in the light of information theory, the PDFs of Figure 7 are the best that can be inferred. The next step involves the model evaluation in each pair (X1;X2) previously generated, which gives rise to the set of possible values for the braking force Fh, shown in the top part of Figure 8. In the bottom part of the same gure the reader can observe a histogram that es- timates the FhPDF form, as well as a nonparametric 8 P. Wolszczak et al. 0 1000 2000 3000 4000 sample index-50000500010000 braking force (N) sample mean mean std 95% prob. -5000 0 5000 10000 braking force (N)012 probability density function10-4 PDF mean mean std 95% prob. Figure 8 Generated samples (top) and probability density function (bottom), with statistical measures, for the braking force. tting obtained by a smooth curve. Mean, standard de- viation, and a 95% con dence interval can be seen in both, top and bottom gures. To prove that these es- timates are reliable, the authors also show the conver- gence of the mean and standard deviation estimators, as a function of the number of samples, in Figure 9. It may be noted that the PDFs of FhandFshave a very similar shape, suggesting that the mechanical model preserves the shape of the spring reaction force distribution. This result is at the least curious and un- expected, since the angular dependencies introduced in the mechanical model by Eqs.(15) and (17) de ne a structure of multiplicative uncertainty between and Fs, what should make Fhnot invariant with respect to the input distribution. This result suggests that the nonlinearity associated with the alpha parameter is very weak, which causes Fhto behave as an ane map of Fs, and thus to pre- serve the form of its distribution. This hypothesis is 0 1000 2000 3000 4000 number of samples400050006000700080009000 braking force mean 0 1000 2000 3000 4000 number of samples010002000300040005000 braking force std deviation Figure 9 Statistical estimators convergence for the braking force mean (top) and standard deviation (bottom). reinforced by analyzing the system response by keeping distribution and Fssupport xed, while the mean value of the latter parameter is varied, assuming the values equal to for 14 kN, 28 kN1and 42 kN. The prob- ability densities corresponding to these di erent inputs, and the corresponding outputs of the mechanical sys- tem can be seen in Figures 10 and 11, respectively. In all cases the input and output PDFs have the same shape. The results of this study allow one to conclude that uncertainties in parameter does not have signi cant in uence on the braking force behavior. Simulations propagating only uncertainties, not included here be- cause of space limitation, demonstrate such an asser- tion. However, the uncertainty propagation study also shows that the variability of Fscannot be ignored, since it has great in uence on the statistical behavior of Fh. 1For this value, which corresponds to the midpoint of the support, the distribution degenerates into an uniform. Robust optimization and uncertainty quanti cation in the nonlinear mechanics of an elevator brake system 9 1 2 3 4 5 spring force (N) 10401234567 probability density function10-5 PDF mean mean std 95% prob. 1 2 3 4 5 spring force (N) 10400.511.52 probability density function10-5 PDF mean mean std 95% prob. 1 2 3 4 5 spring force (N) 10401234567 probability density function10-5 PDF mean mean std 95% prob. Figure 10 Probability density functions and statistical mea- sures for the spring force reaction with di erent mean values: 14 kN (top); 28 kN (middle); 42 kN (bottom). -5000 0 5000 10000 braking force (N)00.511.5 probability density function10-4 PDF mean mean std 95% prob. -5000 0 5000 10000 braking force (N)01234567 probability density function10-5 PDF mean mean std 95% prob. -5000 0 5000 10000 braking force (N)012 probability density function10-4 PDF mean mean std 95% prob.Figure 11 Probability density functions and statistical mea- sures for the braking force with di erent input mean values forFs: 14 kN (top); 28 kN (middle); 42 kN (bottom). 10 P. Wolszczak et al. classical optimization 50 52 54 56 58 60 geometric dimension a (mm)505152535455 geometric dimension c (mm) 650070007500800085009000 Figure 12 Contour map for the classical objective function and with the optimum point is indicated by a cross. 5.2 Optimization In this section the problem of optimum design of the brake system is addressed. The geometric dimensions s= (a;c) are used as design variables, considering as admissible region 50 a60 mm and 50c55 mm. The optimization problem is solved using the stan- dard sequential programming quadratic (SQP) algo- rithm obtained from Matlab (see chap. 18 of [27]), being the contour map of the objective function (35) illus- trated in Figure 12, which also highlights the optimum point found. Despite the fact that this result o ers a starting point for an optimal project for the brake system, it does not take into account the e ect of the uncertainties underlying the operating conditions, which can consid- erably a ect the system response, as shown in the pre- vious section. In this way, robust optimization presents itself as a natural alternative. For the robust optimization problem the design vari- ables s= (a;c) are considered once more, with the same ranges of admissible values used above. The uncertain- ties in andFsare modeled as in section 3, and the probabilistic constraint is characterized by the param- etersy= 0:5 kN andPr= 5%. The convex weights 1= 2= 3= 2=10 and 4= 4=10 are adopted in the robust objective function. This second problem is much more complex because the constraint to be satis ed is nonconvex, o ering ad- ditional challenges to the numerical solution procedure. But for the values described above the SPQ algorithm is able to nd a solution. The reader can see the contour map of the prob- abilistic constraint (38) in Figure 13, while Figure 14 presents the contour map for the robust objective func- probabilistic constraint 50 52 54 56 58 60 geometric dimension a (mm)505152535455 geometric dimension c (mm) -0.032-0.03-0.028-0.026-0.024Figure 13 Contour map for probabilistic constraint with the optimum point is indicated by a cross. robust optimization 50 52 54 56 58 60 geometric dimension a (mm)505152535455 geometric dimension c (mm) 1800200022002400260028003000 Figure 14 Contour map for the robust objective function with the optimum point is indicated by a cross. tion (37). Although the objective function still main- tains a smooth appearance, the problem gains a non- convex status by the irregular forms of constraint. Once the robust objective function takes into ac- count other design criteria than in (35), its behavior is di erent from the classical objective function shown in Figure 12, thus having a di erent optimal point. It is also worth noting that this second formula- tion of the optimization problem considers the e ects of uncertainties, which in a realistic system are always present, thus o ering a design option more suitable for projects that cannot ignore such variabilities. 6 Summary and conclusions This work presents a study regarding the optimization and uncertainty quanti cation of an elevator brake sys- tem. The paper starts from an original construction of a safety gear for the brake, for which a mechanical- Robust optimization and uncertainty quanti cation in the nonlinear mechanics of an elevator brake system 11 mathematical model is constructed. Studies involving the quanti cation of the braking force uncertainties due to the variability in the brake cam angle and the spring reaction force are presented, showing that spring force uncertainties are more signi cant. The paper also fo- cuses on the optimal design of an elevator brake system, showing through the solution of a robust optimization problem that operating conditions uncertainties can sig- ni cantly in uence its eciency. Funding This research was nanced in the framework of the project Lublin University of Technology-Regional Ex- cellence Initiative, funded by the Polish Ministry of Sci- ence and Higher Education (contract no. 030/RID/2018/19). The fourth author acknowledge the nancial support given by the Brazilian agencies Carlos Chagas Filho Re- search Foundation of Rio de Janeiro State (FAPERJ) under grants E-26/010.002.178/2015 and E-26/010.000.805/2018 and Coordena c~ ao de Aperfei coamento de Pessoal de N vel Superior - Brasil (CAPES) - Finance Code 001. Compliance with ethical standards Con ict of Interest The authors declare that they have no con ict of inter- est.