Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 2, pp. 473-482, Warsaw 2009 SENSITIVITY ANALYSIS OF PLANAR MECHANISMS Krystyna Romaniak Department of Architecture, Cracow University of Technology, Cracow, Poland e-mail: kroman@usk.pk.edu.pl When designing a new mechanism, the designer has to know its kine- matic parameters: position, velocity, acceleration, the number of design options, specific positions, motion ranges of particular links, effects of manufacturing tolerances of links and clearances in kinematic pairs on functioning of the mechanism. The latter parameter is associated with sensitivity of themechanism,understoodas its ability to respond to even minimal variationsof the driving linkposition.The sourceof information about these parameters becomes the objective function obtained thro- ugh the application of themodificationmethod. This study explores the effects of clearances in kinematic pairs and structure of the mechanism on its sensitivity. Key words: kinematics, planar mechanisms, sensitivity 1. Introduction Sensitivity (kinematic efficiency) of a mechanism is understood as its ability to respond to evenminimal variations of the driving link position. Sensitivity thus defined is affected by several factors (Młynarski and Romaniak, 2002). There are certain aspects of sensitivity to be considered: • structural sensitivity-determining factors that include structure of ame- chanism, coordinates of external kinematic pairs, links lengths, • technical (constructional, technological) sensitivity affected by rigidity or flexibility of links, clearances in kinematic pairs and manufacturing tolerances of links, • operational sensitivity, depending on service conditions, load types and wearing of the kinematic pairs. 474 K. Romaniak This study investigates the structural sensitivity ofmechanisms and certa- in aspects of constructional sensitivity, focusing on the effects of structure and clearances in kinematic pairs. Themodificationmethod is presented (Młynar- ski andRomaniak, 2001),whichfinally yields theobjective functionunderlying the sensitivity analysis of planar mechanisms. 2. Modification method The modification method is applied to kinematic analysis of high class me- chanisms, enabling us to determine positions, velocity and accelerations of particular links and kinematic pairs. The method uses transformation of the kinematic chain such that the high class unit should become a mechanism of the II class. That is achieved by disconnecting the unit at one ormore points, which involves one or more decision variables. Further analysis is performed to check whether the links can be then reconnected. This relationship for the mechanism disconnected by the link lk is given by the objective function of the form fc =∆lk = lk− √ (xi+1−xi)2+(yi+1−yi)2 → 0 (2.1) where (xi,yi), (xi+1,yi+1) are coordinates of kinematic pairs of the link lk. Zeroing of the objective function implies such a value of the decision variable for which the mechanism can be reconnected. Modification of a high class mechanism yields amechanism of the II class, incorporating k groups of the II class. Depending on the number of k, k ite- rative steps are required to obtain objective functions, the number of which is 2k. Figure 1a shows a mechanism of the III class analysed using the modi- fication method. A kinematic unit of the III class, series 3 is extracted and the modification procedure is applied accordingly. When all external pairs are connected to the base, the angle α2 is chosen as the decision variable (the driving link) and the mechanism is disconnected by excluding link 5 (Fig.1b). For the thus obtainedmechanism of the II class, comprising a single kine- matic unit of the II class, we get the coordinates of kinematic pairs C,D, F and angles α3, α4 in accordance with the formula Sensitivity analysis of planar mechanisms 475 Fig. 1. Modified kinematic unit of the III class, series 3 (b), derived from the mechanism of the III class (a) xC =xB + l2cosα2 yC = yB + l2 sinα2 xC + lCD cosα3+ l4cosα4−xE =0 yC + lCD sinα3+ l4 sinα4−yE =0 (2.2) xF =xC + lCF cos(α3+κ3C) yF = yC + lCF sin(α3+κ3C) Two objective functions are obtained, in accordance with the formula f5c = l5− √ (xG−xF)2+(yG−yF)2 (2.3) The following data: xA = 40, yA = 50, xE = 170, yE = 50, xG = 170, yG = 230, l1 = 63.25, l2 = 72.11, lCD = 94.87, lCF = 94.87, l4 = 72.8, l5 = 120, κ3C = 0.927rd , yield a plot of the objective function graphed in Fig.2. Zeros of the objective functions imply those values of the decision varia- ble α2 for which the mechanism can be reconnected. Two design options are obtained for α2 =−0.5317rd and α2 =0.54583rd shown in Fig.3. 3. Sensitivity of planar mechanisms Thefirstmotion of the driving links eliminates the clearances in kinematic pa- irs. During thismovement the position of the driving link does not lead to any 476 K. Romaniak Fig. 2. Plot of an objective function Fig. 3. Third class mechanisms obtained for α2 =−0.5317rd (a) and α2 =0.54583rd (b) variations of the driven link position. Sensitivity analysis uses the sensitivity factor expressed as (Romaniak, 1998) µ=1− ∆αi ∆ =1− b ∆ √ 1+ ( d dαi fc(αi) )2 d dαi fc(αi) (3.1) where ∆αi is the distance between the corresponding zeros of the extreme objective functions, ∆ is the operating range of the driven link in a given design option, b is the distance between the extreme objective functions and fc(αi) is the objective function corresponding to nominal dimensions. Sensitivity analysis of planar mechanisms 477 The following denotation is used βk = b ∆ βs = √ 1+(y′)2 y′ (3.2) where y′ = dfc(αi)/dαi. There are several determinants of βk, including the design, manufactu- ring technology, mode of operation, wearing of kinematic pairs. This study investigates only the prognosticated constructional clearances. The factor βk also depends on the structure of the mechanism. Furthermore, the condition is imposed 0¬βkβs < 1 (3.3) to preclude situations when the objective function is tangent to the abscissa axis. Because of the clearance between the shaft and the bearing bushing, the shaft might occupy various positions in the opening. The extreme shaft posi- tions correspond to the objective function forming a band. Actually, the shaft assumes themiddle position and the real objective function is contained inside the band. The plot of the function is ambiguous, mainly because of the acting forces and the presence of friction. If initially the shaft occupied the mini- mum position, and motion was performed such that the shaft should assume another extreme position due to eliminated clearances, then the insensitivity range should stretch from one to another extreme function. That is the gre- atest insensitivity level to be achieved by the given mechanism. The quotient present in the formula becomes the measure of maximal insensitivity of the mechanism, depending on the type of fit applied in kinematic pairs. Actu- ally, it expresses the minimal sensitivity of the mechanism, which in fact is greater. The objective function was analysed for two design options of the third class mechanism shown in Fig.3, taking into account the clearances in kine- matic pairs (8H8/f9). In the case of design option I (Fig.3a), the motion range of the driven link 5 is ∆=0.6529rd and for option II (Fig.3b) is ∆=0.4914rd . For the angle α1 varyingby 0.2rd , the followingparameterswere computed:angle α5, derivatives of the objective function at its zeros, the width of the band of the objective function ∆α5, sensitivity factor µ.Results are summarised inTable1 (the value f ′ c1(α5) is denoted by y ′ in equation (3.2)). Tests were performed for those values of the angle α1 for which all the ob- jective functions have zeros. Taking into account the clearances in kinematic pairs, the motion range will be reduced. If one regarded those values of the 478 K. Romaniak Table 1. Angles α1, α5, derivative of the objective function, width of the band of the objective function ∆α5, sensitivity factor µ Design option I No. α1 α5 f ′c1(α5) ∆α5 µ 1 5.23622 3.7992 −12.517 0.01986 0.98318 2 5.4 3.9745 −80.7428 0.00065 0.99945 3 5.6 4.113 −95.9276 0.00013 0.99989 4 5.8 4.2249 −94.4389 0.00065 0.99945 5 6 4.3015 −89.1443 0.0011 0.99907 6 6.2 4.3274 −92.3389 0.0013 0.9989 7 0 4.3226 −97.1505 0.00128 0.99891 8 0.2 4.2833 −112.822 0.00113 0.99904 9 0.4 4.2182 −127.76 0.00095 0.99919 10 0.6 4.1391 −138.774 0.00082 0.9993 11 0.8 4.0540 −147.212 0.00075 0.99937 12 1 3.9709 −156.376 0.00073 0.99938 13 1.2 3.8964 −169.766 0.00077 0.99935 14 1.4 3.8348 −190.257 0.00084 0.99929 15 1.6 3.7871 −219.938 0.00095 0.9992 16 1.8 3.7515 −260.472 0.00107 0.9991 17 2 3.7255 −314.031 0.00119 0.99899 18 2.2 3.70663 −385.245 0.00133 0.99888 19 2.4 3.693 −486.257 0.00147 0.99875 Design option II No. α1 α5 f ′c1(α5) ∆α5 µ 1 −0.6185 4.9647 10.14966 0.0269 0.94522 2 −0.6 4.924 29.3017 0.0063 0.9872 3 −0.4 4.81262 70.9574 0.00089 0.9982 4 −0.2 4.79756 74.6982 0.00007 0.9999 5 0 4.844 76.9359 0.00011 0.9998 6 0.2 4.9393 74.6412 0.00012 0.9998 7 0.4 5.0822 55.84301 0.00072 0.9985 8 0.53685 5.2867 5.65659 0.0221 0.9551 angle α1 for which the objective function expressing the minimal dimension had zeros, the mechanism might assume the singular position much earlier. The tangent point of the objective function to the abscissa axis determines Sensitivity analysis of planar mechanisms 479 those values of the decision variable for which the mechanism should assume the singular position. Figure 4 shows the band of the objective function for α1 = 0.53797rd (design option II), when one of the extremal objective func- tions does not vanish anywhere. Fig. 4. Band of the objective function for α1 =0.53797rd (design option II) Testswereperformedto checkhowthe typesof fit inkinematic pairs should affect the width of the function band and sensitivity of the mechanism. For comparative purposes, according to the fixed hole principle, the basicmovable fit 8H7/g6 and movable loose fit 8H11/a11 were assumed. Figures 5 and 6 show bands of the objective function for the considered fit types used in the mechanism in Fig.3b. The band width is changed and so is the motion range of the driving link and, hence, the sensitivity factor. Fig. 5. Objective function band for α1 =0.5604rd (8H7/g6) 480 K. Romaniak Fig. 6. Objective function band for α1 =0.5676rd (8H11/a11) 4. The effects of mechanism structure on its sensitivity The structure of the investigated mechanism is modified, assuming that only external kinematic pairs A,G,J, driving link1anddriven7 shouldnot change (Fig.7). Fig. 7. Mechanism of the IV class Themotion range of driven link 7 for one of the design options is 0.511rd. For α1 varying stepwise by 0.2rd, the following parameters were obtained: angle α7, derivatives of the objective function at its zeros, distance ∆α7, sensitivity factor µ. Results are summarised in Table 2. Sensitivity analysis of planar mechanisms 481 Table 2.Derivatives of the objective function for various angles α1 Design option I No. α1 α7 f ′c1(α7) ∆α7 µ 1 −0.4647 4.4706 17.5807 0.02556 0.0759 2 −0.4 4.447 64.7012 0.0051 0.81536 3 −0.2 4.4464 117.6593 0.0031 0.88878 4 0 4.4565 139.2007 0.00266 0.90396 5 0.2 4.4593 143.9309 0.00253 0.90867 6 0.4 4.4182 124.0576 0.00286 0.89655 7 0.585 4.4727 17.0135 0.0267 0.03349 5. Final remarks The analysis of sensitivity data for various design options reveals that sensi- tivity is affected by the fit type and, hence, the clearances in kinematic pairs, mechanism structure and the length of particular links. That is associated with the width and inclination of the objective function band. The motion range of the driven link appears to be themajor determinant of the sensitivity factor. The larger the rotation angle of the driven link is, the weaker are the effects of clearances in kinematic pairs.When the motion range of the driven link is rather small, the type of fit in kinematic pairs will strongly affect the mechanism sensitivity. The motion range of the driven link is also associated with lengths of particular links. It appears that the width of the band of the objective function, which changes when the driving link is in motion, depends on the inclination of the objective function, and hence is closely associated with the derivative of the objective function. If the inclination angle of the tangent line to the plot of the objective function at its zero is close to 0 (i.e., the derivative of the objective function at this point is near zero), then the distance ∆α1 shall be the largest whilst sensitivity of the mechanism–the lowest. Besides, the width of theobjective functionbanddependson the typeof fit assumed forkinematic pairs. Modification of the mechanism structure leads to variations of themotion ranges of particular links and, hence, to changes in the sensitivity of the me- chanism. It also affects the inclination angle of the objective function, which in turn causes changes of the objective function band width and, finally, the mechanism sensitivity. 482 K. Romaniak References 1. Młynarski T., Romaniak K., 2002, Analiza wrażliwości mechanizmów pła- skich, Prace Naukowe Instytutu Konstrukcji i Eksploatacji Politechniki Wro- cławkiej, 297-302 2. MłynarskiT.,RomaniakK., 2001,Themethodization of examinationof the mechanismsofhigh structural complexity,Machine andMechanismTheory,36, 6, 709-715 3. PN-89/M-02102,Układ tolerancji i pasowań 4. Romaniak K., 1998, Ocena wrażliwości złożonych mechanizmów, Materiały XVIOgólnopolskiej Konferencji Naukowo-Dydaktycznej TMM, Rzeszów-Jawor, 119-124 5. Wierzbicki A., 1997, Modele i wrażliwość układów sterowania, WNT, War- szawa Badania wrażliwości mechanizmów płaskich Streszczenie Konstruktor już na etapie projektowania mechanizmu chce wiedzieć, jakie bę- dą jegowłaściwości kinematyczne. Interesują go położenia, prędkości, przyspieszenia, liczba opcji montażowych, położenia osobliwe, zakresy ruchu poszczególnych ogniw, wpływ tolerancji wykonania ogniw oraz luzówwparach kinematycznych na działanie mechanizmu. Ostatnia z wymienionych właściwości łączy się z pojęciem wrażliwości mechanizmu, rozumianej jako jego zdolności do reagowania na najmniejszą zmianę położenia ogniwa napędzającego. Źródłem informacji na tematwszystkichwymienio- nychwłaściwości jest funkcja celu otrzymanawwyniku stosowaniametodymodyfika- cji.Wniniejszymopracowaniuprzedstawionowpływ luzówwparachkinematycznych i strukturymechanizmu na jego wrażliwość. Manuscript received November 20, 2008; accepted for print December 29, 2008