The paper deals with a study case regarding the determination of adequate assembly tolerances. The study concerned the assembly of two bearings on  an axle. Determining the adequate tolerances of the sizes chains is based on Lagrange’s multipliers use.  The optimization criterion for determining the assembly tolerances is the minimization of the product achievement cost. The relation between tolerances and cost will be revealed in a graphic componen

Constantin MILITARU

Constanţa RĂDULESCU

Liviu Marius CÎRŢÎNĂ

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                                                  Fiabilitate si Durabilitate - Fiability & Durability     nr.1/2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  109 STUDIES REGARDING THE DETERMINATION OF THE ADEQUATE ASSEMBLY TOLERANCES DEPENDING ON THE MANUFACTURING COSTS OF THE PARTS PROCESSED  ON CLASSIC MACHINE TOOLS  Lecturer Eng.Constanţa RĂDULESCU1 Prof.  Dr.Eng. Liviu Marius CÎRŢÎNĂ1 Prof. Dr. Eng Constantin MILITARU2 1Constantin Brancusi University of Targu Jiu             2Polytechnic University of Bucharest  Abstract:  The paper deals with a study case regarding the determination of adequate assembly tolerances. The study concerned the assembly of two bearings on an axle. Determining the adequate tolerances of the sizes chains  is  based  on  Lagrange’s  multipliers  use.    The  optimization  criterion  for  determining  the  assembly tolerances is the minimization of the product achievement cost. The relation between tolerances and cost will be revealed in a graphic component.   Keywords: assembly, tolerances, costs.  1.  Introductory notions. In order to determine and allocate the adequate assembly tolerances and not only, a significant  factor  is  the  function  between  cost  and  tolerance.  In  time,  several  algebraic functions that make the connection between cost and tolerance have been described. The mathematic models for approximating the tolerance – cost relation are: - energetic pattern (1969):                                    C = A+ B Tk - exponential pattern (1972):                                C = A∙e-B T    - mutual quadratic pattern (1973):                        C = A+ B /Tk  - mutual energetic pattern (1975):                        C = A+ B T-k - mutual pattern (1990):                                        C = A+ B /T   - combined exponential-linear pattern (1991):      C = A+ B +B1T+B2T2 +B312T3    All the relations described above could not consider all the factors that contribute to manufacturing costs, referring here to the cost of materials, excise costs, cutting tool costs, etc. Therefore, in time various objective functions have been proposed as they include various types of costs.    Unfortunately, all the information regarding a part manufacturing cost is not published by companies, because even the companies that use the same machine may have various  costs  from  one  section  to  another  (manpower,  materials,  devices,  etc).  A  study between  cost  and  tolerance  is  very  necessary  for  cutting  processes  for  various  values  of nominal sizes.  The study regarding the processing cost depending on tolerances is approached in this paper for classic machine tools and this is why it cannot be applied for modern machine tools.      In order to be able to determine the minimal cost depending on elements tolerances,  Lagrange’s method has been developed, supposing a cost function of the form: C= A+ B /T2                                                                      (1) It was extended, using the cost function as follows:                 n i tol funcTt funcT i i,..., 1 0 . . cos .                           (2)                                                   Fiabilitate si Durabilitate - Fiability & Durability     nr.1/2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  110                            ) n ,..., 1 i ( 0 T TT TBAT2asm2jikjjjij                                                  ) ,..., 1 (22 n iTB ki kii i                                       Eliminating λ, expressed by term T1 (arbitrarily selected):                                          221211 11iikk ki ii TB kB kT                                            (3)                   Replacing for every Ti in the assembly tolerance sum, we get:                                          22 21221 12121iikk ki iasm TB kB kT T                               (4)                     A similar derivation based on the algebraic method, that is the product of allocated tolerances sum, is the equation:                                                         111111 111iikk ki iasm TB kB kT T                         (5) The unknown in this relation is T1. The value of T1 will repeat itself until both parts of the equation are equal, achieving tolerances with minimal costs.   2.  Case study.  In order to determine adequate tolerances using Lagrange’s method, we present, as a case  study,  the  assembly method of bearings on an axle, fig.1. Two brackets maintain the distance between the bearings in order for them to be able to fix on  the  axle.  Tolerances accumulation  in  the  assembly results  in  the  end  in  the variation  of  the  compensating element. In the case sizes chain solving  is  made  through  the compensation  method.  For  this case  positive  compensation  is necessary.                                Fig. 1.  Bearings assembly on an axle.  The initial tolerances for parts B, D, E, and F are selected from tolerance guides, like for instance those presented in table 1 (tolerances taken from the specialized literature or from the STAS for certain accuracy classes and various values of nominal sizes).  The histogram indicated the typical tolerance field for several technological operations for making the parts: facing, boring, enlarging, reaming, milling, broaching, rectification, etc. Every row of the table corresponds to a different field of values of the nominal size. For instance, a part is faced at a nominal size of Ø30mm it can be created at a tolerance field from 0,03 mm  to 0,20 mm. The way these tolerances are achieved depends on the number of passing, the type of the machine tool, its roughness, the fastening devices of the part, the cutting tool, etc.                                                    Fiabilitate si Durabilitate - Fiability & Durability     nr.1/2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  111 It is indicated that tolerances be initially chosen at the middle of the tolerance field for every  processed  size,  then  to  be  adopted  within  the  design  limitations  suitable,  therefore reducing manufacturing costs. Table 2 presents the information on this problem. The safety ring (A) and the two bearings (C) and (G) assembled on the axle are purchased parts, therefore their tolerances are fixed and cannot be changed during the tolerances allocation process.  The other parts are processed in the factory. Depending on parts role and importance, during their design process, tolerances values are initially given for parts B, D, E and F.    Table 1.  Sizes field of nominal values and their tolerances. Sizes field, in mm  Tolerance, in mm from  to 0  15  0,004  0,005  0,008  0,013  0,020  0,030  0,050  0,075  0,13 15  25  0,004  0,006  0,010  0,015  0,025  0,040  0,065  0,12  0,15 25  40  0,005  0,008  0,013  0,020  0,030  0,050  0,075  0,13  0,20 40  70  0,006  0,010  0,015  0,025  0,040  0,065  0,12  0,15  0,25 70  110  0,008  0,013  0,020  0,030  0,050  0,075  0,13  0,20  0,30 110  190  0,010  0,015  0,025  0,040  0,065  0,12  0,15  0,25  0,40 190  355  0,013  0,020  0,030  0,050  0,075  0,13  0,20  0,30  0,50 355  500  0,015  0,025  0,040  0,065  0,12  0,15  0,25  0,40  0,65 Processing type Lapping and honing   xxxxxxxxxxxxxxxxxx         Diamond rectification and facing   xxxxxxxxxxxxxxxxxxxxxxxxx       Broaching    xxxxxxxxxxxxxxxxxxxxxxx     Reaming    xxxxxxxxxxxxxxxxxxxxxxxxxxxxx   Facing, enlarging, planing      xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx Milling    xxxxxxxxxxxxxxxxxxxxxxxxxx Boring     xxxxxxxxxxxxxxxxxxx These tolerances are initially selected from table 1, supposing a value that is at the middle of the tolerance field they are part of. The compensation that will appear will be represented by the clearance between the axle and the bearing support ring which is determined with the tolerance accumulation in the assembly.   Tab. 2.  Description of initial tolerances  Component elements Nominal  size, mm Initial tolerance, in mm Processed tolerances limitations, in mm Minimal tolerance  Maximal tolerance A  1,3  0,038– fixed tolerance  -  - B  203,2  0,2  0,075  0,50 C  12,95  0,06– fixed tolerance  -  - D  10,2  0,05  0,012  0,130 E  195,9  0,15  0,065  0,40 F  10,2  0,05  0,012  0,130 G  12,95  0,06– fixed tolerance  -  -  The average compensation is given by the sum of nominal sizes of the parts included in the assembly cycle and is given by the relation: Average  compensation=-A+B-C+D-E+F–G=-1,3+203,2-12,95+10,2–195,9+10,2-12,95=0,5mm Necessary compensation = 0,5±0,2 mm                                                   Fiabilitate si Durabilitate - Fiability & Durability     nr.1/2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  112 The algebraic method is  used to  achieve the compensated tolerance by  adding the initial tolerances of the chain elements, as follows:  608 . 0 06 . 0 05 . 0 15 . 0 05 . 0 06 . 0 2 . 0 038 . 0 T T T T T T T T G F E D C B A mm                            In order to apply the minimal cost algorithm we have to:    Fix Asm T T T                                              And replacing TD, TE and TF in   TB, we have the equation: 1 / 1111 / 1111 / 111kF kBkB BF F k kBkB BE Ek kBkB BD DB G C A AsmBFEEDDTB kB kTB kB kTB kB kT T T T TBB                            (6)                                       46823 . 1 / 43899 . 1B46823 . 1 / 146537 . 1 / 43899 . 1B46537 . 1 / 146823 , 1 / 43899 . 1B46823 . 1 / 1BT15997 . 0 43899 . 007202 . 0 46823 . 0T15997 . 0 43899 . 012576 . 0 46537 . 0T15997 . 0 43899 . 007202 . 0 46823 . 0T 06 . 0 06 . 0 038 . 0 35 . 0          (7)     The values of k and B for every nominal size were achieved from the cost-tolerance functions  for  every  operation,  being  described  in  tables  from  the  specialized  literature. Following the mathematic solution, of the above relation, we get the value of TB. The resulted value of TB will be replaced in the individual expressions in order to get the related values of TD, TE and TF and forecasted cost.          mm 056 . 0 T1599 . 0 43899 . 012576 . 0 46537 . 0Tmm 038 . 0 T15997 . 0 43899 . 007202 . 0 46823 . 0T Tmm 06 . 0 T46537 . 1 / 43899 . 1B46537 . 1 / 1E46823 . 1 / 43899 . 1B46823 . 1 / 1F DB                           (8) Table 3. Allocated tolerances and their cost. Data on the tolerance cost     Tolerances allocation Size  Assembly      A Coefficient B Exponent k Initial tolerance, in mm Algebraic method Statistic method A   -  -  -  0,038  0,038  0,038 B  1  0,15997  0,43899  0,2  0,06  0,18 C  -  -  -  0,06  0,06  0,06 D  1  0,07202  0,46823  0,05  0,038  0,12 E  1  0,12576  0,46537  0,15  0,056  0,18 F  1  0,07202  0,46823  0,05  0,038  0,12 G  -  -  -  0,06  0,06  0,06 Tolerances variation during assembly   0,608 0,276 0,35  0,35  Cost  9,34UM  11,07UM  8,06UM Acceptance ratio    1,00  0,99737 Real cost    11,07UM  8,08UM The assembly cost is given by coefficient A. The installation cost does not influence optimization. For this example, assembly costs are chosen equal, in this way they will not mask the effect of the allocated tolerance. In this case, only the cost of 4,00 monetary units is added to the assembly for every case, in order not to influence the final cost.                                                   Fiabilitate si Durabilitate - Fiability & Durability     nr.1/2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  113    Parts A, C and G are supplied (purchased), and then their tolerances are fixed and their price cannot be changed by reallocating the tolerances, and therefore no cost information is included in the table. The allocated tolerance resulted by calculation with the help of the statistic method was similarly achieved, using equation (1). In this assembly example, as we can notice in table 3, the cost grows when tolerances allocation is calculated through the algebraic method.  The initial tolerances, when added through the algebraic method, give an assembly variation of 0,608 mm. This  exceeds  the  specific  tolerance assembly  limitation  of  0.35mm. Therefore, it was necessary to decrease the allocated tolerances, by increasing costs.  When the tolerance is achieved through the  statistic  method,  the  assembly variation was 0,275mm.  This value is below the specified assembly tolerance limitation.  Therefore,  the  allocated tolerances  increased  by  decreasing costs.  A  graphic  comparison  is presented  in  fig.  2.  The  real  cost,  in table 4, is defined as being the total cost of an assembly divided to benefits.   Fig. 2. Comparison of the results between tolerance and cost   Tab. 4.  Minimal real cost. Cost model  Assembly cost   z  Optimal acceptance ratio  Real cost A + B/tolk             4UM  2,03  0,9576  7,67UM A + B/tolk             8UM  2,25  0, 9756  11,82UM  Therefore, the total adopted cost also includes a part of the assembly cost. The total cost is adjusted in order for a part of it to include the cost of assembly waste. Anyway, it does not  include  some  parts  that  can  be  recovered  or  the  cost  of  waste  parts  as  individual component parts.   A  very  important  thing  is  to  calculate  the  optimal  acceptance  ratio,  meaning  waste  rate resulting in the minimal real cost. This requires an iterative solution. For the presented case, the results are recorded in table 4. The results indicate that the increase of tolerances will save manufacturing costs money, but will increase waste cost. The iterative solution applied to the acceptance  ratio  allows  to  find  the  value  that  minimizes  the  combined  cost  between manufacturing cost and waste cost. As we can notice, the main costs were established as being low. If they double, as in the second row of the table, waste costs would be higher, requiring a higher acceptance level.       Compararea rezultatelor cost - toleranţã Toleranţa el.B Toleranţa el.D Toleranţa el.E Toleranţa el.F0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20 0,22CostCost minimCost minimCost original 9,34UM11,06UM8,06UM                                                  Fiabilitate si Durabilitate - Fiability & Durability     nr.1/2010                                 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X  114 3.  Conclusions  The  optimal  determination  of  assembly  tolerances  depending  on  the  parts manufacturing cost was made by developing Lagrange’s multipliers method.  The advantages for developing this method are:   -  it can easily handle the algebraic calculation method or the statistic calculation method.   - it allows alternative cost-tolerance methods.  Disadvantages: -  calculations apply only for the parts processed on classic machine tools; -  admitted limitations cannot be imposed to technological operations;  - we cannot promptly deal with the simultaneity problem by optimizing the interdependent design characteristics. The problems that have such characteristics can be optimized by using a non-linear technical program.  The total cost is adopted and includes a part of the assembly cost. The total cost is adjusted, in order for a part of it to include the cost of assembly waste. A very important thing is  that  the  optimal  acceptance  ratio  was  calculated,  that  is  the  waste  ratio  results  in  the minimal  real  cost.  It  requires  an  iterative  solution.    The  iterative  solution  applied  to  the acceptance  ratio  allows  to  find  the  value  that  minimizes  the  combined  cost  between manufacturing cost and waste cost. As we can notice, the main costs were established as being low. If they double, as in the second row of the table, waste costs would be higher, requiring a higher acceptance level.   4. References [1].  Rădulescu, C.  – Contributions regarding the technical and economic aspects  when solving  sizes  chains  in  industrial  machines  and  equipments.  Doctoral  thesis.  Polytechnic University, Bucharest, 2009. [2]. Rădulescu, C.,  Militaru, C.,  Designing a theoretical pattern for the relation between  dimensional tolerance and product manufacturing cost. Quality  - access to success, no. 5 / May 2009, CNCSIS, B+ category, ISSN: 1582-2559, www.srac.ro/srac.php?id=170. [3]. C.Militaru, L.C.Cirtina, C. Radulescu, Aspecte privind influenta tipului de repartitii a dimensiunilor  asupra  realizarii  preciziei  lanturilor  de  dimensiuni  tehnologice-Revista Fiabilitate si durabilitate, nr.1/2008, ISSN 1844-640X  [4].  L.Luca,  C.  Rădulescu  –  Metoda  fus  –  coardă,  metodă  computeriyată  pentru desfasurarea sferei Revista Fiabilitate si durabilitate, nr.1/2008, ISSN 1844-640X [5] C Radulescu, C. Militaru, LM Cirtina, Elaborarea unei metodologii de rezolvare a lanturilor de dimensiuni complexe Revista Fiabilitate si durabilitate, nr.2/2008, ISSN 1844-640X [6].  C.  Radulescu,  LM  Cirtina,  The  study  of  chains  of  dimensions  specific  to  reveting assemblies and their significance, Revista Fiabilitate si durabilitate, nr.1/2009, ISSN 1844-640X           

STUDIES REGARDING THE DETERMINATION OF THE  ADEQUATE ASSEMBLY TOLERANCES DEPENDING ON THE  MANUFACTURING COSTS OF THE PARTS PROCESSED   ON CLASSIC MACHINE TOOLS

STUDIES REGARDING THE DETERMINATION OF THE ADEQUATE ASSEMBLY TOLERANCES DEPENDING ON THE MANUFACTURING COSTS OF THE PARTS PROCESSED ON CLASSIC MACHINE TOOLS

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