Aim of Speed and Feed Regulation •
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A machining operation should be conducted at such values of cutting parameters (speed, feed, depth of cut, etc.) that ensure the minimum cost price of the machined component The machi achin ning ing cost ost can be exp expres ressed by the the equat quatio ion: n:
C = C mt + C npt npt + C tc + C t C mt =
(W+ E)t m represents the cost of machining time; W is the wage wage rate rate,, E is cost of operating the machine tool per unit time and t m , the the mach machin inin ing g tim time
C npt npt =
(W + E )t npt npt represents the cost of non-productive time; t npt npt is
the total time of non-productive operations, such as loading and unlo unload adin ing, g, idle idle trav travel el of cutt cuttin ing g tool tool
C tc = (W +E ) t tc /Q represents the tool changing cost per component; t tc is the time required for replacing a blunt tool and setting the new one and Q the number of components machined during the period of tool tool life life..
C tc =
T/Q T/Q repre epressents nts the the cos cost of the the tool tool per com componen onent; t; T is the
of the tool for a period equal to the tool life
cost
Amo
pee an Fee Regulation
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If the machining cost is optimized, it yields a particular value of tool life life whic which h corr corres espo pond ndss to minim inimum um machi achini ning ng cost cost..
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This optimum tool life can be achieved on a particular operation only by working at optimum values of cutting speed v , feed s , and depth of cut t .
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In order to machine a part of arbitrary diameter, the spindle rpm must be set as n = 1000v/ π D , i.e., there must be a stepless regulation of v so that any desired value of the spindle rpm may be set corr corres espo pond ndin ing g to the the opti optim mum cutt cuttin ing g spee speed. d.
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By a similar logic, the machine tool should have provision for step tepless less vari variaation tion of the the feed eed rate rate..
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Economically, viable systems of stepless speed and feed rate regulation have, however, not yet been designed for a majority of machi machine ne tools. tools.
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On such machine tools only certain discrete values of the spindle rpm and feed rate are available.
Determination of Maximum & Minimum Speed •
The maximum and minimum cutting and feed speeds are specified by a number of extreme conditions under which the machine is supp suppos osed ed to oper operat ate. e.
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Thes Thesee condi ndition tionss incl nclude ude mater ateria ialls of the work workpi piec ecee and tool, ool, cutti utting ng enviro environm nment ent etc. etc.
Various Laws of Stepped Regulation •
It was stated above that in stepped regulation of speed only certain discrete values of the spin spindl dlee rpm rpm are are avai availa labl blee on the the mach machin inee tool tool..
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A pertinent question that arises is what should be the criterion for choosing these discrete steps?
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Betw Betweeen two two extre xtrem me avail vailaable ble val values ues n1 and n z of the spindle rpm, the same number of z inte interm rmed edia iate te step stepss may be plac placeed in a numb numbeer of way ways.
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The The vario various us serie seriess of rpm rpm valu values es will will have have diff differ eren entt opera operatio tiona nall char charac acte teris risti tics cs..
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Let Let us anal analyz yzee four four case casess and and sele select ct the the most most suit suitab able le law law of spee speed d rang rangee dist distri ribu buti tion on •
Arithm Ar ithme eti c Pr ogre gr ession ssion
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G eometr i c Pr ogre gr ession ssion
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H armonic rmonic Pr ogre gr ession ssion
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L ogarith gari thm mic Pr ogre gr ession ssion
Arithmetic Progression
Geometric Progression
Harmonic Progression
Logarithmic Progression
Kinematic advantages of GP series 1.
Constant Loss of Economic Cutting Speed in the Whole rpm Range:
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Suppose the spindle rpm values constitute the following series:
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Consider that the optimum cutting speed vopt v opt is is such that it lies between the rpm values nj and n j+1, i.e.
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Obviously, Obviously, of the two rpm rpm values, nj and nj+1 we select the one which gives a cutting speed closer to vopt.
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The difference between the actual cutting speed and the optimum cutting speed is known as the loss of economic cutting speed.
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The difference between the actual cutting speed and vopt, and hence the loss of economic cutting speed is maximum maximum when the optimum optimum cutting speed lies at the middle of two speeds speeds provided provided by nj and nj+1.
Kinematic advantages of GP series
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The maximum loss of economic cutting speed is then:
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From the above two equations, we obtain:
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It may be concluded that in order that may be constant in the whole range of exploitation of the machine tool, the ratio must be constant.
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This means that the spindle rpm values must lie in a geometric progression.
Kinematic advantages of GP series 2. •
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Constant Loss of Productivity in the Whole rpm Range:
The productivity of a machining operation can be expressed as the surface area of metal removed in unit time, i.e., by the quantity:
where v is the cutting speed, m/min s is the feed, mm/rev
3.
Better Design Features:
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When all the rpm values of a spindle are obtained ob tained from a single transmission, then any of the series discussed above can be utilized for designing the speed box;
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The desired rpm values are obtained by using appropriate transmission ratios of various gear pairs.
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If the rpm values are changed by mounting a new pair of gears on the shaft every time, then changing of speeds becomes time consuming, inconvenient and economically infeasible.
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If the rpm values are obtained by mounting gear pairs of the appropriate transmission ratio on the shafts permanently, permanently, then the axial dimensions of
Kinematic advantages of GP series
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These considerations underline the fact that speed steps in a speed box should be obtained not through a single transmission between two shafts but through a group of transmissions between a number of shafts.
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This features can be realized in an actual speed box design only if the rpm values lie in a geometric progression and may be explained by the following properties of a geometric progression:
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Initial Information Required for Designing a Speed Box: Selection of Range Ratio
The following information is essentially required before we can start designing a stepped drive. drive.
1. 2. 3. 4.
The highest output rpm, nmax The lowest output rpm, nmin The number of steps z into which the range nmi n is divided. between nmax and nmi The number of stages in which the required number of speed steps are to be achieved.
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There is a tendency of assigning higher cutting speeds for machining operations as new tool materials permitting higher cutting speeds are developed.
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While selecting the lowest and highest speed limits for a new machine tool, we must take into account its exploitation in actual production conditions.
Initial Information Required for Designing a Speed Box: Selection of Range Ratio •
An important parameter in designing speed boxes is the range ratio Rn ratio Rn given by:
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Here Rv Here Rv represents the range of cutting speeds employed on the machine tool and Rd and Rd the range of workpiece diameters machined.
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A very wide speed range is generally neither neither practicable nor economically feasible, thus value of Rv of Rv should, therefore, be kept within reasonable limits.
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The range of diameters should also be selected on the basis of the statistical study of the working of similar machine tools.
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Investigations conducted by ENIMS (Machine Tool Tool Research Institute, Moscow, Moscow, Russia) reveal that a ratio of Rd of Rd = dmax/ dmax/dm dmin in = 4 covers covers more than 85% of the workpieces, while Rd while Rd = 6 covers 92% of the workpieces.
n a n orma on equ re or es gn ng a Speed Box: Selection of Range Ratio •
Typical values of Rn of Rn for some groups of machine tools are given in the following followi ng Table: Table:
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A variety variety of cutting tools of different shapes and compositions are used on general-purpose machine tools, and therefore, Rn therefore, Rn values are relatively large.
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On the other hand, in grinding machines the grinding wheel diameter generally varies in the range Rd range Rd < 2 and the wheel material being the
Initial Information Required for Designing a Speed Box: Selection of Speed Step •
The number of speed steps in which the total range of rpm values available on the machine tool is divided is determined as follows: Suppose the rpm values n1, n2,……nz constitute nz constitute a geometric progression. Then,
Which can also be written as:
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The value of z can be determined provided R n and φ are known.
Standard Values of Geometric Progression Ratio and Guidelines Guidelines for Selecting Selecting a Proper Value •
The standard values of φ are established from the following two main considerations:
1.
In machine tool drives, two speed motors are often used; the ratio of the two speeds is generally equal to 2 (e.g., motors have rpm values of 3000 and 1500, or 1500 and 750, etc.). If the spindle rpm values constitute a geometric progression for the lower rpm of the motor, motor, then according to the property of geometric progression, the spindle speeds should increase two times when the motor speed is switched to the higher one.
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The geometric progression should be developed by keeping the standards of preferred numbers and preferred series in mind. The geometric progression should then satisfy the condition:
Standard Values of Geometric Progression Ratio and Guidelines Guidelines for Selecting Selecting a Proper Value The standard values of φ are obtained from the condition that they must simultaneously satisfy the previous two equations:
where S' is an arbitrary whole number, then
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The standard values of S2 from the series of preferred numbers are 40, 20, 10 and 5. Correspondingly Correspondingly S'= S2/l0 = 4, 2, 1 and 0.5 0.5 and S1 = 3S' = 12, 6, 3 and 1.5. The corresponding standard values of φ are:
Standard Values of Geometric Progression Ratio and Guidelines Guidelines for Selecting Selecting a Proper Value •
The standard values of φ, their characteristics and the specific loss of economic cutting speed:
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It is desirable to select a small value of φ so that the loss of economic cutting speed and, hence, productivity loss is low.
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Also, for a particular value of the range ratio, the number of speed steps increases with a reduction in the value of φ.
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On the other hand, a large number of speed steps make the drive complicated and expensive.
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The proper value of φ must be selected by weighing these contradictory factors and striking a judicial balance.
Standard Values of Geometric Progression Ratio and Guidelines Guidelines for Selecting Selecting a Proper Value •
Recommended values values of φ in machine tools:
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The most commonly used values of φ are 1.26, 1.41 and 1.58.
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Smaller values of φ( φ = 1.12 and 1.06) complicate the drive to an extent where it cannot cannot normally normally compete with stepless regulation.
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Large values of φ (φ = 1.78 and 2.0) result in very rough regulation leading to large productivity losses. They are rarely used and that too only in special purpose machine machine tools
Break up of Speed Steps •
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The value of the number of speed steps z found is rounded off to the nearest whole number, preference being given to the number which can be broken into multiples of 2 and 3. For example, Numbers between 5 and 7 are rounded off to z to z = 6 Numbers between 7 and 8.5 are are rounded off off to z to z = 8 Numbers between 8.5 and 10 are rounded off to z = 9 Numbers between 10 and 11 11 are rounded off to z to z = 10 Numbers between 11 and 13 are rounded rounded off to z = 12 Numbers between 13 and 15 are rounded off off to z = 14
Structural Diagrams and Their Analysis Analysis to Select Select the Best One •
Suppose a speed on one shaft yields two speed values on the next shaft, i.e., the number of speed steps of the particular transmission group is p is p = 2.
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The maximum reduction of speed is limited to four times to keep the radial dimensions of the speed box within reasonable limits, while the maximum increase of speed is restricted to two times due to limitations of the pitch line velocity. The transmission range of the group is given by
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Suppose there are z speed steps n1, n2, n3…….nz in a particular transmission group such that
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Structural Diagrams and Their Analysis Analysis to Select Select the Best One Since the speeds on the last shaft of the speed box must constitute a geometric progression, progression, the following relationship must be satisfied: satisfied:
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Here X Here X is known as the characteristic of the transmission group and it denotes the number of steps of the spindle rpm geometric progression by which two adjacent adjacent rpm values of the particular group group are separated. As earlier explained, the number of speed steps is represented in the form This expression can be written in a number of ways by arranging p1, p2, p3,….. pu pu in different positions. The total number of possible combinations = u!/ u!/q! , where q is the number of groups with identical transmissions.
Structural Diagrams and Their Analysis Analysis to Select Select the Best One •
Since the rpm values of the output shaft of the speed box ought to be b e in a geometric progression, there must be one transmission transmission group that has a characteristic characteristic X1 = 1; this group is known as the main transmission transmission group group and it has a progression ratio = φ1 = φ
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The next transmission group has a characteristic X2 = p1 and a progression ratio = φ X2 = φ p1 where p1 is the number number of speed steps in the first group; similarly, similarly, the third transmission group has a characteristic X3 = p1 * p2 and a progression ratio φ p1p2 .
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Thus, a more elaborate expression for z may be written as follows:
Where, The following equation is known as the structural formula of the speed box.
Structural Diagrams and Their Analysis Analysis to Select Select the the Best One Let us consider an example. Suppose the number of speed steps z = 12 of a speed box are to be realized in three stages, i.e., u = 3. The number 12 may be written as a multiplication of 2 and 3 in three different ways. Let us consider one such combination, combination, z = 2 x 3 x 2
Select Select the the Best Best One •
The structural formulae are represented in the form of special graphs known as structural as structural diagrams. diagrams.
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For plotting the structural diagram, we draw u + 1 vertical lines at a convenient distance from each other; the first vertical line represents the transmission from the motor shaft and the rest represent the transmission groups of the speed box
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We now draw an array of horizontal lines intersecting the vertical lines at a distance of log φ from each other.
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The number of horizontal lines is equal to the number of speed steps z of the speed box.
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The structural diagram gives information about
1. the number of shafts in the speed box. bo x. 2. the number of gears on each shaft. 3. the order of changing transmissions in individual groups to get the desired spindle speed.
4. the transmission range and characteristic of each group
Structural Diagrams and Their Analysis Analysis to Select Select the Best One
e se ec on o e es s ruc ura diagram
The selection of the best version is guided by the following two factors:
1.
Trans ransmi miss ssio ion n rati ratio o restr estric icti tion on:: This requirement has already been discussed and it need only be emphasized that the maximum value of the transmission range of a group is ig < ig < 8.
2.
Minimum total shaft size: The dimensions of a shaft are determined for the lowest rpm at which it rotates, because the lower the rpm of the shaft, the higher is the torque that the shaft has to transmit and hence the larger its diameter. The required accuracy of machining of a shaft depends upon the highest rpm at which it rotates; the higher the rpm of the shaft the higher should be its accuracy and surface finish. In a speed box the nmax and nmin values on the last shaft (spindle) must necessarily be the same in all the versions of structural diagrams. However, the nmin and nmax values for the intermediate shafts will differ for each version. The best version is the one which ensures,
The selection of the best structural diagram •
Besides factors (a) and (b) discussed above the following guidelines, which stem from rationality of the speed box design and its exploitation can also be helpful in selecting the best diagram: 1. The number of gears on the last shaft (spindle) should be the minimum possible. 2. The transmission ratio between the spindle and the shaft preceding it should be the maximum possible, possible, i.e., i.e., speed reduction reduction should be the maximum possible. 3. The number of gears on the shafts should not generally be more than three, though in exceptional cases it may be four. four. 4. imax imax * imin = 1 favors the least least radial radial dimen dimensions sions of of the gear gear box.
The selection of the best structural diagram •
Let us now analyze structural diagrams a and e that we have drawn.
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Diagram e
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The selection of the best structural diagram We can similarly analyze the remaining four structural diagrams also. Upon analysis, we find that,
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Hence, if the selected value of the progression ratio is φ = 1.26, then all the six structural diagrams qualify for selection as far as consideration of factor (a) is concerned. If φ = 1.41, then diagrams b and e are ruled out. If φ > 1.41, then none of the structural diagrams is suitable for designing the speed box and an attempt must be made with a different arrangement of the speed step distribution, e.g., z e.g., z = 3 x 2 x 2
The selection of the best structural diagram •
If more than one structural diagram satisfies the transmission-range transmission-range constraint, then these must be analyzed keeping factor (b) and the additional guidelines in mind.
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A comparison of the six structural diagrams reveals that diagrams a and c are better than the rest because nmin values values of shaft shaft III in both both these diagrams are maximum.
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However, diagram a scores over diagram c when shaft II is compared; the nmin value of shaft shaft II in diagram a is higher as compared to the corresponding values of diagram c.
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It may be thus seen that structural diagram a is the best among all the considered versions.
General Recommendations for Developing the Gearing Diagram •
All requirements that are essential for the proper functioning of a gear transmission must be satisfied and gear-box dimensions kept minimum.
1.
These essential requirements for proper functioning of a gear transmission may be summed up as: (i) The number of teeth on the smallest gear of a transmission should be such that there is no undercutting of gear teeth; generally for gears gears with an uncorrected profile and 20° 20° pressure angle, angle, Zmin > 17 17. (ii) If gear pairs on parallel shafts have the same module, the sum of the number of teeth of mating mating gear pairs must be the same. (iii) The spacing between adjacent gears on a shaft shaft should be such that one one gear pair pair gets completely disengaged before the next begins to mesh mesh (Fig. 2.18).
General Recommendations for Developing the Gearing Diagram (iv) The number of teeth of adjacent gears must differ by at least four. four. This point may may be proved by considering the the following example (Fig. 2.19). The sliding block mounted on shaft I provides three speeds on shaft II depending upon which of the gear pairs is used for transmission. The centre distance between the shafts is
where m is the module of all the gears. The sum of the addendum radii of gears Z5 and Z4 is When the sliding gear is moved rightward to make gear Z1 gear Z1 mesh with Z2, gear Z5 will pass over Z4 without interference interference only if
General Recommendations for Developing the Gearing Diagram •
Besides the four essential requirements, a major consideration is how to achieve minimum possible dimensions of the gear box.
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It is helpful to remember that the radial dimensions of a gear box are minimum when the maximum maximum speed reduction reduction and maximum maximum speed increase in a transmission group are equal.
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The radial dimensions can be reduced by making coincident the axes of shafts of adjacent transmission groups, e.g., e.g., shafts I and III in Fig. 2.20 have been made coincident.
General Recommendations for Developing the Gearing Diagram
2.
Specific features peculiar to the functioning of the machine tool for which the gear box is being designed should be taken into account. Some of these features are discussed below:
(i) In machine tools with large inertia of the driven member, such as vertical turret lathes, planers, etc., a friction clutch and brake should be provided on the input shaft. (ii) Reversing devices with friction clutches should be provided on turret lathes, thread-cutting lathes, radial drilling machines, etc., so that after cutting the thread, the tool can be returned to its initial position. The reversal speed should be 1.3-1.5 times greater than the cutting speed. However, However, in light machine tools having a drive motor rating of up to = 3.5 kW, kW, spindle rotation may be reversed by applying an opposing current, e.g., by reversing two connections of the stator winding.
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(iii) If the spindle head traverses during the working operation, the electric motor should be mounted on the speed box and the transmission from motor shaft to the input shaft of the speed box bo x obtained through a clutch or gear pair.
General Recommendations for Developing the Gearing Diagram (iv) If the spindle is kinematically linked to the feed mechanism, mechanism, the transmission from the spindle to the feed train must be shown on the gearing diagram. (v) Gear transmissions that do not slide during speed changing should be made helical to provide smooth running of the spindle; this is especially desirable of gears mounted on the spindle.
