Progettazione Funzionale di Sistemi Meccanici e Meccatronici

Transcription

1 Camme - Progettazione di massima prof. Paolo Righettini paolo.righettini@unibg.it Università degli Studi di Bergamo Mechatronics And Mechanical Dynamics Labs November 3, 2013

2 Timing for more coordinated movements we must define the timing diagram of all movements using the same abscissa axis y 1 h 1 c 1 c 2 y 2 c 3 h 2 0 2π we may have some design constraints to satisfy (c 1, c 2,c 3 )

3 Timing y 1 h 1 c 1 c 2 y 2 c 3 h 2 0 2π each movement has its own stroke h 1 and h 2 the end of the rise part of the first movement triggers the beginning of the second movement the second movement must be completed in the angular interval specified by the constraint c 3

4 Cam design y 1 h 1 c 1 c 2 y 2 c 3 h 2 0 2π each movement may depend on other movement or design constraints the timing diagram of a complex machine is a critical stage in this stage we may adjust the angular interval or the stroke of the movements in this stage we may also take into account the feasibility of the cam each cam is designed on its resulting timing diagram

5 Cam design y 1h1 0 α sri α sret 2π to design the entire cam we must extend the concept of motion law to all the follower movements the follower movement is defined for cam rotation starting from 0 to 2π radians in this interval we may have more than one motion law (rise, return,...) we need a function that gives the y, y, y of the follower as a function of the angular cam rotation α, 0 α 2π we need the geometric velocity and acceleration profiles to correctly design the cam profile this function may use the adimensional form of the motion law to easily build the required profile

6 Cam design y 1h1 0 α sri α sret 2π for example we may use the same shape of motion law both for the rise and return part of the follower motion let the adimensional form of the motion law described by the adimensional function f, F and G we have the same displacement h 1 but different angular interval, for example α rise and α return the rise starts at α sri while the return starts at α sret

7 Cam design y 1h1 0 α sri α sret 2π for the rise part α sri α α sri + α rise we may have x = α α sri α rise y = h 1 G(x) y = y = h 1 α rise F(x) h 1 α 2 f(x) rise

8 Cam design y 1h1 0 α sri α sret 2π for the return part α sret α α sret + α return we may have x = α αsret α return y = h 1 (1 G(x)) y y = = h 1 α return F(x) h 1 α 2 f(x) return

9 Cam design y 1h1 0 α sri α sret 2π the method is very simple we have the following differences: the calculation of the adimensional time interval x the sign of the adimensional function for the return part we must also introduce some selection procedures to distinguish between the rise, upper dwell, return and lower dwell

10 Preliminary design if we have more coordinated movements the resulting cams must have comparable dimensions comparable dimensions are required as the cams must be assembled in the same machine (or shaft) we require the smallest dimension of the cam we need a tool for the preliminary design of the cam to outline which design parameters limit the dimension of the cam the cam dimension is represented by the pitch base circle, introduced by means of R b0

11 Preliminary design we must choose in the first steps of the design process the nominal dimension of the cam (R b0 ) the pitch base radius limits the pressure angle θ and the curvature profile ρ these cam characteristics also depend on the amplitude h of the movement and on the angular interval α d in which realize the stroke h for any type of cam we have all the analytical/numerical methods to calculate the profile, pressure angle and curvature I introduce a simple mathematical tool to get a preliminary design of the pitch base radius

12 Preliminary design starting from a tapped cam, we may take into account the radius R that represents the average cam dimension the pitch cam profile stands across the circle of radius R we may represent the pitch cam profile as a radial displacement y at the curvilinear abscissa x = Rα we may represent the profile in a cartesian plane

13 Preliminary design we have the displacement x = Rα in abscissa and the follower displacement along the coordinate axis n y(x) t θ θ the cam profile is called equivalent translating shape; the follower realizes the same translation

14 Preliminary design - pressure angle the pressure angle is equal to the slope of the profile in the contact point according to the derivative definition for a planar curve we have dy(x) dx = tanθ y(x) n θ t h remembering that x = Rα, it results θ dy(x) dx = dy(α) dα dα dx = y 1 R x tanθ = y 1 R the maximum value of the angular pressure depends on the maximum geometric acceleration

15 Preliminary design - pressure angle it results tanθ max = y max R the maximum value of the geometric velocity depends on the shape of the motion law, it results y(x) n θ θ t h finally y max = Cv h α d tanθ max = Cvh Rα d x

16 Preliminary design - pressure angle to limit the maximum value of the pressure angle to θ lim to limit the side thrust, it must result n tanθ max tanθ lim C vh Rα d tanθ lim from the previous equation we may obtain a minimum value for R y(x) θ θ t h hc v R α d tanθ lim x this equation shows all the design parameters the minimum value of R depends linearly on h and C v

17 Preliminary design - pressure angle hc v R α d tanθ lim the value of C v may be reduced using acceleration profile with low acceleration in the middle part for symmetric constant acceleration laws C v may have the feasible interval 1.2 C v 2 the choice of a motion law with C v = 1.2 respect to one with C v = 2 leads to a reduction of the cam dimension of 40% the cam dimension may be reduced by means of the increasing of α d y(x) n θ θ t x h

18 Preliminary design - pressure angle C v R h α d tanθ lim the radius of the pith base circle may be calculated remembering that the maximum value of θ depends on the maximum value of the velocity we choose the circle of radius R that intersect the point profile with the maximum value of the velocity f(x) x v C v F(x) 1 x R = R b0 +y(x vα d ) R b0 h α d C v tanθ lim y(x vα d )

19 Preliminary design - pressure angle taking into account a tapped cam with C v = 2, θ lim = 30, it results α d is measured in radians R h 3.4 α d the ratio R/h gives us indication of the cam dimension respect to the requested stroke we have bulky cams for R/h 5 similar considerations may be stated for the return path of the cam; in this case we have θ lim = 50 we must choose the higher values for R

20 Preliminary design - curvature the curvature of the cam profile also depends on the pitch base circle we must design the cam profile avoiding undercut phenomena and low value of the curvature the effective radius of curvature ρ depends on the pitch profile curvature and on the roller diameter ρ = ρ 0 R r the undercut phenomena appears when ρ and ρ 0 have opposite sign starting from convex pitch profile (ρ 0 > 0) the undercut appears if R r > ρ 0 ; in this case the center of curvature K lies in the region covered by the roller during its path we use a preliminary design approach to take into account this situation to outline the effects of the design parameters

21 Preliminary design - curvature taking into account the equivalent translating shape the curvature of line in cartesian frame is expressed by the equation 1 R c = d 2 y/dx 2 [ ( ) ] 2 3/2 1+ dy dx y(x) n θ θ t h where the sign has been introduced to take into account positive curvature for convex profile remembering that x = Rα and dy/dx = tanθ, it results x 1 R c = cosθ 3y R 2

22 Preliminary design - curvature the effective curvature of the pith profile depends on the reference radius R; winding the equivalent translating shape around the circle of radius R we have curvature 1 ρ 0 = 1 R cos3 θ y R 2 ρ 0 = R 2 R y cos 3 θ we have the minimum value of ρ for the maximum value of the denominator cosθ 0 for π/2 θ π/2

23 Preliminary design - curvature the maximum value of the denominator occurs for the minimum value of the negative acceleration the modulo of the minimum value of the negative acceleration is y max = C a h α(d 2 taking into account the maximum value of the denominator and assuming cosθ = 1 it results R 2 ρ 0min R + Ca this equation shows that the minimum value of the radius of curvature depends on the coefficient C a which indicates the peak of the negative acceleration we may limit the radius of curvature of the pitch profile limiting C a of the motion law we limit C a using asymmetric motion law h α 2 d

24 Preliminary design - curvature to avoid undercut phenomena it must result ρ 0 R r R 2 R + C a h α 2 d R r solving the previous expression respect to R we obtain R Rr 2 [ C a h α 2 d Rr ]

25 Preliminary design - curvature we may also set limitation on the minimum value of the effective radius of curvature of the cam profile setting the minimum value of the radius of curvature to ρ lim, it results ρ min = ρ 0min R r ρ lim it is a practical design guide the choice ρ lim = 1.5R r for this condition we have ρ 0min 2.5R r we may use the previous equations substituting 2.5R r to R r conventionally we use R = R b0 +h, therefore R b0 = R h

26 Preliminary design - considerations we have determined two limitations on the dimension of a roller contact cam R R hc v α d tanθ lim [ 1+ R r 2 ] 1+4 C a h α 2 d Rr pressure angle limitation curvature limitation these limitations must be applied to both the rise and return parts of the whole follower motion in practical application the dimension of roller contact cams is often limited by the limitation on the pressure angle rather than the limitation on the curvature

27 Preliminary design - offset we may limit the pressure angle for the rise or return part of the follower motion by means of the relative initial position of the follower respect to the cam taking into account tapped cam we may use offset cam; for this type of cam the follower has an offset e respect to the radial direction the follower axis is tangent to the circle of radius e for all the positions starting form the base pitch radius the follower direction has the angle ξ 0 respect the radial direction through the same contact point P 0

28 Preliminary design - offset in this configuration the pressure angle for the contact point P 0 on the base circle is not equal zero this configuration will give an increasing or decreasing of the pressure angle in the rise part respect to the centered tapped type the follower translates of s = s b0 +y(α) for this configuration we have tanξ 0 = e s b0 R b0 = s b0 /cosξ 0 θ 0 = ξ 0 = arctane/s b0

29 Preliminary design - offset the effect of the offset may be introduced at the preliminary design level of the cam by means of the equivalent translating shape taking into account the curvilinear coordinate x = Rα where R is the radius of a circle representing the average size of the cam the point P 1 has abscissa x and has null distance along the follower direction from the reference circle the point P 2 is not on the radial direction, starting from the point P 1 we must follow the straight line tilted in clockwise direction of the angle ξ (follower direction)

30 Preliminary design - offset the point P 2 is not on the radial direction, starting from the point P 1 we must follow the straight line tilted in clockwise direction of the angle ξ (follower direction) the distance between P 2 and P 1 is s s r where s r is the follower displacement when P 2 coincides with P 1 we may draw the equivalent translating shape taking into account the angle ξ respect to the radial direction; we use a cartesian plane with x on the abscissa axis, radial cam direction along the ordinate axis the point P must be drown along the direction tilted of ξ in counterclockwise direction respect to radial direction (the ordinate axis)

31 Preliminary design - offset n r ξ t θ h P θ + ξ ȳ s sr x x x the points P draws the equivalent translating shape on a new cartesian plane whose coordinates are x = x (s s r)sinξ = Rα (s s r)sinξ ȳ = (s s r)cosξ

32 Preliminary design - offset x = x (s s r)sinξ = Rα (s s r)sinξ ȳ = (s s r)cosξ in this case the first derivative of the equivalent translating curve is r n θ ξ t h tan(θ +ξ) = dȳ d x = d((s s r)cosξ) d(rα (s s r)sinξ) the angle ξ and s r are constant, s = s b0 +y, it results x x P θ + ξ s sr ȳ x tan(θ +ξ) = y cosξ R y sinξ

33 Preliminary design - offset n tan(θ +ξ) = y cosξ R y sinξ r θ ξ t h in this case we have lower values of the pressure angle θ in the rise part of the cam; we have higher values of the pressure angle θ in the return part of the cam we also have tanθ +tanξ = y R cosξ x x P θ + ξ ȳ s sr x

34 Preliminary design - offset maximum value of the pressure angle depends again on the maximum value of the velocity r n ξ t tanθ max = y max R cosξ tanξ tanθ lim we must limit the maximum value starting from this inequality we have new expression for the minimum cam dimension θ x P θ + ξ s sr ȳ x h R y max h cos(ξ)(tanθ max +tanξ) = C v α d cos(ξ)(tanθ max +tanξ) x

35 Preliminary design - offset starting the design from a centered tapped cam (null offset) if we have high values of the pressure angle in the rise part of the cam, we may choose the angle ξ so that θ max ξ θ lim we should use an average cam dimension R so that R C v h α d cos(ξ)(tanθ max +tanξ) knowing R, the cam offeset e results e = R sinξ

36 Preliminary design - offset assuming that the reference circle with radius R intersects the follower direction at half stroke, it results R cosξ = s r = s b0 + h 2, s b 0 = R cosξ h 2 finally, the base radius R b0 results R b0 = sb 2 0 +e 2 = ( Rcosξ h ) 2 +e 2 2

37 Preliminary design - offset blue line follower for null offset; red line follower with offset; green cam profile with offset; the two cam mechanisms realizes the same motion law the follower positions represent the same point of the motion law the followers translate parallel together the normal to the profile in the contact point have different slope; the cam with offset has the normal in the contact point tilted in counterclockwise direction the pressure angle for the cam with offset is lower than the other

38 Preliminary design - offset

39 Preliminary design - offset to reduce the pressure angle on the rise part we must introduce a cam offset so that the follower moves towards the cam rotation; we must use opposite offset for the return part of the cam we also must check the pressure angle at the minimum and maximum stroke; for follower on the base circle we have θ b0 = arctan e S b0 at the maximum stroke it results [ ] e θ h = arctan s b0 +h

40 Preliminary design - offset

41 Preliminary design - offset also for rocked arm cams we may introduce the offset concept the figure shows a follower configuration that leads to a null pressure angle at the beginning of the rise; the pressure angle increases starting from 0 during the lift the pressure angle at the stroke h results θ h = h ξ = h arctan b(1 cosh) R b0 +bsinh

42 Preliminary design - offset we may reduce the pressure angle during the rise part using a different configuration of the follower; rotating the follower clockwise around the center of the roller we will start the rise part with a not null pressure angle during the rise part the pressure angle increases; it will have smaller values during the rise, higher value at the end of the stroke

43 Preliminary design - offset

44 Preliminary design - offset we may reduce the pressure angle during the return part using a different configuration of the follower; rotating the follower counterclockwise around the center of the roller we will start the rise part with a not null pressure angle clockwise and counterclockwise rotation of the follower are relative to the cam rotation; if the cam rotates in opposite direction we must work in opposite mode

45 Preliminary design - offset

46 Preliminary design - rocked arm the equation introduced for the preliminary design of the tapped cam may be used for the preliminary design of rocked arm cam using an equivalent linear displacement h = bh

47 Preliminary design - input torque if we have pure dynamic load the input torque on the cam shaft depends on the motion law profile used to design the cam remembering the expression introduced for the peak power coefficient the maximum value of the power (peak power W p) may be obtained by the power balance W inp = W motp = m h2 td 3 max(f(t/t d )F(t/t d )) the maximum value of the peak power depends on the maximum value of the function T(x) = f(x)f(x) the product between the adimensional acceleration f(x) and the adimensional velocity F(x)

48 Preliminary design - input torque T(x) = f(x) F(x) the maximum value of the function T(x) depends on the profile acceleration f(x) (F(x) is the primitive of f(x)) the maximum value of T(x), 0 x 1, is called power coefficient C w C w = max(t(x)) = max(f(x)f(x))

49 Preliminary design - input torque for a cam the input power on the cam shaft is W inp = ωt p where T P is the peak torque on the cam shaft it must result where C w is the power coefficient W inp = W outp ωt p = m h2 td 3 max(f(t/t d )F(t/t d )) ωt p = m h2 td 3 C w

50 Preliminary design - input torque remembering that α = ωt we have ωt p = m h2 α 3 ω 3 C w d T p = m h2 α 3 ω 2 C w d for this type of mechanisms the peak power coefficient is also the peak torque coefficient this equation let us the preliminary design of the cam shaft for constant symmetric acceleration law we have C w = 8 and for cubic motion law we have C w = 3.4