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Cam mechanisms based on a double roller translating follower of negative radius
Mechanism and Machine Theory 95 (2016) 93–101
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Cam mechanisms based on a double roller translating follower of negative radius E. Sanmiguel-Rojas a,⁎, M. Hidalgo-Martínez b a b
Área de Mecánica de Fluidos, Universidad de Jaén, Campus de Las Lagunillas, 23071 Jaén, Spain Área de Ingeniería Mecánica, Universidad de Córdoba, Campus de Rabanales, 14071 Córdoba, Spain
a r t i c l e
i n f o
Article history: Received 22 January 2015 Received in revised form 24 August 2015 Accepted 30 August 2015 Available online xxxx Keywords: Constant-breadth cams Roller follower Negative-radius Bézier curves
a b s t r a c t Constant-breadth cam mechanisms are characterized by the fact that the closure of the camfollower contact is guaranteed thanks to the geometry and, therefore, auxiliary elements as springs are not necessary. On the other hand, cam mechanisms with a negative radius roller follower allow designing simple machines exerting remarkably high forces when the space is restricted. This work proposes a new constant-breadth cam mechanism with a double roller translating follower of negative radius. The cam profile is designed with an optimizing method based on Bézier curves. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction The goal of the constant-breadth cam mechanism is the elimination of the extra preloading constrains to prevent the loss of the contact between cam and followers. By the type of follower, constant-breadth cam mechanisms can be classified as of flat-faced follower or roller follower. Fig. 1(a) shows a cam mechanism with a parallel flat-faced double translating follower. In these mechanisms, the distance between the parallel flat faces of the follower is constant. They can operate either translating or oscillating follower if the appropriate desmodromic conditions are established [1]. Rothbart [2] shows for a constant-breadth cam with a flat-faced double translating follower that the displacement function of the follower can be defined freely in the interval between 0o and 180o, however, the remaining interval of the displacement function is imposed by the first one because the distance between the faces of the follower is constant. Koloc and Václavik [3] analyze a constant-breadth cam as a case of conjugated cam mechanisms. Zayas et al. [4] describe a generating process by means of circular arcs to design cams with both translating and oscillating followers. Using non-parametric Bézier curves, Cardona et al. [5] show a procedure which guarantees automatically the global continuity of the law of displacement for both parallel flat-faced double translating follower and parallel flat-faced double oscillating follower. Recently, these authors [6] have analyzed the influence of the inclination and offset of the translating follower in a constant-breadth cam mechanism. They also calculate the sliding velocities for both translating and oscillating followers. Regarding constant-diameter cam mechanisms with a double roller follower of positive radius (see Fig. 1(b)), Rothbart [10] develops the equations of the radius of curvature in force-closed cam mechanisms which use both flat-faced and roller followers. Qian [7] studies a constant-diameter cam which operates a double roller follower. By the motion of the follower, the relations between the cam angle and the different geometric parameters are established. Lin et al. [8] and Shin et al. [9] develop design solutions for breadth cams with double roller followers with positive radius of curvature. This new method is based on calculating the coordinates ⁎ Corresponding author. E-mail address: [email protected] (E. Sanmiguel-Rojas).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.08.018 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
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(a)
(b)
O O
Fig. 1. (a) A constant-breadth cam mechanism with a parallel flat-faced double translating follower, and (b) a constant-diameter cam mechanism with a double roller follower of positive radius. The center of rotation of the cam (O) is also shown.
to determine the profile through each contact point. These authors use polynomial functions until seventh degree to ensure continuity in the cam profile. On the other hand, the literature is sparse about cam mechanisms with negative radius roller followers. Carra et al. [11] develop a synthesis procedure, based on the modified trapezoidal curves, in order to generate the correct motion law for these cam mechanisms without undercutting problems. Recently, Hidalgo et al. [12] have proposed a numerical method for optimizing the design cams using a Bézier ordinate as an optimization parameter. As application example, they find the maximum of the follower lift, avoiding the undercutting problem, for the particular case of cam mechanisms with followers of negative radius. There are many ways to express a cam profile mathematically. The procedures to generate the cam profile are deeply explained in the technical literature (see for example [10,13]). The functions to design the cam profile include splines, harmonic, cycloid, modified harmonic, trapezoidal, modified trapezoidal, polynomial, etc [10,14–18]. Other authors have shown that the Bézier curves are a powerful tool for designing cams [5,12,19–21]. However, these curves are much less widespread in this engineering field than the modified trapezoidal curves or the spline functions. In the current work, a constant-breadth cam mechanism that operates a double roller follower of the negative radius is proposed as a new form-closed mechanism. As far as the authors know, this cam mechanism has not been analyzed before. The procedure to
Fig. 2. Scheme and nomenclature used to generate the cam profile. The Oxy reference system is considered fixed to the frame while the Ox1y1 reference system is attached to the cam. Also shown the pitch curve (dash–dot red line), the prime circle (dashed black line) with radius Ra and both radii Rf of the follower. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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design the cam profile, based on the curvature theory, is described in the next section. Section 3 is devoted to maximize the lift follower using an optimization method based on the Bézier curves. The sliding velocities are calculated in Section 4 for the particular case of sliding followers of negative radius. Finally, some conclusions are drawn in the last section. 2. Description of a constant-breadth cam mechanism with a double roller follower of negative radius It is known that a cam mechanism with a roller follower of negative radius has certain advantages over the classical roller follower of positive radius [11,12]. In particular, this mechanism is useful in applications where space is constrained and very high forces are involved. Furthermore, this cam-follower mechanism does not present such severe restrictions with the maximum pressure angle, ϕmax, which can reach 41o. Finally, this kind of mechanism brings to a reduction of the problems of excessive Hertz contact pressure. In order to describe the procedure to obtain the cam profile with a double roller follower of negative radius, we have included in Fig. 2 a sketch of such mechanism. A 3D view of the mechanism is also included in Fig. 3. Notice that the double follower has a double negative radius Rf (solid black/thin line). The prime circle radius Ra is also shown. The reference system Oxy is fixed, i.e. it is attached to the frame of the mechanism, with its origin O placed at the cam axis of rotation. In addition, on its axis Oy are placed both centers C1 and C2 of the follower. However, the reference system Ox1y1 is attached to the cam. The motion of the cam with respect to the frame is a rotation of angle θ with angular velocity ω ¼θ k. Thus, the motion of the follower with respect to the reference system Oxy is a translation in the direction Oy, and the points O, C1 and C2 remain always aligned in such direction. Notice that the points C1 and C2 are the intersection between the axis Oy and the pitch curve. The lengths OC1 and OC2 can be written as,
OC 1 ¼ Ra −f ðθÞ; OC 2 ¼ Ra −f ðθ þ πÞ;
ð1Þ
where f(θ) is the displacement function of the follower. Since the length between the centers of the follower C1 and C2 or breadth of the cam mechanism must remain constant, namely dc, summing up the expressions in Eq. (1) yields, OC 1 þ OC 2 ¼ 2Ra −f ðθÞ− f ðθ þ πÞ ¼ dc :
ð2Þ
If we assume that f(0) = 0 and f(π) = h*, where h* is the maximum follower displacement, from Eq. (2) we found,
dc ¼ 2Ra −h :
ð3Þ
Therefore, inserting Eq. (3) in Eq. (1) we obtain the desmodromic condition for this cam mechanism, i.e.
f ðθÞ þ f ðθ þ πÞ ¼ h :
ð4Þ
Fig. 3. 3D view of the mechanism.
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Thus, the displacement function f(θ) can be defined only in the interval 0 ≤ θ ≤ π (designed segment), whereas the displacement function in the interval π b θ b 2π (calculated segment) is given by Eq. (4). Using the Bézier curves, the designed segment f(θ) can be written as 8 <
θ 0 ⩽ θ ⩽ β ðrising phaseÞ; f ðθÞ ¼ β : h β b θ ⩽ π;
h P
ð5Þ
where P is an nth-degree Bézier curve and β is the cam angle required to reach h* (see Hidalgo et al. [12] for further details). To establish the continuity conditions between the designed and calculated segments, we derivate twice Eq. (4) with respect to θ yielding, f θ ðθÞ þ f θ ðθ þ πÞ ¼ 0;
ð6aÞ
f θθ ðθÞ þ f θθ ðθ þ πÞ ¼ 0:
ð6bÞ
To obtain the cam profile and the pressure angle, we will focus on the contact point P1 between the cam and follower. Note from Fig. 2 that the follower has only one instantaneous center I, because if we assume two different instantaneous centers I1 and I2, the abscissa of both centers with respect to the system Oxy must fulfill, xI1 ¼ −f θ ðθÞ ¼ f θ ðθ þ πÞ ¼ −xI2 :
ð7Þ
Therefore, the position vector of the contact point P1 is rI −r C 1 ; rP1 ¼ rC 1 −jR f j rI −r C 1
ð8Þ
where rC1 ¼ rI ¼
0 ; Ra − f ðθÞ
ð9Þ
− f θ ðθÞ : 0
ð10Þ
On the other hand, the pressure angle ϕ is the angle (at any point) between the normal to the pitch curve and the direction of the follower motion [12], j r −r C1 I R −f ð θ Þ a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ¼ arccos ϕ ¼ arccos 2 2 r C −r I ½ ð Þ ½ ð Þ θ þ R − f θ f θ a 1
ð11Þ
where j is the unit vector in y direction. Finally, with the help of the Euler–Savary equation [22–24], the radius of curvature of the pitch curve is given by Hidalgo et al. [12] as, ! ρ ¼ jC 10 C 1 j ¼
n o3 ½ f θ ðθÞ2 þ ½Ra −f ðθÞ2 2 2½ f θ ðθÞ2 þ ½Ra − f ðθÞ½Ra −f ðθÞ þ fθθ ðθÞ
;
ð12Þ
where C10 is a common center of curvature. Therefore, the radius of curvature of the cam ρc is defined as, ρc ¼ jR f j−ρ;
ð13Þ
being the no-undercutting condition, ρc N 0
⇒
ρ b jR f j:
ð14Þ
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3. Optimization of the maximum follower displacement In this section, we use the optimization method originated by Hidalgo et al. [12] to find the maximum reachable follower displacement avoiding the undercutting and preserving the pressure angle ϕ b 41o, for a given Rf and β. In this method, the displacement function of the follower f(θ) is defined by a Bézier curve, see Eq. (5), and its shape is modified using an ordinate bi as optimization parameter. To present the results, we define the maximum dimensionless follower displacement h = h*/|Rf |, the dimensionless prime circle radius α = Ra/|Rf | and the maximum reachable dimensionless follower displacement hmax = max(h) for ϕ b 41o and without undercutting. Fig. 4(a) displays the parametric study performed numerically (see flow chart in [12]) to find hmax fulfilling the no-undercutting condition and ϕ b 41o, for a Bézier curve of degree n = 6 and β = π. Notice that hmax = hmax(α, bi) for a given β [12]. As in the case of a follower with one negative radius [12], the highest values of hmax are reached when β = π and the dimensionless prime circle radius α = Ra/|Rf | is increased. The first remarkable feature is that, independently of the value α, hmax reaches its maximum at b3 = 0.5. It is also interesting to point out that all the graphs are symmetric with respect to the value b3 = 0.5. To find an explanation to this behavior, we have plotted in Fig. 4(b) the dimensionless displacement function of the follower f(θ)/h* and its first two derivatives, for the values b3 = 0.1 (continuous line) and b3 = 0.9 (dashed line). Notice that these values of the ordinate b3 are equidistant with respect to b3 = 0.5 in Fig. 4(a). As can be observed, the rising phase of the follower for b3 = 0.1 (continuous line for 0 ≤ θ ≤ π) is perfectly symmetric, with respect to θ = π, to the return phase of the follower for b3 = 0.9 (dashed line for π b θ b 2π), and vice versa. Furthermore, the same happens to the second derivative. However, the first derivative shows, for both equidistant values of b3, an antisymmetric behavior in the rising and return phases with respect to θ = π. This behavior in curves of degree n = 6 evaluated at ordinates equidistant from b3 = 0.5 is due to the desmodromic condition (4). Notice from the equations of the pressure angle (11) and radius of curvature of the pitch curve (12) that, for a given h = h*/|Rf |, these properties of symmetry and antisymmetry between curves of values b3 equidistant from b3 = 0.5, have no effect on the maximum of ϕ or on the minimum of ρc. To show this fact, we have plotted in Fig. 5 the dimensionless radius of curvature ρc/|Rf | and the pressure angle ϕ, for α = 0.4 and hmax(b3 = 0.1) = hmax(b3 = 0.9) ≈ 0.35. Notice that the minimum of the dimensionless radius of curvature for both values of b3 is the same, i.e. ρc/|Rf | ≈ 0, and the maximum of the pressure angle is also the same in both cases, i.e. ϕ ≈ 28.7o. Furthermore, the curves corresponding to the rising phase for b3 = 0.1 and the return phase for b3 = 0.9, and vice versa, are also perfectly symmetric with respect to θ = π. Therefore, the numerical algorithm finds the same hmax for equidistant ordinates with respect to b3 = 0.5. It is also interesting to point out that, for all the values of α and b3, the radius of curvature is more restrictive than the pressure angle, which is always clearly lower than 41o. The same happens for a Bézier curve of degree n = 8 with the ordinate b4 as optimization parameter. On the other hand, Fig. 6 shows the dimensionless displacement function f(θ)/h* and its first two derivatives with respect to the cam angle θ, corresponding to a Bézier curve n = 6 for b3 = 0.5 (continuous line). Notice that the curves f(θ)/h*, fθ(θ)/h* and fθθ(θ)/ h* fulfill Eqs. (4), (6a) and (6b), respectively, as in the case of the curves plotted in Fig. 4(b). As a Bézier curve of degree n = 6 guarantees continuity C2, the jerk-curve is discontinuous as can be appreciated from the slope of the curve fθθ(θ)/h* at θ = 0 and θ = π. Moreover, it is worth noting that in the particular case of a Bézier curve of degree n = 6 evaluated at the ordinates ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 Þ ¼ ð0; 0; 0; 0:5; 1; 1; 1Þ;
ð15Þ
one obtain the curve of degree n = 5 evaluated at the ordinates ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 Þ ¼ ð0; 0; 0; 1; 1; 1Þ:
ð16Þ
(a)
(b) α α
*
f/ h
α
f θ /h
h m ax
*
f θθ /h
*
b3
θ/ π
Fig. 4. (a) Maximum reachable values of h without undercutting and ϕ b 41o using a Bézier curve of degree n = 6, β = π and different values of α as indicated. (b) Dimensionless displacement function f(θ)/h* and its first two derivatives, for a Bézier curve of degree n = 6 with b3 = 0.1 (continuous line) and b3 = 0.9 (dashed line).
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(b)
φ
ρc / | R f |
(a)
θ/π
θ/π
Fig. 5. For α = 0.4, h = hmax and β = π, (a) dimensionless radius of curvature ρc/|Rf | and (b) pressure angle ϕ (in degrees), for a Bézier curve of degree n = 6 with b3 = 0.1 (continuous line) and b3 = 0.9 (dashed line).
To show that, we have also included in Fig. 6 (circles) the dimensionless displacement and its first two derivatives for a Bézier curve of degree n = 5. Therefore, preserving continuity C2 and independently of the dimensionless prime circle radius α, the maximum reachable values of h without undercutting and ϕ b 41o is reached with a Bézier curve of degree n = 5. In addition, we have performed numerically the same parametric study but with a Bézier curve of degree n = 8, which is summarized in Fig. 7(a). Notice that the graphs show the same properties discussed above. However, for each α, the maximum reachable values of h without undercutting are clearly lower than those obtained for n = 6. We have plotted in Fig. 7 the dimensionless displacement function f(θ)/h* and its first two derivatives for a Bézier curve of degree n = 8 and ordinate b4 = 0.5 (continuous line). Notice that in this case, the slope in the second derivative is continuous. Therefore, the continuity C3 is guaranteed and we will have a continuous jerk-curve. Finally, we have also included in Fig. 7(b) (circles) the dimensionless displacement and its first two derivatives but for a Bézier curve of degree n = 7, which for the ordinates ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 Þ ¼ ð0; 0; 0; 0; 1; 1; 1; 1Þ;
ð17Þ
is the particular case of a Bézier curve of degree n = 8 evaluated at the ordinates ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 ; b8 Þ ¼ ð0; 0; 0; 0; 0:5; 1; 1; 1; 1Þ:
ð18Þ
f/ h
*
f θ /h
*
f θθ /h
*
θ/ π Fig. 6. Dimensionless displacement function f(θ)/h* and its first two derivatives corresponding to the Bézier curves n = 5 (circles) and n = 6 at b3 = 0.5 (continuous line).
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(b) α α
f θθ /h
∗
h m ax
f θ /h ∗
α
f/ h ∗
(a)
b4
θ/ π
Fig. 7. (a) As in Fig. 4(a) but for Bézier curves of degree n = 8. (b) As in Fig. 5 but for Bézier curves of degree n = 7 (circles) and n = 8 at b4 = 0.5 (continuous line).
Thus, independently of the dimensionless prime circle radius α, the maximum reachable values of h without undercutting and ϕ b 41o is reached preserving continuity C3 with a Bézier curve of degree n = 7. 4. Sliding velocities in constant-breadth cams with a double sliding follower of negative radius Friction opposes the relative movement of contacting bodies in all machinery. In cam-follower mechanisms, both sliding and rolling frictions are manifested. The sliding velocity plays an important role on the energetic efficiency and longer life of these mechanisms. Cardona et al. [6] calculate the sliding velocities at the two contact points of a constant-breadth cam mechanism with a parallel flat-faced double translating follower. They show that the sum of both sliding velocities is constant for a given angle of inclination and offset of the follower. Furthermore, they also demonstrate that the choice of such design parameters may ensure lower sliding velocities. On the other hand, to manufacture constant-breadth cam mechanisms with a double roller follower of negative radius, using the inner rings of commercial roller bearings as roller followers [11], minimizes practically to zero the sliding velocity. However, we think that it is interesting to analyze the sliding velocities in the particular case of a constant-breadth cam mechanism with a double sliding follower of negative radius. Some advantages of a double sliding follower over a double roller follower, for the same application, could be: lower weight; simplicity of maintenance; ease of lubrication; and lower manufacturing costs. Thus, with the help of Fig. 2, the modulus of the sliding velocities at the contact points P1 and P2 between the cam and the follower, can be determined as ! vP 1 ¼θ jIP 1 j;
ð19aÞ
! vP 2 ¼θ jIP 2 j;
ð19bÞ
! ! where θ is the angular velocity of the cam. The values of the vectors jIP 1 j and jIP 2 j are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! jIP 1 j ¼ R f −jIC 1 j ¼ R f − ½ f θ ðθÞ2 þ ½Ra − f ðθÞ2 ;
ð20aÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! jIP 2 j ¼ R f −jIC 2 j ¼ R f − ½ f θ ðθ þ πÞ2 þ ½Ra −f ðθ þ π Þ2 :
ð20bÞ
From Eqs. (4) and (6a) we know that
f ðθ þ πÞ ¼ h −f ðθÞ;
ð21aÞ
f θ ðθ þ πÞ ¼ −f θ ðθÞ:
ð21bÞ
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! Therefore, the vector jIP 2 j can be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! jIP 2 j ¼ R f − ½ f θ ðθÞ2 þ ½Ra −h þ f ðθÞ2 :
ð22Þ
Fig. 8(a) shows the dimensionless sliding velocities defined as
λP 1
λP 2
! IP 1 ¼ ; ¼ θ jR f j R f vP1
ð23aÞ
! IP 2 ¼ ; ¼ θ jR f j R f vP2
ð23bÞ
for different values of the dimensionless prime circle radius α. In addition, we have plotted in Fig. 8(b) the sum of the sliding velocities Λ ¼ λP 1 þ λP 2 ;
ð24Þ
for the same values of α. Unlike what happens in the case of a parallel flat-faced double translating follower [6], the sum of the sliding velocities is not constant. Moreover, the amplitude of the fluctuations in both sliding velocities and in their sum is increased for higher values of α. These fluctuations could lead to fatigue in the material of the cam or the follower. However, the modulus of the sliding velocity decreases when α is increased. To show more clearly of this feature, Fig. 9 displays the maximum and minimum of the sum of the dimensionless sliding velocities as a function of α, i.e. Λmax = max(Λ) and Λmin = min(Λ), calculated for h = hmax with Bézier curves of degree n = 5, Fig. 9(a), and n = 7, Fig. 9(b). Notice that for both degrees n = 5 and n = 7, such maximum and minimum of the sum tend to decrease as α increases. Furthermore, such values are slightly lower for n = 7 than for n = 5. Finally, it is worth mentioning that this tend is the opposite of that corresponding to the case of a parallel flat-faced double translating follower, for which the sliding velocities are increased for higher values of the prime circle radius Ra. 5. Conclusions This work introduces and studies a new constant-breadth cam mechanism with a double roller follower of negative radius, which belongs to the category of form-closed cam mechanisms. This new mechanism allows higher forces into restricted spaces than the equivalent mechanism of parallel flat-faced double follower. In particular, we deduce the desmodromic condition of this cam mechanism, and the continuous displacement function of the follower is defined with Bézier curves. Using the optimization method introduced by Hidalgo et al. [12], the follower lift is maximized avoiding the undercutting problem but keeping the pressure angle lower than 41o, given the total rise angle β and the double negative radius of the follower |Rf |. On the other hand, we have also performed a
(b)
Λ
λP 1, λ P 2
(a)
α
α
α
α
α
α
θ/ π
θ/ π
Fig. 8. For different values of α: (a) Dimensionless sliding velocities λP1 (continuous lines) and λP 2 (discontinuous lines); (b) sum of the dimensionless sliding velocities Λ ¼ λP1 þ λP2 . Calculations carried out with a Bézier curve of degree n = 5 and h = hmax for each α.
E. Sanmiguel-Rojas, M. Hidalgo-Martínez / Mechanism and Machine Theory 95 (2016) 93–101
(a)
α
101
(b) Λ
Λ
Λ
Λ
α
Fig. 9. Maximum, Λmax = max(Λ), and minimum, Λmin = min(Λ), of the sum of the dimensionless sliding velocities, for cams designed with Bézier curves of degree n = 5 (a) and n = 7 (b). For each value of α, the maximum dimensionless follower displacement is hmax.
study of the sliding velocities at the two contact points for the particular case of a double sliding follower of negative radius. We find that, unlike what happens in the case of a parallel flat-faced double translating follower, the sum of the sliding velocities is not constant and exhibits fluctuations. Furthermore, the amplitude of such fluctuations is increased for higher values of the dimensionless prime circle radius α, although the mean value of the sliding velocities decreases when α is increased. Finally, taking advantage of inner rings of commercial bearings to manufacture the roller follower, the sliding velocity could be minimize practically to zero, which is not possible in cams with a parallel flat-faced double translating follower.
Acknowledgments This work was supported by the Universidad de Jaén (Spain) and the Universidad de Córdoba (Spain).
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E.E. Zayas, Aportación al estudio de levas desmodrómicas(Doctoral Thesis) Universitat Politècnica de Catalunya, Barcelona, 2001. H. Rothbart, Cam design, Dynamic and Accuracy. John Wiley and Sons, New York, 1956. Z. Koloc, M. Václavik, Studies in mechanical engineering 14, Cam Mechanism. Polygrafia, Praga, 1993. E.E. Zayas, S. Cardona, L. Jordi, Analysis and synthesis of displacement functions of the follower in constant-breadth cams mechanisms, Mech. Mach. Theory 44 (2009) 1938–1949. S. Cardona, E.E. Zayas, L. Jordi, P. Català, Synthesis of displacement functions by Bézier curves in constant-breadth cams with parallel flat-faced double translating and oscillating followers, Mech. Mach. Theory 62 (2013) 51–62. S. Cardona, E.E. Zayas, L. Jordi, Radius of curvature and sliding velocity in constant-breadth cam mechanisms, Mech. Mach. Theory 81 (2014) 181–192. Z. Quian, Research on a constant-diameter cam mechanism with a planar motion follower, Mech. Mach. Theory 42 (2007) 1017–1028. L. Lin, D.Y. Hyun, E.A. Seong, H.S. Joong, M.K. Soon, Breadth cam profile design with oscillating roller follower, Proceedings of the 2010 IEEE International Conference on Mechatronics and Automation 2010, pp. 389–394 (Xi'an China). J. Shin, L. Li, H. Yang, S. Kwon, Kinematical system of breadth cam profile design, J. Cent. S. Univ. Technol. 18 (2) (2011) 451–457. H. Rothbart, Cam Design Handbook, McGraw-Hill, 2004. S. Carra, R. Garziera, M. Pellegrini, Synthesis of cams with negative radius follower and evaluation of the pressure angle, Mech. Mach. Theory 39 (2004) 1017–1032. M. Hidalgo-Martínez, E. Sanmiguel-Rojas, M.A. Burgos, Design of cams with negative radius follower using Bézier curves, Mech. Mach. Theory 82 (2014) 87–96. F.Y. Chen, Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982. J. Angeles, C.S. López-Cajún, Optimal synthesis of cam mechanisms with oscillating flat face followers, Mech. Mach. Theory 23 (1) (1988) 1–6. H. Qiu, C.J. Lin, Z.Y. Li, H. Ozaki, J. Wang, Y. Yue, A universal optimal approach to cam curve design and its applications, Mech. Mach. Theory 40 (6) (2005) 669–692. V.T. Nguyen, D.J. Kim, Flexible cam profile synthesis method using smoothing spline curves, Mech. Mach. Theory 42 (2007) 825–838. M. Mandal, T.K. Naskar, Introduction of control points in splines for synthesis of optimized cam motion program, Mech. Mach. Theory 44 (2009) 255–271. U. Chavan, S. Joshi, Synthesis of cam profile using classical splines and the effect of knot locations on the acceleration, jump, and interface force of cam follower system, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 225 (2011) 3019–3030. H.S. Yan, W.J. Tsa, Motion adaptation of cam-follower systems by varying input speeds, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 222 (2008) 459–472. W.J. Qin, J.Q. He, Optimum design of local cam profile of a valve train, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 224 (2010) 2487–2492. K.L. Ting, N.L. Lee, G.H. Bradan, Synthesis of polynomial and other curves with the Bézier technique, Mech. Mach. Theory 29 (1994) 887–903. R. Hartenberg, J. Denavit, Kinematic Synthesis of Linkages, McGraw-Hill, 1964. G.N. Sandor, A.G. Erdman, E. Raghavacharyulu, Double-valued solutions of the Euler–Savary equations and its counterpart in Bobillier's construction, Mech. Mach. Theory 20 (1985) 145–148. L.O. Barton, A new graphical technique for solving the Euler–Savary equation, Mech. Mach. Theory 30 (1995) 1305–1309.
Contents lists available at ScienceDirect
Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt
Cam mechanisms based on a double roller translating follower of negative radius E. Sanmiguel-Rojas a,⁎, M. Hidalgo-Martínez b a b
Área de Mecánica de Fluidos, Universidad de Jaén, Campus de Las Lagunillas, 23071 Jaén, Spain Área de Ingeniería Mecánica, Universidad de Córdoba, Campus de Rabanales, 14071 Córdoba, Spain
a r t i c l e
i n f o
Article history: Received 22 January 2015 Received in revised form 24 August 2015 Accepted 30 August 2015 Available online xxxx Keywords: Constant-breadth cams Roller follower Negative-radius Bézier curves
a b s t r a c t Constant-breadth cam mechanisms are characterized by the fact that the closure of the camfollower contact is guaranteed thanks to the geometry and, therefore, auxiliary elements as springs are not necessary. On the other hand, cam mechanisms with a negative radius roller follower allow designing simple machines exerting remarkably high forces when the space is restricted. This work proposes a new constant-breadth cam mechanism with a double roller translating follower of negative radius. The cam profile is designed with an optimizing method based on Bézier curves. © 2015 Elsevier Ltd. All rights reserved.
1. Introduction The goal of the constant-breadth cam mechanism is the elimination of the extra preloading constrains to prevent the loss of the contact between cam and followers. By the type of follower, constant-breadth cam mechanisms can be classified as of flat-faced follower or roller follower. Fig. 1(a) shows a cam mechanism with a parallel flat-faced double translating follower. In these mechanisms, the distance between the parallel flat faces of the follower is constant. They can operate either translating or oscillating follower if the appropriate desmodromic conditions are established [1]. Rothbart [2] shows for a constant-breadth cam with a flat-faced double translating follower that the displacement function of the follower can be defined freely in the interval between 0o and 180o, however, the remaining interval of the displacement function is imposed by the first one because the distance between the faces of the follower is constant. Koloc and Václavik [3] analyze a constant-breadth cam as a case of conjugated cam mechanisms. Zayas et al. [4] describe a generating process by means of circular arcs to design cams with both translating and oscillating followers. Using non-parametric Bézier curves, Cardona et al. [5] show a procedure which guarantees automatically the global continuity of the law of displacement for both parallel flat-faced double translating follower and parallel flat-faced double oscillating follower. Recently, these authors [6] have analyzed the influence of the inclination and offset of the translating follower in a constant-breadth cam mechanism. They also calculate the sliding velocities for both translating and oscillating followers. Regarding constant-diameter cam mechanisms with a double roller follower of positive radius (see Fig. 1(b)), Rothbart [10] develops the equations of the radius of curvature in force-closed cam mechanisms which use both flat-faced and roller followers. Qian [7] studies a constant-diameter cam which operates a double roller follower. By the motion of the follower, the relations between the cam angle and the different geometric parameters are established. Lin et al. [8] and Shin et al. [9] develop design solutions for breadth cams with double roller followers with positive radius of curvature. This new method is based on calculating the coordinates ⁎ Corresponding author. E-mail address: [email protected] (E. Sanmiguel-Rojas).
http://dx.doi.org/10.1016/j.mechmachtheory.2015.08.018 0094-114X/© 2015 Elsevier Ltd. All rights reserved.
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(a)
(b)
O O
Fig. 1. (a) A constant-breadth cam mechanism with a parallel flat-faced double translating follower, and (b) a constant-diameter cam mechanism with a double roller follower of positive radius. The center of rotation of the cam (O) is also shown.
to determine the profile through each contact point. These authors use polynomial functions until seventh degree to ensure continuity in the cam profile. On the other hand, the literature is sparse about cam mechanisms with negative radius roller followers. Carra et al. [11] develop a synthesis procedure, based on the modified trapezoidal curves, in order to generate the correct motion law for these cam mechanisms without undercutting problems. Recently, Hidalgo et al. [12] have proposed a numerical method for optimizing the design cams using a Bézier ordinate as an optimization parameter. As application example, they find the maximum of the follower lift, avoiding the undercutting problem, for the particular case of cam mechanisms with followers of negative radius. There are many ways to express a cam profile mathematically. The procedures to generate the cam profile are deeply explained in the technical literature (see for example [10,13]). The functions to design the cam profile include splines, harmonic, cycloid, modified harmonic, trapezoidal, modified trapezoidal, polynomial, etc [10,14–18]. Other authors have shown that the Bézier curves are a powerful tool for designing cams [5,12,19–21]. However, these curves are much less widespread in this engineering field than the modified trapezoidal curves or the spline functions. In the current work, a constant-breadth cam mechanism that operates a double roller follower of the negative radius is proposed as a new form-closed mechanism. As far as the authors know, this cam mechanism has not been analyzed before. The procedure to
Fig. 2. Scheme and nomenclature used to generate the cam profile. The Oxy reference system is considered fixed to the frame while the Ox1y1 reference system is attached to the cam. Also shown the pitch curve (dash–dot red line), the prime circle (dashed black line) with radius Ra and both radii Rf of the follower. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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design the cam profile, based on the curvature theory, is described in the next section. Section 3 is devoted to maximize the lift follower using an optimization method based on the Bézier curves. The sliding velocities are calculated in Section 4 for the particular case of sliding followers of negative radius. Finally, some conclusions are drawn in the last section. 2. Description of a constant-breadth cam mechanism with a double roller follower of negative radius It is known that a cam mechanism with a roller follower of negative radius has certain advantages over the classical roller follower of positive radius [11,12]. In particular, this mechanism is useful in applications where space is constrained and very high forces are involved. Furthermore, this cam-follower mechanism does not present such severe restrictions with the maximum pressure angle, ϕmax, which can reach 41o. Finally, this kind of mechanism brings to a reduction of the problems of excessive Hertz contact pressure. In order to describe the procedure to obtain the cam profile with a double roller follower of negative radius, we have included in Fig. 2 a sketch of such mechanism. A 3D view of the mechanism is also included in Fig. 3. Notice that the double follower has a double negative radius Rf (solid black/thin line). The prime circle radius Ra is also shown. The reference system Oxy is fixed, i.e. it is attached to the frame of the mechanism, with its origin O placed at the cam axis of rotation. In addition, on its axis Oy are placed both centers C1 and C2 of the follower. However, the reference system Ox1y1 is attached to the cam. The motion of the cam with respect to the frame is a rotation of angle θ with angular velocity ω ¼θ k. Thus, the motion of the follower with respect to the reference system Oxy is a translation in the direction Oy, and the points O, C1 and C2 remain always aligned in such direction. Notice that the points C1 and C2 are the intersection between the axis Oy and the pitch curve. The lengths OC1 and OC2 can be written as,
OC 1 ¼ Ra −f ðθÞ; OC 2 ¼ Ra −f ðθ þ πÞ;
ð1Þ
where f(θ) is the displacement function of the follower. Since the length between the centers of the follower C1 and C2 or breadth of the cam mechanism must remain constant, namely dc, summing up the expressions in Eq. (1) yields, OC 1 þ OC 2 ¼ 2Ra −f ðθÞ− f ðθ þ πÞ ¼ dc :
ð2Þ
If we assume that f(0) = 0 and f(π) = h*, where h* is the maximum follower displacement, from Eq. (2) we found,
dc ¼ 2Ra −h :
ð3Þ
Therefore, inserting Eq. (3) in Eq. (1) we obtain the desmodromic condition for this cam mechanism, i.e.
f ðθÞ þ f ðθ þ πÞ ¼ h :
ð4Þ
Fig. 3. 3D view of the mechanism.
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Thus, the displacement function f(θ) can be defined only in the interval 0 ≤ θ ≤ π (designed segment), whereas the displacement function in the interval π b θ b 2π (calculated segment) is given by Eq. (4). Using the Bézier curves, the designed segment f(θ) can be written as 8 <
θ 0 ⩽ θ ⩽ β ðrising phaseÞ; f ðθÞ ¼ β : h β b θ ⩽ π;
h P
ð5Þ
where P is an nth-degree Bézier curve and β is the cam angle required to reach h* (see Hidalgo et al. [12] for further details). To establish the continuity conditions between the designed and calculated segments, we derivate twice Eq. (4) with respect to θ yielding, f θ ðθÞ þ f θ ðθ þ πÞ ¼ 0;
ð6aÞ
f θθ ðθÞ þ f θθ ðθ þ πÞ ¼ 0:
ð6bÞ
To obtain the cam profile and the pressure angle, we will focus on the contact point P1 between the cam and follower. Note from Fig. 2 that the follower has only one instantaneous center I, because if we assume two different instantaneous centers I1 and I2, the abscissa of both centers with respect to the system Oxy must fulfill, xI1 ¼ −f θ ðθÞ ¼ f θ ðθ þ πÞ ¼ −xI2 :
ð7Þ
Therefore, the position vector of the contact point P1 is rI −r C 1 ; rP1 ¼ rC 1 −jR f j rI −r C 1
ð8Þ
where rC1 ¼ rI ¼
0 ; Ra − f ðθÞ
ð9Þ
− f θ ðθÞ : 0
ð10Þ
On the other hand, the pressure angle ϕ is the angle (at any point) between the normal to the pitch curve and the direction of the follower motion [12], j r −r C1 I R −f ð θ Þ a qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi; ¼ arccos ϕ ¼ arccos 2 2 r C −r I ½ ð Þ ½ ð Þ θ þ R − f θ f θ a 1
ð11Þ
where j is the unit vector in y direction. Finally, with the help of the Euler–Savary equation [22–24], the radius of curvature of the pitch curve is given by Hidalgo et al. [12] as, ! ρ ¼ jC 10 C 1 j ¼
n o3 ½ f θ ðθÞ2 þ ½Ra −f ðθÞ2 2 2½ f θ ðθÞ2 þ ½Ra − f ðθÞ½Ra −f ðθÞ þ fθθ ðθÞ
;
ð12Þ
where C10 is a common center of curvature. Therefore, the radius of curvature of the cam ρc is defined as, ρc ¼ jR f j−ρ;
ð13Þ
being the no-undercutting condition, ρc N 0
⇒
ρ b jR f j:
ð14Þ
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3. Optimization of the maximum follower displacement In this section, we use the optimization method originated by Hidalgo et al. [12] to find the maximum reachable follower displacement avoiding the undercutting and preserving the pressure angle ϕ b 41o, for a given Rf and β. In this method, the displacement function of the follower f(θ) is defined by a Bézier curve, see Eq. (5), and its shape is modified using an ordinate bi as optimization parameter. To present the results, we define the maximum dimensionless follower displacement h = h*/|Rf |, the dimensionless prime circle radius α = Ra/|Rf | and the maximum reachable dimensionless follower displacement hmax = max(h) for ϕ b 41o and without undercutting. Fig. 4(a) displays the parametric study performed numerically (see flow chart in [12]) to find hmax fulfilling the no-undercutting condition and ϕ b 41o, for a Bézier curve of degree n = 6 and β = π. Notice that hmax = hmax(α, bi) for a given β [12]. As in the case of a follower with one negative radius [12], the highest values of hmax are reached when β = π and the dimensionless prime circle radius α = Ra/|Rf | is increased. The first remarkable feature is that, independently of the value α, hmax reaches its maximum at b3 = 0.5. It is also interesting to point out that all the graphs are symmetric with respect to the value b3 = 0.5. To find an explanation to this behavior, we have plotted in Fig. 4(b) the dimensionless displacement function of the follower f(θ)/h* and its first two derivatives, for the values b3 = 0.1 (continuous line) and b3 = 0.9 (dashed line). Notice that these values of the ordinate b3 are equidistant with respect to b3 = 0.5 in Fig. 4(a). As can be observed, the rising phase of the follower for b3 = 0.1 (continuous line for 0 ≤ θ ≤ π) is perfectly symmetric, with respect to θ = π, to the return phase of the follower for b3 = 0.9 (dashed line for π b θ b 2π), and vice versa. Furthermore, the same happens to the second derivative. However, the first derivative shows, for both equidistant values of b3, an antisymmetric behavior in the rising and return phases with respect to θ = π. This behavior in curves of degree n = 6 evaluated at ordinates equidistant from b3 = 0.5 is due to the desmodromic condition (4). Notice from the equations of the pressure angle (11) and radius of curvature of the pitch curve (12) that, for a given h = h*/|Rf |, these properties of symmetry and antisymmetry between curves of values b3 equidistant from b3 = 0.5, have no effect on the maximum of ϕ or on the minimum of ρc. To show this fact, we have plotted in Fig. 5 the dimensionless radius of curvature ρc/|Rf | and the pressure angle ϕ, for α = 0.4 and hmax(b3 = 0.1) = hmax(b3 = 0.9) ≈ 0.35. Notice that the minimum of the dimensionless radius of curvature for both values of b3 is the same, i.e. ρc/|Rf | ≈ 0, and the maximum of the pressure angle is also the same in both cases, i.e. ϕ ≈ 28.7o. Furthermore, the curves corresponding to the rising phase for b3 = 0.1 and the return phase for b3 = 0.9, and vice versa, are also perfectly symmetric with respect to θ = π. Therefore, the numerical algorithm finds the same hmax for equidistant ordinates with respect to b3 = 0.5. It is also interesting to point out that, for all the values of α and b3, the radius of curvature is more restrictive than the pressure angle, which is always clearly lower than 41o. The same happens for a Bézier curve of degree n = 8 with the ordinate b4 as optimization parameter. On the other hand, Fig. 6 shows the dimensionless displacement function f(θ)/h* and its first two derivatives with respect to the cam angle θ, corresponding to a Bézier curve n = 6 for b3 = 0.5 (continuous line). Notice that the curves f(θ)/h*, fθ(θ)/h* and fθθ(θ)/ h* fulfill Eqs. (4), (6a) and (6b), respectively, as in the case of the curves plotted in Fig. 4(b). As a Bézier curve of degree n = 6 guarantees continuity C2, the jerk-curve is discontinuous as can be appreciated from the slope of the curve fθθ(θ)/h* at θ = 0 and θ = π. Moreover, it is worth noting that in the particular case of a Bézier curve of degree n = 6 evaluated at the ordinates ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 Þ ¼ ð0; 0; 0; 0:5; 1; 1; 1Þ;
ð15Þ
one obtain the curve of degree n = 5 evaluated at the ordinates ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 Þ ¼ ð0; 0; 0; 1; 1; 1Þ:
ð16Þ
(a)
(b) α α
*
f/ h
α
f θ /h
h m ax
*
f θθ /h
*
b3
θ/ π
Fig. 4. (a) Maximum reachable values of h without undercutting and ϕ b 41o using a Bézier curve of degree n = 6, β = π and different values of α as indicated. (b) Dimensionless displacement function f(θ)/h* and its first two derivatives, for a Bézier curve of degree n = 6 with b3 = 0.1 (continuous line) and b3 = 0.9 (dashed line).
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(b)
φ
ρc / | R f |
(a)
θ/π
θ/π
Fig. 5. For α = 0.4, h = hmax and β = π, (a) dimensionless radius of curvature ρc/|Rf | and (b) pressure angle ϕ (in degrees), for a Bézier curve of degree n = 6 with b3 = 0.1 (continuous line) and b3 = 0.9 (dashed line).
To show that, we have also included in Fig. 6 (circles) the dimensionless displacement and its first two derivatives for a Bézier curve of degree n = 5. Therefore, preserving continuity C2 and independently of the dimensionless prime circle radius α, the maximum reachable values of h without undercutting and ϕ b 41o is reached with a Bézier curve of degree n = 5. In addition, we have performed numerically the same parametric study but with a Bézier curve of degree n = 8, which is summarized in Fig. 7(a). Notice that the graphs show the same properties discussed above. However, for each α, the maximum reachable values of h without undercutting are clearly lower than those obtained for n = 6. We have plotted in Fig. 7 the dimensionless displacement function f(θ)/h* and its first two derivatives for a Bézier curve of degree n = 8 and ordinate b4 = 0.5 (continuous line). Notice that in this case, the slope in the second derivative is continuous. Therefore, the continuity C3 is guaranteed and we will have a continuous jerk-curve. Finally, we have also included in Fig. 7(b) (circles) the dimensionless displacement and its first two derivatives but for a Bézier curve of degree n = 7, which for the ordinates ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 Þ ¼ ð0; 0; 0; 0; 1; 1; 1; 1Þ;
ð17Þ
is the particular case of a Bézier curve of degree n = 8 evaluated at the ordinates ðb0 ; b1 ; b2 ; b3 ; b4 ; b5 ; b6 ; b7 ; b8 Þ ¼ ð0; 0; 0; 0; 0:5; 1; 1; 1; 1Þ:
ð18Þ
f/ h
*
f θ /h
*
f θθ /h
*
θ/ π Fig. 6. Dimensionless displacement function f(θ)/h* and its first two derivatives corresponding to the Bézier curves n = 5 (circles) and n = 6 at b3 = 0.5 (continuous line).
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(b) α α
f θθ /h
∗
h m ax
f θ /h ∗
α
f/ h ∗
(a)
b4
θ/ π
Fig. 7. (a) As in Fig. 4(a) but for Bézier curves of degree n = 8. (b) As in Fig. 5 but for Bézier curves of degree n = 7 (circles) and n = 8 at b4 = 0.5 (continuous line).
Thus, independently of the dimensionless prime circle radius α, the maximum reachable values of h without undercutting and ϕ b 41o is reached preserving continuity C3 with a Bézier curve of degree n = 7. 4. Sliding velocities in constant-breadth cams with a double sliding follower of negative radius Friction opposes the relative movement of contacting bodies in all machinery. In cam-follower mechanisms, both sliding and rolling frictions are manifested. The sliding velocity plays an important role on the energetic efficiency and longer life of these mechanisms. Cardona et al. [6] calculate the sliding velocities at the two contact points of a constant-breadth cam mechanism with a parallel flat-faced double translating follower. They show that the sum of both sliding velocities is constant for a given angle of inclination and offset of the follower. Furthermore, they also demonstrate that the choice of such design parameters may ensure lower sliding velocities. On the other hand, to manufacture constant-breadth cam mechanisms with a double roller follower of negative radius, using the inner rings of commercial roller bearings as roller followers [11], minimizes practically to zero the sliding velocity. However, we think that it is interesting to analyze the sliding velocities in the particular case of a constant-breadth cam mechanism with a double sliding follower of negative radius. Some advantages of a double sliding follower over a double roller follower, for the same application, could be: lower weight; simplicity of maintenance; ease of lubrication; and lower manufacturing costs. Thus, with the help of Fig. 2, the modulus of the sliding velocities at the contact points P1 and P2 between the cam and the follower, can be determined as ! vP 1 ¼θ jIP 1 j;
ð19aÞ
! vP 2 ¼θ jIP 2 j;
ð19bÞ
! ! where θ is the angular velocity of the cam. The values of the vectors jIP 1 j and jIP 2 j are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! jIP 1 j ¼ R f −jIC 1 j ¼ R f − ½ f θ ðθÞ2 þ ½Ra − f ðθÞ2 ;
ð20aÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! jIP 2 j ¼ R f −jIC 2 j ¼ R f − ½ f θ ðθ þ πÞ2 þ ½Ra −f ðθ þ π Þ2 :
ð20bÞ
From Eqs. (4) and (6a) we know that
f ðθ þ πÞ ¼ h −f ðθÞ;
ð21aÞ
f θ ðθ þ πÞ ¼ −f θ ðθÞ:
ð21bÞ
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! Therefore, the vector jIP 2 j can be rewritten as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! jIP 2 j ¼ R f − ½ f θ ðθÞ2 þ ½Ra −h þ f ðθÞ2 :
ð22Þ
Fig. 8(a) shows the dimensionless sliding velocities defined as
λP 1
λP 2
! IP 1 ¼ ; ¼ θ jR f j R f vP1
ð23aÞ
! IP 2 ¼ ; ¼ θ jR f j R f vP2
ð23bÞ
for different values of the dimensionless prime circle radius α. In addition, we have plotted in Fig. 8(b) the sum of the sliding velocities Λ ¼ λP 1 þ λP 2 ;
ð24Þ
for the same values of α. Unlike what happens in the case of a parallel flat-faced double translating follower [6], the sum of the sliding velocities is not constant. Moreover, the amplitude of the fluctuations in both sliding velocities and in their sum is increased for higher values of α. These fluctuations could lead to fatigue in the material of the cam or the follower. However, the modulus of the sliding velocity decreases when α is increased. To show more clearly of this feature, Fig. 9 displays the maximum and minimum of the sum of the dimensionless sliding velocities as a function of α, i.e. Λmax = max(Λ) and Λmin = min(Λ), calculated for h = hmax with Bézier curves of degree n = 5, Fig. 9(a), and n = 7, Fig. 9(b). Notice that for both degrees n = 5 and n = 7, such maximum and minimum of the sum tend to decrease as α increases. Furthermore, such values are slightly lower for n = 7 than for n = 5. Finally, it is worth mentioning that this tend is the opposite of that corresponding to the case of a parallel flat-faced double translating follower, for which the sliding velocities are increased for higher values of the prime circle radius Ra. 5. Conclusions This work introduces and studies a new constant-breadth cam mechanism with a double roller follower of negative radius, which belongs to the category of form-closed cam mechanisms. This new mechanism allows higher forces into restricted spaces than the equivalent mechanism of parallel flat-faced double follower. In particular, we deduce the desmodromic condition of this cam mechanism, and the continuous displacement function of the follower is defined with Bézier curves. Using the optimization method introduced by Hidalgo et al. [12], the follower lift is maximized avoiding the undercutting problem but keeping the pressure angle lower than 41o, given the total rise angle β and the double negative radius of the follower |Rf |. On the other hand, we have also performed a
(b)
Λ
λP 1, λ P 2
(a)
α
α
α
α
α
α
θ/ π
θ/ π
Fig. 8. For different values of α: (a) Dimensionless sliding velocities λP1 (continuous lines) and λP 2 (discontinuous lines); (b) sum of the dimensionless sliding velocities Λ ¼ λP1 þ λP2 . Calculations carried out with a Bézier curve of degree n = 5 and h = hmax for each α.
E. Sanmiguel-Rojas, M. Hidalgo-Martínez / Mechanism and Machine Theory 95 (2016) 93–101
(a)
α
101
(b) Λ
Λ
Λ
Λ
α
Fig. 9. Maximum, Λmax = max(Λ), and minimum, Λmin = min(Λ), of the sum of the dimensionless sliding velocities, for cams designed with Bézier curves of degree n = 5 (a) and n = 7 (b). For each value of α, the maximum dimensionless follower displacement is hmax.
study of the sliding velocities at the two contact points for the particular case of a double sliding follower of negative radius. We find that, unlike what happens in the case of a parallel flat-faced double translating follower, the sum of the sliding velocities is not constant and exhibits fluctuations. Furthermore, the amplitude of such fluctuations is increased for higher values of the dimensionless prime circle radius α, although the mean value of the sliding velocities decreases when α is increased. Finally, taking advantage of inner rings of commercial bearings to manufacture the roller follower, the sliding velocity could be minimize practically to zero, which is not possible in cams with a parallel flat-faced double translating follower.
Acknowledgments This work was supported by the Universidad de Jaén (Spain) and the Universidad de Córdoba (Spain).
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