Progettazione Funzionale di Sistemi Meccanici e Meccatronici Camme - Introduzione prof. Paolo Righettini paolo.righettini@unibg.it Dipartimento di Progettazione e Tecnologie Mechatronics and Mechanical Dynamics Laboratories prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 1/40
Introduction cam mechanisms may be represented by the basic plane configuration constituted by three main parts: the driver part: the cam profile the driven part: the follower; the follower moves the payload in the working zone the fixed part: the frame the cam transmits the desired motion to the follower by direct contact prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 2/40
Cam mechanisms - introduction cam mechanisms may be used to realize both progressive or alternative motion it may be classified on the follower motion linear motion (translating follower) rotary motion (oscillating follower) prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 3/40
Cam mechanisms - introduction the type of output motion fixes the type of the follower: tapped for translating follower rocked arm for oscillating follower prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 4/40
Cam mechanisms - introduction we may have three types of contact surfaces between cam profile and follower knife edge roller flat faced prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 5/40
Cam mechanisms - introduction we may have plane and spatial cams in plane cams the follower moves on the same plane of the cam profile in spatial cams the follower does not move on the same plane of the cam profile prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 6/40
Cam mechanisms - introduction the choice of the type of follower depends on the type of motion required by the payload in the working zone the type of contact depends on the working force the choice between plane and spatial cams depends on the amplitude of the motion required, on the stiffness and on the machine configuration spatial cams are widely used for intermittent progressive motion generation prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 7/40
Cam profile - introduction the synthesis process of cam profile requires the geometrical motion law of the follower (y, y, yy ) as a function of the cam shaft rotation α some geometric parameters of the cam mechanism taking into account a rocked arm cam at the initial configuration, we have the size of the cam depends on the primitive base radius R b0, on the length of frame and rocked arm (we need three geometric parameters to define the initial configuration) prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 8/40
Cam profile - introduction starting from the initial configuration we define the direction of rotation of the follower so that β(α) = β 0 +y(α) the point P of the follower must follow the cam profile while the cam rotates of α and the follower oscillates according to β(α) prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 9/40
Cam profile - introduction taking into account the relative motion of the cam profile respect to the frame, we may fix the cam profile and than we rotate in opposite direction the frame of the cam mechanism while the frame rotates of α the rocked arm oscillates respect to it of β(α) the point P 2 describe the pitch profile of the cam we obtain the primitive profile in the absolute reference frame prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 10/40
Cam profile - introduction the pitch profile (black line) has been determined by means of the knife-edge follower if we use a roller or a flat face contact the effective profile (blue line) will be determined by means of envelope of the subsequent positions of the follower contact part along the pitch profile all the roller positions must take part in the determination of the effective profile the roller and the enveloped line have the same tangent at the contact point prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 11/40
Cam profile - introduction we may have undercutting phenomenon during the design of the effective profile the same problem may occur during the machining phase of the cam profile with a milling or grinding machine prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 12/40
Contact force cam profile and follower are between them in contact at the contact point (or line contact) the force required for the follower motion is applied by the cam profile at the contact point; this is the contact force S neglecting the friction between cam profile and follower the direction of the contact force direction corresponds to the direction of the normal (perpendicular) to the cam profile at the contact point prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 13/40
Contact force generally, only a component of the contact force works for the follower motion we introduce the pressure angle θ as the angle between the contact force and the direction of the contact point velocity of the follower the contact force component that works for the follower motion is therefore F = Scosθ prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 14/40
Pressure angle all the mechanical parts will be designed according to the contact force S higher values of the pressure angle lead to bigger mechanical parts for the same follower motion during the design process we must limit the pressure angle starting from S = F/cosθ to limit the contact force S lower the about 1.4 times the work force F we limit the pressure angle to 45 prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 15/40
Cam design - introduction the external forces F e = F a +F i depends on the application and on the inertial action (follower mass and motion acceleration) the size of the mechanical parts depends on the contact force S, therefore on F e and on pressure angle θ S = S(F e, θ ) the diameter of the roller depends on S the design process of the cam profile takes into account the conprog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 16/40
Cam design - introduction the contact pressure p depends on the geometry of the contact surfaces (profile curvature ρ, roller radius R r, cam thickness b), on the material (YoungŠs modulus E) p = p(s,r r, ρ,e,b) for any contact point it must result p p am where p am is an admissible contact pressure for the cam material prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 17/40
Cam design - introduction the profile curvature ρ depends on the motion law and on the base dimension of the cam R b (base radius) ρ = ρ(r b,y,y ) the profile curvature increases with the increment of R b the pressure angle θ decreases with the increment of R b prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 18/40
Cam design - introduction the pressure angle influences the design of the mechanical components of the mechanisms the minimum cam dimension depends on the maximum value of the pressure angle θ and on the maximum value of the contact pressure p the contact pressure depends on the minimum value of the profile curvature ρ the acceleration profile of the required follower motion gives us the opportunity to limit the pres- prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 19/40
Pressure angle - tapped the effect of the contact force S on the follower depends on its type taking into account the tapped follower in the rise motion considering the contact of the follower in two points of the guide bearing of the follower; we have the normal contact forces N 1, N 2 T 1 and T 2 are the frictional forces opposing the follower motion (T = fn, f sliding friction) F e eternal forces, θ pressure angle prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 20/40
Pressure angle - tapped along the vertical direction we have Fv = Scosθ F e T 1 T 2 = 0 along the horizontal direction we have Fh = Ssinθ +N 1 N 2 = 0 neglecting the transverse dimension of the follower the sum of the moments around O results MO = SsinθA N 1 B = 0 the contact force results S = F e cosθ f(1+2a/b)sinθ prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 21/40
Pressure angle - tapped the contact force S = F e cosθ f(1+2a/b)sinθ is infinity (which means that the follower will jam) when the the denominator equals zero it results cosθ max f(1+2a/b)sinθ max tanθ max = B f(2a+b) for f = 0.15 and A = B/2 we have [ ] 1 θ max = arctan = 73.5 0.15 2 prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 22/40
Pressure angle - tapped we may express the contact force as a function of θ max S = F e sinθ max cosθsinθ max sinθcosθ max = F esinθ max sin(θ max θ) F e 1 sin(θ max θ) if we would like the limitation S 1.4F e introduced before, it results θ max θ 45, θ θ lim = θ max 45 for f = 0.15 and A = B/2 we have, for the rise part θ lim 30 for the return phase of the movement the friction forces have opposite direction; we have not jamming (the follower moves in the same direction of the external forces) and we may assume again θ lim = 45 prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 23/40
Pressure angle - rocked arm in this the pressure angle is the maximum value for the pressure angle is θ lim = 45 prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 24/40
Cams configuration followers are held in contact to cam profile with: external forces due to spring: force coupling double contact point: positive drive cam or shape coupling; in this case we have two cam profiles, one for positive motion force, one for negative motion force positive drive systems may use two complementary (conjugate) cam profiles; double or complex follower is needed force coupling systems use one cam profile and a simple follower prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 25/40
Cams configuration positive drive systems are flexible, accurate and reliable for high speed movements prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 26/40
Rocked arm cams b rocked arm d frame R b0 pitch base radius β 0 starting angle between frame and rocked arm P, P 2 points on the pitch profile β(α) = β 0 + y(α) prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 27/40
Rocked arm cams y(α) must be defined for 0 α 2π y(α) h D R R D α d 2π α prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 28/40
Rocked arm cams the polar coordinates ρ p and α p = α+α c of the pitch profile are ρ p = b 2 +d 2 2bdcosβ ( bsinβ α p = α+α c = α+arctan d bcosβ ) prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 29/40
Rocked arm cams pitch profile and follower are subject to the contact force S whose direction is orthogonal to the pitch profile in the contact point; we must determine the direction γ of the normal to profile respect to the frame considering unitary efficiency, the power balance applied to the follower and the cam gives SMOω = SNO 2 β = SNO2 ωy (α) MO = NO 2 y (α) where ω is the velocity of rotation of the cam prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 30/40
Rocked arm cams it also results it results NO 2 = bsin(β +γ) N 2 O 2 = dsinγ MO = NO 2 N 2 O 2 = bsin(β +γ) dsinγ finally from MO = NO 2 y (α) it results bsin(β +γ) dsinγ = bsin(β +γ)y (α) tanγ = b(1 y )sinβ d bcosβ(1 y ) prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 31/40
Rocked arm cams γ gives the direction of the normal to profile from the figure it results θ +β +γ = π 2 θ = π 2 β γ for y = 0 ( dy = 0) the normal to profile at the contact dα point pass through the point O, the axis of rotation of the cam prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 32/40
Rocked arm cams to obtain the point P of the effective cam profile, we must start from the correspondent pint C on the pith profile and subtract the roller radius R r along the direction of the normal the direction of the normal is indicated by the angle γ from the figure results ρ p = OH 3 2 +PH3 2 prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 33/40
Rocked arm cams OH 3 results OH 3 = d H 3 H 1 bcosβ OH 3 = d R r cosγ bcosβ PH 3 results PH 3 = bsinβ R r sinγ prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 34/40
Rocked arm cams the polar coordinates of the cam profile are ρ p = (d R r cosγ bcosβ) 2 +(bsinβ R r sinγ) 2 ϕ = α+α p = α+arctan ( ) bsinβ Rr sinγ d R r cosγ bcosβ prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 35/40
Rocked arm cams the point K on the normal is the center of curvature of the profile around the contact point to determine the curvature radius ρ 0 we take into account the vectorial closure KO + CK + O 2 C + OO 2 = 0 its projections along and orthogonal to the frame are r k cosα k +ρ 0 cosγ +bcosβ d = 0, // r k sinα k +ρ 0 sinγ bsinβ = 0, prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 36/40
Rocked arm cams taking into account a second order approximation of the profile, it results dα k dα = 1 (the position on the center of curvature K does not change for dα) the derivative respect to α of the projection orthogonal to the frame results r k cosα k ( 1)+ρ 0 cosγ dγ dα bcosβdβ dα = 0 where dγ dα = γ, and dβ dα = y prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 37/40
Rocked arm cams it results r k cosα k ( 1)+ρ 0 cosγγ bcosβy = 0 substituting this expression in the projection along the frame, it results ρ 0 = d bcos(β)(1 y ) (1+γ )cosγ where γ is γ = b(1 y )y cos(β +γ) by sin(β +γ) dcosγ b(1 y )cos(β +γ) prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 38/40
Rocked arm cams - considerations the proposed equations consider clockwise rotation of cam and follower; we must introduce some modification if the mechanism does not work according to this schema if the follower rotates counterclockwise, the angle of the rocked arm respect to the frame is β = β 0 y(α) therefore we must change the sign to the geometric acceleration y if the cams rotates counterclockwise we have α instead of α therefore we must change the sign to the first derivative respect to α we may have both the above cases prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 39/40
Rocked arm cams - considerations the proposed equations let us design of positive drive mechanisms we must use the same motion law for the two cams we must define the geometry of both the followers prog. funz. sis. Meccanici e Meccatronici - prof. Paolo Righettini p. 40/40