Relly Victoria PETRESCU, Adriana COMĂNESCU, Florian Ion PETRESCU

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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DYNAMICS OF CAM GEARS ILLUSTRATED AT THE CLASSIC DISTRIBUTION MECHANISM Relly Victoria PETRESCU, Adriana COMÄ‚NESCU, Florian Ion PETRESCU Abstract: The paper presents an original method in determining the general dynamic of the mechanisms with rotation cams and followers, particularized to the plate translated follower. First, one presents the dynamics kinematics. Then one solves the Lagrange equation and with an original dynamic model (see [1]) with one degree of freedom, with variable internal amortization, one makes the dynamic analyze of two models.

1 Introduction The paper proposes an original dynamic model illustrated for the rotating cam with plate translate follower. One presents the dynamics kinematics (the original kinematics); the variable velocity of the camshaft obtained by an approximately method has been used with an original dynamic system having one grade of mobility and an variable internal amortization [1]; one tests two movement laws, one classics and one original.

2 Dynamics of the classic distribution mechanism 2.1 Precision kinematics at the classic distribution mechanism In the picture number one, one presents the kinematics schema of the classic distribution mechanism, in two consecutive positions; with a interrupted line is represented the particular position when the follower is situated in the most down plane, (s=0), and the cam which has an orally rotation, with constant angular velocity, ω, one is situated in the point A0, (the recordation point between the base profile and the up profile), particular point that mark the up begin of the follower, imposed by the cam-profile; with a continue line is represented the superior couple in someone position of the up phase.

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Al III-lea Seminar National de Mecanisme, Craiova, 2008

r v1 Ai

s’

τ

C

r r v2 v12 B

D

θ&

θ

A0 rA

τ

s

ϕ

r0=s0

A0i O

ω

Fig. 1. The kinematics of the classic distribution mechanism The point A0, which marks the initial couple position, represents in the same time the contact point between the cam and follower in the first position.

The cam has an angular velocity, ω, (the camshaft angular velocity).

Cam is rotating with the velocity, ω, describing the angle ϕ, which show how the base circle has rotated in the orally sense, (with the camshaft together); this rotation can be seen on the base circle between the two particular points, A0 and A0i.

In this time the vector rA=OA (which represents the distance between the centre of cam O, and the contact point A), has rotating (trigonometric) with the angle, τ. If one measures the angle, θ, which position the general vector, rA in function of the particular vector, rA0, one obtains the relation (0): (0) θ =ϕ +τ r Where rA is the module of the vector, rA , and θA represents the phase r angle of the vector, rA . r The rotating velocity of the vector rA is θ&A which it’s a function of the angular velocity of the camshaft, ω, and a rotating angle, ϕ, (by the movement laws s(ϕ), s’(ϕ), s’’(ϕ)).

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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The follower isn’t acted directly by the cam, by the angle, ϕ, and by the r angular velocity ω; it’s acted by the vector rA , which has the module rA, the position angle θA and the angular velocity θ& . From here result a particular A

(dynamic) kinematics, the classical kinematics being just a static and approximate kinematics.

Kinematics one defines the next velocities (see the picture 1). r r v1 =the cam velocity; which is the velocity of the vector, rA , in the point A; now the classical relation (1) become an approximately relation, and the real relation take the form (2). v1 = rA .ω (1) v1 = rA .θ& A (2) r r The velocity v1 == AC is separating in the velocity v2 =BC (the r follower velocity which act in its axe, on a vertical direction) and v12 =AB (the slide velocity between the two profiles, the sliding velocity between the cam and the follower, which works by the direction of the commune tangent line of the two profiles in the contact point).

Usually the cam profile is synthesis with the AD=s’ knew, for the classical module C, and one can write the relations: (3) rA2 = ( r0 + s ) 2 + s ' 2

rA = (r0 + s) 2 + s' 2 cos τ =

r0 + s r0 + s = rA (r0 + s ) 2 + s ' 2

(4) (5)

AD s' s' = = (6) rA rA (r0 + s) 2 + s' 2 s' (7) v 2 = v1 . sin τ = rA .θ&A . = s'.θ&A rA Now, the follower velocity isn’t s& ( v 2 ≠ s& ≡ s '⋅ω ), but it’s given by the relation (9). At the classical distribution mechanism the transmitting function D, is given by the relations (8): θ&A = D.ω (8) θ& v D= A = 2 ω s& & (9) v 2 = s '⋅θ A = s '⋅D ⋅ ω = s& ⋅ D Determining of the sliding velocity between the profiles is made with the relation (10): sin τ =

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Al III-lea Seminar National de Mecanisme, Craiova, 2008

r +s v12 = v1 . cos τ = rA .θ&A . 0 = (r0 + s).θ&A (10) rA The angles τ and θA will be determined, with him first and second derivatives.

The τ angle has been determined from the triangle ODAi (see fig.1) with the relations (11-13): s' sin τ = (11) (r0 + s) 2 + s' 2 cos τ =

r0 + s

(r0 + s ) + s' s' tgτ = r0 + s One derives (11) in function of ϕ angle and one obtains (14): 2

2

(r0 + s ).s '+ s '.s ' ' rA τ '. cos τ = 2 (r0 + s) + s ' 2 The relation (14) will be written in the form (15):

(12) (13)

s ' '.rA − s '.

τ '. cos τ =

s ' '.(r0 + s ) 2 + s ' '.s' 2 − s ' 2 .( r0 + s ) − s ' 2 .s ' ' [(r0 + s) 2 + s ' 2 ]. (r0 + s ) 2 + s' 2

(14)

(15)

From the relation (12) one extracts the value of cosτ, which will be introduced in the left term of the expression (15); then one reduces s’’.s’2 from the right term of the expression (15) and one obtains the relation (16): r0 + s (r0 + s ).[ s ' '.(r0 + s ) − s ' 2 ] (16) τ '. = (r0 + s ) 2 + s ' 2 [(r0 + s ) 2 + s ' 2 ]. (r0 + s ) 2 + s ' 2 After some simplifications one obtains finally the relation (17) which represents the expression of τ’: s' '.(r0 + s ) − s' 2 τ'= (17) (r0 + s ) 2 + s ' 2 Now, when one has, τ’ explicitly, one can determine the next derivatives. The expression (17) will be derived directly and one obtains for begin the relation (18):

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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τ ''= [ s ' ' ' (r0 + s ) + s ' ' s '−2 s ' s ' ' ][(r0 + s ) 2 + s ' 2 ] − 2[ s ' ' (r0 + s ) − s ' 2 ][(r0 + s ) s '+ s ' s ' ' ] [(r0 + s ) 2 + s ' 2 ] 2 (18) One reduces the terms from the first bracket of the numerator (s’.s’’), and then one draws out s’ from the fourth bracket of the numerator and one obtains the expression (19): [ s ' ' '.(r0 + s ) − s '.s ' ' ].[(r0 + s ) 2 + s ' 2 ] − 2.s '.[s ' '.(r0 + s ) − s ' 2 ].[r0 + s + s ' ' ] τ ''= [(r0 + s ) 2 + s ' 2 ] 2 (19) Now one can calculate θA, with its first two derivatives, θ&A şi θ&&A . One writes, θ, and not θA, to simplify the notation. Now one can determine (20), or (0): (20) θ =τ + ϕ One derives (20) and one obtains the relation (21): θ& = τ& + ϕ& = τ '.ω + ω = ω.(1 + τ ' ) = D.ω (21) One makes the second derivative of (20), and the first derivative of (21) and one obtains (22): θ&& = τ&& + ϕ&& = τ ' '⋅ω 2 = D'⋅ω 2 (22) One can write now the transmitting functions, D and D’ (at the classical module, C), in the forms (23-24): D = τ '+1 (23) (24) D' = τ ' ' The follower velocity (25), need the expression of the transmitted function, D. v2 = s '⋅w = s '⋅θ&A = s '⋅θ& = s '⋅D ⋅ ω = s& ⋅ D (25) Where, w = D.ω (26) For the classical distribution mechanism (Module C), the variable w is the same with θ&A (see the relation 25). But at the B and F modules (at the cam gears where the follower has roll), the transmitted function D, and w, take some complex forms.

Now one can determine the acceleration of the follower (27). &y& ≡ a2 = ( s ' '⋅D + s '⋅D' ) ⋅ ω 2 (27) In the picture 2, one can see the kinematics classic and dynamic; the velocities (a), and the accelerations (b).

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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Vclasic[m/s] Vprecis[m/s]

4 3

4000 3000

2

2000 1000

1 0

-3

0 -1000 0 -2000 -3000

-4

-4000

-1 0 -2

a2clasic[m/s2] a2precis[m/s2]

5000

50

100

150

200

50

100

150

200

a) b) Fig. 2. The classical and dynamic kinematics; a-velocities and b-accelerations of the follower To determine the acceleration of the follower, are necessary s’ and s’’, D and D’, τ’ and τ’’.

The kinematics dynamic diagrams of v2 (obtained with relation 25, see fig. 2, a), and a2 (obtained with relation 27, see fig. 2, b), have a dynamic aspect more than one kinematics. One has used the movement law SIN, a rotation velocity at the crankshaft, n=5500 [rpm], an up angle, ϕu=75 [grad], a down angle ϕd=75 [grad] (identically with the up angle), a ray at the basic circle of the cam, r0=17 [mm] and a maxim stroke of the follower, hT=6[mm].

Anyway, the dynamic is more complex, having in view the masses and the inertia moments, the resistant and motor forces, the elasticity constants and the amortization coefficient of the kinematics chain, the inertia forces of the system, the rotation velocity of the camshaft and the variation of the camshaft velocity, ω with the cam position, ϕ, and with the rotation speed of the crankshaft, n.

2.2 Solving approximately the Lagrange movement equation In the kinematics and the static forces study of the mechanisms one considers the shaft velocity constant, ϕ& = ω =constant, and the angular speed null, ϕ&& = ω& = ε = 0 . In reality, this velocity ω it isn’t constant, but it is variable with the camshaft position, ϕ.

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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The mechanisms with cam and follower have the variable velocity of the camshaft as well. Then, one shall see the Lagrange equation, written in the differentiate mode and its general solve.

The differentiate Lagrange equation, has the form (28): J * .ϕ&& +

1 *I 2 .J .ϕ& = M * 2

(28)

Where J* is the inertia moment (mass moment, or mechanic moment) of the mechanism, reduced at the crank, and M* represents the difference between the motor moment reduced at the crank and the resistant moment reduced at the crank; the angle ϕ represents the rotation angle of the crank (crankshaft). J*I represents the derivative of the mechanic moment in function of the rotation angle ϕ of the crank (29).

1 *I 1 dJ * .J = . =L 2 2 dϕ

(29)

If one use the notation (29), the equation (28) will be written in the form (30): J * .ϕ&& + L.ϕ& 2 = M *

(30)

One divide the terms at J* and (30) takes the form (31):

ϕ&& +

L 2 M* .ϕ& = * J* J

(31)

The term with ϕ& 2 will be move in the right and one obtains (32):

ϕ&& =

M* L − * .ϕ& 2 * J J

(32)

One change the left term of the expression (32) with (33), and one obtains the relation (34):

ϕ&& =

dϕ& dϕ& dϕ dϕ& dω = . = .ϕ& = .ω dt dϕ dt dϕ dϕ

dω M * L 2 M * − L.ω 2 = * − * .ω = ω. dϕ J J J*

(33) (34)

Because, for an angle ϕ, ω vary from the nominal constant value ωn to the variable value ω, one can write the relation (35), where dω represents the

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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momentary variation for an angle ϕ; the variable dω with the constant ωn drive us to the needed variable, ω:

ω = ω n + dω

(35)

In the relation (35), ω and dω are functions of the angle, ϕ, and ωn is a constant parameter, which can take different values in function of the rotation velocity of the drive-shaft, n. To a moment, n is a constant and ωn is a constant as well (because ωn is a function of n). The rotation velocity, ω become a function of n too (see the relation 36):

ω (ϕ , n) = ω n (n) + dω (ϕ , ω n (n))

(36)

With (35) in (34), one obtains the equation (37): (ω n + dω ).dω = [

M* L − * .(ω n + dω ) 2 ].dϕ * J J

(37)

The relation (37) takes the form (38):

ω n .dω + (dω ) 2 =

M* L .dϕ − * .dϕ .[ω n2 + (dω ) 2 + 2.ω n .dω ] * J J

(38)

The equation (38) will be written in the form (39): M* L .dϕ + * .dϕ .ω n2 + * J J L L + * .dϕ .( dω ) 2 + 2. * .dϕ .ω n .dω = 0 J J

ω n .dω + (dω ) 2 −

(39)

The relation (39), take the form (40): L L 1 .dϕ + 1).( dω ) 2 + 2.( * .dϕ + ).ω n .dω − * 2 J J * M L − ( * .dϕ − * .dϕ .ω n2 ) = 0 J J

(

(40)

The relation (40) is a equation of the second degree in (dω). The discriminate (discriminative) of the equation (40) can be written in the forms (41) and (42):

ω n2 L2 L L.M * 2 2 2 d d .( ϕ ) . ω + + . ϕ . ω + .(dϕ ) 2 n n *2 * *2 4 J J J * 2 M L L + * .dϕ − *2 .(dϕ ) 2 .ω n2 − * .dϕ .ω n2 J J J Δ=

(41)

Al III-lea Seminar National de Mecanisme, Craiova, 2008

Δ=

ω n2 4

+

L.M * M* 2 .( d ϕ ) + .dϕ J* J *2

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(42)

One keeps for dω, just the positive solution, which can generate positives and negatives normal values (43), and in this mode one obtains for ω only normal values; for Δ < 0 one consider dω=0 (this case must to be not seeing if the equation is correct). dω =

−

ω L .dϕ .ω n − n + Δ * 2 J L .dϕ + 1 J*

(43)

Observations: For the mechanisms with rotate cam and follower, using the new relations with M* (the reduced moment of the mechanism) obtained by the writing of the reduced resistant moment knew and with the determine of the reduced motor moment by the integration of the resistant moment one obtains frequently some bigger values for dω, or zones with Δ negative, with complex solutions for dω. This fact gives us the obligation to reconsider the method to determine the reduced moment.

If one considers knew M*r and M*m and one calculates them independently (without integration) one obtains at the mechanisms with cam and follower, normal values for dω, with, Δ ≥ 0 .

In paper [1] one presents the relations to determining the resistant force (44) reduced at valve, and the motor force (45) reduced at the ax of valve: Fr* = k .( x 0 + x)

(44)

Fm* = K .( y − x)

(45)

The reduced resistant moment (46) or the reduced motor moment (47), can be obtains by the force resistant or motor, amplified with the reduced velocity, x’.

M r* = k .( x 0 + x).x'

(46)

M m* = K .( y − x).x'

(47)

2.3 The dynamic relations used The dynamics relations used (48-49), have been deduced in the paper [1]:

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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ΔX = (−1) ⋅

(k 2 + 2 ⋅ k ⋅ K ) ⋅ s 2 + 2 ⋅ k ⋅ x0 ⋅ ( K + k ) ⋅ s + [ 2 ⋅ (s +

K2 ⋅ mS* + ( K + k ) ⋅ mT* ] ⋅ ω 2 ⋅ ( Ds' ) 2 K +k

k ⋅ x0 ) ⋅ ( K + k )2 K +k

K2 [ ⋅ mS* + ( K + k ) ⋅ mT* ] ⋅ ω 2 ⋅ ( Ds' ) 2 X =s− K +k k ⋅ x0 2 ⋅ (s + ) ⋅ (K + k ) 2 K +k 2 2 (k + 2 ⋅ k ⋅ K ) ⋅ s + 2 ⋅ k ⋅ x0 ⋅ ( K + k ) ⋅ s − k ⋅ x0 2 ⋅ (s + ) ⋅ (K + k ) 2 K +k

(48)

(49)

2.4 The dynamic analysis The dynamic analysis or the classical movement law sin, can be seen in the diagram from the picture 3, and in the picture 4 one can see the diagram of an original movement law (C4P) (module C).

Analiza dinamicã la cama rotativã cu tachet n=5000[rot/min] translant plat - A10 ϕu=75 [grad] amax =4900 k=20 [N/mm] 5000 s max =5.78 r0=14 [mm] 4000 x0=40 [mm] 6000

3000

hs =6 [mm] hT=6 [mm] i=1;η=8.9% legea: sin-0 y=x-sin(2π x)/(2π )

2000 1000 0 -1000 -2000

0

50

100

150

200 a[m/s2]

amin= -1400

s*k[mm] k=

673.05

Fig. 3. The dynamic analysis of the law sin, Module C, ϕu=75 [grad], n=5000 [r/m]

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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n=10000[rot/min] Analiza dinamicã la cama rotativã cu tachet translant plat - A10 ϕu=45 [grad] amax =39000 k=200 [N/mm] 40000 r0=17 [mm] 50000

s max =4.10

x0=50 [mm]

30000

hs =6 [mm] hT=6 [mm] i=1;η=15.7% legea:C4P1-1

20000 10000

2

y=2x-x 2 yc =1-x

0 0

20

40

60

80

amin= -8000

-10000

100 s*k[mm] k=

a[m/s2] 7531.65

Fig. 4. The dynamic analysis of the new law, C4P, Module C, ϕu=45 [grad], n=10000 [r/m] 120000 100000

Analiza dinamicã la cama rotativã cu tachet n=40000[rot/min] translant cu rolã ϕu=80 [grad] amax=97000 k=400 [N/mm] r0=13 [mm] s max =3.88

80000

x0=150 [mm]

60000

hs =10 [mm] hT=10 [mm] i=1;η=12.7%

40000

rb=2 [mm] e=0 [mm] legea: C4P1-5

20000 0

2

-20000 0

50

-40000

100

150

y=2x-x

200

s*k[mm] k=

amin= -33000

a[m/s2] 19963,94

Fig. 5. Law C4P1-5, Module B, ϕu=80 [grad], n=40000 [rot/min] Analiza dinamicã la cama rotativã cu tachet n=40000[rot/min] balansier cu rolã (Modul F) - A12 ϕu=85 [grad] amax =80600 k=800 [N/mm] s max =4.28 r0=10 [mm]

100000 80000 60000 40000

hT=15.70 [mm]

20000 0 0

50

100

-20000 -40000 -60000

150

rb=3 [mm] b=30 [mm] d=30 [mm] x0=200 [mm] i=1;η=16.5% legea: C4P3-2 200 y=2x-x2 2 yc =1-x

amin= -40600

a[m/s2] s*k[mm] k=

15044,81

Fig. 6. Law C4P3-2, Module F, ϕu=85 [grad], n=40000 [rot/min]

Al III-lea Seminar National de Mecanisme, Craiova, 2008

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3 Conclusions Using the classical movement laws, the dynamics of the distribution cam-gears, spoils rapidly at the increasing of the rotation velocity of the shaft. To support a high rotation velocity is necessary the synthesis of the cam-profile by new movement laws, and at new Modules. A new and original movement law is presented in the pictures number 4, 5 and 6; it allows the increasing of the rotation velocity at the values: 1000020000 [rot/min], at the classical module C presented (fig. 4). With others modules (B, F) one can obtains 30000-40000 [rot/min] (see the fig. 5, 6).

References 1- Petrescu F.I., Petrescu R.V., Contributions at the dynamics of cams. In the Ninth IFToMM International Sympozium on Theory of Machines and Mechanisms, SYROM 2005, Bucharest, Romania, 2005, Vol. I, p. 123-128.

Relly Victoria PETRESCU Universitatea Politehnica din Bucureşti, Departamentul GDGI Splaiul Independenţei 313, Bucureşti, Sector 6, cod 060042 petrescuvictoria@yahoo.com Adriana COMĂNESCU Universitatea Politehnica din Bucureşti, Departamentul TMR Splaiul Independenţei 313, Bucureşti, Sector 6, cod 060042 adrianacomanescu@yahoo.com Florian Ion PETRESCU Universitatea Politehnica din Bucureşti, Departamentul TMR Splaiul Independenţei 313, Bucureşti, Sector 6, cod 060042 petrescuflorian@yahoo.com