Mechanical design elements by Ion Tiberiu

The 3rd International Conference on ″Computational Mechanics and Virtual Engineering″ COMEC 2009 29 – 30 OCTOBER 2009, Brasov, Romania

SOME MECHANICAL DESIGN ELEMENTS 1

F.I. Petrescu1, B. Grecu1, A. Comănescu1, R.V. Petrescu1, Bucharest Polytechnic University, Bucharest, ROMANIA, petrescuflorian@yahoo.com

Abstract: The paper presents first (1.) an original geometric method for the study of geometry and determining positions of RRR dyad. One uses one geometric relation and a second trigonometric expression as well. The advantage of this original method consists in the direct determination of mechanism (dyad) position. One must indicate only the position of internal couple C of the dyad (North or South). Second (2.) one presents shortly the MP-3R inverse kinematics solved by an original method. Then the V engine (3.) and geared transmission (4.) design. Keywords: RRR dyad geometry, robot inverse kinematics, V engine, geared transmission

1. THE GEOMETRY OF RRR DYAD One presents shortly an original geometric method to determine the positions of the RRR dyad (see the figure 1). Vector equations are written in the form (1, 2) and put in the form (3, 4). One square up and gather the equations of systems 3 respectively 4, and resulting the forms 5 and 6 which is dissolved in 5’ and 6’ (system 7).

⎧ xC = xB + d ⋅ cosψ (1) ⎨ ⎩ yC = yB + d ⋅ sinψ ( xC − xB ) 2 + ( yC − y B ) 2

⎧ xC = xD + b ⋅ cosθ ⎧ xC − xB = d ⋅ cosψ ⎧ xC − xD = b ⋅ cosθ (2) ⎨ (3) ⎨ (4) ⎨ ⎩ yC = yD + b ⋅ sin θ ⎩ yC − y B = d ⋅ sinψ ⎩ yC − yD = b ⋅ sin θ = d2 (5) ( xC − xD ) 2 + ( yC − yD ) 2 = b 2 (6)

⎧ 2 2 2 2 2 ⎪⎪ xC + xB − 2 ⋅ xB ⋅ xC + yC + y B − 2 ⋅ y B ⋅ yC = d (5' ) ⎨ ⎪ x 2 + x 2 − 2 ⋅ x ⋅ x + y 2 + y 2 − 2 ⋅ y ⋅ y = b 2 (6' ) ⎪⎩ C D D C C D D C

(7)

By reduced equation 5’ from 6’ one obtains the relation 8, which takes the form 9.

xD2 − xB2 + 2 ⋅ ( xB − xD ) ⋅ xC + y D2 − y B2 + 2 ⋅ ( y B − y D ) ⋅ yC = b 2 − d 2 ( xB − xD ) ⋅ xC + ( y B − y D ) ⋅ yC =

b −d + x −x + y − y 2 2

2 B

2 D

2 B

(8)

2 D

(9)

On the other hand we can write the relations 10 and 11:

yC − y D xC − xB yC − yB xC − xD ⋅ − ⋅ b d d b (b + c + d ) ⋅ (b + c − d ) ⋅ (b + d − c) ⋅ (c + d − b) = 2⋅b ⋅ d 2⋅b ⋅ d

sin(θ −ψ ) = sin θ ⋅ cosψ − sinψ ⋅ cosθ =

(10)

sin C =

(11)

When the point C is located at north (see the figure 1), C = θ -ψ , sin C = sin(θ −ψ ) , and equaling 10 and 11 one obtains the forms 12, 13, 14 and 15.

xC ⋅ yC − yD ⋅ xC − xB ⋅ yC + xB ⋅ yD − xC ⋅ yC + y B ⋅ xC + xD ⋅ yC − xD ⋅ yB =

(12)

( yB − y D ) ⋅ xC + ( xD − xB ) ⋅ yC + xB ⋅ y D − xD ⋅ yB =

(13)

C (at North)

d b

ψ B

θ c D

C (at South)

Figure 1: The RRR dyad geometry When the point C is located at south,

C = ψ − θ , sin C = sin(ψ − θ ) , and the relation (13) takes the form

(13’):

( yB − yD ) ⋅ xC + ( xD − xB ) ⋅ yC + xB ⋅ yD − xD ⋅ y B = − Now, one can write the generalized relationship (14), with and (–) when C is located at South.

(13’)

± sign; one takes the sign (+) when C is located at North

( yB − yD ) ⋅ xC + ( xD − xB ) ⋅ yC + xB ⋅ yD − xD ⋅ y B = ± ( yB − y D ) ⋅ xC + ( xD − xB ) ⋅ yC = ±

− xB ⋅ y D + xD ⋅ y B

(14) (15)

We maintain only the relations 9 and 15, which form the system 16.

⎧ b 2 − d 2 + xB2 − xD2 + y B2 − y D2 − ⋅ + − ⋅ = ( x x ) x ( y y ) y D C B D C ⎪⎪ B 2 ⎨ ⎪( y − y ) ⋅ x − ( x − x ) ⋅ y = ± + x D ⋅ y B − xB ⋅ y D D C B D C ⎪⎩ B 2

(16)

With the notation (17) the system (16) takes the form (18).

⎧ A = xB − x D ⎪B = y − y B D ⎪ 2 ⎪ b − d 2 + xB2 − xD2 + y B2 − y D2 ⎨C = 2 ⎪ ⎪ (b + c + d ) ⋅ (b + c − d ) ⋅ (b + d − c) ⋅ (c + d − b) ⎪D = ± + xD ⋅ y B − x B ⋅ y D 2 ⎩ ⎧ A ⋅ xC + B ⋅ yC = C ⎨ ⎩ B ⋅ xC − A ⋅ yC = D

(17)

(18)

The linear system (18) will be solved directly (19-22):

Δ=

B −A

= − A2 − B 2 = −( A2 + B 2 )

(19)

Δx =

= − A ⋅ C − B ⋅ D = −( A ⋅ C + B ⋅ D )

(20)

= A ⋅ D − B ⋅ C = −( B ⋅ C − A ⋅ D )

(21)

D −A

Δy =

A B

xC =

C D

Δx A⋅C + B ⋅ D = A2 + B 2 Δ

Δy

yC =

B ⋅C − A⋅ D A2 + B 2

(22)

One has determined first the Cartesian coordinates of the point C, and then we can find the two angles,

ψ and θ

(see the relations 23).

x − xB y − yB x − xD y − yD ⎧ ; sinψ = C ; cosθ = C ; sinψ = C ⎪cosψ = C d d b b ⎨ ⎪⎩ψ = sign(sinψ ) ⋅ arccos(cosψ );θ = sign(sin θ ) ⋅ arccos(cosθ );

(23)

2. THE MP-3R INVERSE KINEMATICS One presents shortly an original method to solve the robot inverse kinematics exemplified at the 3R-Robots (MP3R). The system which must be solved (2.4) has three equations (2.1-2.3) and three independent parameters ( ϕ10 ,

ϕ 20 , ϕ30 ) to determine. See the figure 2 and [1]. y3 d3

ϕ30

⎡ xM ⎤

M ⎢⎢ y M ⎥⎥ ⎢⎣ z M ⎥⎦

z3 d2 z0, z1

y2 y1

ϕ20

d1 A

a1 O0

a2 O2 x1

ϕ10

ϕ10 = ϕ10 ϕ 21 = ϕ 20 ϕ32 = ϕ30 − ϕ 20

Figure 2: The geometry of 3R Robot (MP)

⎧ xM = d1 ⋅ cos ϕ10 − a2 ⋅ sin ϕ10 + d 2 ⋅ cos ϕ 20 ⋅ cos ϕ10 − a3 ⋅ sin ϕ10 + d 3 ⋅ cos ϕ30 ⋅ cos ϕ10 (2.1) ⎪ ⎨ yM = d1 ⋅ sin ϕ10 + a2 ⋅ cos ϕ10 + d 2 ⋅ cos ϕ 20 ⋅ sin ϕ10 + a3 ⋅ cos ϕ10 + d 3 ⋅ cos ϕ30 ⋅ sin ϕ10 (2.2) (2.4) ⎪ z = a + d ⋅ sin ϕ + d ⋅ sin ϕ (2.3) 1 2 20 3 30 ⎩ M We aim to solve the system directly obtaining accurate solutions. At first step one multiplies the equation (2.1) with

− sin ϕ10 and the relation (2.2) with cos ϕ10 ; then add the two resulting relations and one obtains the relation (2.5) with solutions (2.6) for the first independent parameter ϕ10 .

− xM ⋅ sinϕ10 + yM ⋅ cos ϕ10 = a2 + a3

(2.5)

⎧ (a + a ) ⋅ y ± xM ⋅ xM2 + yM2 − (a2 + a3 ) 2 ⎪cos ϕ10 = 2 3 M xM2 + yM2 ⎪ (2.6) ⎨ − (a2 + a3 ) ⋅ xM ± yM ⋅ xM2 + yM2 − (a2 + a3 ) 2 ⎪ ⎪sin ϕ10 = xM2 + yM2 ⎩ Now one multiply the equation (2.1) with cos ϕ10 and the relation (2.2) with sin ϕ10 ; one add the two resulting relations and obtains the relation (2.7), which form with (2.3) a new system (2.8) who generate the last two independent parameters

ϕ 20 and ϕ30 .

⎧ ⎪⎪ xM ⋅ cos ϕ10 + yM ⋅ sin ϕ10 − d1 = d 2 ⋅ cos ϕ 20 + d 3 ⋅ cos ϕ30 (2.7) ⎨ ⎪ z − a = d ⋅ sin ϕ + d ⋅ sin ϕ (2.3) ⎪⎩ M 1 2 20 3 30

(2.8)

One use the notations (2.9) and obtains for the system (2.8) the exactly solutions (2.10).

⎧C1 = xM ⋅ cos ϕ10 + yM ⋅ sin ϕ10 − d1 ⎪ ⎨C2 = z M − a1 ⎪ 2 2 2 2 ⎩k = C1 + C2 + d 2 − d 3 ⎧ k ⋅ C1 ± C2 ⋅ 4 ⋅ C12 ⋅ d 22 + 4 ⋅ C22 ⋅ d 22 − k 2 ⎪cos ϕ 20 = 2 ⋅ (C12 + C22 ) ⋅ d 2 ⎪ ⎪ k ⋅ C2 ± C1 ⋅ 4 ⋅ C12 ⋅ d 22 + 4 ⋅ C22 ⋅ d 22 − k 2 ⎪ ⎨sin ϕ 20 = 2 ⋅ (C12 + C22 ) ⋅ d 2 ⎪ ⎪ C1 − d 2 ⋅ cos ϕ 20 ⎪cos ϕ30 = d3 ⎪ ⎩

(2.9)

(2.10)

Finally one keeps the three solutions (2.11):

⎧ (a2 + a3 ) ⋅ yM ± xM ⋅ xM2 + yM2 − (a2 + a3 ) 2 ⎪cos ϕ10 = xM2 + yM2 ⎪ ⎪ k ⋅ C1 ± C2 ⋅ 4 ⋅ C12 ⋅ d 22 + 4 ⋅ C22 ⋅ d 22 − k 2 ⎪ ⎨cos ϕ 20 = 2 ⋅ (C12 + C22 ) ⋅ d 2 ⎪ ⎪ C1 − d 2 ⋅ cos ϕ 20 ⎪cos ϕ30 = d3 ⎪ ⎩

(2.11)

3. THE V ENGINE DESIGN One just remembers about an original method to solve the kinematics and dynamics of V engines. The calculations can be seen in [2] and the issues in [3]. The geometry of V engine is presented in figure 3. The V Motors’ kinematics and dynamics synthesis can be made optimally by the value of constructive angle (α). For this reason, as generally constructive value angle was chosen randomly, after various technical requirements constructive or otherwise, inherited or calculated by various factors (more or less essential), but never got to discuss crucial factor (which takes account of the intimate physiology of the mechanism) angle that is constructive with his immediate influence on the overall dynamics of the mechanism, the actual dynamics of the mechanism with the main engine in the V suffered, the noise and vibration are generally higher compared with the similar engines in line. This paper aims to make a major contribution to remedy this problem so that the engine in V can be optimally designed and its dynamic behavior in the operation to become blameless, higher than that of similar engines in line. In the picture number 3 one can see the kinematics schema of the V Engine. The crank 1 has a trigonometric rotation (ω) and

actions the connecting-rod 2 which moves the piston 3 along the slide bar ΔB and actions the second connecting-rod 4, which moves the second piston 5 along the slide bar ΔD. There is a constructive angle α between the two axes ΔB and ΔD.

FCm

ΔD

4 b

α-β

γ+β

ΔB

FCn B

FCn

a FBm

FCm α

π/2+ϕ+β-α

π/2-ϕ-β

FB A

FBm

||ΔB

α/2

r ϕ

ω O

V Motors’ Kinematics and Dynamics Synthesis by the Constructive Angle Value (α); Forces Distribution, Angles, Elements and Couples (Joints) Positions; a+b=l

Figure 3: The geometry of V engine The same constructive angle (α) is formed by the two arms of the connecting-rod 2; first arm has the length l, and the second (which transmits the movement to the second connecting-rod 4) has the length a; this length a, add with the length b of the second connecting-rod 4 must gives the length l of the first connecting-rod. The crank motor force Fm is perpendicular at the crank length r, in A. A part of it (FBm) is transmitted to the first arm of connecting-rod 2 (along l) towards the first piston 3. Another part of the motor force, (FCm) is transmitted towards the second piston 5, by (along) the second arm of first connecting-rod 2 (a). A percent (of motor force Fm) x is transmitted towards the first piston (element 3) and the percent y is transmitted towards the second piston (element 5); the sum between x and y is 1 or 100%. The dynamic velocities have the same direction like forces. From the element 2 (first arm) to the first piston (element 3) one transmits the force FB and the dynamic velocity vBD. To force the first piston velocity equalises the dynamic value, one introduces a dynamic coefficient DB. The second Motor’ outline can be solved now. In C, FCm and vCm are projected in FCn and vCn. The transmitted force along of the second connecting-rod (FCn) is projected in D on the ΔD axe in FD. One determines the dynamic coefficient in D, DD. One put the condition to have a single dynamic coefficient of the mechanism, D=DB=DD. The value of x was determined from the imposed condition to have a single dynamic coefficient for the mechanism. The dynamic analysis made with the presented systems indicates some good values for the constructive angle (α), which allow the motor in V works normally without vibrations, noises and shocks (see the table 1): Table 1: Alfa angle values in grad

α [grad] 0–8 12 – 17 23 – 25

α [grad] 155 – 156 164 – 167 173 – 179

With α indicate in the table 1 one can make V Engine work without vibrations. The values presented in the table are not convenient for the motor makers; one can correct them with the relations presented in [2].

4. GEARED TRANSMISSIONS DESIGN One just remembers about an original method to solve the kinematics and dynamics of geared transmissions (see [4], figure 4, and the relation 4.1). In this paper one makes a brief presentation of an original method to obtain the efficiency of the geared transmissions in function of the cover grade. With the presented relations one can make the dynamic synthesis of the geared transmissions having in view increasing the efficiency of gearing mechanisms in work [4]. © 2005 Florian Ion PETRESCU The Copyright – Law Of March 01, 1989 U.S. Copyright Library of Congress Washington, DC 20559-6000 202-707-3000

O2 Fτl, vτl rb2 Fml, vml

Fτi, vτi

K2 l k

Fmi, vmi

rl1 rk1

ri1αj αi

rj1

rb1 O1

Figure 4: Four pairs of teeth in contact concomitantly.

ηm =

1 2π ⋅ tgα 0 2π ⋅ (ε12 − 1) ⋅ (2 ⋅ ε12 − 1) ± ⋅ (ε12 − 1) 1 + tg α 0 + 2 z1 3 ⋅ z1 2

(4.1)

5. CONCLUSION Today industrial machines construction requires new technologies of manufacturing which require a permanently renewed fundamental research. The presented elements of industrial machines (mechanical) design are trying to fit these requirements.

REFERENCES [1] Antonescu P.: Mecanisme şi manipulatoare, Editura Printech, Bucharest, 2000, p. 103-104. [2] Petrescu F.I., Petrescu R.V.: V Engine Design, ICGD2009, Vol. Ib, p. 533-536, ISSN 1221-5872, Cluj-Napoca, 2009. [3] Petrescu F.I., Petrescu R.V.: Designul motoarelor în V, Revista Ingineria Automobilului, Nr. 11, iunie 2009, p. 1112, ISSN 1842-4074, 2009. Petrescu F.I., Petrescu R.V.: Designul motoarelor în V, Revista Ingineria Automobilului, Nr. 11, iunie 2009, p. 11-12, ISSN 1842-4074, 2009. [4] Petrescu R.V., Petrescu F.I.: Geared Transmissions Design, ICGD2009, Vol. Ib, p. 541-544, ISSN 1221-5872, ClujNapoca, 2009.