Some engineering design ion petrescu

SOME ELEMENTS IN ENGINEERING DESIGN

Ion PETRESCU, PhD. Eng. Lecturer at TMR, UPB Victoria PETRESCU, PhD. Eng. Lecturer at GDGI, UPB

ABSTRACT: The paper presents first the MP-3R inverse kinematics solved directly by an original method. Second one presents the V engine kinematics and dynamics design by an original method. Third one trate shortly the dynamics design of geared transmission. Fourth one presents the cams design. Last it presents the Otto Engine Design. 1. The MP-3R Inverse Kinematics

One presents shortly an original method to solve the robot inverse kinematics exemplified at the 3R-Robots (MP-3R).

The system which must be solved (1.4) has three equations (1.1-1.3) and three

independent parameters ( ) to determine. See the figure 1 and [1]. 302010 ,, ϕϕϕ

x1

y1

z0, z1

O1

O0

x0

y0

ϕ10

a1

d1

y2

x2

O2

z2

a2

d3

d2

ϕ20

A

z3

x3

y3

O3Ba3 M

ϕ30 ⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

M

M

M

zyx

203032

2021

1010

ϕϕϕϕϕϕϕ

−===

Figure 1: The geometry of 3R Robot (MP)

⎪⎩

⎪⎨

⎧

⋅+⋅+=⋅⋅+⋅+⋅⋅+⋅+⋅=⋅⋅+⋅−⋅⋅+⋅−⋅=

)3.1(sinsin)2.1(sincoscossincoscossin)1.1(coscossincoscossincos

3032021

1030310310202102101

1030310310202102101M

ϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕϕ

ddazdadadydadadx

M

M (1.4)

We aim to solve the system directly obtaining accurate solutions. At first step one

multiplies the equation (1.1) with 10sinϕ− 10cosϕ and the relation (1.2) with , then add the two resulting relations and one obtains the relation (1.5) with solutions (1.6) for the first independent parameter 10ϕ .

321010 cossin aayx MM +=⋅+⋅− ϕϕ (1.5)

⎪⎪

⎩

⎪⎪

⎨

⎧

++−+⋅±⋅+−

=

++−+⋅±⋅+

=

22

232

2232

10

22

232

2232

10

)()(sin

)()(cos

MM

MMMM

MM

MMMM

yxaayxyxaa

yxaayxxyaa

ϕ

ϕ (1.6)

10cosϕ 10sinϕNow one multiply the equation (1.1) with and the relation (1.2) with , one add the two resulting relations and obtains the relation (1.7), which form with (1.3) a new

system (1.8) who generate the last two independent parameters . 3020 ϕϕ and

⎪⎪⎩

⎪⎪⎨

⎧

⋅+⋅=−

⋅+⋅=−⋅+⋅

)3.1(sinsin

)7.1(coscossincos

3032021

30320211010

ϕϕ

ϕϕϕϕ

ddaz

dddyx

M

MM (1.8)

One use the notations (1.9) and it obtains for the system (1.8) the exactly solutions

(1.10).

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⋅−=

⋅+⋅−⋅⋅+⋅⋅⋅±⋅

=

⋅+⋅−⋅⋅+⋅⋅⋅±⋅

=

3

202130

222

21

222

22

22

2112

20

222

21

222

22

22

2121

20

coscos

)(244

sin

)(244

cos

ddC

dCCkdCdCCCk

dCCkdCdCCCk

ϕϕ

ϕ

ϕ

⎪⎩

⎪⎨

⎧

−++=

−=−⋅+⋅=

23

22

22

21

12

110101 sincos

ddCCk

azCdyxC

M

MM ϕϕ(1.9) (1.10)

Finally one keeps the three solutions (1.11):

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

⋅−=

⋅+⋅−⋅⋅+⋅⋅⋅±⋅

=

++−+⋅±⋅+

=

3

202130

222

21

222

22

22

2121

20

22

232

2232

10

coscos

)(244

cos

)()(cos

ddC

dCCkdCdCCCk

yxaayxxyaa

MM

MMMM

ϕϕ

ϕ

ϕ

(1.11)

2. 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 2.

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 2 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.

β

ϕ

γ

γ+β

αα

α-β

O

B

D

A

C

Fm

FBm

FCmFBm

FB

FCm

FCn

FCn

FD

π/2-ϕ-β

π/2+ϕ+β-α

© 2006 Florian PETRESCUThe Copyright-LawOf March, 01, 1989,U.S. Copyright OfficeLibrary of CongressWashington, DC 20559-6000202-707-3000

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

1

24

35

ω

r

l

a

b

α/2 α/2

ΔBΔD

||ΔB

Figure 2: 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 D . B

The second Motor’ outline can be solved now. In C, FCm and vCm are projected in FCn and

v . The transmitted force along of the second connecting-rod (FCn Cn) 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=D =DB D. 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: The alfa angle values in grad α [grad] α [grad]

0 – 8 155 – 156 12 – 17 164 – 167 23 – 25 173 – 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].

3. Geared Transmissions Design

One just remembers about an original method to solve the kinematics and dynamics of geared transmissions (see [4], figure 3, and the relation 3.1). 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].

αi

O1

O2

K1

K2

j

A

rb1

rb2

i

αj

kl

ri1rj1

rl1

rk1

Fτl, vτl

Fml, vml Fτi, vτi

Fmi, vmi

© 2005 Florian Ion PETRESCUThe Copyright – LawOf March 01, 1989U.S. CopyrightLibrary of CongressWashington, DC 20559-6000202-707-3000

Figure 3: Four pairs of teeth in contact concomitantly

)1(2)12()1(321

1

121

012122

1

2

02 −⋅

⋅±−⋅⋅−⋅

⋅++

=εαπεεπα

η

ztg

ztg

m (3.1)

4. Cams Design

In the figure 4 one presents shortly four models of cams mechanisms [5].

τ

O

A

r0

s

s’

rA

1vr2vr

12vrB

C

Dτ

ω

δ

δ

δ

ψFr mF

rcFr F

E

© 2002 Florian PETRESCUThe Copyright-Law Of March, 01, 1989U.S. Copyright OfficeLibrary of CongressWashington, DC 20559-6000202-707-3000

α0αA

ϕθA

θB

δ

μ

γ

αA-δ

Fn, vn

Fm, vm

Fa, va

Fi, viFn, vn

Fu, v2

B

B0

A0

A

O

x

e

s 0

r0

rA

rB

s

n

C

rb


a-Cam with plate translated follower b-Cam with translated follower with roll

α0

αA

ϕθA

ψ2

μ

αB

Fn, vn

Fm, vmFa, va

Fc, vc

Fn, vn

Fu, v2

B

B0

A0

x

rbr0

rA

rB

Aδ

α B

γ

O D

ψ

ψ0d

b

b


r0

G δ

B

O D

d

A

A0

B0

H

I

ρ l

bG0

l.ψ’

ρ.ψ’

r

τ

ψ

ψ

θ

β

αMαm

xϕ

γ ψ

1

2

Fm;vm

Fa;va

Fn;vn

τ

α


c-Cam and rocking follower with roll d-Cam and general plate rocking follower

Figure 4: Cams’ kinematics and dynamics

The cams design (geometry, efficiency, forces, dynamics) can be followed in the paper [5]. 5. Otto Engine Design

In the figure 5 one presents shortly the Otto Engine Design [6].

(c)

2

222 )cos(1cossin

sin1)sin(

)sin(sin

lre

rF

rFPP

m

m

c

ui

ϕαψ

ψϕψω

ϕψψωη

⋅+−===

=⋅−⋅⋅⋅

−⋅⋅⋅⋅==

(5.1) (d)

2

222

22

]sin)cos(cos)cos([

)(sin)(sin

l

rerel

rFrF

PP

m

m

c

ui

ϕϕϕϕ

ϕψω

ϕψωη

⋅⋅++⋅⋅+−=

=−=⋅⋅

−⋅⋅⋅== (5.2)

0

0

O

A

B

l

r

e

yB

x

y

P

1

2

3

ϕ

α

ψ

ω

1

y

0

0

O

AI

BI

l

r

eP

1

2

3

ϕI

αI ψI

ω

y

0

0

O

AII

BII

l

reP

2

3

ϕII

ψII

ωl-r

xx

l+r

αII

l

near dead point

distant dead point

a - the crank is in prolonging with the connecting-rod

b - the crank is overlapped on the connecting-rod

a-The kinematical schema of Otto b-Extremely positions.

0

O

A

B

l

r

e

yB

x

y

P

ϕ

α

ψ

ω

ψ-ϕα

α

Fm

Fn

Fτ

Fn

ϕ

Fu

Fc ψ-ϕψ-ϕ

0

O

A

B

l

r

e

yB

x

y

P

ϕ

α

ψ

ω

ψ-ϕα

α

Fm

Fn

Fτ

Fn

ϕ

Fu

Fr

ψ-ϕ

ψ-ϕ

α

c-The forces of Otto-mechanism, when the d-The forces of Otto-mechanism, piston works like a motor mechanism when piston works like a steam roller

Fig. 5. The Otto Engine Design

6. Conclusions

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. BIBLIOGRAPHY [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. 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, Cluj-Napoca, 2009. [5] Popescu N., Petrescu R.V., Petrescu F.I.: Cam Gear Design, ICGD2009, Vol. Ia, p.

215-220, ISSN 1221-5872, Cluj-Napoca, 2009. [6] Petrescu R.V., Petrescu F.I.: Otto Engines Design, ICGD2009, Vol. Ib, p. 537-540,

ISSN 1221-5872, Cluj-Napoca, 2009.

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Some engineering design ion petrescu