The constraint functions of the elements of mechanisms

(1)

Academy of Romania A. Mechanics, Mechanical and Industrial Engineering,

www.jesi.astr.ro Mechatronics

Received 3 November 2016 Accepted 7 March 2017 Received in revised from 13 January 2017

The constraint functions of the elements of mechanisms

STĂNESCU NICOLAE–DORU^*

University of Piteúti, Faculty of Mechanics and Technology, Piteúti, 1, Târgul din Vale, 110040, Argeú, Romania

Abstract: In the present paper we purpose a discussion concerning the constraints which may appear at one arbitrary element of a mechanism. We also determine the corresponding constraint functions. The obtained results may be applied in practical situations. An application highlights the theory.

Keywords: constraint, function of constraint, mechanism, multibody

1. Introduction

It is well known that in the multibody approach for the rigid solid dynamics, the equation of motion may be written as

> @ > @

> @ > @ ^ `

^ ` ^ `

^ ` > @ ^ `

^»_¼^º

«¬ ª

»

¼

« º

¬ ª

»»

¼ º

««

¬

ª

q B C

F Ȝ

q 0 B

B M

T

, (1)

the notations having the significance given in [1].

In this way, the problem reduces to the determination of the matrix of constraints

> @

B for an arbitrary rigid body.

In the particular case of the mechanisms, the constraints may appear either from the connections with the external system or from the connections between the elements of the mechanism.

A classical approach of the analysis and synthesis of the mechanisms is presented in [2]. Some applications for simple mechanisms may be found in [3, 4, 5, 6, 7, 8, 9, 10, 11].

* Correspondence addres: [email protected]

(2)

Further on, we will analyze different types of constraints that may appear at mechanisms determining, for each case, the function or functions of constraints.

2. Notations

We will use the following notations:

– OXYZ – the fixed reference system;

– O_ix_iy_iz_i – mobile reference system rigidly linked to the element i of the mechanism;

– \ , _i T , _i M – Bryan’s rotational angles of the system _i O_ix_iy_iz_i with respect to the fixed reference system OXYZ ;

–

> @

ȥ , i

> @

ș , i

> @

ĳ – the square matrices given by i

> @

»»

»

¼ º

««

«

¬ ª

\

i i

i

cos sin

0

sin cos

0

0 0

1

ȥ ,

> @

»»

»

¼ º

««

«

¬ ª

T T

i i

i

cos 0 sin

0 1 0

sin 0 cos

ș ,

> @

»»

»

¼ º

««

«

¬ ª

M M

M

1 0 0

0 cos sin

0 sin cos

i i

ĳi ;

(2)

–

> @

A – the rotational matrix of the element i i

> @ > @> @> @

Ai ȥi și ĳi ; (3)

– X , _i Y , _i Z – the coordinates of the point _i O with respect to the fixed _i reference system;

– P , _j Q – points of one element of the mechanism; this points will be defined _j in each case;

–

^ `

RO_i – the column matrix

^ `

O

>

Xi Yi Zi

@

^T

R i ; (4)

– x_Pⁱ , y_Pⁱ , z_Pⁱ – the coordinates of a point P belonging to the element i with respect to the mobile reference system O_ix_iy_iz_i;

–

^ `

rPⁱ – the column matrix

^ `

>

Pⁱ

@

^T

i P i P i

P x y z

r ; (5)

– X_Pⁱ , Y_Pⁱ , Z_Pⁱ – the coordinates of the point P of the element i with respect to the fixed reference system O₀XYZ;

–

^ `

RPⁱ – the column matrix

^ `

>

Pⁱ

@

^T

i P i P i

P X Y Z

R ; (6)

– f , g – the equations of the surface or curve;

(3)

–

> @

Ai – the rotational matrix in the planar case

> @

_»

¼

« º

¬ ª

M M

M

i i

i sin cos

sin

A cos . (7)

3. The case of a point constrained to move on a fixed surface

Let be

X,Y,Z

0

f (8)

the equation of the fixed surface

6 relative to the fixed reference system (Fig. 1).

P i

O O_i x_i

z_i

y_i

X Y

Z

(6)

Fig. 1. A point constrained to move on a fixed surface.

The coordinates of the point P that belongs to the element i , constrained to move on the surface

6 may be written as

^ `

^ ` > @ ^ `

»»»

¼ º

««

«

¬ ª

i P i P

i P i

P O

i P

Z Y X

i A r

R

R ; (9)

it results the constraint function

, ,

0

1

i P i P i P i

P i P i

P Y Z f X Y Z

X

f . (10)

In conclusion, the system looses one degree of freedom.

4. The case of a point constrained to move on a fixed curve

4.1. The general case

Proceeding as in the paragraph 3, one obtains the constraint functions

^, ^,

⁰

1

i P i P i P i

P i P i

P Y Z f X Y Z

X

f , (11)

(4)

, ,

0

2

i P i P i P i

P i P i

P Y Z g X Y Z

X

f ,

where

X,Y,Z

0

f , g

X,Y,Z

0 (12)

are the equations of the curve in the fixed reference system.

We have now two functions of constraints and, consequently, the system looses two degrees of freedom.

4.2. The planar case

The situation is presented in Fig. 2.

P

O

O_i x_i y_i

X Y

M_i (*)

Fig. 2. A point constrained to move on a fixed curve.

Let be

X,Y

0

f (13)

the equation of the curve

* in the fixed reference frame.

We may write

> @

»

¼

« º

¬

»ª

¼

« º

¬ ª

M M

M

»¼

« º

¬

» ª

¼

« º

¬

ª

»¼

« º

¬

» ª

¼

« º

¬ ª

i P i P i i

i i

O Oi i

P i P i O

Oi i

P i P

y x Y

X y

x Y

X Y

X

i

i sin cos

sin

A cos , (14)

wherefrom

i i P i i P O i

P X x y

X i cosM sinM , Y_Pⁱ Y_O x_Pⁱ _i y_Pⁱ _i

i sinM cosM ; (15)

it results the constraint function

^cos^M ¹ ^,^sin^M^, ^, ^sin^M ^cos^Mi

⁰^.

i P i i P O i i P i i P O

i P i P i

P i P

y x

Y y

x X f

Y X f Y X f

i i

(16) The system looses one degree of freedom.

(5)

5. The case of the spherical joint

The common point P of the spherical joint has the coordinates x , _Pⁱ y , _Pⁱ z _Pⁱ relative to the mobile reference system O_ix_iy_iz_i, and the coordinates x_Pⁱ¹ , y_Pⁱ¹ ,

i1

zP relative to the mobile reference system O_i₁x_i₁y_i₁z_i₁ (Fig. 3).

It results

^ `

O

> @

i

^ `

Pⁱ i

P R _i A r

R , (17)

^ `

^ ` >

1

@ ^ `

¹

1

_O _i _Pⁱ

i

P R _i A r

R (18)

and from the equality

^ `

Pⁱ¹ i

P R

R (19)

one obtains three functions of constraints, that is, the system consisting in the elements i and i1 looses three degrees of freedom.

X O Y

Z Oi ⁱ

y zi

xi

i

O xi+1

yi+1 i+1

zi+1

i+1

P

Fig. 3. The spherical joint.

X O Y

Z Oi ⁱ

y zi

xi

i

O xi+1

yi+1 i+1

zi+1

P₁ i+1 P2

Q1 Q₂

Fig. 4. The cylindrical joint.

(6)

6. The cylindrical joint

Let be P and ₁ P two points situated on the axis of the cylindrical joint and ₂ which are rigidly linked to the element i , and let be Q and ₁ Q two other points ₂ situated on the same axis and rigidly linked to the element i1 (Fig. 4). The four points can be real or virtual.

At an arbitrary moment of time, one may write the relations

^ `

O

^{> @}

i

^ `

Pⁱ i

P₁ R _i A r₁

R , (20)

^ `

O

> @

i

^ `

Pⁱ i

P₂ R _i A r₂

R , (21)

^ `

^ ` ^>

1

^@ ^ `

¹

1

1 1

1

_O _i _Qⁱ

i

Q R _i A r

R , (22)

^ `

^ ` >

1

@ ^ `

¹

1

1 2 2

_O _i _Qⁱ

i

Q R _i A r

R . (23)

From the previous relations one deduces the coordinates X_Pⁱ

1 , Y_Pⁱ

1 , Z_Pⁱ

1 , X_Pⁱ

2 ,

ⁱ

YP 2 , Z_Pⁱ

2 , ¹

1

i

XQ , ¹

1

i

YQ , ¹

1

i

ZQ , and ¹

2

i

XQ , ¹

2

i

YQ , ¹

2

i

ZQ of the points P , ₁ P , ₂ Q , ₁ and Q , respectively, relative to the fixed reference system. The points ₂ P and ₁ P ₂ define a straight line, the equations of which with respect to the fixed reference system read

X,Y,Z

0

f , g

X,Y,Z

0. (24)

The constraint functions require the appurtenance of the points Q₁ and Q₂ to this straight line. One obtains four functions of constraints

₁¹^, ₁¹^, Qⁱ₁¹

⁰

i Q i

Q Y Z

X

f ,

₁¹^, ₁¹^, Qⁱ₁¹

⁰

i Q i

Q Y Z

X

g ,

₂¹^, ₂¹^, Qⁱ₂¹

⁰

i Q i

Q Y Z

X

f ,

₂¹^, ₂¹^, Qⁱ₂¹

⁰

i Q i

Q Y Z

X

g , (25)

the number of the degrees of freedom of the system diminishing by four.

7. The planar revolute joint

One obtains the relations (Fig. 5)

i+1

O

Oi+1

xi

y_i+1

xi+1

X Y

O yi

i

Fig. 5. The planar revolute joint.

(7)

^ `

> @

»

¼

« º

¬

»ª

¼

« º

¬ ª

M M

M

»¼

« º

¬

» ª

¼

« º

¬

ª

»¼

« º

¬

» ª

¼

« º

¬ ª

i P i P i i

i i

O O i

P i P i O

O i

P i P i

P y

x Y

X y

x Y

X Y

X

i i i

i

cos sin

sin A cos

R , (26)

^ `

> @

»

¼

« º

¬

»ª

¼

« º

¬ ª

M M

M

»¼

« º

¬ ª

»¼

« º

¬

ª

»¼

« º

¬

» ª

¼

« º

¬ ª

1 1

1 1 1 1

1 1

cos sin

sin cos

1 1

i P i P i i

i i

O O

i P i P i O

O i

P i P i

P

y x Y

X

y x Y

X Y

X

i i

i

i A

R

(27)

and from the condition

^ `

Pⁱ¹ i

P R

R (28)

one deduces two functions of constraints

cos sin 0,

sin cos

, , , , ,

1 1

1

1 1

M

i i

P i i

P O

i i p i i P O i

O O i O O

y x

X

y x

X Y

X f

i

i i

i (29)

sin cos 0.

cos sin

, , , , ,

1 1

1 2

1

1 1

M

i i

P i i

P O

i i p i i P O i

O O i O O

y x

Y

y x

Y Y

X Y

X f

i

i i

i (30)

In this situation, the system looses two degrees of freedom.

8. The prismatic joint

8.1. The general case

We choose the points P , ₁ P , ₂ Q and ₁ Q as in the case of the cylindrical joint ₂ (paragraph 6). Similarly, one gets the functions of constraints (25) (Fig. 6).

O Y

X Z

xi+1

yi+1

Oi+1

Oi

zi

x_i

zi+1

y_i Q3

Q2

Q1 P⁴ P₃ P1 P2 P5

Fig. 6. The spatial prismatic joint

(8)

In addition, we choose the points P , ₃ P and ₄ P belonging to the element i , ₅ situated on a face of the prism, and the point Q , belonging to the element ₃ i 1 and situated in the plane defined by the points P , ₃ P and ₄ P . One may write ₅

^ `

O

> @

i

^ `

Pⁱ i

P₃_,₄_,₅ R _i A r₃_,₄_,₅

R , (31)

^ `

^ ` >

1

@ ^ `

¹

1

3 1

3

_O _i _Qⁱ

i

Q R _i A r

R . (32)

The condition which we are looking for requires the appurtenance of the point Q to the plane defined by the points 3 P , ₃ P and ₄ P , that is, ₅

0 1

1 1 1

3 3 3

5 5 5

4 4 4

3 3 3

i

Q i Q i Q

i P i P i P

Z Y X

. (33)

We have five functions of constraints and, consequently, the system looses five degrees of freedom.

8.2. The planar case

The problem is more difficult in this case (Fig. 7).

O

X Y

Oi

yi xi

i

O xi+1

yi+1

i+1

i+1 Mi

M_i+1

Fig. 7. The planar prismatic joint

The condition which states that the elements i and i1 do not rotate one about other (in plane) leads to the function of constraints

1 ct

1 M_i M_i

f . (34)

The rotation about the axis of the joint can be vanished directly by an obvious function of constraints. It results from the conditions which state that the points O _i and O_i₁ (if these points do not belong to the rotational axis, otherwise one has to choose other two points) belong to the fixed plane OXY .

(9)

9. Example

O₂ O₁

g g

1 1

1 2

2

2 2

1

x

z m y y

m A B

z C

Y Z

X

O0

Fig. 8. Example

Let us consider two bars AC and CB of lengths l and ₁ l , masses ₂ m and ₁ m , ₂ respectively, linked one to another by a spherical joint at the point C . The mobile reference frames O₁x₁y₁z₁, O₂x₂y₂z₂ are principal central systems of inertia for which one knows the values

x1

J , J , y1 z1

J , Jx2 ,

y2

J ,

z2

J . For the fixed reference system O₀XYZ the axis O₀Z is vertical ascendant. Obviously, knowing all these parameters, one may continue the example in order to obtain the matrix equation of motion [12].

Choosing the Bryan angles as rotational parameters for each bar, we have

> @

»»

»

¼ º

««

«

¬ ª

\

i i

i

cos sin

0

sin cos

0

0 0

1

ȥ ,

> @

»»

»

¼ º

««

«

¬ ª

T T

i i

i

cos 0 sin

0 1 0

sin 0 cos

ș ,

> @

»»

»

¼ º

««

«

¬ ª

M M

M

1 0 0

0 cos sin

0 sin cos

i i

ĳi , i 1,2,

(i)

> @ > @> @> @

, c c c s s s c s s c s c

c s c c s s s s c c s s

s s

c c

c

»»

»

¼ º

««

«

¬ ª

T

\ M

\

M T

\ M

\

M T

\

T

\

M

\

M T

\

M

\

M T

\

T M

T

M

T

i i i i i i i i i i i i

i i

i

i i i

i ȥ ș ĳ

A

2 , 1

i , (ii)

We will the consider the following order of the parameters X_O₁, Y_O₁, Z_O₁, \ , ₁ T , 1 M , ₁

O2

X , Y_O₂, Z_O₂ , \ , ₂ T , ₂ M . ₂

(10)

We have only one constraint given by the appurtenance of the point C to the two bars.

For the bar AC one may write 2 l1

x_C , y_C 0, z_C 0, while for the bar BC one has

2 l2

x_C , 0y_C , 0z_C . From the relations

> @

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

0 0 2

1

1 1 1

l

Z Y X

O O O

C C C

A , (iii)

> @

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

0 02

2

2 2 2

l

Z Y X

O O O

C C C

A (iv)

one gets the expressions

2 2 2 1

1

1 cos cos

cos 2

2cos ²

1 T M l T M

l X

X_O _O ,

sin sin cos cos sin

,

2

sin cos cos

sin 2 sin

2 2 2

1 1 1

2 1

M

\

M T

\

M

\

M T

\

Y l Y l

O O

cos sin cos sin sin

,

2

sin sin cos

sin 2 cos

2 2 2

1 1 1

2 1

M

\

M T

\

M

\

M T

\

Z l Z l

O O

(v)

so that the functions of constraints read

^cos ^cos ⁰

cos 2 2 cos

...,

, ₂ ¹ ₁ ₁ ² ₂ ₂

1 ₁ M ₁ ₂ l T M l T M

X X X

f _O _O _O ,

^sin ^sin ^cos ^cos ^sin

⁰^,

2

sin cos cos

sin 2 sin

..., ,

2 2 2

1 1 1

1 1 1 2

2 ₁ ₁ ₂

M

\

M T

\

M

\

M T

\

M

l

Y l Y X

f _O _O _O (vi)

(11)

cos sin cos cos sin

0,

2

sin sin cos

sin 2 cos

..., ,

2 2 2

1 1 1

2

3 ₁ ₁ ₂

M

\

M T

\

M

\

M T

\

M

l

Z l Z X

f _O _O _O

wherefrom one obtains the components of the matrix of constraints as follows 1

1 1

11 w

w XO

B f , 0

1 1

12 w

w YO

B f , 0

1 1

13 w

w ZO

B f , 0

1 1

14 w\

wf

B ,

1 1 1 1 1

15 sin cos

2 T M

T w

wf l

B , ¹ ₁ ₁

1 1

16 cos sin

2 T M

M w

wf l

B ,

1

2 1

17

w w XO

B f , 0

2 1

18 w

w YO

B f , 0

2 1

19 w

w ZO

B f , 0

2 1

110 w\

wf

B ,

2 2 2 2 1

111 sin cos

2 T M

T w

wf l

B , ² ₂ ₂

2 1

112 cos sin

2 T M

M w

wf l

B

(vii)

0

1 2

21 w

w XO

B f , 1

1 2

22 w

w YO

B f , 0

1 2

23 w

w ZO

B f ,

1 1 1 1

1 1 2

24 cos sin sin sin

2 \ T \ M

\ w

wf l

B ,

1 1 1 1 1 2

25 sin cos cos

2 \ T M

T w wf l

B ,

₁ ₁ ₁ ₁ ₁

1 1 2

26 sin sin sin cos cos

2 \ T M \ M

M w wf l

B , 0

2 2

27 w

w XO

B f ,

1

2 2

28

w w

YO

B f , 0

2 2

29 w

w ZO

B f ,

2 2 2 2 2

2 2 2

210 cos sin cos sin sin

2 \ T M \ M

\ w

wf l

B ,

2 2 2 2 2 2

211 sin cos cos

2 \ T M

T w wf l

B ,

2 2 2 2 2

2 2 2

212 sin sin sin cos cos

2 \ T M \ M

M w

wf l

B ,

(viii)

0

1 3

31 w

w XO

B f , 0

1 3

32 w

w YO

B f , 1

1 3

33 w

w ZO

B f ,

1 1 1 1 1

1 1 3

34 sin sin cos cos sin

2 \ T M \ M

\ w

wf l

B ,

(ix)

(12)

1 1 1 1 1 2

35 cos cos cos

2 \ T M

T w

wf l

B ,

1 1 1 1 1

1 1 3

36 cos sin sin sin cos

2 \ T M \ M

M w

wf l

B , 0

2 3

37 w

w XO

B f ,

0

2 3

38 w

w YO

B f , 1

2 3

39

w w ZO

B f ,

2 2 2 2 2

2 2 3

2 \ T M \ M

\ w

wf l

B ,

2 2 2 2 2 3

311 cos cos cos

2 \ T M

T w

wf l

B ,

2 2 2 2 2

2 2 3

312 cos sin sin sin cos

2 \ T M \ M

M w

wf l

B .

Let us consider now that at the point C we have a cylindrical joint.

In this situation we have to add two new functions of constraints

^,^..., 2

2 1 ⁰

4 X_O₁ M \ \

f , f5

X_O₁^,^...,M2

T2 T1 ⁰, (x) the matrix of constraints having now 5 rows and 12 columns.

The first three rows were calculated previously; the next two rows have the expressions

41 0

B , 0B₄₂ , 0B₄₃ , 1B₄₄ , 0B₄₅ , 0B₄₆ , 0B₄₇ ,

48 0

B , 0B₄₉ , 1B₄₁₀ , 0B₄₁₁ , 0B₄₁₂ , (xi)

51 0

B , B₅₂ 0, B₅₃ 0, B₅₄ 0, B₅₅ 1, B₅₆ 0, B₅₇ 0,

58 0

B , B₅₉ 0, B₅₁₀ 0, B₅₁₁ 1, B₅₁₂ 0. (xii) Let us impose now the supplementary conditions which state that the points A , and B are situated on the spheres of equations

, ,

² ² ² ² 0

1 X Y Z X Y Z R

g , (xiii)

and g₂

X,Y,Z

X 2R

² Y² Z² R² 0, (xiv) respectively.

The points A and B have the coordinates

2 l1

x_A , y_A 0 , z_A 0 ,

2 l2

x_B , 0y_B , 0z_B ; it results

> @

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

0 02

1

1 1 1

l

Z Y X

O O O

A A A

A , (xv)

(13)

> @

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

0 0 2

2

2 2 2

l

Z Y X

O O O

B B B

A , (xvi)

that is,

1 1

1 cos cos

1 l2 T M

X

X_A _O ,

1 1 1 1 1

1 l2 \ T M \ M

Y

Y_A _O ,

1 1 1 1 1

1 l2 \ T M \ M

Z

Z_A _O ,

(xvii)

2 2

2 cos cos

2 l2 T M

X

X_B _O ,

2 2 2 2 2

2 l2 \ T M \ M

Y

Y_B _O ,

₂ ₂ ₂ ₂ ₂

2 l2 \ T M \ M

Z

Z_B _O .

(xviii)

One gets the functions of constraints

cos sin cos sin sin

0,

2

sin cos cos

sin 2 sin

cos 2cos

,...,

2 2 1 1 1

1 1 1

2 1 1 1

1 1 1

2 1 1 1 2

4

1 1

»¼

«¬ º

ª \ T M \ M

»¼

«¬ º

ª \ T M \ M

¸

¹

¨ ·

©

§ T M

M

l R Z

Y l

X l X

f

O O

(xix)

cos sin cos sin sin

0.

2

sin cos cos

sin 2 sin

cos 2 cos

2 ,...,

2 2 2 2 2

2 2 2

2 2 2 2

2 2 2

2 2 2 2 2

5

2 2

2 1

»¼

«¬ º

ª \ T M \ M

»¼

«¬ º

ª \ T M \ M

¸

¹

¨ ·

©

§ T M

M

l R Z

Y l

R l X

X f

O O

(xx)

The matrix of constraints has now 5 rows and 12 columns.

Let us impose the condition which states that the points A and B are situated in the plane Z 0, while at the point C we have a cylindrical joint.

(14)

We may successively write (the conditions for the appurtenance of the points A and B to the plane Z 0; the rest of the functions of constraints were previously determined):

> @

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

0 02 0

1

1 1 1

l

Z Y X Y

X

O O O A

A

A , (xxi)

cos sin cos sin sin

0

2 ¹ ¹ ¹ ¹ ¹

1

1 l \ T M \ M

Z_O . (xxii)

> @

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

»»

»

¼ º

««

«

¬ ª

0 0 2 0

2

2 2 2

l

Z Y X Y

X

O O O B

B

A , (xxiii)

cos sin cos sin sin

0

2 ² ² ² ² ²

2

2 l \ T M \ M

Z_O . (xxiv)

The matrix of constraints has now 7 rows and 12 columns.

10. Conclusions

This paper presents a multibody approach for the determination of the functions of constraints and, consequently, the matrix of constraints in the case of a general mechanism. The functions of constraints are determined in each particular case. An example shows the application of the theory.

References

[1] Pandrea N., Stănescu N.-D., Dynamics of the Rigid Solid with General Constraints by a Multibody Approach, John Wiley & Sons, Chichester, UK, 2016.

[2] Pandrea N., Popa D., Stănescu N.-D., Classical and Modern Approaches in the Theory of Mechanisms, John Wiley & Sons, Chichester, UK, 2017.

[3] Shabana A. A, Dynamics of Multibody Systems, 4^th ed., Cambridge University Press, Cambridge, 2013.

[4] Amirouche F., Fundamentals of Multibody Dynamics. Theory and Applications, Birkhäuser, 2016.

[5] Coutinho M. G., Dynamic Symulation of Multibody Systems, Springer, 2001.

[6] Wittenburg J., Dynamics of Multibody Systems, Springer, 2008.

[7] Gattringer H., Gerstmayr J., Multibody System Dynamics, Robotics and Control, Springer, 2013.

[8] Nikravesh P. E., Planar Multibody Dynamics: Formulation, Programming and Applications, CRC Press, 2007.

[9] de Jalon J. G., Bayo E., Kinematic and Dynamic Simulation of Multibody Systems: The Real- Time Challenge, Springer-Verlag, 1994.

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[10] Stănescu N.-D., Dynamics of a System of Rigid Bodies with General Constraints by a Multibody Approach, Proceedings of the 6^th International Conference „Computational Mechanics and Virtual Engineering“ – COMET 2015, 15-16 October 2015, Braúov, Romania

[11] Stănescu N.-D., Dynamics of the Four-Bar Mechanisms with RRR or RTR Dyads by a Multibody Approach, Proceedings of the 6^th International Conference „Computational Mechanics and Virtual Engineering “ – COMET 2015, 15-16 October 2015, Braúov, Romania.

[12] Stănescu N.-D., Mecanica sistemelor, Editura Didactică úi Pedagogică, Bucureúti, 2013.