Kinematics of the Planar Quadrilateral Mechanism

KINEMATICS OF THE PLANAR QUADRILATERAL MECHANISM

Florian Ion Petrescu1 Relly Victoria Petrescu2

Abstract: This paper presents an original method to determine the kinematic parameters at the linked quadrilateral mechanism. It is starting with a trigonometric method, which has the advantage to determine very quickly the position angles. The velocities can be determined faster using a geometric method. This method is developed and for the accelerations determinations. The (proposed) geometric method, determines first the kinematic parameters of the internal couple (B) and then the rotation angles with their derivatives. Secondary, the paper presents the determination of the efficiency of this mechanism. Determines and dynamic coefficient, D. With this one proposes two yields; the mechanical efficiency and the dynamic efficiency.

Keywords: 3R dyad, kinematic parameters, efficiency, dynamic coefficient

Introduction ; where the

positions parameters (xO ; yO ) of the couple O are The paper presents an original method to zero (if we set the rectangular system in O); l1 is determine the kinematic parameters at the linked the length of the crank 1; l2 is the length of the quadrilateral mechanism [3-4]. connecting rod 2; l3 is the length of the rocker 3; It is starting with a trigonometric method, j1 is the crank angle position, and ω1 is its an- which has the advantage to determine very qui- gular velocity. The C couple coordinates (xC ; yC ) ckly the position angles. are also known. The velocities can be determined faster We must determine now the cinematic pa- using a geometric method. This method is develo- rameters which give the positions of the rod (j2 ) ped and for the accelerations determinations. The and the crank (j3 ). (proposed) geometric method, determines first the kinematic parameters of the internal couple y j (B) and then the rotation angles with their deri- A j vatives. A 2

2 l2 Secondary, the paper presents the determi- A l1 B j j nation of the efficiency of this mechanism. Deter- 2 ω 1 1 x mines and dynamic coefficient, D. l l3 O With this one proposes two yields; the me- j 1 3 l j chanical efficiency and the dynamic efficiency. 3 0 C C The kinematics of the planar quadrilateral mechanism

The kinematic schema of a planar quadrila- Fig. 1. Kinematic schema of a planar quadrilat- teral mechanism [1-2] can be seen in the Figure 1. eral mechanism The following kinematic parameters consi- dered known:

1 Senior Lecturer Ph.D., Bucharest Polytechnic University, [email protected] 2 Senior Lecturer Ph.D., Bucharest Polytechnic University, [email protected]

ENGEVISTA, V. 14, n. 3. p. 345-348, dezembro 2012 345 (1)

(2)

Determining the positions

It is starting with a trigonometric method, which has the advantage to determine very quickly the position angles (the system 1). First it determines the cinematic parameters of the A couple ( xA; yA ), and their first two derivatives ( xA; y A;xA; yA ). Second, one finds the variable length (l) between C and A, and it determines the position angle (j) of the CA vector. We also determine the angles A and C from the triangle Determining the velocities of ABC. Finally we found the positions of the rod ( the couple B j2 ) and the crank (j3 ).

The velocities ( xB ; yB ) can be determined faster using a geometric method (the system 2). First, we write the equations of the circles which

have the radius l3 and l2. They intersect in B. This system does not have to be solve because the solu- tions are already known. It will be derivative two times to obtain the systems of velocity and accelerations. These systems are linear and are solved by determinants. At 2.2 we obtain the velocities (

xB ; yB ), and at 2.3 determine the accelerations (

xB ; yB ).

Determining the accelerations of the couple B

346 ENGEVISTA, V. 14, n. 3. p. 345-348, dezembro 2012 The accelerations ( xB ; yB ) can be determined faster using a geometric method (the system 3).

(4)

(3)

The efficiency of the planar quadrilateral mechanism

The efficiency of a planar quadrilateral mechanism can be determined starting from the forces and velocities repartition, (Figure 2). Determining the angular velocities and accelerations j1 −j2 y Ft, vt j Fm, vm The angular velocities (ω2;ω3 ) and acce- A 180 − (j3 −j2 ) j2 lerations (ε 2;ε 3 ) can be determined now, faster, 2 l2 A bt bt l using the vectorial method (system 4). F , v Fn, vn A 1 j2 j1 ω1 Fn, vn B l x l3 O Fbn, vbn j 1 3 l j 3 0 C C

Fig. 2. Forces and velocities repartition of a planar quadrilateral mechanism

The system (5) presents the relationships which give the forces and the velocities on the planar quadrilateral mechanism. The driving for-

ce Fm is perpendicular on the crank 1 in A. Its component along the connecting rod

ENGEVISTA, V. 14, n. 3. p. 345-348, dezembro 2012 347 (the bar 2) F , gives the normal component F . n bn ω D = ⋅ω F is perpendicular on the rocker 3 in B. 1 D 1 . We consider that the rotation speed bn ω These forces give the dynamic velocities ( 1 ) of the crank is a constant for a constant rota- which are similar with the forces. tion speed (n) (relation 6). (6) The forces are always the same, but the velocities (the dynamic velocities) are different than In fact ω1 varies with the position of the the kinematics velocities. crank angle (j) (system 7, [3-4]). For this reason the dynamic efficiency will be different than the mechanical yield.

Fn = Fm ⋅sin(j1 −j2 )  j j vn = vm ⋅sin( 1 − 2 )  FB ≡ Fbn = Fn ⋅sin[π − (j3 −j2 )] =  = Fm ⋅sin(j1 −j2 )⋅sin(j3 −j2 )  D vB ≡ vbn = vn ⋅sin[π − (j3 −j2 )] = = v ⋅sin(j −j )⋅sin(j −j )  m 1 2 3 2  (7)  ⋅ j −j ⋅ω  l1 sin( 1 2 ) 1 ω3 = ⇒  l3 ⋅sin(j3 −j2 )  ⋅ω ⋅ j −j  l1 1 sin( 1 2 ) vB = l3 ⋅ω3 = =  sin(j3 −j2 )  v ⋅sin(j −j ) = m 1 2  sin(j3 −j2 )  D vB = D ⋅ vB ⇔  vm ⋅sin(j1 −j2 )⋅sin(j3 −j2 ) =  (5) *  v ⋅sin(j −j ) = D ⋅ m 1 2 ⇒ J is the mechanical moment of inertia re-  * * sin(j3 −j2 )  duced to crank; J m = J medium . D = sin 2 (j −j )  3 2   References P3 FB ⋅ vB ηi = = =  P1 Fm ⋅ vm  v ⋅sin(j −j ) Erdman, A.G., Three and four precision point kine-  F ⋅sin(j −j )⋅sin(j −j )⋅ m 1 2  m 1 2 3 2 sin(j −j ) matic synthesis of planar linkages = 3 2 = , Elsevier, Mecha-  F ⋅ v  m m nism and Machine Theory,V olume 16, Issue 3, = sin 2 (j −j )  1 2 1981, pages 227-245.   PD F ⋅ vD η D = 3 = B B = Gibson, C.G., Newstead, P.E., On the geometry i P F ⋅ v  1 m m of the planar 4-bar mechanism, Acta Applicandae  F ⋅sin(j −j )⋅sin(j −j )⋅ v ⋅sin(j −j )⋅sin(j −j ) = m 1 2 3 2 m 1 2 3 2 = Mathematicae, Volume 7, Number 2, pages 113-  Fm ⋅ vm  = sin 2 (j −j )⋅sin 2 (j −j ) = D ⋅η 135.  3 2 1 2 i Petrescu, F.I., Teoria Mecanismelor si a Masinilor, Conclusions Create Space Publisher, USA, 2011, ISBN/EAN 13: 1468015826 / 978-1-4680-1582-9, page The presented method is the most elegant count 432. and direct method to determine the kinematics planar quadrilateral mechanism. Petrescu, F.I., Bazele Analizei si Optimizarii Siste- Relationships used by this method allow melor cu Memorie Rigida, Create Space Publisher, and the determination of the dynamic system USA, 2012, ISBN/EAN 13: 1470024365 / 978- vibration. In the dynamic kinematics the cons- 1-4700-2436-9, page count 164. tant rotation speed ω1 = ct . gets a variable value

Artigo submetido em 14/11/2011, aceito em 15/03/2012

348 ENGEVISTA, V. 14, n. 3. p. 345-348, dezembro 2012