1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
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CALCULATION OF THE UNDETERMINED STATIC REACTIONS
FOR THE ARTICULATED PLAN QUADRILATERAL MECHANISM
Jan-Cristian Grigore1
, Nicolae Pandrea2
1
(University of Piteşti, str. Targul din Vale nr.1, Romania)
2
(University of Piteşti, str. Targul din Vale nr.1, Romania)
ABSTRACT
Spatial mechanisms of the non-zero families constitute statically undetermined
systems, the undetermination order is given by the number representing the family of the
mechanism. The articulated plan quadrilateral mechanism, shown in this paper, is a third
family mechanism, an undetermined static third order mechanism. This paper uses the
relative displacement method and it establishes the mathematical model that allows the linear
elastic calculation in order to determine the statically undetermined reactions.
Keywords: coordinates pl ckeriene, matrix flexibility, stiffness matrix
I. INTRODUCTION
If the plane mechanisms are stressed by vector components forces perpendicular to
the motion plane or by vector component moment placed in the plane of motion, they are
statically undetermined systems. In these cases, in order to determine the components of the
reaction forces perpendicular to the motion plane as well as the components of the reaction
moments in the motion plane, the linear elastic calculation shall be used. This paper shows
these components using the relative displacement method [3], [4] and the pl ckeriene
coordinates.
II. NOTATIONS, REFERENCE SYSTEMS, TRANSFORMATION RELATIONS
Forces acting on a rigid point, the velocities of the points of a rigid, the small
movements of the points of the rigid are systems reduced to a point O ( Fig.1)at a torsion
vector consisting of mainly f and moment vector m .
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ISSN 0976 – 6340 (Print)
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Volume 4, Issue 3, May - June (2013), pp. 400-408
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2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
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If the reduction is in point 0O , then equivalent torsion has components F , M
satisfying the conditions
fF = ; fxOOmM 0+= (1)
Considering reference systems with origins in points 0,OO (Fig.1) and noting with
( )zyxzyx mmmfff ,,,,, , ( )zyxzyx MMMFFF ,,,,, vector projections ( )mf , , ( )MF,
respective on the axis of the systems Oxyz , XYZO0 then these scalar components are
pl ckeriene coordinates [4] of the torsion with representation by matrices column so;
{ } [ ]T
zyxzyx mmmffff = ; { } [ ]T
zMyMxMzFyFxFF = (2)
For a system of forces, f is the resultant force vector and m is moment resulting in
O , for rigid speeds f is the angular velocity of the rigid and m is the velocity of point O
and for small displacements of rigid, f is small rotation vector, and m is small movement
of the point O .
Fig. 1. System of forces
With notations:
( )000 ,, ZYX - point coordinates O ; 3,2,1,,, =iiii γβα , direction cosines of axes
OzOyOx ,, ;[ ]G , [ ]R , [ ]T translation matrices, position respectively
[ ]










−
−
−
=
0
0
0
00
00
00
XY
XZ
YZ
G ; [ ]










=
321
321
321
γγγ
βββ
ααα
R ; [ ]
[ ] [ ]
[ ] [ ] [ ]





⋅
=
RRG
R
T
0
(3)
Obtaining [4] transformation relations between the matrices column { }{ }fF ,
{ } [ ] { } { } [ ] { }FTffTF ⋅=⋅=
−1
; (4)
where
[ ] [ ] [ ]
[ ] [ ] [ ] 





=
−
TTT
T
RGR
T
T
01
(5)

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
402
III. RELATIONS BETWEEN RELATIVE MOVEMENTS AND EFFORTS AT THE
ENDS OF A BAR
Consider the straight bar AB (see Fig. 2), with length l , constant section, the area A ,
modules of elasticity GE, and either Axyz local reference system, AyAx, the central
principal axes of inertia of the normal section A.
Fig. 2. Straight bar AB , under the influence of efforts
Noting with ( )AA mf , torsion effort from A ; with ( )BB mf , torsion effort from B ;
with BBA mdd ,, vectors defined by the relations:
BfxABBmBmABxABBBBdAAAd +=+′=′=
*
;; θ (6)
and noting the projections on the axes trihedral Axyz of vectors
BdBAdABmBfAmAf ,,,,,,, θθ , respectively with ( )AzAyAx fff ,, , ( )AzAyAx mmm ,, ,
( )BzByBx fff ,, , ( )BzByBx mmm ,, , ( )AzAyAx ddd ,, , ( )AzAyAx θθθ ,, , ( )BzByBx θθθ ,, ,
( )BzByBx ddd ,, , we obtain column matrix of pl ckeriene coordinates in the local
system Axyz
{ } [ ]T
AzAyAxAzAyAxA mmmffff = ; { } [ ]T
BzByBxBzByBxB mmmffff = (7)
{ } [ ]T
AzAyAxAzAyAxA dddd θθθ= ; { } [ ]T
BzByBxBzByBxB dddd θθθ= (8)
and equality resulting from the equilibrium condition
{ } { } { }0=+ BA ff (9)
Stiffness matrix [ ]ABk and matrix flexibility [ ] [ ] 1−
= ABAB kh [4] expressed in the
reference system Axyz are given by the equalities

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
403
[ ]




















−
−
⋅=
060
2
400
6000
2
40
00000
2
1200060
0120600
00
2
000
3
lzIlzI
lyIlzI
lzI
E
G
zIlyI
zIlzI
Al
l
E
ABk ;
[ ]


































−
−
⋅=
0
3
0
2
2
00
3
000
2
2
0
00000
6
6
000
3
0
0
6
0
3
00
006000
6
yI
l
yI
l
zI
l
zI
l
A
zIzI
l
yIyI
l
G
E
E
l
ABh
(10)
where zy II , are the principal central moments of inertia of normal areas on axis
Ax and xI is defined by equality
zyx III += (11)
With these notations [4] to obtain equalities
{ } [ ] { } { } [ ] { }AABABABABA fkddkf ⋅=⋅= ; (12)
where { }ABd is the relative displacement
{ } { } { }BAAB ddd −= (13)
Switching to the reference OXYZ is done using relations (3), (4), (5) and these
equalities are obtained
{ } [ ]{ } { } [ ]{ } { } { } { }
{ } [ ]{ } { } [ ]{ }
{ } [ ]{ }[ ] [ ] [ ]{ }[ ] 11
;
;
;;
−−
==
==
−===
ABABABABABABABAB
BABBAABA
BAABBABBAABA
TkTHTkTK
fTFfTF
DDDdTDdTD
(14)
{ } [ ]{ } { } [ ]{ }AABABABABA FHDDkF == ; (15)

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
404
IV. CALCULATION OF THE REACTIONS
Considering articulated plan mechanism ABCD from Fig. 3 acted by external forces
and moments [4] which are distributed in points DCB ,, and are marked by column matrix
of pl ckeriene coordinate. Noting generally with { }E efforts at the ends of the bars, and
with{ }R reactions by isolating bars and nodes DCB ,, obtain formal scheme from fig. 4, for
which the following equations of equilibrium can be written
Fig. 3. Articulated quadrilateral plan mechanism
{ } { } { } { } { } { } { } { } { }
{ } { } { } { } { } { }0;
;0;;0
3332
22211
=+=+
=+=+=+
DCCCC
CBBBBBA
EEPEE
EEPEEEE
(16)
from which resulting
{ } { } { } { } { } { } { }CPBPAECEBPAEBE ++=+= 3;2 (17)
{ } { } { } { } { } { } { } { } { } { } { } { }AEDPCPBPDRAECPBPCRAEBPBR +++=++=+= ;; (18)
and using relations (15) the following expressions are derived:
{ } [ ]{ } { }{ } { } { } [ ]{ } { }{ }
{ } { } { } [ ]{ } { }{ }33
221
;
DCCDCBA
CBBCBABAABA
DDKPPE
DDKPEDDKE
−=++
−=+−=
(19)
or
{ } { } [ ]{ } { } { } [ ]{ } { }{ }
{ } { } [ ]{ } { } { }{ }CBACDDC
BABCCBAABBA
PPEHDD
PEHDDEHDD
++=−
+=−=−
33
221
;
(20)
where{ }iBD ,{ }iCD ,{ }iDD , 2,1=i , are the movements at the ends of indexed bars (Fig. 3)
with indices 3,2,1 .

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
405
Fig. 4. Formal mechanism quadrilateral scheme, the representations efforts
and reaction forces
Consider the insertion of A, { } { }( )0=AD in the linear elastic calculation and in this
way the movements of the other sections relate to the section of A . Noting with
{ } { } { }DCB UUU ,, column matrices attached to kinematic couplings of rotation [6] to obtain
expressions
{ } [ ]
{ } [ ]
{ } [ ]T
DD
CCC
T
BBB
XU
XYU
XYU
00100
,0100
,0100
−=
−=
−=
(21)
and noting with DCB ξξξ ,, rotations of the joints, the following equalities are derived:
{ } { } { }
{ } { } { }
{ } { } { }DDD
CCCC
BBBB
UD
UDD
UDD
ξ
ξ
ξ
+=
+=
+=
3
23
12
0
;
;
(22)
By adding relations (20), taking into account the equations (22) and using notations
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] { } { } { }[ ] [ ]
{ } [ ]{ } [ ]{ } { }{ }CBCDBBC
D
C
B
DCBADAD
CDBCABAD
PPHPH
UUUUHK
HHHH
++=∆










===
++=
−
~
;;
;
1
ξ
ξ
ξ
ξ (23)
obtain the equation
[ ] [ ][ ]{ } [ ]{ }∆−=
~
ADKUADKAE ξ (24)
Using the notations

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
406
{ } [ ]
{ } [ ]
{ } [ ];1000
~
;1000
~
;0000
~
DD
T
D
CC
T
C
BB
T
B
XYU
XYU
XYU
−=
−=
−=
(25)
{ } { } { } { }[ ] ;
~~~~ T
DCB UUUU = (26)
knowing [6] that reactions satisfy the relations
{ } { } { } { } { } { } 0
~
;0
~
;0
~
=== D
T
DC
T
CB
T
B RURURU (27)
taking into account the equality relations (18) and the notation
{ }
{ } { }
{ } { } { }{ }
{ } { } { } { }{ }











++
+−=
DCB
T
D
CB
T
C
B
T
B
T
PPPU
PPU
PU
P
~
~
~ (28)
obtain the expression
[ ]{ } { }TA PEU =
~
(29)
with that of (24) the matrix of rotations in the joints is deduced
{ } [ ] { } [ ][ ] { }{ }∆⋅+⋅⋅=
− ~~~ 1
ADTAD KUPUKUξ (30)
and then from (24) the reaction is deduced form { } { }AA ERA =,
Relations (30), (29), (24) can be expressed in a simpler form if the following
notations are made
{ } [ ]
{ } [ ]T
zyxzyx
T
AzMAyMAxMAzEAyEAxEAE
∆∆∆=∆
=
~~~~~~~
θθθ
(31)
[ ] [ ] [ ] 































===
565251
464241
363231)2(
;
656463
252423
151413)1(
;
065646300
560005251
460004241
360003231
025242300
015141300
KKK
KKK
KKK
ADK
KKK
KKK
KKK
ADK
KKK
KKK
KKK
KKK
KKK
KKK
ADK (32)
[ ] [ ]










−−−
=










−
−
−
=
DCB
DCB
DD
CC
BB
XXX
YYYB
XY
XY
XY
A
111
;
1
1
1
(33)
and then it follows
{ } [ ] [ ] [ ] { } [ ]










∆
∆+⋅⋅⋅=
−−−−
y
x
z
TAD BPAKB
~
~
~
111)1(1
θ
ξ (34)

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
407
[ ] { }
( )
[ ]










∆
⋅−=










⋅=










−
z
y
x
AD
AY
AX
AZ
T
AZ
AY
AX
K
M
M
E
PA
M
E
E
~
~
~
;
2
1
θ
θ
(35)
Calculation of the other reactions { } { } { }DCB RRR ,, are made using the relations (18).
V. CONCLUSION
The matrix { }TP defined by the relation (28) depends only on the components of forces
{ } { }{ }DCB PPP ,, compatible with the movement of the mechanism, respectively on the
components BZBYBX PPP ,, and the analogues ones. The statically determined components of
the reaction { } { }AA ER = are given by the first relation (35) and as expected they depend on
the components compatible with external forces movement and they do not depend on the
stiffness of the elements of the mechanism.
The matrix { }∆
~
is the result matrix partitions
{ } [ ]T
yxz ∆∆=∆
~~~~
1 θ ; { } [ ]T
zyx ∆=∆
~~~~
2 θθ (36)
The matrix { }1
~
∆ with components in the plane of motion and depending on the
components compatible with movement BZBYBX MPP ,, and the analogues, and { }2
~
∆ with
incompatible components with moving parts and is not compatible with motion-dependent
BYBXBZ MMP ,, and analogues.
It follows from this and from the second relation (35) that statically indeterminate
reactions depend exclusively on the components of the external forces incompatible with
moving parts.
Concerning the matrix{ }ξ , movements of kinematic couplings resulting from the set
and relation (35) it depends on the stiffness of elements as well as on the components of the
external forces compatible with movement.
Based on the relations established in this paper we can develop an algorithm and a
program for numerical calculation of statically indeterminate components, an objective that
will result in a subsequent paper.
VI. ACKNOWLEDGEMENTS
This paper is a continuation of research conducted under the grant "PD -683 / 2010",
and we want to thanked the Romanian Government (UEFISCDI), which certainly those
funding research.

9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 3, May - June (2013) © IAEME
408
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