Multi-Objective Parametric Optimization of an Equilibrator Mechanism
3.2 Modelling of the System
3.2.1 Equilibrator System Components
The major system components of the equilibrator mechanism are as follows;
• Compression Springs contained in housing
• Pivot point (usually it is a pulley in actual applications)
• Hinge Point (mechanically attached to the rotary structure)
• Cable for transmission of the equilibrator force
These components are schematically illustrated in Fig.3.2below
Basically, the cable loaded by the springs is wrapped around a pulley which is located at a fixed pivot point. The springs are housed in a container which is also fixed to the ground. The force induced on the cable by the spring is transmitted through the hinge point which is mechanically fixed to rotor structure. The force applied at the hinge point creates a moment in the counterclockwise direction and balances the moment created by CM of the rotary structure, which acts in the clockwise direction.
3.2.2 System Variables
The system described in Fig.3.2is schematized geometrically and kinematically in Fig.3.3.
The variables depicted in Fig.3.3above can be briefly described as follows:
œ: The angle between the position vector of the CM of rotary mass and x-axis in counter clockwise (CCW) direction
’: Angle of elevation of the rotatory system measured from its initial configuration (0ı) in CCW direction
™: Orientation of the position vector of the CM measured from the x-axis in CCW direction at’
”: The angle between the position vector of the CM and the position vector of the Hinge Point (H)
“: The angle between the position vector of the Hinge Point (H) and the position vector of the Pivot Point (P) in CCW direction
Fig. 3.2 Schematic illustration of equilibrator mechanism
Fig. 3.3 Kinematic and geometric variables of the system
r: The magnitude of the position vector of CM and it is defined in Eq. (3.3) R: The magnitude of the position vector of Hinge Point (H)
c: The magnitude of the position vector of Pivot Point (P)
U(™):The distance between the point H to the point P as a function of™
F:The magnitude of the force applied by the equilibrator cable to the rotary structure at the Hinge Point (H) in the direction of the vectorHP
The variables which define the overall layout of the equilibrator mechanism above can be classified into the design parameters, the system parameters, the independent variables and the dependent variables. From mechanical point of view, the rotary system is assumed to be free to attain any elevation angle between some defined limits, i.e.,˛2[˛min,˛max]. This automatically makes’the independent variable. Since the munb, r andœare the properties of the rotary system to be balanced, they are considered to be the system parameters. Observing Figs.3.2and3.3, it is seen that the parameters”,“, R, c are properties of the equilibrator mechanism layout yet to be designed. Thus, they are considered to be the design parameters.
Since U(™) is the distance between the points H and P, and the F is defined by the mechanism layout and the springs, they are the dependent variables of the system.
A typical compression spring and its housing depicted in Fig.3.2is also a part of the system and its detailed illustration is given in Fig.3.4below.
In this system, the spring force F is applied to the compression spring contained in the housing by a retainer at one end, while the other end of the spring is supported and fixed. Here, due to some space limitations, the spring housing is constrained by the maximum allowable length Lalwand maximum allowable diameter Dalw. On the other hand, the spring itself has its own parameters and properties which must be taken into account during the design of the overall system.
Parameters related with spring and force F applied to the spring is depicted in Fig.3.5below
Spring parameters, its properties and their relation to each other in terms of strength, performance and mechanical behavior are explained and can be found in [3]. The ones taken into consideration and related with our system are explained briefly in this study as follows. The spring stiffness or the spring rate k is defined as
kD d4G 8D3na
(3.4) where d is wire diameter, D is the mean diameter, G is the shear modulus of the spring material and nais the active number of coils and for the squared and ground compression springs is given by
naDnt2 (3.5)
and ntis the total number coils in the spring. The solid length of the spring Lsis given as
LsD.nt0:75/d (3.6)
Fig. 3.4 Compression spring, its housing and spring force representation
Fig. 3.5 Spring parameters and the spring force F
The critical shear stress in the spring is given by
DKw8FD
d3 (3.7)
where Kwis the Wahl factor and is given by
KwD 4C1
4C4 C0:615
C (3.8)
and C is the spring index and given by
CD D
d (3.9)
For squared and ground steel springs, free length must satisfy the following relation in order avoid buckling [3]
L05:26D (3.10)
The maximum allowable torsional yield stress Ssyis given in [3,4]as a widely accepted and conservative relation
SsyDalwD0:56Sut (3.11)
where Sutis the ultimate tensile strength of the material. Also the first fundamental frequency of the spring is [3]
!D 1 2
s k ms
(3.12) where msis the mass of the spring and is given by
msD 2d2Dna
4 (3.13)
where¡is the density of the spring material.
3.2.3 Balance Equations of the System
Observing Figs.3.2and3.3, the static balance equation of the overall system is
!MeqC!
MunbD0 (3.14)
where!
Meq is the moment vector created by the equilibrator force F around the axis of rotation and!
Munbis the moment vector created by the CM and they are given as
!MeqD
Rcos.C/OiCRsin.C/Oj ˝!
F (3.15)
and
MunbD
rcos./OiCrsin./Oj ˝
munbgOj
D munbgrcos./kO (3.16)
If we let!
f be the vector pointing from the point H to point P, then it is given by
!f Dh
.ccos.CCˇ/Rcos.C//OiC.csin.CCˇ/Rsin.C//Oj i
(3.17) The magnitude of!
f , which equals U(™), is given by ˇˇˇ!
f ˇˇˇDU./Dh
Œccos.CCˇ/Rcos.C/ 2CŒcsin.CCˇ/Rsin.C/ 2i1=2
(3.18) Then the spring force!
F with magnitude F is given by
!F D F
hŒccos.CCˇ/Rcos.C/ OiCŒcsin.CCˇ/Rsin.C/ Oj i ˇˇˇ!
f ˇˇˇ (3.19)
Then!
Meqbecomes
!MeqDh
Rcos.C/OiCRsin.C/Oj i˝F
hŒccos.CCˇ/Rcos.C/ OiCŒcsin.CCˇ/Rsin.C/ Oj i ˇˇˇ!
f ˇˇˇ
(3.20)
After carrying out the cross product multiplication we obtain
!MeqD cFRŒsin.CCˇ/cos.C/cos.CCˇ/sin.C/ Ok ˇˇˇ!
f ˇˇˇ (3.21)
Since the nominator of Eq. (3.21) is a trigonometric identity, we obtain
!MeqD cF./Rsin.ˇ/
ˇˇˇ! f ˇˇˇ
kO (3.22)
The overall moment balance equation becomes
!Mres D! MeqC!
MunbD cF./Rsin.ˇ/
ˇˇˇ! f ˇˇˇ
kOmunbgrcos./kOD 2
4cF./Rsin.ˇ/
ˇˇˇ!
f ˇˇˇ munbgrcos./
3
5kOD0 (3.23)
where!
Mres is the residual moment. In order for this equation to hold independent of™, the relation sin(“)Dcos(™) must hold. This means
CˇD90ı (3.24)
which is the same result obtained in [2].
If Eq. (3.24) holds, it implies
F.ˇ; /D munbgr Rc
ˇˇˇ!
f ˇˇˇD munbgr
Rc U.ˇ; / (3.25)
Since
F.ˇ2; 2/F.ˇ1; 1/D munbgr
Rc ŒU.ˇ2; 2/U.ˇ1; 1/ DksL.ˇ/ (3.26) The Eq. (3.26) implies that in order for the equilibrator system to perfectly balance, the stiffness of the spring k must satisfy the equation
kD munbgr
Rc (3.27)
If Eqs. (3.24) and (3.27) are satisfied then the system is perfectly balanced. However, in actual implementations there may be constraints which prevent the designer from satisfying these equations. In these cases, an optimization process must be employed in order to minimize!
Mres and at the same time, some other possible objective functions while satisfying the system constraints.