Multi-Objective Parametric Optimization of an Equilibrator Mechanism
3.3 Optimization of the System
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.
3.3.1 Constraints and Objective Function of the Optimization Problem
Equation (3.24) can be satisfied is the sum of system parameterœand the design parameter“is made 90ısince
CˇD900!CˇD9008˛ (3.28)
However, depending on the value ofœ,“may not attain the value 90ıœ. On the other hand, the maximum total length of the spring or springs employed in the housing cannot exceed Lalwor the outer diameter D0of the spring should be lower than Dalw. Also the factor of safety at solid length should be higher than 1.2 [3]. The spring(s) ideally should operate on some portion of its available stroke for the validity of assumption of linear behavior. Also there can be some constraints on the layout design parameters”,“, R, c. The overall constraints on the system can be expressed as
1/LmaxLalw
2/DODalw
3/18FsF 78Fs
4/2Œmin; max
5/ˇ2Œˇmin; ˇmax
6/R2ŒRmin;Rmax
7/c2Œcmin;cmax
8/Lmin0:02na.DCd/CLs
9/L05:26D
(3.29)
where eighth constraint it is due to [5]. There are many approaches in the literature to estimate the fatigue life of a spring. In this study, we use a parametric life estimation formula as expressed in [6]
LifeDe
h1 ylnKE
KC i
(3.30) where
KED KU.KS2KS1/ 2KU.KS2CKS1/ KS1D min
Sut
KS2D max
Sut
(3.31)
It is evident that, the total system is composed two subsystems interacting with each other via the force F. The first one is the equilibration mechanism layout which is governed by Eqs. (3.18) and (3.23). The second is the compression spring(s) whose internal state is governed by Eqs.(3.4)–(3.12). Also, since for’> 0,U() <U()andL() <L(), we know that the spring force F(™) is
F./DF./CŒU./U./ k (3.32)
and at the same time
F./DF./CŒL./L./ k (3.33)
which means the two subsystems are coupled through the relation L./D U./CL./CU./D U./C
L0F./
k
CU./ (3.34)
It is natural to require!
Mresto be zero at zero elevation angle’D0, which means that™Dœ. Then solving for F(œ) in Eq. (3.23), we obtain
F./D munbgrcos./
cRsin.ˇ/ U./ (3.35)
Also we know that,
L0DLmaxC F.max/
k (3.36)
whereLmaxDL(max). Substituting the Eq. (3.35) into the Eq. (3.32), and then substituting the Eq. (3.32) into the Eq. (3.36) yields
L0DLmaxCF./
k CU.max/U./ (3.37)
Substituting the Eq. (3.37) into the Eq. (3.34) yields,
L./D U./CLmaxCU.max/ (3.38)
where U(™) was given by Eq. (3.18). The Eq. (3.38) is actually expected and it could directly be derived from the Eqs. (3.32) and (3.33). Thus, if the design parameters for the layout”,“,R, cand the design parameters of the springD, nt,Lmaxalong with the coupler design parameterkare selected, then the rest of the variables in the system are completely defined. The two subsystems are interacting with each other through the force F and hence the spring constant k.
Moreover, the total system state defined in the design parameter xD[R,,ˇ,c,k,D,nt,Lmax]2R8 for 2[min,max] shall should be optimized according to some performance measure(s) which is related with both the layout and the spring, while obeying the constraints. We define the performance measure related with the layout as
J1DRMS !
Mres
D vu uu ut 1
maxmin Zmax min
cF./Rsin.ˇ/
U./ munbgrcos./
2
d (3.39)
One ideally would desire to minimize J1, since it is root mean square of the residual moments over the operational range minmaxof the system. On the other hand, the performance measure related with the spring is
J2D a1ln.Life/a2!Ca3ns (3.40)
where an’s are the weighting coefficients and nsis the load factor (reciprocal of safety factor) at the solid length Lsand it is defined by
nsD Kw8FsD d3Ssy
(3.41) Then J2becomes
J2D a1e
h1 yln
KE KC
i
a2 d 2naD2
s G
2 Ca3 d3Ssy
Kw8FsD (3.42)
One ideally would desire to minimize J2 in order to maximize the operational life, the natural frequency and the static safety factor of the spring. The natural frequency of a spring should be as high as possible in order to avoid resonance with the operational frequency of the applied force F. If Eq. (3.42) is observed, it is obvious that there should already be some trade-offs in selection of design parameters in order to maximize J2.
Thus the overall optimization problem can described as follows minimize hJDb1J1Cb2J2i
subjectto hconstraints given by inequalities.29/i formin max
(3.43)
where bn’s are the weighting coefficients.
3.3.2 Solution Approach
Solution to the problem defined by (3.43) has been implemented in Python programming language. First, the system has been mathematically modeled and has been optimized by the SLSQP (Sequential Least Square Programming) routine [7].
The solution approach to the problem has been schematized in Fig.3.6below.
As shown Fig. 3.6, the optimal solution is being searched in the search space (state space) defined by xD[R,,ˇ,c,k,D,nt,Lmax]2R8 for2[min,max]. The state x is optimized according to the measure defined in (3.43) and the successful termination of the optimization algorithm yields the optimal statexD[R,,ˇ,c,k,D,nt,Lmax].
The selection of the weighting coefficients a1, a2, a3, b1 and b2 is somewhat dependent on the specific application and may be subjective to some extent.
Fig. 3.6 Diagram of the implemented optimization procedure