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Optimal design of flexible mechanisms using a parametric approach
Computers It Sfrucrwes Vol. 45, No. 5/6, pp. 965-971, 1992 Printed in Great Britain.
OPTIMAL
004s7949p2ss.lm + 0.00
0 I!?92 Fqamonpress Ltd
DESIGN OF FLEXIBLE MECHANISMS A PARAMETRIC APPROACH
USING
F. W. LIOU and J. D. LIU Department of Mechanical and Aerospace Engineering, and Engineering Mechanics, Intelligent Systems Center, University of Missouri, Rolla, MI 65401-0249, U.S.A. (Received 19 September 1991) Abstract-In this paper, an effort to develop a more generic design procedure for flexible mechanisms is initiated. Based on a parametric study, the important parameters which a!kct the stress, displacement, and natural frequency can be closely related. An approach using these relations to obtain the optimal design of flexible mechanisms subject to stress, displacement and natural frequency constraints is presented. Optimality conditions are derived to moderate the conflicting constraints, and a recursion approach is developed to show the application of this study. Two examples using this approach to design flexible mechanisms are presented to demonstrate the idea.
INTRODUCTION
BACKGROUND
synthesis of flexible mechanisms can be treated as a nonlinear programming problem [l, 21, and can be solved using the optimality criterion technique [3,4]. Cleghom et al. [5] presented a new procedure in which the finite difference method was used between cross-sectional diameter and maximum stress in each link to derive a Jacobian matrix. This approach can account for the force effect from changing the inertia (cross-sectional area) of other links to improve efficiency. Zhang and Grandin [6] improved their formulas to develop a design procedure which uses kinematic refinement to compensate the error due to elastic deformation. Kishore and Ke.efe [7l developed a procedure to synthesize elastic four-bar linkage for four precision-point motion generation. Optimization was preceded by a kinematic resynthesis. The above references indicate that most of the formulation done in this area deals with very limited constraint(s) and design parameter(s). Since the design of a flexible mechanism is a very complex procedure which involves many design parameters such as geometry, material property, and the input speed, and may be subject to several different constraints such as stress, displacement, and natural frequency. A more general and robust design approach is needed in this area. In this paper, an effort to develop a more generic design procedure is initiated. A parametric study of flexible mechanisms is conducted to relate the stress, displacement and natural frequency with some important parameters, such as the geometry and material property of a mechanism. A synthesis approach based on the parametric study of flexible mechanisms is presented, and optimality conditions are derived to negotiate between the constraints. Some numerical examples are used to demonstrate the idea.
Based on the previous studies, the transverse deformation of a link of a general industrial four-bar linkage can be considered as linear and decoupled from the remainder of the deformation phenomena in the system [8]. The normalized steady-state amplitude x, of vibration corresponding to the kth mode and jth harmonic have been developed, and can be written as
The
tx8.3)jk (llk31(F,IFa) - XSI = 1 - ( j2/k4)Q2 ’
(1)
where F, is the jth harmonic of the effective point load of inertia force, F(t), acting at the midpoint of a link; F. is the maximum value of inertia force, F(t) during the cycle; x, is the steady-state amplitude of vibration; X = w/o. = (input crack speed)/(natural frequency of the corresponding link for the first mode); x,, is the maximum state deflection; and x, = F,I&
(2)
where K, is the equivalent spring constant of a link. The corresponding transient amplitude can be written as
(x,,>,k O’lksX’WF,/Fo 1 -=X,1 1 - ( j2/k4)Q2 ’
(3)
The above amplitude can be combined to form the total deformation response as [8]
+ 965
W3)(4/Fa) . (j2/k4~2 sln(ior) ’ 1
_
1
(4)
F. W. LIOUand J. D. LIIJ
966
Since a well-designed mechanism is made up of straight links, the k = 1 mode dominates the response, and can be written as
F./F --““sin(jwt) + 1 - (j#
where rBis the radius of gyration of the cross-section of the link, I is the moment of inertia of the crosssectional area of a link, and E is the Young’s modulus of the link. Similarly, strain can be derived as follows:
1. (5)
If R is small in eqn (5), the steady-state term dominates the response
PARAMETRIC
STUDY
In this parametric study, the correlation between link geometry, material, and maximum stress, displacement, or natural frequency will be studied. From eqns (2) and (6), the following equations can be obtained
x(t)
-=-=
X31
F(t) FO
1
(7)
x(r)=Ix,,=1$ e where 1 is a coefficient relating F(t) dimensional analysis, one can 6nd force of a mechanism is proportional link length, and square of the crank i.e. F, = CMLw =,
O( ) L’y yz
E
w2 ’
(12)
g
o2 _ K, “--=M
48EIIL’ PAL
48EI =48 =pAL4
(13)
Therefore 0,=4J5
J( ) E - ‘B. P L2
(14)
The eqns (1 l), (12), and (14) can be used to relate the general characteristics of the flexible mechanisms, such as strain, displacement, and natural frequency, as a function of input speed, material property, and the geometry of the mechanism. Since certain assumptions are made to obtain these equations, the direct use of these equations may not be appropriate. However, the following sections shows how to apply these equations as recursive formulas in the design of flexible mechanisms.
(8) RECURSIVE
and F.. From a that the inertia to the link mass, angular velocity, (9)
where M is the mass of the link = pAL (p is the mass density), L is the link length of the corresponding link, w is the crank input speed, and C is a coefficient which relates the maximum inertia force and the above parameters in eqn (9). The stiffness of the corresponding link of the first mode can be written as 48EI K e=-. L’
=b!&Cyl !I!
where y is the coordinate axis normal to the link length along the thickness direction with its origin at the midpoint of the cross-section. The natural frequency of the link can be derived as follows:
(6) Equation (6) implies that the inertia force F(r) can be treated as a quasi-static force applied to the link, if n is small (say R < 0.05 [8]), and k = 1 mode dominates the response.
6
FORMULAS
According to the above equations, the following relations could be developed, assuming the radius of gyration of the cross-sectional area, rg, is the only design parameter, and rg is proportional to A2 for a link with symmetric cross-section. The equations for other parameters can be used by following the similar derivation. (A) Stress constraint From eqn (12), the stress equation can be obtained as u = p(C2Y)(a2)(L3/r:). In this section,
Y, w, p, and L are considered
(15) as
(10) constants, and only the absolute value of the maxi-
Therefore, substituting eqns (9) and (10) into (8), the following developments can be obtained CpALZw2 48EI L’ (11)
mum stress, 16, (, which is proportional to l/r:, is to be considered during the design procedure. For mechanisms with symmetric cross-sectional area A, the maximum stress ]a,,1 is proportional to l/A 2. Therefore, the following recursion formula can be derived from eqn (15) (A,),+, = {IQi.maxIv/~oij~v(Ai)u,
(16)
Optimal design of flexible mechanisms
where (Ai),+ 1 = cross-sectional area in the ith member during the v + lth iteration, (A,), = cross-sectional area in the ith member during the vth iteration, lQi,-ie = the maximum absolute value of stress in the ith member during the vth iteration, a,, = specified allowable stress, qy= stress elation factor during the vth iteration.
adjustable relaxation coefbcients [5] using the following iterative equations
tl: = {log(Ai)c,+I -
ttC =
Displacement constraint equations can be obtained in a similar way. Since only the absolute value of the maximum displacement, }X,,I, was considered, and the displacement is proportional to ri from eqn (1 l), the following recursion equation can be obtained
where (A,),+ I = cross-sectional area in the ith member during the v + Ith iteration, (A,), = cross-sectional area in the ith member during the vtb iteration, IX,,,,], = the maximum displacement in the ith member during the o th iteration, Xaj= specified allowable displa~ment, s:= displacement relaxation coefficient during the 0th iteration. (C) Natural frequency constraint Similarly, according to eqn (14), the following reIationship can be obtained
where (A)v+r =cross-sectional area during the v + Ith iteration, (A), = cross-sectional area during the vth iteration, w, = specified natural frequency, (o,,,& = the lowest natural frequency of system during the vth iteration, 9.” = natural frequency relaxation coefficient during the vth iteration. In order to gain a better convergence rate, q. in eqn (16), a: in eqn (17), and qi in eqn (18), can be recalculated in each iteration and become selfCAS 43 : W-K
lOg(Ai),}l{lOS(~i,-)”
-
h&4),+
Note that formula (16) is the same as that Khan and co-workers [3,4] have obtained. (B) D~~la~~rn~ntco~traint
967
1 -
W4L
~/~lOg(~~~
-
ORALS
l”gtxi,max)~
f I>
tzo)
10
Il.
hwuninl*+
(21)
COND~O~S FOR MUL~PLE CONSTRAINTS
In order to determine the active constr~nt(s) among different constraints, the optimality conditions are derived in this section. For simplicity, a mechanism composed of N members with stress and displacement constraints is considered in this derivation. Other cases can be derived using the similar approach, The design goal is to find the crosssectional sizes of each member, characterized by the design variable Yi to minimize the total volume V V = 2 A,& inl
(22)
subject to lQnaxlg @I?0 i=1,2,3 ,...,
N
(23)
where A, is the cross-sectional area in the ith member, Li is the length in the ith member, ]cT~,_Iis the maximum absolute stress in the ith member, cai is the spec%ed allowable stress in the ith member, IX,,, 1is the maximum absolute displacement at thejth point, A’, is the specified allowable displa~m~t at the jth point, N is the number of members (possibly links or finite elements), and K is the number of points with specified deflection. Since variable Yican be the cross-sectional area A,, the diameter of a circular member, or the other similar quantity, the area Aj, can be expressed as a function of the design variabh Yi Aj= BY;,
(25)
where B and b are constants. Considering the mechanism moves periodically, the maximnm stress in the ith member can be written as
F. W. LIOUand J. D. LKJ
968
where lP,,_ 1is the maximum absolute force in the ith member within a cycle. Considering any member i which contains the deflection constraint at pointj, IX;,,, 1is proportional to l/A,, i.e.
Because the stress constraint can be applied to every member of the mechanism under the strength consideration, and the deflection constraint can be applied to certain point(s), two optimality conditions can be discussed by solving eqns (32)-(34). Condition 1
(27)
If any member i does not include the deflection constraint (i.e., only stress constraint gi is active in this member), then the deflection constraint can be ignored in eqns (32)-(34). After solving eqns (32)-(34), it is found that only when Iaj,_l = a, is the optimality condition satisfied.
where D, is a positive constant. After arranging eqns (22)-(27), the following can be obtained V = 5 BY;L,
Condition 2
Ipi1
If any member i contains the deflection constraints at point i, then the following three cases can be discussed: (a) If both g, and g, are active, the optimality condition is when both la,,_ I = alri and IX,, I = &,. (b) If g, is active but g, is inactive, then only when lai,mPxl= a,, is the optimality condition satisfied. (c) If gi is inactive but gj is active, then only when I_X’+.,I= X, is the optimality condition satisfied.
where V is the objective function, g, and gj are the constraint functions. The above functions can be redefined as follows:
These conditions can be used to determine whether the current design has reached the optimum or not. For example, if a design satisfies the stress constraint (i.e., the maximum stress equals the allowable stress in each member of the mechanism), and if all the deflections in the specified points are below the allowable deflections, then this case satisfies the optimality condition (b). The solution obtained under the stress constraint is the optimum solution.
c$ = f BY;Li+d. ,(s-aai) i-1
a4
By;-&!!&___
s__:
I
I 2.
aAi.
SDi -0 BY!+’
(32)
lpimaxl a,_O BY:
m
APPLICATIONS
Based on the above discussion, two design procedures are suggested to design high-speed mechan-
(34)
,Xd,
---------t
I
lull.
I
IULI, Maxlmum
I
(0.1)
I
1ull11,
Stnrr Fig. 1. Numerical interpretation in the critical member for the second design procedure (line A-linear interpolation function, line B-actual function).
Optimal
design of flexible mechanisms
969
Table 1. Designed results for Case (A) Designed
Max. deflection in
Link
thickness (in)
specified point (in)
Maximum stress (Psi)
Full vibration
Crank Coupler Follower
1.718 1.053 0.652
0.081 -
4987.17 2683.87 5001.56
Quasi-static
Crank Coupler Follower
1.010 0.800 0.414
0.080 _-
4993.11 2014.37 4999.97
Method
isms. The first procedure is the improvement of Thornton’s design process [4], which uses eqns (16)-(21) to find (A,),,, and uses the optimal conditions suggested above to find the active (A,), + , . The second procedure is to use interpolation technique as shown in Fig. 1. Briefly, ‘Data 1’ and ‘Data 2’ in Fig. 1 are two runs from the finite element analysis program, and are employed to find ‘Data 3’ using linear interpolation. These three sets of data points are then used to find the optimal design (point F), with either X, or rrai active.
‘Quasi-static method’ refers to the dynamic solution ignoring the effect of mass inertia, and ‘full dynamic method’ refers to the complete dynamic solution. The design results are shown in Table 1. Only six iterations are needed for the full dynamic analysis and five iterations are needed for the quasi-static analysis. It is found that both the stress and deflection constraints can be satisfied. (B) Another four-bar mechanism is similar to the previous example, except that the allowable deflection at the midpoint of the follower link equals 0.02 in, and the link specifications are as follows:
CASE STUDIES
(A) The design variables of a four-bar mechanism with rectangular cross-sectional shape (constant width equals 1 in) are the thicknesses of the individual link. Assuming aluminum has been chosen as the material with modulus of elasticity = 1.03 x 10’ psi, mass density = 0.1 lbf/in’. The allowable stress for each link = 5OOOpsi, and the allowable deflection at the midpoint of the coupler link = 0.08 in. The crank speed is 3 1.4 rad/sec, and the link specifications of the mechanism are as follows: Length Length Length Length
of of of of
OA AB BC OC
= = = =
12.0 in 38.0 in 38.0 in 48.0 in.
Both the quasi-static and full dynamic methods are used for the design with the first design procedure.
Length Length Length Length
of of of of
OA AB BC QC
= = = =
12.0 in 36.0 in 36.0 in 48.0 in.
This problem cannot easily be solved by using the first design procedure, although the relaxation coefficients have been reduced quite close to zero. The second design procedure was then used to solve this problem. Both the full dynamic and the quasi-static methods are used for the design. The design history for full dynamic analysis is shown in Table 2 and quasi-static analysis is shown in Table 3. Twenty-six iterations are needed by using full dynamic method and 16 iterations are needed for the quasi-static method. It can be found that both the stress and deflection constraints are satisfied.
Table 2. Design history for Case (B) (full dynamic response) Specified stress @si)
Designed thickness (in)
Max. stress (psi)
Max. disp. in specified point (in)
Iteration No.
Step
Link
1
2 3 4
5000 5000 5000
1.115 0.554 0.579
4983.94 4945.08 4993.54
0.241
6
2
2 3 4
5000 5000 415.0
3.184 0.700 9.215
4965.26 4838.09 403.29
0.009
6
3
2 3 4
5000 5000 621.24
2.588 0.676 5.626
4996.61 4751.10 614.04
0%
2 3 4
5000 5000 1157.52
2.000 0.702 2.810
4917.52 4821.06 1168.35
0.019
4
-
6
8
F. W. Lieu and J. D. Ltu
970
Table 3. Design history for Case (B) (quasi-static response) Specified stress @si)
Designed thickness (in)
Max. stress (Psi)
Max. disp. in specified point (in)
Iteration No.
Step
Link
1
2 3 4
5000 5000 5000
0.650 0.294 0.359
5150.75 5~.~ 5000.00
0.366
2
2 3 4
5000 5000 300.0
1.893 0.302 6.495
5002.15 5000.04 300,oo
0.013
5
3
2 3 4
5090 5000 401.86
1.629 0.299 4.735
5043.07 4999.69 401.86
0.015
4
4
2 3 4
5000 5000 615.20
1.311 0.297 3.024
5210.67 5000.75 615.46
0.020
3
In both of the examples, the initial values of the cross-sectional areas A, for iteration are equal to unity. DISCU~ION
This parametric study of elastic mechanisms is conducted based on the following assumptions: 1. w/w, is small (co.05 [8]), where o is the crank input speed, and CD,is the natural frequency of the system component being studied. While most of the ~alisti~lly proportioned m~hanisms are quite stiff, this assumption can be valid for most of the practical cases. 2. Links are treated with classical beam theory, and the axial degree-of-freedom is neglected. From the previous research [9], the assumption of neglecting the axial degree-of-f~dom is valid for a welldesigned beam with small deformation. 3. The first mode of the response dominates the vibration. If a mechanism is made up of straight links, then the assumption is generally valid [S]. 4. No external load is applied to the system. Based on these assumptions, a group of nondimensional parameters such as those in eqns (ll), (12) and (14) can be discovered. These equations can relate the general characteristics of the flexible mechanisms, such as strain, displacement and natural frequency, as a function of input speed, material property, and the geometry of the mechanism. This can provide the inexperienced designer with useful guidelines as to how the designs can be improved. Further research can be conducted to release these assumptions. Although these equations are derived based on certain assumptions, and therefore the direct use of these equations in design may not be appropriate, this paper has shown the usefulness of these by rearranging them into an iterative form as in eqns (16)-(18). The examples have shown that they are valid to help to find the optimal solution of a mechanism because the dimensional analysis of these equations is correct.
4
Although this paper only shows the result of treating the thickness as a design parameter, other parameters such as the material properties can be used for design consideration. The optim~ity conditions to determine the active constr~nt(s) are also derived in this paper when both the stress and deflection constraints are considered. Two design procedures are suggested. The first procedure is the improved Thornton’s design procedure where the interating equation is much simphtied and the relaxation factor can be automati~lly updated. In the second design procedure, without using complicated procedure to compromise stress and deflection constraints, interpolation technique can easily be used to obtain the optimal design. In most of the cases, the first design procedure can be used to solve the problem quite efficiently. The second design procedure consumes more CPU time yet is more stable. Therefore, it can be used to solve those problems that cannot be solved with the first design procedure, Acknowledgements-The authors would like to express their appreciation for the financial support of Weldon Spring Endo~ent Fund (grant number R-3-42024), and the Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla. REFERENCES
1. 1. Imam, G. N. Sandor and S. N. Kramer, D&k&ion and strain analysis in high-speed planar mechanisms with elastic links. ASME J. &gng Indzatry 95B, 541-548 (1973). 2. I. Imam and G. N. Sandor, High-speed mechanism design-a general analytical approach. ASME J. Engng Industry 97, 609-628 (1975). 3. M. R. Khan, W. A. Thornton and K. D. Willmert, Optimality criterion techniques applied to rn~h~~ design. ASME J. Mech. Design 100, 319-327 (1978). 4. W. A. Thornton, K. D. Willmert and M. R. Khan, Mechanism optimization via optimality criterion techniques. ASME J. Mech. Design 101, 392-391 (1979). 5. W. L. Cleghorn, R. G. Fenton and B. Tabarrok, Optimum design of high-speed flexible mechan-
Optimal design of flexible mechanisms isms. Mechanism and Machine Theory 16, 399-406 (1981). Ce. Zhang and H. T. Grandin, Optimum design of flexible mechanisms. ASME J. Mech., Trans, and Automation in Design 105, 267-272 (1983). A. Kishore and M. Keefe, Synthesis of an elastic mechanism. Mechanism and Machine Theory 23, 305-312 (1988). J. R. Sanders and D. Tesar, The analytical and experimental evaluation of vibratory oscillations in realistically proportioned mechanisms. J. Mech. Design 100, 762-768 (1978).
9. W. L. Claghom
971
and C. J. Konze~an, Comparative analysis of finite element types used in flexible mechanism models. Proceedings of the 8th OSU Applied Mechanisms Conference, pp. 30-l-30-7 (1983). 10. F. W. Liou, A. G. Erdman and K. Stelson, General design rules for high-speed, flexible mechanisms. 2&/r Biennial Mechanisms Conference, 1988, Kissimmee, FL (1988). 11. D. J. Liu, Optimal design rules applied to the design of high-speed mechanisms under deflection and stress constraints. Master’s thesis, University of Missouri-Rolla (1989).
OPTIMAL
004s7949p2ss.lm + 0.00
0 I!?92 Fqamonpress Ltd
DESIGN OF FLEXIBLE MECHANISMS A PARAMETRIC APPROACH
USING
F. W. LIOU and J. D. LIU Department of Mechanical and Aerospace Engineering, and Engineering Mechanics, Intelligent Systems Center, University of Missouri, Rolla, MI 65401-0249, U.S.A. (Received 19 September 1991) Abstract-In this paper, an effort to develop a more generic design procedure for flexible mechanisms is initiated. Based on a parametric study, the important parameters which a!kct the stress, displacement, and natural frequency can be closely related. An approach using these relations to obtain the optimal design of flexible mechanisms subject to stress, displacement and natural frequency constraints is presented. Optimality conditions are derived to moderate the conflicting constraints, and a recursion approach is developed to show the application of this study. Two examples using this approach to design flexible mechanisms are presented to demonstrate the idea.
INTRODUCTION
BACKGROUND
synthesis of flexible mechanisms can be treated as a nonlinear programming problem [l, 21, and can be solved using the optimality criterion technique [3,4]. Cleghom et al. [5] presented a new procedure in which the finite difference method was used between cross-sectional diameter and maximum stress in each link to derive a Jacobian matrix. This approach can account for the force effect from changing the inertia (cross-sectional area) of other links to improve efficiency. Zhang and Grandin [6] improved their formulas to develop a design procedure which uses kinematic refinement to compensate the error due to elastic deformation. Kishore and Ke.efe [7l developed a procedure to synthesize elastic four-bar linkage for four precision-point motion generation. Optimization was preceded by a kinematic resynthesis. The above references indicate that most of the formulation done in this area deals with very limited constraint(s) and design parameter(s). Since the design of a flexible mechanism is a very complex procedure which involves many design parameters such as geometry, material property, and the input speed, and may be subject to several different constraints such as stress, displacement, and natural frequency. A more general and robust design approach is needed in this area. In this paper, an effort to develop a more generic design procedure is initiated. A parametric study of flexible mechanisms is conducted to relate the stress, displacement and natural frequency with some important parameters, such as the geometry and material property of a mechanism. A synthesis approach based on the parametric study of flexible mechanisms is presented, and optimality conditions are derived to negotiate between the constraints. Some numerical examples are used to demonstrate the idea.
Based on the previous studies, the transverse deformation of a link of a general industrial four-bar linkage can be considered as linear and decoupled from the remainder of the deformation phenomena in the system [8]. The normalized steady-state amplitude x, of vibration corresponding to the kth mode and jth harmonic have been developed, and can be written as
The
tx8.3)jk (llk31(F,IFa) - XSI = 1 - ( j2/k4)Q2 ’
(1)
where F, is the jth harmonic of the effective point load of inertia force, F(t), acting at the midpoint of a link; F. is the maximum value of inertia force, F(t) during the cycle; x, is the steady-state amplitude of vibration; X = w/o. = (input crack speed)/(natural frequency of the corresponding link for the first mode); x,, is the maximum state deflection; and x, = F,I&
(2)
where K, is the equivalent spring constant of a link. The corresponding transient amplitude can be written as
(x,,>,k O’lksX’WF,/Fo 1 -=X,1 1 - ( j2/k4)Q2 ’
(3)
The above amplitude can be combined to form the total deformation response as [8]
+ 965
W3)(4/Fa) . (j2/k4~2 sln(ior) ’ 1
_
1
(4)
F. W. LIOUand J. D. LIIJ
966
Since a well-designed mechanism is made up of straight links, the k = 1 mode dominates the response, and can be written as
F./F --““sin(jwt) + 1 - (j#
where rBis the radius of gyration of the cross-section of the link, I is the moment of inertia of the crosssectional area of a link, and E is the Young’s modulus of the link. Similarly, strain can be derived as follows:
1. (5)
If R is small in eqn (5), the steady-state term dominates the response
PARAMETRIC
STUDY
In this parametric study, the correlation between link geometry, material, and maximum stress, displacement, or natural frequency will be studied. From eqns (2) and (6), the following equations can be obtained
x(t)
-=-=
X31
F(t) FO
1
(7)
x(r)=Ix,,=1$ e where 1 is a coefficient relating F(t) dimensional analysis, one can 6nd force of a mechanism is proportional link length, and square of the crank i.e. F, = CMLw =,
O( ) L’y yz
E
w2 ’
(12)
g
o2 _ K, “--=M
48EIIL’ PAL
48EI =48 =pAL4
(13)
Therefore 0,=4J5
J( ) E - ‘B. P L2
(14)
The eqns (1 l), (12), and (14) can be used to relate the general characteristics of the flexible mechanisms, such as strain, displacement, and natural frequency, as a function of input speed, material property, and the geometry of the mechanism. Since certain assumptions are made to obtain these equations, the direct use of these equations may not be appropriate. However, the following sections shows how to apply these equations as recursive formulas in the design of flexible mechanisms.
(8) RECURSIVE
and F.. From a that the inertia to the link mass, angular velocity, (9)
where M is the mass of the link = pAL (p is the mass density), L is the link length of the corresponding link, w is the crank input speed, and C is a coefficient which relates the maximum inertia force and the above parameters in eqn (9). The stiffness of the corresponding link of the first mode can be written as 48EI K e=-. L’
=b!&Cyl !I!
where y is the coordinate axis normal to the link length along the thickness direction with its origin at the midpoint of the cross-section. The natural frequency of the link can be derived as follows:
(6) Equation (6) implies that the inertia force F(r) can be treated as a quasi-static force applied to the link, if n is small (say R < 0.05 [8]), and k = 1 mode dominates the response.
6
FORMULAS
According to the above equations, the following relations could be developed, assuming the radius of gyration of the cross-sectional area, rg, is the only design parameter, and rg is proportional to A2 for a link with symmetric cross-section. The equations for other parameters can be used by following the similar derivation. (A) Stress constraint From eqn (12), the stress equation can be obtained as u = p(C2Y)(a2)(L3/r:). In this section,
Y, w, p, and L are considered
(15) as
(10) constants, and only the absolute value of the maxi-
Therefore, substituting eqns (9) and (10) into (8), the following developments can be obtained CpALZw2 48EI L’ (11)
mum stress, 16, (, which is proportional to l/r:, is to be considered during the design procedure. For mechanisms with symmetric cross-sectional area A, the maximum stress ]a,,1 is proportional to l/A 2. Therefore, the following recursion formula can be derived from eqn (15) (A,),+, = {IQi.maxIv/~oij~v(Ai)u,
(16)
Optimal design of flexible mechanisms
where (Ai),+ 1 = cross-sectional area in the ith member during the v + lth iteration, (A,), = cross-sectional area in the ith member during the vth iteration, lQi,-ie = the maximum absolute value of stress in the ith member during the vth iteration, a,, = specified allowable stress, qy= stress elation factor during the vth iteration.
adjustable relaxation coefbcients [5] using the following iterative equations
tl: = {log(Ai)c,+I -
ttC =
Displacement constraint equations can be obtained in a similar way. Since only the absolute value of the maximum displacement, }X,,I, was considered, and the displacement is proportional to ri from eqn (1 l), the following recursion equation can be obtained
where (A,),+ I = cross-sectional area in the ith member during the v + Ith iteration, (A,), = cross-sectional area in the ith member during the vtb iteration, IX,,,,], = the maximum displacement in the ith member during the o th iteration, Xaj= specified allowable displa~ment, s:= displacement relaxation coefficient during the 0th iteration. (C) Natural frequency constraint Similarly, according to eqn (14), the following reIationship can be obtained
where (A)v+r =cross-sectional area during the v + Ith iteration, (A), = cross-sectional area during the vth iteration, w, = specified natural frequency, (o,,,& = the lowest natural frequency of system during the vth iteration, 9.” = natural frequency relaxation coefficient during the vth iteration. In order to gain a better convergence rate, q. in eqn (16), a: in eqn (17), and qi in eqn (18), can be recalculated in each iteration and become selfCAS 43 : W-K
lOg(Ai),}l{lOS(~i,-)”
-
h&4),+
Note that formula (16) is the same as that Khan and co-workers [3,4] have obtained. (B) D~~la~~rn~ntco~traint
967
1 -
W4L
~/~lOg(~~~
-
ORALS
l”gtxi,max)~
f I>
tzo)
10
Il.
hwuninl*+
(21)
COND~O~S FOR MUL~PLE CONSTRAINTS
In order to determine the active constr~nt(s) among different constraints, the optimality conditions are derived in this section. For simplicity, a mechanism composed of N members with stress and displacement constraints is considered in this derivation. Other cases can be derived using the similar approach, The design goal is to find the crosssectional sizes of each member, characterized by the design variable Yi to minimize the total volume V V = 2 A,& inl
(22)
subject to lQnaxlg @I?0 i=1,2,3 ,...,
N
(23)
where A, is the cross-sectional area in the ith member, Li is the length in the ith member, ]cT~,_Iis the maximum absolute stress in the ith member, cai is the spec%ed allowable stress in the ith member, IX,,, 1is the maximum absolute displacement at thejth point, A’, is the specified allowable displa~m~t at the jth point, N is the number of members (possibly links or finite elements), and K is the number of points with specified deflection. Since variable Yican be the cross-sectional area A,, the diameter of a circular member, or the other similar quantity, the area Aj, can be expressed as a function of the design variabh Yi Aj= BY;,
(25)
where B and b are constants. Considering the mechanism moves periodically, the maximnm stress in the ith member can be written as
F. W. LIOUand J. D. LKJ
968
where lP,,_ 1is the maximum absolute force in the ith member within a cycle. Considering any member i which contains the deflection constraint at pointj, IX;,,, 1is proportional to l/A,, i.e.
Because the stress constraint can be applied to every member of the mechanism under the strength consideration, and the deflection constraint can be applied to certain point(s), two optimality conditions can be discussed by solving eqns (32)-(34). Condition 1
(27)
If any member i does not include the deflection constraint (i.e., only stress constraint gi is active in this member), then the deflection constraint can be ignored in eqns (32)-(34). After solving eqns (32)-(34), it is found that only when Iaj,_l = a, is the optimality condition satisfied.
where D, is a positive constant. After arranging eqns (22)-(27), the following can be obtained V = 5 BY;L,
Condition 2
Ipi1
If any member i contains the deflection constraints at point i, then the following three cases can be discussed: (a) If both g, and g, are active, the optimality condition is when both la,,_ I = alri and IX,, I = &,. (b) If g, is active but g, is inactive, then only when lai,mPxl= a,, is the optimality condition satisfied. (c) If gi is inactive but gj is active, then only when I_X’+.,I= X, is the optimality condition satisfied.
where V is the objective function, g, and gj are the constraint functions. The above functions can be redefined as follows:
These conditions can be used to determine whether the current design has reached the optimum or not. For example, if a design satisfies the stress constraint (i.e., the maximum stress equals the allowable stress in each member of the mechanism), and if all the deflections in the specified points are below the allowable deflections, then this case satisfies the optimality condition (b). The solution obtained under the stress constraint is the optimum solution.
c$ = f BY;Li+d. ,(s-aai) i-1
a4
By;-&!!&___
s__:
I
I 2.
aAi.
SDi -0 BY!+’
(32)
lpimaxl a,_O BY:
m
APPLICATIONS
Based on the above discussion, two design procedures are suggested to design high-speed mechan-
(34)
,Xd,
---------t
I
lull.
I
IULI, Maxlmum
I
(0.1)
I
1ull11,
Stnrr Fig. 1. Numerical interpretation in the critical member for the second design procedure (line A-linear interpolation function, line B-actual function).
Optimal
design of flexible mechanisms
969
Table 1. Designed results for Case (A) Designed
Max. deflection in
Link
thickness (in)
specified point (in)
Maximum stress (Psi)
Full vibration
Crank Coupler Follower
1.718 1.053 0.652
0.081 -
4987.17 2683.87 5001.56
Quasi-static
Crank Coupler Follower
1.010 0.800 0.414
0.080 _-
4993.11 2014.37 4999.97
Method
isms. The first procedure is the improvement of Thornton’s design process [4], which uses eqns (16)-(21) to find (A,),,, and uses the optimal conditions suggested above to find the active (A,), + , . The second procedure is to use interpolation technique as shown in Fig. 1. Briefly, ‘Data 1’ and ‘Data 2’ in Fig. 1 are two runs from the finite element analysis program, and are employed to find ‘Data 3’ using linear interpolation. These three sets of data points are then used to find the optimal design (point F), with either X, or rrai active.
‘Quasi-static method’ refers to the dynamic solution ignoring the effect of mass inertia, and ‘full dynamic method’ refers to the complete dynamic solution. The design results are shown in Table 1. Only six iterations are needed for the full dynamic analysis and five iterations are needed for the quasi-static analysis. It is found that both the stress and deflection constraints can be satisfied. (B) Another four-bar mechanism is similar to the previous example, except that the allowable deflection at the midpoint of the follower link equals 0.02 in, and the link specifications are as follows:
CASE STUDIES
(A) The design variables of a four-bar mechanism with rectangular cross-sectional shape (constant width equals 1 in) are the thicknesses of the individual link. Assuming aluminum has been chosen as the material with modulus of elasticity = 1.03 x 10’ psi, mass density = 0.1 lbf/in’. The allowable stress for each link = 5OOOpsi, and the allowable deflection at the midpoint of the coupler link = 0.08 in. The crank speed is 3 1.4 rad/sec, and the link specifications of the mechanism are as follows: Length Length Length Length
of of of of
OA AB BC OC
= = = =
12.0 in 38.0 in 38.0 in 48.0 in.
Both the quasi-static and full dynamic methods are used for the design with the first design procedure.
Length Length Length Length
of of of of
OA AB BC QC
= = = =
12.0 in 36.0 in 36.0 in 48.0 in.
This problem cannot easily be solved by using the first design procedure, although the relaxation coefficients have been reduced quite close to zero. The second design procedure was then used to solve this problem. Both the full dynamic and the quasi-static methods are used for the design. The design history for full dynamic analysis is shown in Table 2 and quasi-static analysis is shown in Table 3. Twenty-six iterations are needed by using full dynamic method and 16 iterations are needed for the quasi-static method. It can be found that both the stress and deflection constraints are satisfied.
Table 2. Design history for Case (B) (full dynamic response) Specified stress @si)
Designed thickness (in)
Max. stress (psi)
Max. disp. in specified point (in)
Iteration No.
Step
Link
1
2 3 4
5000 5000 5000
1.115 0.554 0.579
4983.94 4945.08 4993.54
0.241
6
2
2 3 4
5000 5000 415.0
3.184 0.700 9.215
4965.26 4838.09 403.29
0.009
6
3
2 3 4
5000 5000 621.24
2.588 0.676 5.626
4996.61 4751.10 614.04
0%
2 3 4
5000 5000 1157.52
2.000 0.702 2.810
4917.52 4821.06 1168.35
0.019
4
-
6
8
F. W. Lieu and J. D. Ltu
970
Table 3. Design history for Case (B) (quasi-static response) Specified stress @si)
Designed thickness (in)
Max. stress (Psi)
Max. disp. in specified point (in)
Iteration No.
Step
Link
1
2 3 4
5000 5000 5000
0.650 0.294 0.359
5150.75 5~.~ 5000.00
0.366
2
2 3 4
5000 5000 300.0
1.893 0.302 6.495
5002.15 5000.04 300,oo
0.013
5
3
2 3 4
5090 5000 401.86
1.629 0.299 4.735
5043.07 4999.69 401.86
0.015
4
4
2 3 4
5000 5000 615.20
1.311 0.297 3.024
5210.67 5000.75 615.46
0.020
3
In both of the examples, the initial values of the cross-sectional areas A, for iteration are equal to unity. DISCU~ION
This parametric study of elastic mechanisms is conducted based on the following assumptions: 1. w/w, is small (co.05 [8]), where o is the crank input speed, and CD,is the natural frequency of the system component being studied. While most of the ~alisti~lly proportioned m~hanisms are quite stiff, this assumption can be valid for most of the practical cases. 2. Links are treated with classical beam theory, and the axial degree-of-freedom is neglected. From the previous research [9], the assumption of neglecting the axial degree-of-f~dom is valid for a welldesigned beam with small deformation. 3. The first mode of the response dominates the vibration. If a mechanism is made up of straight links, then the assumption is generally valid [S]. 4. No external load is applied to the system. Based on these assumptions, a group of nondimensional parameters such as those in eqns (ll), (12) and (14) can be discovered. These equations can relate the general characteristics of the flexible mechanisms, such as strain, displacement and natural frequency, as a function of input speed, material property, and the geometry of the mechanism. This can provide the inexperienced designer with useful guidelines as to how the designs can be improved. Further research can be conducted to release these assumptions. Although these equations are derived based on certain assumptions, and therefore the direct use of these equations in design may not be appropriate, this paper has shown the usefulness of these by rearranging them into an iterative form as in eqns (16)-(18). The examples have shown that they are valid to help to find the optimal solution of a mechanism because the dimensional analysis of these equations is correct.
4
Although this paper only shows the result of treating the thickness as a design parameter, other parameters such as the material properties can be used for design consideration. The optim~ity conditions to determine the active constr~nt(s) are also derived in this paper when both the stress and deflection constraints are considered. Two design procedures are suggested. The first procedure is the improved Thornton’s design procedure where the interating equation is much simphtied and the relaxation factor can be automati~lly updated. In the second design procedure, without using complicated procedure to compromise stress and deflection constraints, interpolation technique can easily be used to obtain the optimal design. In most of the cases, the first design procedure can be used to solve the problem quite efficiently. The second design procedure consumes more CPU time yet is more stable. Therefore, it can be used to solve those problems that cannot be solved with the first design procedure, Acknowledgements-The authors would like to express their appreciation for the financial support of Weldon Spring Endo~ent Fund (grant number R-3-42024), and the Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla. REFERENCES
1. 1. Imam, G. N. Sandor and S. N. Kramer, D&k&ion and strain analysis in high-speed planar mechanisms with elastic links. ASME J. &gng Indzatry 95B, 541-548 (1973). 2. I. Imam and G. N. Sandor, High-speed mechanism design-a general analytical approach. ASME J. Engng Industry 97, 609-628 (1975). 3. M. R. Khan, W. A. Thornton and K. D. Willmert, Optimality criterion techniques applied to rn~h~~ design. ASME J. Mech. Design 100, 319-327 (1978). 4. W. A. Thornton, K. D. Willmert and M. R. Khan, Mechanism optimization via optimality criterion techniques. ASME J. Mech. Design 101, 392-391 (1979). 5. W. L. Cleghorn, R. G. Fenton and B. Tabarrok, Optimum design of high-speed flexible mechan-
Optimal design of flexible mechanisms isms. Mechanism and Machine Theory 16, 399-406 (1981). Ce. Zhang and H. T. Grandin, Optimum design of flexible mechanisms. ASME J. Mech., Trans, and Automation in Design 105, 267-272 (1983). A. Kishore and M. Keefe, Synthesis of an elastic mechanism. Mechanism and Machine Theory 23, 305-312 (1988). J. R. Sanders and D. Tesar, The analytical and experimental evaluation of vibratory oscillations in realistically proportioned mechanisms. J. Mech. Design 100, 762-768 (1978).
9. W. L. Claghom
971
and C. J. Konze~an, Comparative analysis of finite element types used in flexible mechanism models. Proceedings of the 8th OSU Applied Mechanisms Conference, pp. 30-l-30-7 (1983). 10. F. W. Liou, A. G. Erdman and K. Stelson, General design rules for high-speed, flexible mechanisms. 2&/r Biennial Mechanisms Conference, 1988, Kissimmee, FL (1988). 11. D. J. Liu, Optimal design rules applied to the design of high-speed mechanisms under deflection and stress constraints. Master’s thesis, University of Missouri-Rolla (1989).