B. A.

Salamon
An Introduction to Mechanical
Dow Chemical Company
Midland, Ml 48667 Advantage in Compliant
A. Midha
Mechanisms
Department of Mechanical & Aerospace
Engineering & Engineering Mechanics An energy approach is utilized to determine mechanical advantage in compliant
University of Missourl-Roila mechanisms by duly accounting for lost work due to deformation. Three mechanical
Rolia, MO 65409 advantage types are then defined which examine the isolated influences of various
parameters. Finally, a case study is investigated to exemplify these definitions and
demonstrate resulting trends in mechanical advantage.

Introduction For any structural system, the total energy (11) of the system
in any given state can be expressed by the following relation-
The mechanical advantage of single-input and single-output ship:
port, rigid-link mechanisms is well understood and readily eval-
uated. There are numerous references, e.g., Shigley and Uicker n =u+ V (2)
(1980) and Erdman and Sandor (1991), which discuss the me-
chanical advantage of conventional single-input and single-out- where U is the strain energy of the system and V the potential
put port mechanisms. Midha et al. (1984) presented a discussion energy with respect to the zero potential energy reference. The
of mechanical advantage concepts for a more general case of potential energy of the system is also equal to the negative of
single-input and multiple-output port, rigid-link mechanisms. the work {W) done on the system by the external forces. Thus,
More recently, Howell and Midha (1995) considered the effects
of a compliant workpiece on the input and output characteristics y= -w (3)
of rigid-link toggle mechanisms. A more recent treatise on com-
For the system to be in equilibrium, the energy function must
pliant mechanisms may be found in Howell (1993).
assume a stationary value. This occurs when
In general, for rigid-link mechanisms, e.g., the slider-crank 5n = 0 (4)
mechanism shown in Fig. 1, the links are assumed to be infi-
nitely rigid, and if friction and inertia forces are neglected, work Using Eqs. (2), (3), and (4) yields the following expression
(or power) will be conserved between the input and output
ports. The mechanical advantage of rigid-link mechanisms can Q = SU-&W (5)
be shown to be a function of the geometry of the given position Equation (5) states that the differential change in work SW is
of the mechanism. For example, using the instant center method, equal to the differential change in the strain energy 8U. This
the mechanical advantage (MA) of the mechanism in Fig. 1 is equation holds for any incremental change in the system from
given as one equilibrium condition to another nearby equilibrium condi-
tion. Equation (5) is general, and is applicable to any structural
luhA dj_ system, including compliant and rigid-body mechanisms. For
MA = (1)
inhi d„ the degenerate case of a rigid-body mechanism, the differential
strain energy is assumed to be zero, and thus the differential
where //, is the instant center of rotation of link j about link ;, external work is conserved. As stated earlier, this is not true
and di and d^ are the perpendicular distances to the input and for compliant mechanisms.
output forces (F, and F„) from the instant centers /13 and / u ,
respectively. For this single-degree-of-freedom mechanism, it The general force-deflection characteristics of a compliant
is then simple to plot the variation in mechanical advantage mechanism over its total range of operation are nonlinear. For
with position. an incremental change in position, however, the mechanism
force-deflection behavior may be approximated as linear. If
then, for a given state of the mechanism, the input force f, is
Generalized Meclianical Advantage increased by an amount (5F,, the output force F„ will increase
by an amount SF^,. Assuming that these incremental changes
In the case of compliant mechanisms, due to member compli-
in the forces occur linearly with respect to the corresponding
ance, energy is absorbed with deformation, and thus may not
displacements, the incremental work at the input and output
be assumed to be conserved between the input and output ports.
ports, 6Wi and SW^, respectively, are given as
Not only does member deformation lessen the available energy
at the output, it also affects the kinematics by varying effective
6W: = (F, + \6Fi)6di
link lengths. The dependence of mobility on applied forces and
their locations is discussed by Her (1986). Considering all
these factors, to quantify mechanical advantage in compliant 6W, = {Fo + k&F„)6d, (6)
mechanisms is a rather complex procedure. Using the energy
method then, general relations for mechanical advantage of sin- where 6di and bd„ are incremental displacements of the input
gle-input and single-output port mechanisms are developed. and output ports in the directions of the input and output forces,
respectively. Neglecting the higher-order terms in Eq. (6) gives

Contributed by the Design Automation Committee for publication in the JOUR- 6Wi = FMi
NAL OF MECHANICAL DESIGN. Manuscript received Mar. 1998. Revised Apr. 1998.
Associate Technical Editor: David A, Hoeltzel. 8W, = FM„ (7)

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 For a single-input and single-output port compliant mecha- 0 = 6U - FM + FMo (11)
nism, the externally applied input force and a reactive output
force are assumed to be the only forces that do work on the Defining mechanical advantage MA as the instantaneous ratio
system. All other forces are assumed to be reaction forces which of the output force (F^) to the input force (F,), equation (11)
correspond to displacement boundary conditions. The differen- is rearranged to give
tial external work 6W done on the system is then given as
SW = 8W, - 6W„ (8)
MA = ^ = J - U - ^ (12)
F> Sd, V F,
The minus sign associated with the differential work at the
output (6W„) indicates that the mechanism is doing work on a This general relation is vahd for any mechanism, compliant or
workpiece, or the workpiece is doing negative work on the otherwise, provided it has a single-input and a single-output
mechanism. port (Midha et al., 1984). For example, if SU is zero, as for a
Strain energy is usually written in one of two forms. For a rigid-body mechanism, Eq. (12) becomes
system with a finite number of discrete compliances, strain en-
ergy (U) takes the form
MA = ^ = ^ (13)
Fi 6d„
U=Y.\kx^ (9)
Eq. (12) may be used to develop an insight into the mechanical
where k represents the stiffness, or the reciprocal of the value advantage characteristics of compliant mechanisms.
of the discrete compliance, and x the amount of deformation Consider the following rearrangement:
associated with the given compliance. Equation (9) cannot de-
scribe the strain energy for a compliant mechanism, however, 6U
since the corresponding compliance is distributed rather than MA = ^ = MA, - MA, (14)
6d„ 6d,Fi
discrete.
It is possible, when the compliance distribution is known, to The first term in Eq. (14) takes the form of a rigid-body me-
represent the strain energy as an integral of the distributed strain chanical advantage. This term would result if an instant center
energy of the internal forces over the entire continuum. How- analysis for mechanical advantage (Shigley and Uicker, 1980)
ever, to do so requires the equilibrium geometry of the contin- could be applied to the compliant mechanism in any instanta-
uum to be known. For the case of small deflections, the final neous position. It would be a function of several parameters
equilibrium geometry is approximated by the original unde- including those defining the original mechanism geometry as
formed geometry. For a compliant mechanism which may expe- well as the externally apphed loads. The effective link lengths
rience large deflections, the final equilibrium geometry is not thus change with the load, and the "rigid-body" mechanical
known, and thus this method of denoting strain energy lacks advantage of the compliant mechanism (MA,) cannot be repre-
applicability. sented by a single rigid-body counterpart for the entire range
In general, the strain energy can be considered as the summa- of operation of the compliant mechanism.
tion of the individual strain energies of a finite number of seg- The second term in Eq. (14) also resembles a mechanical
ments which idealize the continuum of the compliant mecha- advantage term. It is referred to as the compliant component of
nism. Incremental strain energy may also be represented as the mechanical advantage ( M A J , and it accounts for the energy
stored in the mechanism. The single-input and single-output
8U = X 6U, (10) port compliant mechanism may be considered to have two out-
put ports, the actual physical output port and an internal port
which performs work by elastically deforming the mechanism
where 6Ui is the incremental change in the strain energy of the members. The mechanical advantage is thus maximized at a
(•"' segment, and N the total number of segments representing given instant when the compliant component of mechanical
the mechanism. Each SUi can be considered as resulting from advantage (MA,-) becomes zero. When this occurs, the compli-
either a discrete or distributed compliance. ant mechanism behaves identically as a representative rigid-
Combining the results in Eqs. (5), (7) and (8) gives the body mechanism.
relation
Another useful form of Eq. (12) is given as

6U
MA = ^ 1 = MA, 1 - ^ (15)
&d„ 6d,Fi F,

where F, is the compliant component of the input force (compli-

ance force), or that part of the input force which is needed just
to deform the mechanism members, Thus, the actual mechanical
advantage is some fraction of the rigid-body mechanical advan-
tage (MAr) associated with a given mechanism position. Again,
if the work of elastic deformation is minimized, the mechanical
advantage is maximized.
Note that when 6d„ is zero, Eq. (12) is still valid but the
mechanical advantage is not necessarily infinite since it is also
true in this instance that

..,.-^ = 0 (16)
Fi

This is obtained by letting the last term in Eq. (11) be zero.

Equation (16) can also be expressed as a mechanical advantage
Fig. 1 Rigid-body slider cranl< mechanism by introducing F„ and rean-anging to give

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 F„6d; Table 1 Geometric properties of the compliant crimping meciianism in
MA = (17) Fig. 2
f,- 6U

For this case, since the output displacement is fixed, i.e., 6d„ = i Xi (in) yi (in) li (in*)
0, the output force may be considered to be a reaction force
and is a nonlinear function of the input force.
1 0.000 0.080 1.350 X 10"^
Defining Mechanical Advantage Types
2 0.000 0.160 1.350 X 10-^
Because mechanical advantage of a rigid-body mechanism is
3 0.000 0.260 7.813 X 10"^
a function of the linkage position only, a plot of its variation 4 -1.000 0.400 6.250 X 10-2
over the mobility range of the mechanism is readily constructed. 5 1.340 1.100 6.250 X 10-2
As stated earlier, the mobility of a compliant mechanism is also
a function of the applied forces. It would therefore be not possi- 6 1.510 1.160 5.788 X 10-^
ble to construct one single, two-dimensional plot describing the 7 1.660 1.210 1.725 X 10"^
variation of mechanical advantage for a compliant mechanism.
Three mechanical advantage types are defined herein, which in
8 1.780 1.250 4.556 X 10-^
turn also help alleviate this problem. 9 1.920 1.270 2.637 X 10-^
These definitions are based on the assumption that there is 10 2.080 1.285 2.637 X 10"^
only one input force, and that no applied loads other than the
input force are changing. All forces that change as a result of
11 2.240 1.280 6.250 X 10-^
changes in the input force are considered as reaction forces 12 2.710 1.000 6.250 X 10-2
(including the output force) which correspond to given dis- 13 5.000 1.570 6.250 X 10-2
placement boundary conditions. Only one of these reaction
forces is treated as the output force. Thus, the following defini- 14 3.200 0.150 6.250 X 10-2
tions of the mechanical advantage types (Types 1, 2 and 3)
are forwarded for single-input and single-output port compliant
mechanisms. Type 2 (or output-port-displacement-dependent) mechanical
Type 1 (or input-force-dependent) mechanical advantage is advantage is measured when the input force is held constant.
measured by fixing the output port displacement at a given The output force then varies as a function of the output port
constant value. The output force then varies with the input force. displacement.
Type 3 mechanical advantage is a result of an interaction
between the mechanism and the workpiece. It may appropriately
be termed as workpiece-dependent mechanical advantage. For
this type, the input force is determined based on the require-
ments at the output port. These are requirements of both force
and displacement and result from the force-displacement char-
acteristics of the workpiece.
Types 1 and 2 mechanical advantages are more easily con-
structed and give more direct insight to mechanical advantage
of compliant mechanisms than does Type 3. Type 3 mechanical
advantage, however, is expected to be the most useful and prev-
alent of the three types in evaluating the overall performance
of a compliant mechanism.

(a) A Compliant Mechanism Case Study

Fig. 2 (a) A compliant crimping mechanism To illustrate the definitions in the previous section, various
mechanical advantage plots for a compliant mechanism are pre-
sented. The specific mechanism considered (Midha 1983) is
shown in Fig. 2a. Due to its symmetry, only one-half of the
mechanism is analyzed. Figures 2b and 2 c show the nodal
distribution of a simply discretized model used for this example.
The corresponding geometric properties are listed in Table 1.
The flexural modulus of elasticity is 0.9 X 10' psi, and the chain
algorithm with a shooting method is employed, as a method of
large-deflection analysis described in Her (1986), using 10 load
Fig. 2 (/)) Discretized half-model increments.
For this mechanism, the input port is at node 13 and the
13 output port at node 4. The input force acts in the negative y-
8 10
direction and the output force in the positive })-direction. Node
14 is attached to a slider (Fig. 2c) which does not permit a.y-
direction displacement.
Type 1 mechanical advantage curves for this mechanism are
shown in Fig. 3a. Each curve corresponds to an output port
displacement d„ between 0.00 in. and 0.13 in., in increments of
0.01 in. The input force (F,) is varied between 1 and 20 lb in
1-lb steps.
(C)
For the range of the data shown, the Type 1 curves are
Fig. 2 (c) Finite element model bounded above (Fig. 3a) by the curve corresponding to zero

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 a function of two variables, i.e., the input force ( F , ) and the
output port displacement {d„), it is appropriate to construct a
three-dimensional surface plot of the mechanical advantage for
this mechanism. This plot is shown in Fig. Ab, and it fully
describes the mechanical advantage characteristics of this mech-
Mechanical
anism. Note that the Type 1 and Type 2 mechanical advantage
Advantage,
curves are the intersection of this surface and planes parallel to
MA
the M A — Fj and M A — do planes, respectively. The F^ versus
do curve discussed earlier and shown in Fig. 3i> is found as the
intersection of the mechanical advantage surface with the F, -
do plane.
The Type 3 mechanical advantage curves are plotted in the
6 9 12 15 18 M A — Fj plane in Fig. 5a. The curves shown assume a work-
(a) Input Force, F; (lb) piece having a linear force-deflection relation. Each curve corre-
sponds to a different stiffness value. Points on these curves are
Fig. 3 (a) Type 1 mechanical advantage plot
determined numerically by applying the output force to the
18- mechanism, and then finding the input force that will provide
the corresponding output port displacement as per the force-
deflection behavior of the workpiece.
For the mechanism under consideration, the mechanical ad-
Compliance vantage increases slightly (Fig. 5 a ) with the input force for a
Force, constant stiffness workpiece. The performance of this mecha-
Fc (lb) nism increases with increased workpiece stiffness as shown in
Fig. 5b. Assuming the constant stiffness curves to have constant
M A , this relation may be shown to take the general form

M A = MA, (18)
0.05 .1 15 a + K
(b) Output Port Displacement, dg (in)
where MA^ is the bounding mechanical advantage as discussed
Fig. 3 (b) Compliance force (/%,) variation with output port displace- with regard to Eq. ( 1 7 ) , and a is the sensitivity index which
ment {do)

output port displacement. The value of the mechanical advan- 3-|

tage for this curve is nearly constant at 2.55. All Type 1 curves
Input Force,
shown in Fig. 3fl may be approximated as
2- Fi (lbs)

MA = MA, 1 (17) Mechanical

F, Advantage, 1-
MA 5:20
where MA, is the mechanical advantage associated with the
0-
bounding curve, and it corresponds to the rigid-body mechanical 1 3 5 7 9 11 13 15 •"17
advantage (MA^) of the initial mechanism position. F^ is the
input force required to displace the output port a distance do •1- 1 1 1 I—r —1— 1 1 1 1 1 1 1 1
without generating an output force; contact is then made with
the workpiece. In other words, this is the input force required
(a) Output Port Displacement, dg (in)
to overcome compliance in moving the mechanism to a given
position, and it is therefore called the compliance force. Having Fig. 4 (a) Type 2 mechanical advantage plot
reached this position of constant output port displacement {do),
further increasing the input force will yield useful output and
the mechanical advantage increases. Equation (17) takes the
same form as Eq. (15).
The value of F^ for a given Type 1 curve is easily obtained.
It is the input force value that corresponds to an output port
displacement do and zero mechanical advantage. Figure 3/?
shows the variation of Fc with do for the mechanism under
consideration. This curve illustrates the input force versus out-
put port deflection characteristic of the mechanism when there
is no output force present. The area under this curve represents MA
the energy stored in the mechanism.
The Type 2 mechanical advantage curves are shown in Fig.
4a. In this figure, each curve corresponds to a constant value
of input force. Each of these curves is nearly linear. They show
that as output port displacement {do) increases, more energy is
stored in the mechanism and less force is available at the output.
This is evidenced by the decreasing mechanical advantage Fj (lb) ,;-
(MA). do (10-2 in)
(b) 2 0 ^ 13
Because the mechanical advantage of single-input and single-
output port compliant mechanisms can be suitably described as Fig. 4 (b) IVIechanical advantage surface plot

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 defines the sensitivity of mechanism performance to the stiff-
ness of the workpiece. The sensitivity index ( a ) is minimized
by minimizing the energy stored in the mechanism. This results
in a mechanism with performance having little dependence upon
the stiffness of the workpiece. This, of course, becomes an
important parameter in the design of compliant mechanisms. Mechanical
Another useful plot, shown in Fig. 6, depicts constant output
Advantage,
force curves in the MA - do plane. The ease with which the MA
force-deflection properties of the workpiece are coordinated
with this plot leads to its utility. These curves can be thought of
as a transformed coordinate grid on which the force-deflection
relation of any workpiece may be plotted. Thus, by having this -1—I—I—r ~\ I I i I I I I r
type of a plot for a mechanism, it is simple to manually construct .00 .03 .06 .09 .12
a Type 3 curve, corresponding to a given workpiece stiffness, Output Port Displacement, do (in)
rather than determining it numerically.
Fig. 6 Output port cfiaractsristics

Conclusions
Generalized equations for mechanical advantage in compliant Mechanical advantage types. Types 1, 2 and 3, have been
mechanisms, which duly account for energy stored with mecha- defined which address the typically encountered boundary con-
nism deformation, have been derived. Also forwarded are the ditions of force and displacement. These also aid in simplifying
concepts of the rigid-body and compliant components of me- the understanding of the mechanical advantage property in com-
chanical advantage, and the idea of compliance force. Mechani- pliant mechanisms. A case study has been presented to exem-
cal advantage has been shown to be maximized as the elastic plify these definitions. Type 1 mechanical advantage curves are
deformation is minimized. found to be of a form similar to the generalized mechanical
advantage equation. An elucidative mechanical advantage sur-
face plot has been introduced that incorporates the behavior of
mechanical advantage Types 1 and 2. When acting on a compli-
Workpiece Stiffoess, ky, (lb/in) ant workpiece, the compliant mechanism examined has been
shown to maintain a nearly constant mechanical advantage over
2000 the range of input force considered. In addition, the concept of
2- 1000 a sensitivity index in a compliant mechanism has been intro-
Mechanical duced to show the reliance of its mechanical advantage on the
Advantage, workpiece stiffness.
MA
1- •200
Acknowledgments
• 100
The authors would like to acknowledge the support of the
School of Mechanical Engineering and the use of its facilities.
I I I I I I I I I I I I I 1 I I I I I I I I I
3 6 9 12 15 18 21 24 The support of National Science Foundation for this research,
(a)
through NSF Grant No. MSS-8902777, is gratefully acknowl-
Input Force, Fj (lb)
edged.
Fig. 5 (a) Type 3 mechanical advantage piot

3-
References
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Vol. 1, Second Edition, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1991.
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Purdue University, 1986.
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Advantage, 1993.
MA Howell, L. L., and Midha, A., "The Effects of a Compliant Workpiece on the
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1000 2000 3000 4000 of Single-Input and Multiple-Output Ports Mechanical Device," ASME Journal
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Fig, 5 (b) IVIeclianical advantage variation witli worl<piece stiffness McGraw-Hill Book Company, New York, New York, 1980.

Journal of Mechanical Design JUNE 1998, Vol. 120 / 315

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