Design of Compliant Mechanisms
for Minimizing Input Power
in Dynamic Applications
Tanakorn Tantanawat
e-mail: tanakorn@umich.edu
Sridhar Kota
e-mail: kota@umich.edu
Department of Mechanical Engineering,
The University of Michigan,
Ann Arbor, MI 48109
In this paper, we investigate power flow in compliant mechanisms that are employed in
dynamic applications. More specifically, we identify various elements of the energy storage and transfer between the input, external load, and strain energy stored within the
compliant transmission. The goal is to design compliant mechanisms for dynamic applications by exploiting the inherent energy storage capability of compliant mechanisms in
the most effective manner. We present a detailed case study on a flapping mechanism, in
which we compare the peak input power requirement in a rigid-body mechanism with
attached springs versus a distributed compliant mechanism. Through this case study, we
present two approaches: (1) generative-load exploitation and (2) reactance cancellation,
to describe the role of stored elastic energy in reducing the peak input power requirement. We propose a compliant flapping mechanism and its evaluation using nonlinear
transient analysis. The input power needed to drive the proposed compliant flapping
mechanism is found to be 50% less than a rigid-link four-bar flapping mechanism without
a spring, and 15% less than the one with a spring. This reduction of peak input power is
primarily due to the exploitation of elasticity in compliant members. The results show
that a compliant mechanism can be a better alternative to a rigid-body mechanism with
attached springs. 关DOI: 10.1115/1.2756086兴
Keywords: compliant mechanisms, flapping mechanisms, flapping-wing air vehicles, input power reduction
Introduction
ant mechanisms may be efficient when they are operated under a
quasi-static condition, but not a dynamic one. Analysis and synthesis tools based on a dynamic condition are, therefore, necessary
to extend the applications of compliant mechanisms. While the
purpose of this ongoing research is to develop a systematic
method of designing compliant mechanisms based on dynamic
performance, this paper presents the results, as a preliminary step,
of an attempt to understand the benefits of elasticity in dynamic
systems. This understanding will be fundamental for the development of an efficient and robust design tool during the course of the
research.
Due to its multifunctional structure that combines the function
of a mechanism and an energy storage component together, a
compliant mechanism provides a lightweight and compact system.
Its monolithic structure leads to scalability, no wear, and low
manufacturing cost. In addition, its ability to incorporate unconventional actuators makes it very attractive to several applications.
Nonetheless, if a distributed compliant mechanism is used, further
benefit can be gained. Since stress and strain are more uniformly
distributed than those in lumped compliant mechanisms, there is
less stress concentration but higher energy storage capacity in
distributed compliant mechanisms. Because of all these benefits, a
compliant mechanism is expected to be an excellent alternative to
a rigid-body mechanism, especially for autonomous robots, whose
size and weight are critical. In this study, a compliant flapping
mechanism, which can be applied to flapping-wing micro air vehicles developed by other researchers, will be used as a case study
during the development of the design method.
When a compliant mechanism is deformed to transmit force
and motion, some of the input energy is stored in the mechanism
in the form of strain energy, while the rest is transferred to the
output load. This stored energy is commonly perceived as energy
loss, preventing full energy transfer from the input port to the
output port. One of the approaches to design efficient compliant
mechanisms for a quasi-static condition, therefore, attempts to
minimize this stored strain energy. The scenario is completely
different if a compliant mechanism is operated under a dynamic
condition, in which strain energy is stored and released during a
cycle of operation. Most of the stored strain energy, if not all, can
be recycled. An approach used to design a compliant mechanism
for a dynamic condition should, therefore, be different from the
one used for a quasi-static condition. Khatait et al. 关1兴 suggest the
use of compliant mechanisms to reduce input torque requirement
in a flapping mechanism. The study is, however, limited to the
case of unloaded lumped compliant mechanisms. In addition,
Madangopal et al. 关2兴 optimize a four-bar flapping mechanism
with attached springs in an attempt to minimize the input torque
requirement of a flapping-wing micro air vehicle. However, how
the springs enable input torque reduction has not been clearly
explained. These two examples have demonstrated the benefits of
elasticity in reducing input torque requirement, and lead us to
believe that a distributed compliant mechanism holds a potential
to be an alternative for dynamic systems.
Since most of the early research and development in analysis
and synthesis tools for compliant mechanisms were based on a
quasi-static assumption, the performance of the designed compli-
Related Work
Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 13, 2006; final
manuscript received October 25, 2006. Review conducted by Larry L. Howell. Paper
presented at the ASME 2006 Design Engineering Technical Conferences and Computers and Information in Engineering Conference 共DETC2006兲, Philadelphia, PA,
September 10–13, 2006.
The use of compliant mechanisms to transmit force and motion
has been studied for several decades. This type of mechanism
evolved from work in robotics, where manipulator arms must perform various tasks on workpieces whose locations and dimensions
are not precisely known 关3兴. In this application, the use of compliance is one of the approaches to accommodate such uncertain-
1064 / Vol. 129, OCTOBER 2007
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ties. Work in the development of compliant mechanisms for robotics applications can be found at least as early as in the 1980s
关3–6兴. In 1983, Holl et al. 关7兴 proposed a compliant mechanism
for joints in prosthetic devices. Holl noted the self-stabilizing
characteristics and other benefits of compliance including reduction in wear, material weight, and manufacturing cost. Another
work in compliant joints was reported by Trease et al. 关8兴 in 2005,
where several designs of highly effective and kinematically wellbehaved compliant joints were proposed. Since compliant mechanisms were studied extensively, various applications of compliant
mechanisms have been investigated, including bistable compliant
mechanisms 关9兴, compliant micromechanisms and structures
关10,11兴, micromanipulators and microrobots 关12兴, actuator leverage 关13,14兴, structural shape morphing in smart structures 关15兴,
surgical tools 关16兴, active flow control 关17兴, and vibration isolation
关18兴.
Various methods of analysis and synthesis of compliant mechanisms can be found in the literature including Pseudo-Rigid-Body
Models 共PRBMs兲 and continuum mechanics methods for topology, geometry, and size optimization. Much of the reported work
on analysis and design assumes a quasi-static condition. It was not
until recently that researchers began to incorporate dynamic behaviors of compliant mechanisms, such as Du et al. 关19兴, Nishiwaki et al. 关20兴, and Maddisetty and Frecker 关21兴. Li and Kota
关22兴 proposed a systematic method for dynamic analysis of compliant mechanisms. Dimensional synthesis based on desired mode
shapes is presented in 关23兴 by Lai and Anathasuresh. Boyle et al.
关24兴, Handley 关25兴 et al., and Yu et al. 关26兴 presented the development in dynamic modeling of compliant mechanisms based on
PRBMs. Khatait et al. 关1兴 used dynamic PRBMs to investigate the
reduction of torque requirement in a flapping mechanism. None of
the previous work, however, directly addressed the issue of input
power requirement. Since it is the power requirement that governs
various component designs and affects the overall system’s performance, we use the power requirement as one of the criteria in
the design method. Even though a design approach based on
torque requirement is equivalent to a design approach based on
power requirement when the operating frequency is constant, it is
not the case when the frequency changes during the operation, or
when the frequency is one of the design variables during a system
design. A design approach based on the power requirement will
still be applicable to these situations.
Power Analysis of a Four-Bar Flapping Mechanism
To understand the behavior of input power reduction, we first
perform power analysis on a four-bar flapping mechanism optimized by Madangopal et al. 关2兴. In their work, the aerodynamic
model is based on quasi-steady blade element analysis. The model
includes unsteady wake effects, camber and partial leading edge
suction effects, and post-stall behavior. The grounds for the
mechanism are assumed inertially fixed. For a given set of parameters, the optimal values of spring stiffness k, free length l0, and
location of attachment a were obtained. We model this mechanism
in ADAMS, which is dynamic mechanical system simulation software. The aerodynamic load Fo共t兲 is simplified using a point force
acting perpendicular to the wing at its center. The model for this
study is shown in Fig. 1.
The input displacement ␾in共t兲 at the operating frequency f can
be written as
␾in共t兲 = 2␲ ft
共2兲
The values of the links’ dimensions and operating frequency are
shown in Table 1.
Journal of Mechanical Design
Note that the links’ cross sections are adjusted so that their
masses match those in Madangopal’s model. The operating frequency is adjusted to match the peak values of inertial torque.
Finally, a damping constant and a constant force component are
adjusted to match the peak value of the lift. Since the weights of
the links are small compared to the aerodynamic force on the
wing, the gravity effect is ignored in this study to simplify the
analysis. Once the basic understanding is gained, more accurate
results can be obtained by including the gravity.
The input power can be either positive or negative during a
flapping cycle. A positive input power indicates that the motor is
supplying energy into the system, while a negative input power
indicates that the motor is absorbing energy from the system.
Since the focus of this study is to capture the instantaneous input
power experienced by the motor, which affects the design of motors and electrical components, we ignore the energy loss in the
motor and mechanism. In other words, we assume that all energy
absorbed by the motor can be fully recovered. In addition, we
assume that instantaneous power experienced by the motor is
equivalent in both directions 共supplying and absorbing energy兲.
The quantity being investigated as an objective function to be
minimized in this study is, therefore, the maximum of the absolute
value of instantaneous input power, or a peak input power.
The values of the spring stiffness and free length are optimized
using a Generalized Reduced Gradient 共GRG兲 algorithm provided
by ADAMS. The optimal values of the spring’s properties, along
with some other parameters, are shown in Table 2. Plots of inertial
torque and lift obtained from this simplified model are shown in
Figs. 2 and 3, respectively. Plots of input powers for the mechanism with and without a spring are shown in Fig. 4. Various power
components for the mechanism with a spring are shown in Fig. 5.
Figure 4 shows that adding a spring reduces peak input power
by 42% from 1.00 N m / s to 0.58 N m / s. Figure 5 indicates that
energy flow in and out of the link’s mass 共kinetic energy rate
共KER兲, spring potential energy rate 共PER兲, and aerodynamic force
共Pout兲兲 all significantly contribute to the required input power
共Pin兲. The results themselves, however, do not provide deeper insight into the role of a spring and various components of energy
Table 1 Links’ dimensions and operating frequency
l1 = 6 mm
x = 15 mm
l2 = 20 mm
y = 21 mm
l3 = 330 mm
a = 50 mm
f = 4 Hz
b = 165 mm
共1兲
The simplified aerodynamic force, parametrized by a damping
constant co, constant force component f o, and normal velocity at
the center of the wing vo can be written as:
Fo共t兲 = − covo共t兲 + f o
Fig. 1 Four-bar flapping mechanism adapted from Madangopal et al. †2‡
Table 2 System parameters and optimal values of spring
properties used in the analysis of the simplified four-bar flapping mechanism
mAB = 0.25 g
mBC = 0.50 g
mCD = 2.75 g
co = 0.207 N s / m
f o = 0.27 N
k = 38 N / m
l0 = 50 mm
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Fig. 2 Variation of inertial torque over a cycle of flapping
motion
exchanged within the system components. We, therefore, investigate the effect of a spring when the system is subject to three
different cases:
共I兲 with constant output force only
共II兲 with inertial force only
共III兲 with damping output force only.
In each case, the values of spring stiffness and free length are
optimized for minimum peak input power. The resulting power
variations in the mechanism without a spring and with a spring are
then compared. Based on these results, we attempt to visualize the
path of the energy flow within the system.
Case I. The mechanism subject to constant output force only The
values of parameters are shown in Table 3. In this case, we set the
mass of all links and the damping component of output force to be
zero to investigate the interaction of the spring and the constant
component of output force. The results are shown in Figs. 6–8.
From Fig. 6, adding a spring reduces the peak input power by
84% from 0.45 N m / s to 0.07 N m / s. To understand this power
reduction, we trace power components of the system over a flapping cycle for each case. For the mechanism without a spring
Fig. 5 Various power components of the mechanism with a
spring, including kinetic energy rate „KER…, potential energy
rate „PER…, input power „Pin…, and output power „Pout…
共Fig. 7兲, the constant output force, as being upward, produces
positive work into the mechanism during the upstroke. To maintain the steady-state motion, the motor supplies negative work into
the mechanism. In other words, it has to absorb energy flowing
from the output. During the downstroke, the output force is still
upward but the motion of the wing is downward. The output force
is producing negative work into the system, or extracting the energy from the system. When a spring is added into the system
共Fig. 8兲, the output force still produces positive work into the
mechanism during the upstroke. However, instead of having the
motor absorb this work from the output, the spring stores this
Table 3 Parameters for Case I „constant output force only…
mAB = 0 g
mBC = 0 g
mCD = 0 g
co = 0 N s / m
f o = 0.27 N
k = 0 or 35 N / m
l0 = 41 mm
Fig. 6 For Case I „constant output force only…, adding a spring
reduces peak input power by 84%
Fig. 3 Variation of lift over a cycle of flapping motion
Fig. 4 Input powers of the mechanism with and without a
spring. Adding a spring reduces peak input power by 42%.
1066 / Vol. 129, OCTOBER 2007
Fig. 7 For Case I „constant output force only… without a
spring, the motor has to supply and absorb the full amount of
energy to balance the energy extracted and injected by the output force
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Fig. 8 For Case I „constant output force only… with a spring,
the presence of a spring helps the motor supply and absorb
energy extracted and injected by the output force, thus reducing the effort needed by the motor to balance the system
positive work as elastic energy. The motor, then, needs less effort
in balancing the steady-state motion. During the downstroke, the
output force extracts energy out of the system. During this period,
the spring releases the stored elastic energy to the output. In this
situation, adding a spring to the system reduces the peak input
power requirement.
Case II. The mechanism subject to inertial force only
The values of parameters are shown in Table 4. In this case, the
links’ masses are nonzero. The output force is set to be zero for
both damping and constant components. The results are shown in
Figs. 9–11.
From Fig. 9, adding a spring into the system reduces the required peak input power by 83% from 0.18 N m / s to 0.03 N m / s.
For the mechanism without a spring, the motor has to supply and
absorb the energy required by and released from the system’s
inertia. The required input power is, therefore, the same as that
stored or released by the system’s inertia. Adding a spring into this
type of system will help reduce the input power demand. The
spring will absorb the energy when the links decelerate and release kinetic energy, thus reducing the effort needed by the motor
to absorb the energy. On the other hand, when the links need to
accelerate, the spring releases the stored elastic energy, reducing
the power demand from the motor.
Case III. The mechanism subject to damping output force only
The values of parameters are shown in Table 5. In this case, all
links’ masses are zero. The output force only has a damping component. The results of power analysis are shown in Figs. 12–14.
Table 4 Parameters for Case II „inertial force only…
mAB = 0.25 g
mBC = 0.50 g
mCF = 2.75 g
co = 0 N s / m
fo = 0 N
k = 0 or 226 N / m
l0 = 117 mm
Fig. 9 For Case II „inertial force only…, adding a spring reduces
peak input power by 83%
Journal of Mechanical Design
Fig. 10 For Case II „inertial force only… without a spring, the
motor has to supply and absorb the full amount of energy
stored and released by links’ masses
From Fig. 12, adding a spring into the system increases the
required peak input power by 78% from 0.55 N m / s to
0.98 N m / s. For the mechanism without a spring 共Fig. 13兲, the
damping output force extracts energy from the system both during
upstroke and downstroke. The motor always has to supply this
amount of energy to maintain the steady-state motion. The peak
input power is, therefore, equal to the peak output power. When a
spring is added into the system 共Fig. 14兲, the motor has to supply
energy both to the output and to the spring during the upstroke
and, therefore, requires more power than it does without a spring.
During the downstroke, the spring releases energy to the output,
Fig. 11 For Case II „inertial force only…, the spring added to the
system helps absorb and supply energy as the links decelerate
and accelerate, thus reducing the power requirement from the
motor
Table 5 Parameters for Case III „damping output force only…
mAB = 0 g
mBC = 0 g
mCD = 0 g
co = 0.207 N s / m
fo = 0 N
k = 0 or 38 N / m
l0 ⫽ 50 mm
Fig. 12 For Case III „damping output force only…, adding a
spring increases peak input power by 78%
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Fig. 13 For Case III „damping output force only… without a
spring, the motor has to supply energy equal to the amount
extracted by the output force
reducing input power requirement from the motor. The entire flapping cycle, therefore, requires more input power if a spring is
added to this type of system. In fact, this should always be true for
the following situation. If the only source that provides positive
work into a massless system is the input, then the energy required
to deform the spring will have to be from this source and, therefore, the source needs to provide this extra power to the system in
addition to the power required from the dissipative output. Figures
15 and 16 show that for the case of damping output force only, the
peak input power is minimal when there is no spring added to the
system. The results of all case studies are summarized in Table 6.
If we consider each component of the load acting on the system, this study reveals that the existence of either a constant output force 共Case I兲 or an inertial property of the system 共Case II兲
allows the use of elasticity to reduce the peak input power. These
two types of load provide instantaneous positive work, which can
be stored during one stage of a cycle and reused in another. On the
Fig. 16 For Case III „damping output force only…, adding a
spring of any free length, in this case with a stiffness of
38 N / m, results in undesired increase in peak input power. At
best it keeps the peak input power as low as that resulting from
not adding a spring into the system.
contrary, the existence of a damping force reduces the effectiveness of elasticity to reduce the peak input power. From Table 6,
the optimal values of spring’s properties for the case of a combined constant-inertial-damping force 共stiffness of 38 N / m and
free length of 50 mm兲 are between those of a pure constant output
force 共Case I兲 and a pure inertial force 共Case II兲. It can be further
noticed that these optimal values are closer to those for the case of
a pure constant output force. This corresponds to the fact that the
contribution of the input power due to a pure constant output force
共0.45 N m / s兲 is more significant than that due to a pure inertial
force 共0.18 N m / s兲. The presence of a damping component in the
case of combined force brings the percentage of peak input power
reduction down to 42%. This study provides a visualization of
how each load component affects the optimal values of the
spring’s properties.
Two Approaches to Exploit Elastic Energy
Fig. 14 For Case III „damping output force only… with a spring,
besides supplying energy to the output force, the motor also
has to supply energy to be stored in the spring
Fig. 15 For Case III „damping output force only…, adding a
spring of any stiffness, in this case with a free length of 50 mm,
results in undesired increase in peak input power
1068 / Vol. 129, OCTOBER 2007
The study from the previous section illustrates the function of
an elastic component in a dynamic system as an energy storage
component. In Fig. 17共a兲, a given mechanical system with its
inertia is driven by an input actuator and performs a task at the
output port. The energy flow between these components affects
the required input power. Adding an elastic component to this
system, as shown in Fig. 17共b兲, will split the path of energy flow
in the original system. If the elastic component is added properly,
it will draw some energy from the output or inertia to itself when,
otherwise, the input has to absorb this energy. Similarly, this elastic component will release some energy to the output or inertia
when, otherwise, the input has to supply this energy. If the elastic
component is used to split energy flow via path B, its role is
referred to as generative-load exploitation. This approach of exploiting elastic energy is useful to describe the function of the
elastic component when the output load provides positive work
into the system during a certain portion of the motion cycle. The
aerodynamic load used in this study is an example of generative
load, which can be beneficial for peak input power reduction. If
the elastic component is used to split the energy flow via path C,
its role is referred to as reactance cancellation. Even though KER
and PER are not exactly cancelled, we use this term in referring to
this approach because it involves power reduction between the
elastic component and inertia, which are reactive components
found in a general spring-mass-damper system.
The two approaches of generative-load exploitation and reactance cancellation may be implemented on the design of elastic
components at the same time. In fact, they should be implemented
simultaneously when both generative load and system’s inertia are
significant. A design approach based on resonance frequency,
which corresponds to the reactance cancellation approach as referred in this paper, can result in a suboptimal design when a
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Table 6 Summary of peak input power reduction using optimal springs in four cases. A spring
is useful for peak input power reduction when either constant output force or inertial property
exists, but not when only damping output force exists.
Optimal spring
properties
Load case
Constant output force only 共Case I兲
Combined constant-inertial-damping force
Inertial force only 共Case II兲
Damping output force only 共Case III兲
k
共N/m兲
l0
共mm兲
Pin,peak
without spring
共N m/s兲
Pin,peak reduction
共%兲
35
38
226
0
41
50
117
N/A
0.45
1.00
0.18
0.55
84
42
83
0
generative load is present. In this study, energy flow across the
system’s inertia is small compared to energy flow across the output load 共Fig. 5兲. The optimal design of a spring leans toward the
design obtained by the generative-load exploitation. However, for
the operation at higher frequency, approaching toward resonance,
energy flow across the system’s inertia may become larger than
energy flow across the output load. In this case, the optimal design
of a spring will lean toward the design obtained by the reactance
cancellation.
Even though the distinction in the two approaches described in
this paper may seem unnecessary if a general optimization method
is applied, this distinction is helpful for energy flow visualization.
It will also help set up a design framework and may be useful for
guiding the design optimization process, leading to a more efficient and robust design tool. This is necessary especially for the
design of compliant mechanisms, where nonlinear responses are
computationally expensive.
Power Analysis and Design of a 2DOF System
Two different approaches to describe the role of elasticity,
namely, generative-load exploitation and reactance cancellation,
have been defined based on the investigation of power flow in the
four-bar flapping mechanism with a spring. The concept of these
two approaches cannot be immediately generalized to compliant
mechanisms because compliant mechanisms are elastic systems
consisting of an infinite number of degrees of freedom 共DOF兲.
Unlike rigid-body mechanisms, the kinematics of compliant
mechanisms depends on system’s properties and operating conditions, such as mass, stiffness, and operating frequency. We, therefore, use a 2DOF system, which represents the simplest form of a
compliant mechanism, to demonstrate the validity of the two approaches in exploiting elasticity. The system consists of two
masses: m1 and m2 and three linear springs: k1, k2, and k3, as
shown in Fig. 18. Forces f 1共t兲 and f 2共t兲 are applied to masses m1
and m2, resulting in displacements u1共t兲 and u2共t兲, respectively.
To solve for steady-state responses, we assume that the solutions are:
u1共t兲 = U1 sin共␻t兲 + Ū1
共3兲
u2共t兲 = U2 sin共␻t兲 + Ū2
共4兲
f 1共t兲 = F1 sin共␻t兲 + F̄1
共5兲
f 2共t兲 = F2 sin共␻t兲 + F̄2
共6兲
The equations of motion for the two masses can be written in a
matrix form. The static and harmonic components can be written
separately, yielding the following systems of equations:
冋册冋
F̄1
=
F̄2
冋册冉 冋
F1
F2
= − ␻2
k1 + k2
− k2
− k2
k2 + k3
m1
0
0
m2
册冋
+
册冋
Ū1
Ū2
册
k1 + k2
− k2
− k2
k2 + k3
共7兲
册冊冋 册
U1
U2
共8兲
Assuming that the system is subject to an input displacement 共U1,
Ū1兲 and the harmonic component of an output force 共F2兲 is zero,
we can solve for Ū2, U2, F̄1, and F1. The results are:
F̄2 + k2Ū1
k2 + k3
共9兲
k2
U1
− ␻ m2 + k2 + k3
共10兲
共k1k2 + k1k3 + k2k3兲Ū1 − k2F̄2
k2 + k3
共11兲
Ū2 =
U2 =
F̄1 =
2
Fig. 17 Concept of using elasticity to reduce peak input power requirement in dynamic
systems
Journal of Mechanical Design
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F1 =
␻4m1m2 − ␻2共m1k2 + m1k3 + m2k1 + m2k2兲 + k1k2 + k1k3 + k2k3
U1
− ␻ 2m 2 + k 2 + k 3
共12兲
The requirement of this system is that it has to generate the output amplitude of U2,req. From Eqs. 共10兲–共12兲, to satisfy this requirement,
the amplitude of input displacement, average input force, and amplitude of input force, respectively, must be:
− ␻ 2m 2 + k 2 + k 3
U2,req
k2
共13兲
共k1k2 + k1k3 + k2k3兲Ū1 − k2F̄2
k2 + k3
共14兲
␻4m1m2 − ␻2共m1k2 + m1k3 + m2k1 + m2k2兲 + k1k2 + k1k3 + k2k3
U2,req
k2
共15兲
U1 =
F̄1 =
F1 =
The instantaneous input power can be calculated from:
p1共t兲 = f 1共t兲u̇1共t兲
共16兲
1
p1共t兲 = 关F1 sin共␻t兲 + F̄1兴关␻U1 cos共␻t兲兴 = ␻F1U1 sin共␻t兲cos共␻t兲 + ␻F̄1U1 cos共␻t兲 = ␻F1U1 sin共2␻t兲 + ␻F̄1U1 cos共␻t兲
2
共17兲
Substituting Eqs. 共3兲 and 共5兲 into Eq. 共16兲, we have:
Equation 共17兲, along with Eqs. 共13兲–共15兲, shows that the instantaneous input power is a function of frequency, stiffness, mass, and
average input displacement. It is no longer a function of the input displacement amplitude due to the requirement on the output
displacement amplitude.
Since we are interested in designing a system for given operating frequency, output force, and required output displacement, the
design variables are stiffness, mass, and average input displacement. Equation 共17兲 can be written in the form:
p1共t兲 = G1共k1,k2,k3,m1,m2兲sin共2␻t兲 + G2共k1,k2,k3,Ū1兲cos共␻t兲
共18兲
1 关␻4m1m2 − ␻2共m1k2 + m1k3 + m2k1 + m2k2兲 + k1k2 + k1k3 + k2k3兴关− ␻2m2 + k2 + k3兴 2
G1 = ␻
U2,req
2
k22
共19兲
where
G2 = ␻
关共k1k2 + k1k3 + k2k3兲Ū1 − k2F̄2兴关− ␻2m2 + k2 + k3兴
U2,req
k2共k2 + k3兲
1 共k1k2 + k1k3 + k2k3兲共k2 + k3兲 2
G1 = ␻
U2,req
2
k22
共23兲
共k1k2 + k1k3 + k2k3兲Ū1 − k2F̄2
U2,req
k2
共24兲
共20兲
G2 = ␻
and
k1 ⱖ 0,
k2 ⬎ 0,
k3 ⱖ 0,
m1 ⱖ 0,
m2 ⱖ 0,
− ⬁ ⬍ Ū1 ⬍ ⬁
共21兲
We will determine optimal designs for two cases: one is when
the system has no mass and the other is when the system has no
output force. The first case represents an attempt to use elasticity
based on generative-load exploitation only, while the second case
represents an attempt to use elasticity based on reactance cancellation.
Case I. m1 = m2 = 0 and F̄2 ⫽ 0 共generative-load exploitation兲
Equations 共18兲–共20兲, respectively, reduce to:
p1共t兲 = G1共k1,k2,k3兲sin共2␻t兲 + G2共k1,k2,k3,Ū1兲cos共␻t兲 共22兲
Fig. 18 A 2DOF system used to validate the concept of
generative-load exploitation and reactance cancelation in compliant mechanisms
1070 / Vol. 129, OCTOBER 2007
Since G2 has an extra variable 共Ū1兲, which can be any real
number, G2 may be considered independent of G1. The peak of
p1共t兲, or P1,peak, can be minimized when 兩G1兩 and 兩G2兩 are minimized. Since the minimum value that 兩G1兩 and 兩G2兩 may reach is
zero, we first attempt to find conditions that make G1 and G2 be
zero. From Eq. 共23兲, G1 is set to zero. We have:
共k1k2 + k1k3 + k2k3兲共k2 + k3兲
k22
k 1k 2 + k 1k 3 + k 2k 3 = 0
=0
共25兲
共26兲
Substitute Eq. 共26兲 into Eq. 共24兲 to find conditions that make G2
be zero:
0 − k2F̄2
=0
k2
共27兲
F̄2 = 0
共28兲
This is impossible since we have F̄2 ⫽ 0. Therefore, P1,peak cannot
be made zero.
We then attempt to find conditions that make G1 and G2 approach zero. From Eq. 共23兲, when G1 approaches zero, we have:
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共k1k2 + k1k3 + k2k3兲共k2 + k3兲
→0
共29兲
k1k23 k23
k 1k 3
+ k3 + 2 + → 0
k2
k2
k2
共30兲
k22
k1 + 2
Condition 共30兲 is true only when:
k1 → 0
k3 → 0
k 1k 3
→0
k2
共31兲
Substitute 共31兲 into Eq. 共24兲 and find conditions that make G2
approach zero:
共k1k2 + k1k3 + k2k3兲Ū1 − k2F̄2
→0
k2
共32兲
共k1k2 + k2k3兲Ū1 − k2F̄2
→0
k2
共33兲
Ū1 →
F̄2
k1 + k3
共34兲
Ū1 → ± ⬁
共35兲
This analysis shows that P1,peak → 0 when k1 → 0, k2 ⬎ 0, k3 → 0,
and Ū1 → ± ⬁. By similar analysis, it can also be shown that k1 or
k3, but not both, can be zero to make P1,peak → 0.
However, Ū1 cannot physically be ±⬁. For simplicity, to investigate the case when there is a bound on Ū1, we assume that F̄2
⬎ 0 and the upper bound Ū1,bound ⬎ 0. For the case when k1 = 0,
Eqs. 共23兲 and 共24兲 become:
k3共k2 + k3兲
1
2
G1 = ␻U2,req
2
k2
Fig. 19 A general plot of 円G1円 and 円G2円 as functions of k3 when
k1 is zero
From Eqs. 共36兲 and 共37兲, however, k2 appears only in G1 because k1 has been assumed zero. The value of 兩G1兩 decreases when
k2 increases. The optimal k2 that minimizes the P1,peak, therefore,
approaches ⬁. This result implies, for the case when there is no
input stiffness, that the compliance between the input and the
output should be minimized and a spring should be attached to the
output.
In reality, there is usually stiffness associated with the input,
such as the stiffness of an actuator. In compliant mechanisms,
input stiffness exists due to the members connecting the input to
the ground. In this case, k1 is not zero. G1 and G2 from Eqs. 共36兲
and 共37兲, respectively, are not valid. We have to use Eqs. 共23兲 and
共24兲. From Eq. 共23兲, 兩G1兩 decreases when k2 increases. However,
this is not always true for 兩G2兩 in Eq. 共24兲. We rewrite Eq. 共24兲 as
the following:
G2 = ␻U2,req
冊
k 1k 3
+ k3 Ū1 − F̄2
k2
冉
k1 +
冊
k 1k 3
+ k3 Ū1 − F̄2 = 0
k2
k2 =
k 1k 3
F̄2/Ū1 − 共k1 + k3兲
共38兲
共39兲
共40兲
Even though P1,peak depends on both G1 and G2 in this case,
Eq. 共40兲 shows that the optimal k2 does not necessarily approach
⬁ when there is input stiffness and a bound on average input
displacement. For this situation, compliance between the input
and output will help reduce the peak input power.
Case II. m1 ⬎ 0, m2 ⬎ 0, and F̄2 = 0 共reactance cancellation兲
Equations 共18兲–共20兲 become:
p1共t兲 = G1共k1,k2,k3,m1,m2兲sin共2␻t兲 + G2共k1,k2,k3,Ū1兲cos共␻t兲
共41兲
1 关␻4m1m2 − ␻2共m1k2 + m1k3 + m2k1 + m2k2兲 + k1k2 + k1k3 + k2k3兴关− ␻2m2 + k2 + k3兴 2
G1 = ␻
U2,req
2
k22
G2 = ␻
冎
− F̄2其, which will contribute to P1,peak. 兩G2兩 can be minimized
when:
共37兲
From Eq. 共36兲, 兩G1兩 can be reduced when k3 is reduced. From
Eq. 共37兲, 兩G2兩 can be made zero when k3 ⱖ F̄2 / Ū1,bound. When k3 is
reduced further such that k3 ⬍ F̄2 / Ū1,bound, Ū1 that minimizes 兩G2兩
is Ū1,bound and 兩G2兩 increases to ␻U2,reqF̄2 as k3 approaches zero.
A general plot of function 兩G1兩 and 兩G2兩 is shown in Fig. 19.
Figure 19 implies that an optimal k3 that minimizes P1,peak does
not necessarily approach zero when there is a bound on Ū1. The
analysis for the case when k3 = 0 yields a similar result, in which
the optimal k1 does not necessarily approach zero.
k1 +
When k2 approaches ⬁, G2 will approach ␻U2,req兵共k1 + k3兲Ū1
共36兲
G2 = ␻U2,req共k3Ū1 − F̄2兲
再冉
共k1k2 + k1k3 + k2k3兲Ū1
U2,req
k2
共42兲
共43兲
To minimize P1,peak, Eq. 共43兲 shows that one of the solutions is when Ū1 = 0 and the other one is when k1 = k3 = 0, which will make G2
be zero.
For the case when Ū1 = 0, Eqs. 共41兲 and 共42兲 become:
p1共t兲 = G1共k1,k2,k3,m1,m2兲sin共2␻t兲
Journal of Mechanical Design
共44兲
OCTOBER 2007, Vol. 129 / 1071
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1 关␻4m1m2 − ␻2共m1k2 + m1k3 + m2k1 + m2k2兲 + k1k2 + k1k3 + k2k3兴关− ␻2m2 + k2 + k3兴 2
G1 = ␻
U2,req
2
k22
共45兲
We determine conditions that make G1 be zero. By setting Eq. 共45兲 to be zero, we have:
关␻4m1m2 − ␻2共m1k2 + m1k3 + m2k1 + m2k2兲 + k1k2 + k1k3 + k2k3兴关− ␻2m2 + k2 + k3兴
k22
The solutions to Eq. 共46兲 are:
k2 = ␻ m2 − k3
共47兲
2
or
k2 =
␻4m1m2 − ␻2共m1k3 + m2k1兲 + k1k3
␻2共m1 + m2兲 − k1 − k3
共48兲
Depending on the values of m1, m2, k1, and k3, it is possible that
there is k2 that reduces P1,peak to zero. In addition, it can be proved
that Eqs. 共47兲 and 共48兲 correspond to the natural modes of the
system with m1 fixed and m1 free, respectively.
For the case when k1 = k3 = 0, Eqs. 共41兲 and 共42兲 reduce to:
p1共t兲 = G1共k2,m1,m2兲sin共2␻t兲
共49兲
1 关␻4m1m2 − ␻2共m1k2 + m2k2兲兴关− ␻2m2 + k2兴 2
G1 = ␻
U2,req
2
k22
共50兲
We determine conditions that make G1 be zero. By setting Eq.
共50兲 to be zero, we have:
关␻4m1m2 − ␻2共m1k2 + m2k2兲兴关− ␻2m2 + k2兴
k22
=0
共51兲
The solutions to Eq. 共51兲 are:
共52兲
or
␻ 2m 1m 2
m1 + m2
共53兲
Equations 共52兲 and 共53兲 show that there is always k2 that reduces
P1,peak to zero. The summary of the conditions that minimize the
peak input power is shown in Table 7.
From Table 7, k2 approaching infinity in Case I 共D兲 represents a
rigid-body connection between the input and output. This case
confirms the concept of generative-load exploitation in a rigidbody mechanism. Cases I 共E兲 and I 共F兲, with stiffnesses not approaching infinity, represent a compliant system. These cases
Table 7 Summary of the conditions that minimize peak input
power
F̄2
Case I
共A兲
共B兲
共C兲
共D兲
⫽0
⫽0
⫽0
⫽0
m1
m2
=0
=0
=0
=0
→±⬁
=0 →0 any k2⬎0 →0
=0 =0 any k2 ⬎ 0 →0+ → ± ⬁
=0 →0+ any k2 ⬎ 0 =0
→±⬁
=0 =0
→⬁
→
” 0 Bounded
→
”⬁
=0 →
” 0 any k2 ⬎ 0 =0 Bounded
→
”⬁
=0 →
⬎0
”0
→
” 0 Bounded
→
”⬁
→
”⬁
→
”⬁
共E兲 ⫽0 =0
共F兲 ⫽0 =0
Case II 共G兲 =0 ⬎0 ⬎0
共H兲 =0 ⬎0 ⬎0
共46兲
show that the generative-load exploitation is applicable to a compliant mechanism when an input stiffness and a bound of average
input displacement exist. Since k2 in Case II 共G兲 may approach
infinity, this confirms the concept of reactance cancellation in a
rigid-body mechanism. Both Cases II 共G兲 and II 共H兲 show that the
concept of reactance cancellation is applicable to a compliant
mechanism, regardless of the existence of an input stiffness or a
bound of average input displacement. In addition, Eqs. 共47兲 and
共48兲 indicate that the reactance cancellation corresponds to two
types of resonance, one is associated with a free-input response
and the other is associated with a fixed-input response.
Even though the analysis of a 2DOF system validates the concept of generative-load exploitation and reactance cancellation in
compliant mechanisms, the analysis procedures are not applicable
to analyze and design compliant mechanisms. Most compliant
mechanisms undergo large displacements and require nonlinear
analysis to evaluate their performance. In addition, stiffness and
mass in a compliant mechanism are coupled through geometry
and cannot be designed independently. The analysis of a 2DOF
system in this study, therefore, only provides mathematical understanding in the use of elasticity. The analysis and design of compliant mechanisms with large displacements require more sophisticated methods.
Optimization Strategies
k 2 = ␻ 2m 2
k2 =
=0
k1
k2
+
⬎0
=0
k3
Ū1
+
共47兲, 共48兲
共47兲, 共53兲
1072 / Vol. 129, OCTOBER 2007
⬎0
=0
=0
⫽0
P1,peak
→0
→0+
→0+
⬎0
+
⬎0
⬎0
ⱖ0
=0
Unlike a linear system, most compliant mechanisms undergo
large displacement, and geometric nonlinearity must be taken into
account in the analysis. The amplitude of input displacement required to provide the desired amplitude of output displacement
cannot be determined directly. In addition, a harmonic analysis
and the principle of superposition are no longer applicable. In this
study, we use nonlinear transient analysis based on an implicit
dynamic method to analyze and design a compliant mechanism.
Since the amplitude and mean values of the input displacement
can be controlled during the operation, we do not consider them as
true design variables in the context of a “mechanism design.”
Instead, we use the term “performance parameters” to refer to the
amplitude and mean values of the input displacement. Beam
heights and keypoint locations, on the other hand, cannot be controlled once the system is built and are still considered “design
variables.” Based on the distinction between performance parameters and design variables, there are at least three strategies for
optimizing compliant mechanisms.
共1兲 Performance parameters and design variables are optimized
together in one optimization loop.
共2兲 Performance parameters are optimized as an inner loop for
each evaluation of a set of design variables being optimized
in an outer loop.
共3兲 Performance parameters are fixed and only design variables
are optimized.
Strategy 共1兲 is based on a modern design automation approach,
which does not have to distinguish between performance parameters and design variables. Strategy 共2兲 is based on a traditional
design approach, in which a designer evaluates each design properly before moving and comparing to the next design. Strategy 共3兲
is based on simplicity, which ignores the effects of input on the
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Fig. 20 Front view of the compliant flapping mechanism
showing the initial design before optimization
design. The primary motivation in distinguishing these strategies
is to find the most efficient optimization scheme to search design
space. For example, it is possible that a continuously changing
amplitude or mean value without restarting the analysis from zero
input may reduce computational time by shortening the time during transient state. In this case, Strategy 共2兲 may be preferred.
However, there is no straightforward manner to predict the most
efficient strategy to use.
Design of a Compliant Flapping Mechanism
In this study, we use Strategy 共1兲, optimizing performance parameters and deign variables together in one loop, to design a
compliant flapping mechanism. The initial design, shown in Fig.
20, is an application of a patented compliant stroke amplifier
关27–29兴, in which the wing is directly connected to the output
link. It is modeled and analyzed in ABAQUS, which is commercial
finite element analysis software. Instead of having a uniform beam
width, the wing section of the mechanism has been modified to
redistribute the mass and stiffness, as shown in Fig. 21. The total
mass of the wing section is 2.50 g, the same as that in the four-bar
flapping mechanism, but a significant amount of mass is distributed to the end of the wing section. This will promote the use of
reactance cancellation, in addition to the use of generative-load
exploitation. The wing’s dimensions are shown in Table 8.
The mechanism is modeled using structural beam elements. Its
overall dimensions are approximately the same as those of the
four-bar flapping mechanism studied in the previous section. We
assume that the entire mechanism is made of carbon fiber composites whose Young’s modulus is 520 GPa, Poisson’s ratio is
0.28, and density is 1570 kg/ m3. The output force Fo共t兲, which is
Fig. 22 An example of input displacement with gradually increasing amplitude to facilitate the convergence of nonlinear
transient analysis
applied at the center of the wing, is a simplified aerodynamic
force previously used in the analysis of the four-bar flapping
mechanism, as described by Eq. 共2兲. The values of co and f o are
0.207 N s / m and 0.27 N, respectively. The required flapping
angle ␪flap, measured at the location where the output force is
applied, is 44.8 deg, the same as in the four-bar flapping mechanism. This amount of flapping angle is assumed to provide sufficient lift at the specified frequency to sustain the vehicle in the air
and is the primary functional requirement of the mechanism. The
gravity is ignored in the analysis.
The mechanism is subject to a sinusoidal input displacement
uin共t兲, with amplitude A and average C. However, to facilitate the
convergence in finding solutions of the analysis, we impose the
input displacement with gradually increasing amplitude, expressed in the form:
uin共t兲 = 共1 − e−t/␶H兲A关cos共2␲ ft兲 − 1兴 + 共1 − e−t/␶S兲共C + A兲
共54兲
For this study, ␶H = 1 / 4, ␶S = 1 / 4, and f = 4 Hz. An example of
input displacement is shown in Fig. 22.
There are a total of 37 design variables for this optimization
problem: two are performance parameters including A and C from
Eq. 共54兲, ten are keypoint locations, and 25 are beam heights, as
shown in Fig. 23. The objective of the optimization problem is to
minimize the peak input power requirement, subject to the flapping angle of 44.8 deg.
Each analysis is run until the mechanism reaches a steady state,
which is predicted by comparing responses in one cycle to others.
In doing this, criteria for analysis termination have to be specified.
When the differences in responses 共i.e., flapping angle, input
power, and output power兲 between cycles are within the specified
values, the analysis is terminated and the mechanism is considered
Fig. 21 Top view of the compliant flapping mechanism showing a wing section with nonuniform mass and stiffness
distribution
Table 8 Dimensions of the wing
Wing segment
Height, h 共mm兲
Width, w 共mm兲
Length, l 共mm兲
A
B
C
D
E
1.00
1.00
1.00
1.00
1.00
0.50
1.50
3.50
10.44
18.00
82.50
82.50
82.50
45.83
36.67
Journal of Mechanical Design
Fig. 23 Design variables include „a… ten variables of x and y
locations for five keypoints, „b… 25 variables of beam heights
for nine beam segments, and two variables for amplitude and
average values of input displacement „see Fig. 20…
OCTOBER 2007, Vol. 129 / 1073
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Fig. 24 Optimal design of a compliant flapping mechanism obtained from
the use of NOMADm and SQP algorithms
to reach a steady state. The flapping angle and peak input power
can be determined after the steady-state criteria have been met.
The optimization process is accomplished through the use of
Matlab®. NOMADm 关30兴 algorithm, which is the implementation
Table 9 Keypoints of the optimal mechanism
Keypoint
x-coordinate 共mm兲
y-coordinate 共mm兲
1
2
3
4
5
6
7
8
9
0.00
8.69
21.52
29.98
5.20
31.00
3.00
−2.00
3.00
0.00
12.00
6.01
26.62
23.65
39.97
50.00
50.00
14.00
Table 10 Segments
= 2.00 mm…
of
the
optimal
mechanism
Segment
Height 共mm兲
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
0.23
0.24
0.27
0.24
0.25
0.19
0.26
0.27
0.25
0.24
0.24
0.32
0.25
2.55
0.31
0.24
0.30
0.25
0.24
0.25
0.24
0.25
0.27
0.26
4.90
1074 / Vol. 129, OCTOBER 2007
of a Mesh Adaptive Direct Search 共MADS兲 algorithm, was first
used to search the design space. One of the features of this algorithm that is suitable for this problem is that the algorithm is
intended for a problem whose objective and constraint functions
are computationally expensive to evaluate. The solution was obtained after 3814 analyses, which took 71 h on a computer with
2.8 GHz Pentium® IV processor and 512 MB of RAM. This solution was then used as an initial design for local search using
Sequential Quadratic Programming 共SQP兲. The solution was obtained after 4958 analyses, which took 96 h on the same computer. The optimal design is shown in Fig. 24, along with num-
„width
Fig. 25 The mechanism produces a flapping angle of 44.8 deg,
generating a sufficient lift for the vehicle at 4 Hz
Fig. 26 Wing rotation over a flapping cycle at steady state „after seven cycles…
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关2兴
关3兴
关4兴
关5兴
关6兴
Fig. 27 Variation of different power components over a flapping cycle at steady state
关7兴
关8兴
bering for keypoints and segments, the values of which can be
found in Tables 9 and 10. The optimal amplitude A is 0.612 mm
and average C is 0.770 mm. The extreme positions of the wing are
shown in Fig. 25, confirming that the wing is deformed and rotated as expected. The plot of wing rotation over time is shown in
Fig. 26. The plot of various power components is shown in
Fig. 27.
This design example of a compliant flapping mechanism requires only 0.49 N m / s of input power, compared to 1.0 N m / s
for the four-bar flapping mechanism without a spring, and
0.58 N m / s for the one with a spring 共50% and 15% peak input
power reduction, respectively兲, in performing the same task. Note
that the two peaks of input power are evened out. This is the same
behavior found in the four-bar flapping mechanism when an appropriate spring is added. The peak output power is only
0.75 N m / s, compared to 1 N m / s in the four-bar flapping mechanism. The reduction of peak output power can be explained by the
use of sinusoidal input displacement, which produces more efficient flapping velocity profile, generating the same amount of lift
with lower output power. The 15% reduction of peak input power
in the compliant flapping mechanism over the four-bar flapping
mechanism with a spring can, therefore, be attributed to the use of
elasticity and a more efficient input velocity profile.
Conclusions
In this work, we have investigated the concept of peak input
power reduction through the use of elasticity. From the power
analysis of a four-bar flapping mechanism, we make a distinction
between two different approaches that can be used to describe the
role of an elastic component: 共1兲 generative-load exploitation and
共2兲 reactance cancellation. Only one or both of them may be applied at the same time, depending on the nature of the load. These
two approaches have been shown to be valid to describe the role
of elasticity in compliant mechanisms as well. We then propose a
compliant mechanism as an alternative to a rigid-body mechanism
with attached springs. Through the use of NOMADm and SQP
algorithms, we obtain a compliant flapping mechanism that can
reduce the peak input power requirement by an additional 15%
over the four-bar flapping mechanism with a spring, while generating the same amount of lift.
The computational requirement seems to be too high. Therefore, in an attempt to develop an efficient and robust design tool,
we are currently investigating various problem formulations and
optimization methods to reduce computational time. Other design
issues, such as average power consumption, sensitivity to operating condition and dimensional error, the coupling between system
design and aerodynamic load, and stress constraint will be addressed in the future.
关9兴
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
关20兴
关21兴
关22兴
关23兴
关24兴
关25兴
关26兴
关27兴
关28兴
关29兴
References
关1兴 Khatait, J. P., Mukherjee, S., and Seth, B., 2006, “Compliant Design for Flap-
Journal of Mechanical Design
关30兴
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OCTOBER 2007, Vol. 129 / 1075
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