The 14th IFToMM World Congress, Taipei, Taiwan, October 25-30, 2015
DOI Number: 10.6567/IFToMM.14TH.WC.OS8.008
A Compliant Two-Port Bistable Mechanism with Application to Easy-Chair Design
Darshan S1
T. J. Lassche2
J. L. Herder3
G. K. Ananthasuresh4
Dept. of Mechanical Engineering
Dept. of Mechanical Automation
Dept. of Mechanical Automation
Dept. of Mechanical Engineering
Indian Institute of Science
Bengaluru, India
University of Twente
Enschede, The Netherlands
University of Twente
Enschede, The Netherlands
Indian Institute of Science
Bengaluru, India
Abstract: A click-clack tin-lid mechanism analyzed in this paper
has the feature to be switched between its two stable states by
applying forces along different lines of action, generally at two
different points, called ports here. A mechanism such as this has
an inherent mechanical advantage because the magnitude of the
actuation force at one port is different from that at the other port.
The 3D circular click-clack tin-lid mechanism is modified to
design a planar fully-compliant two-port bistable mechanism. An
analytical kinetoelastic model and finite element analysis are used
to explain the energy behavior of the mechanism and gain insights
in using it in practice. The mechanism is implemented to design a
sit-to-stand easy chair where a small force applied on the armrest
can gently propel and ease the occupant rising from it. Further, a
static balancing spring is combined with the two-port bistable
mechanism to enable adjustment for different weights of the
occupant.
Keywords: Kinetoelastic bistable model, static balancing,
geriatric sit-to-stand chair
1 Introduction
There are about 576 million people aged more than 65
in the world today. Of these, approximately 350 million
people suffer from arthritis [1]. For these people, rising
from a seated position can be a painful experience due to
the inflammation of their joints. While there are several
existing chairs designed to address this problem, most use
electrical actuators [2] or complex hydraulic systems [3].
Since electrical power is not available everywhere for the
entire day, especially in developing countries, there is a
need for an easy-chair that gives support during rising from
the chair without using external power sources. A
mechanism based on energy principles is developed to
fulfil this need.
First, three different energy behaviors are introduced,
which is followed by the analysis of the two-port bistable
compliant mechanism used for the design of the easy-chair.
1.1 Concepts of energy behavior
Various ways are possible to support a person while
sitting in and rising from the chair. The main working
principle of the mechanism can be evaluated using the
energy behavior. Three general concepts will be explained
as represented in Fig. 1 using energy graphs and the
ball-on-the-hill analogy. Fig. 1a shows two states of the
chair, where the ball represents a person on the chair while
the chair is up (1) or down (2). In state 1 the chair is up and
the person is almost standing, state 2 refers to the seated
position.
Balancing the mass exactly over the trajectory will
result in a constant energy level (Fig. 1b), known as static
balancing [4]. As a result, the person can move back from
1
darshan.sheshgiri@gmail.com
t.j.lassche@alumnus.utwente.nl
3
j.l.herder@utwente.nl
4
suresh@mecheng.iisc.ernet.in
2
1a. Two states of the chair
1b. Statically balanced
1c. Monostable
1d. Bistable
Fig. 1. Energy graphs of three general concepts
state 2 to state 1 or vice versa without the need of any
external force.
The monostable configuration (Fig. 1c) shows a
decreasing energy level from state 1 to state 2. A physical
stop can be used to obtain a stable position at state 2. An
easy-chair using this principle was already developed with
a spring system by the company Sprinq [5]. The
monostable energy behavior can also be obtained by using
a partial counterweighted mechanism. The slope of the
energy curve represents the required force to move over the
curve. Fig. 1c. shows a configuration that requires a
constant force to come back from state 2 to state 1.
The third concept shown in Fig. 1d. uses bistable
behavior. Bistability is preferable as it is considered
pleasant to have two stable states for the occupant of the
chair, instead of floating between the two states without
any additional force. The latter is experienced in the
balanced concept when the body mass is fully balanced. In
practice, as it is hard to exactly balance the mass of the
person, sit-to-stand chairs may be designed to have
monostable behavior which compensates for only a part of
the mass, which avoids the floating behavior. However, in
contrast to the monostable concept, a chair with bistable
behavior can also be used in the upper position. For
example, the lower position can be used for reclining while
the upper position can be used for an activity such as eating.
Hence, the advantages of a bistable mechanism in an
easy-chair are twofold: 1) it eliminates the floating feeling
and 2) it creates two useful positions for the user.
1.2 Two-port bistable mechanism
In most planar bistable mechanisms, the point and line
of action of the actuation forces to switch from state 1 to
state 2 is the same as that from state 2 to state 1. In other
words, they are “single-port bistable mechanisms”. This is
not the only possibility as discussed next.
Consider the click-clack tin-lid mechanism shown in
Fig. 2. The hand sketches show how it can be switched
between its two stable states. It has two force-free stable
states (i.e., bistable) and it can be operated at the two ports.
We call this a “two-port bistable mechanism”. The
cross-section of the lid in the two stable states is shown in
Fig. 2 along with the lines of action of the actuation forces.
Fig. 2. Actuation of a two-port bistable lid
Port 1 actuation due to the force applied to the
lid-surface switches the lid from its first stable state to the
second. Port 2 actuation due to a lateral force, whose point
of application and the line of action are different (in this
case perpendicular) to the port 1 force, switches the lid
back to its first state. During port 2 actuation, it was felt
that the force required was less than that of the force
required for port 1 actuation. This will be quantified with
an analytical model in Section 2.
The behavior of the click-clack tin-lid prompted us to
visualize a chair that can gently propel its occupant by
applying a small force on the armrests (port 2 actuation).
Fig. 3 shows the schematic diagram of the easy-chair. The
weight of the person (port 1 force) can be used to bring the
chair to the second stable position. The force applied on the
arm-rests is transferred to port 2 through a mechanism,
which creates a lateral force in the perpendicular direction.
We analysed the click-clack tin-lid mechanism to
design a fully compliant monolithic two-port bistable
mechanism. The weight of the person sitting on the chair
elastically deforms the compliant mechanism and thus
stores strain energy in it. By designing the mechanism
properly, this energy can be used to help the person rise
from the chair. Bistability in compliant mechanisms was
explored in [6]-[8] and static balancing of compliant
mechanisms is already explored in [9]-[13]. The bistable
behavior can be combined with balancing the weight of the
occupant, in order to reduce the required force to switch
between the states [14] and to adjust for the variable
weights of the occupants.
The goal of this paper is to propose a passive
Fig. 3. Concept diagram of easy-chair
mechanical system that facilitates rising from a chair by
employing the features of a two-port bistable mechanism.
The general idea has been introduced and a concept is
selected for the design of an easy chair. In Section 2, we
derive a kinetoelastic model and through it, demonstrate
the possibility of causing bistability by applying forces at
two different points (ports) wherein the lines of action and
magnitudes of the actuation forces are not equal. Section 3
extends the kinetoelastic model to include static balancing,
thereby allowing large varying masses to be both balanced
and bistable. The analytical models are correlated through
simulations and experiments in Section 4. Finally, the
implementation of the mechanism in the design of a
compliant easy-chair is explained in Section 5.
2 Analytical model for two-port bistability
Consider the most general case of two-port bistability
for a system with state 1 and state 2 being the two stable
equilibrium positions at energy levels 𝐸1 and 𝐸2
respectively, shown in Fig. 4. Force 𝐹1 is applied at port 𝑝1
along a displacement path 𝛿1 to switch the system from
state 1 to state 2. To bring the system back to state 1, a force
𝐹2 at port 𝑝2 is applied along a displacement path 𝛿2 . The
system is two-port bistable when 𝑝1 ≠ 𝑝2 , resulting
in 𝛿1 ≠ 𝛿2 . The forces required to switch the system
depend on 1) the difference between 𝐸1 and 𝐸2 and 2) the
derivatives of the potential energy along the displacement
paths. The derivatives are different for both ports of
actuation.
Fig. 4. General representation of two-port bistability
A scaled 2D-cross section of the two-port bistable
click-clack tin-lid, shown in Fig. 5, is used to analyze
two-port bistability. To analytically evaluate this prototype,
Fig. 5. Two stable equilibrium positions of a prototype. The
stress-free first equilibrium (up) configuration is overlaid on the
second equilibrium (down) configuration to visualize the
deformation.
we developed a kinetoelastic model using pseudo-rigid
body modeling. It is well known that an inclined
translational spring, initially in its force-free state results in
bistability if one of its ends is constrained to move
vertically, as shown in Fig. 6a. As the spring is pushed
down (port 1 actuation), it is being compressed and hence
stores strain energy. When the spring is perpendicular to
the sliding direction, maximum strain is stored in it. Any
further motion downwards will cause the spring to release
the energy and snap to its second equilibrium position. In
this, 𝐸1 = 𝐸2 , 𝐹1 = 𝐹2 , 𝑝1 = 𝑝2 and 𝛿1 = 𝛿2 . Hence, it is
a single-port bistable mechanism.
If an appendage is added to this in the form of a flap as
shown in Fig. 6b, the point of application and the line of
action of the force to bring the system from state 2 to state 1
is different from that to bring it from state 1 to state 2. In
port 2 actuation, the lateral displacement of the flap is
transferred to upward motion of the slider. This is a
two-port bistable mechanism. Here, 𝐸1 = 𝐸2 , 𝐹1 ≠ 𝐹2 ,
𝑝1 ≠ 𝑝2 and 𝛿1 ≠ 𝛿2 . In the prototype shown in Fig. 5, the
length of flap is smaller than the length of the translational
spring; and hence the length of 𝛿2 is smaller than the length
of 𝛿1 . Since the system is conservative and 𝐸1 = 𝐸2 , the
input energies to switch between the states are equal,
resulting in 𝐹2 > 𝐹1 .
The highlighted curve of the mechanism rotated about
the pivot as shown in Fig. 5 is examined. In the curve, most
deformation occurs at the center of the mechanism while
there is no deformation at the pivot. This indicates that
energy is stored in the system during its motion from state
1 to state 2. Therefore, a torsional spring is added at the
(a) Bistable
slider in the conceptual model of the two-port bistable
mechanism shown in Fig. 6c. The torsional spring will
continuously store energy during port 1 actuation, i.e.,
𝐸2 > 𝐸1 . During port 2 actuation the stored energy is
released, which reduces the effort. By designing the system
properly, it can be ensured that 𝐹2 is smaller than 𝐹1 while
𝛿2 < 𝛿1 , in contrast to the system shown in Fig. 6b.
A planar prototype of the profile (3D printed with a
Verowhite polymer using an Objet260 Connex) is shown in
Fig. 7. The geometric and material parameters of the
prototype are shown in Table 1.
Fig. 7. Prototype of a planar two-port bistable mechanism
Table 1. Values of constants used in the analytical model
Parameter
Length
Base length
Thickness
Out-of-plane thickness
Initial height
Length of the flap segment
Angle between the flap and the
straight line shown in Fig. 7
Material Young’s modulus
Material density
Symbol
𝑙
2𝑤
𝑡
𝑑
ℎ0
l2
𝛽
Value
62.70 mm
125 mm
1.00 mm
5.00 mm
5.00 mm
9.36 mm
107°
𝐸
𝜌
2.1 GPa
2650 kg/m3
The kinetoelastic model of the symmetric right-half of
the mechanism is shown in Fig. 8. The force at port 1 of the
model represents half of the required force in the prototype.
This model has a single degree of freedom, as is evident
from four bodies (including the fixed body), two revolute
joints (pivots) and two prismatic joints (sliders). The spring
constant of the torsional (or rotational) spring is 𝑘𝑟 and that
of the translational spring is 𝑘𝑡 . The stress-free length of
the translational spring is 𝑙10 and it is placed at an initial
angle 𝜃𝑒𝑞1 . The length of the flap is 𝑙2 which makes an
angle 𝛽 with the translational spring.
(b) Two-port bistable
Fig. 8. Kinetoelastic Model
(c) Two-port bistable including torsional spring, 𝐸2 ≠ 𝐸1
Fig. 6. A conceptual model for a two-port bistable mechanism
Table 2 shows the values of constants in the model
used to evaluate the compliant mechanism shown in Fig. 7.
Determining the numerical values of 𝑙10 , 𝑘𝑡 and 𝑘𝑟 has
been treated using pseudo-rigid body modeling [15].
Table 2. Values of constants used in the kinetoelastic model
Parameter
Translational Spring
Stiffness
Torsional Spring
Stiffness
Length of translational
spring segment
Length of the flap
segment
Angle between the flap
and the spring segment
Angle at first
equilibrium
Symbol
𝑘𝑡
Value
34.7 mN/m
𝑘𝑟
20 kNm
l10
53.1 mm
l2
9.36 mm
𝛽
107°
𝜃𝑒𝑞1
4.57°
stability. We denote the second stable equilibrium position
as 𝜃𝑒𝑞2 . As can be seen in Fig. 9, the addition of the
torsional spring shifts the second stable equilibrium to the
left with respect to the second stable equilibrium of the
system without the torsional spring.
2.2 Port 1 actuation
The required force to bring the mechanism from the
first to the second equilibrium, with port 1 actuation, can
be determined using the first derivative of the energy,
expressed in the displacement 𝛿1 at port 1. Since
𝛿1 = 𝑙10 cos 𝜃𝑒𝑞1 tan 𝜃,
2.1 Strain energy
The total strain energy for the mechanism is given by
the strain energy can be expressed in 𝛿1 by replacing 𝜃 in
the energy equations with
𝜃 = atan
𝑆𝐸𝑡𝑜𝑡𝑎𝑙 = 𝑆𝐸𝑡 + 𝑆𝐸𝑟
(1)
where 𝑆𝐸𝑡 is the strain energy of the translational spring
and 𝑆𝐸𝑟 is the strain energy of the rotating torsional spring.
The energy will be expressed as a function of 𝜃, which
indicates the inclination of the translational spring from the
horizontal.
The length of the translational spring, 𝑙1 (𝜃) is
√(𝑙10 cos 𝜃0 )2 + (𝑙10 sin 𝜃)2 . Using this length, the strain
energy in the translational spring can be written as:
or
1
𝑆𝐸𝑡 = 𝑘(𝑙10 − 𝑙1 (𝜃))2
2
𝑆𝐸𝑡 =
2
1 2 cos 𝜃𝑒𝑞1
𝑘𝑙10 [
− 1] .
2
cos(𝜃)
(5)
𝛿1
.
𝑙10 cos 𝜃𝑒𝑞1
(6)
The strain energy and its first derivative with respect to
the port 1 displacement 𝛿1 , are shown in Fig. 10. The
1
required force 𝐹1 is shown in the figure.
2
(2)
(3)
The strain energy in the torsional spring is given by
𝑆𝐸𝑟 =
1
2
𝑘 (𝜃 + 𝜃𝑒𝑞1 ) .
2 𝑡
(4)
The energy equations are evaluated for values of 𝜃
from −6° to 6°. The plots of the strain energy are shown in
Fig. 9.
Fig. 10. Strain energy, with the first derivative (i.e., the force) for
port 1 displacement
2.3 Port 2 actuation
The required force to bring the mechanism back to the
first equilibrium with port 2 actuation, can be determined
using the first derivative of the energy, expressed in terms
of the displacement 𝛿2 at port 2. First, an expression for 𝛿2
in 𝜃 is given. Using the kinetoelastic model in Fig. 11 it
can be found that
𝛿2 = 𝑙2 sin 𝛾 + 𝑙10 cos 𝜃𝑒𝑞1
(7)
𝛾 = 𝜃 + 𝛽 − 𝜋/2.
(8)
with
By replacing 𝜃 in the energy equations in terms of
𝛿2 using
Fig. 9. Plot of strain energy
Equilibrium positions can be located by searching for
the zeros of the first derivative of the energy. In addition,
the sign of the second derivative at these points determines
𝜃 = asin (
𝛿2 − 𝑙10 cos 𝜃𝑒𝑞1
𝜋
)−𝛽+ ,
𝑙2
2
(9)
actuations are shown in Fig. 13. The energy is plotted for
1
three values of 𝐹1 in Fig. 14 while 𝐹2 is set to zero. Fig.
2
15 shows the energy diagrams for three different values of
1
𝐹2 with 𝐹1 = 0.
2
Fig. 11. Port 2 actuation elements
we obtain energy in terms of 𝛿2 . The strain energy and its
first derivative with respect to the port 2 displacement 𝛿2,
are shown in Fig. 12. The required force 𝐹2 is also shown
in the figure.
Fig. 13. Energy diagrams for two actuations. A: 1st stable
equilibrium position at -4.570, B: 2nd stable equilibrium position
at 3.380, C: Stable equilibrium for port 1 force 𝐹1 at 4.740, D:
Stable equilibrium for port 2 force 𝐹2 at -4.75 0.
Fig. 12. Strain energy, with the first derivative (i.e., the force) for
port 2 displacement
2.4 Evaluation of the energy diagrams
The total strain energy together with the work potential
during the actuation gives the total potential energy of the
system while the system is actuated. Expressed in 𝜃, the
total potential energy is
𝑃𝐸𝑡𝑜𝑡𝑎𝑙 = 𝑆𝐸𝑡 + 𝑆𝐸𝑟 + 𝑊𝑃𝑃1 + 𝑊𝑃𝑃2
(10)
with 𝑊𝑃𝑃1 and 𝑊𝑃𝑃2 , the work potentials due to constant
forces at port 1 and 2 respectively.
1
𝑊𝑃𝑃1 = − 𝐹1 (𝛿1 (𝜃) − 𝛿10 )
2
The strain energy starts at zero energy, marked as ‘A’,
and reaches a stable equilibrium position at 𝜃𝑒𝑞2 = 3.38°,
marked as ‘B’. It represents the potential energy when no
force is applied. When a small force at port 1 is applied, the
energy curve shifts downward, having two minima (the
first near initial stable position and the second near the
final stable position) and one maximum. As the force
magnitude increases, the minimum shifts to the right while
the maximum shifts to the left, as shown in the Fig. 14. At a
critical value (𝐹1 = 0.255 N), the left minimum and the
maximum coincide to form an inflexion point. At this force
value the mechanism is pulled-in to the minimum
occurring at 4.74°, marked with ‘C’ in Fig. 13. Due to loss
of contact with the force, 𝑊𝑃𝑃1 goes to zero and the curve
snaps to the strain energy curve and settles into the second
stable equilibrium state at 3.38°.
(11)
with 𝛿10 the initial distance between the slider and the
horizontal axis
𝛿10 = 𝑙10 sin 𝜃𝑒𝑞1
(12)
𝑊𝑃𝑃2 = 𝐹2 (𝛿2 (𝜃) − 𝛿20 )
(13)
and
with 𝛿20 the initial distance between the slider and the
horizontal axis
𝜋
𝛿20 = 𝑙2 sin (𝜃𝑒𝑞2 + 𝛽 − ) + 𝑙10 cos 𝜃𝑒𝑞1 .
2
(14)
The potential energy diagrams for port 1 and port 2
Fig. 14. Enlarged view of port 1 energy diagram
During port 2 actuation, shown in Fig. 15, the energy
curves begin at the second equilibrium state. As with port 1
actuation, as the force gradually increases, the pull-in
occurs at a critical value (𝐹2 = 0.138 N) at which the
mechanism snaps back to its first equilibrium state. The
first equilibrium state with the force 𝐹2 present is marked
as ‘D’ in Fig. 13. The static critical value for port 2
actuation (𝐹2 = 0.138 N) is 54% of the static critical value
for port 1 actuation (𝐹1 = 0.255 N) for the chosen values
of the parameters in Table 2. It is clear that the force for
port 2 actuation is lower than the force for port 1 actuation,
as a result of the energy stored in the torsional spring and
because 𝛿1 and 𝛿2 are different.
1
𝑃𝐸𝑀 = − 𝑀𝑔(𝛿1 (𝜃) − 𝛿10 ).
2
(17)
While taking the same settings for the stiffnesses, the
bistable behavior of the system is lost, as can be seen in Fig.
16. In the energy plot, the work potentials of both port 1
and port 2 actuations are set to zero. The result is a
mechanism that has one stable equilibrium close to the
second equilibrium state of the mechanism without the
load.
Fig. 15. Enlarged view of port 2 Energy Diagram
Fig. 16. Energy plot when adding a mass
An alternate method to calculate the critical actuation
force is by using the derivatives of the potential energy.
When both the first and second derivatives of potential
energy are set zero and solved simultaneously, the critical
force and critical angle of actuation of the mechanism can
be obtained.
By adjusting the stiffnesses of the compliant
mechanism, the system can be made to become bistable
again. However, the behavior of that new design does only
work for load values close to the load used for the design of
the stiffnesses. It is more convenient to have a mechanism
that can be easily adjusted for different loads by
maintaining the same energy behavior and equilibrium
positions ( 𝜃𝑒𝑞1 and 𝜃𝑒𝑞2 ). Therefore, the use of static
balancing is explored next.
𝜕𝑃𝐸𝑡𝑜𝑡𝑎𝑙
=0
𝜕𝜃
𝜕 2 𝑃𝐸𝑡𝑜𝑡𝑎𝑙
=0
𝜕𝜃 2
(15)
Solving the preceding two equations simultaneously
for 𝐹1 (and 𝐹2 = 0) or 𝐹2 (and 𝐹1 = 0 ) gives 𝜃𝑒𝑞2 and
respectively 𝐹1𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 or 𝐹2𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 . Geometrically, the two
equations solve for the value of 𝐹𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 at which pull-in
occurs and the maximum and minimum of the potential
energy plot coincide. This effect can also be seen in Fig. 14
and Fig. 15: as force increases, the maximum and
minimum of the mechanism come closer, and they merge at
a critical force.
3.1 Static balancing
The concept of static balancing was already
introduced in the first section. First the energy behavior of
the rigid mechanism shown in Fig. 17 will be explored. As
can be seen in the figure, the rigid model is extended with a
zero-free length spring with stiffness 𝑘𝑏 at a distance 𝑎 of
the rotation point of the mechanism [4].
3 Two-port bistability including load
The described model so far did not include the load that
is present at the centre of the compliant mechanism.
1
Therefore the model is extended with a mass 𝑀 at the
2
slider (port 1) in Fig. 8. The influence of an additional load
is demonstrated by adding a mass 𝑀 of 30 g to the
kinetoelastic model whose parameters are listed in Table 2.
To describe the behavior of the mechanism including the
load, Eq. (10) is extended with the potential energy of the
mass, defined as zero in the initial configuration (𝜃𝑒𝑞1 )
𝑃𝐸𝑡𝑜𝑡𝑎𝑙 = 𝑆𝐸𝑡 + 𝑆𝐸𝑟 + 𝑊𝑃𝑃1 + 𝑊𝑃𝑃2
+ 𝑃𝐸𝑀
where
(16)
Fig. 17. Statically balanced rigid model
The potential energy of the mass is given by
1
𝑃𝐸𝑚𝑎𝑠𝑠 = 𝑀𝑔𝑙1 sin 𝜙
2
(18)
and the energy stored in the spring is given by
2
1
𝑆𝐸𝑏 = 𝑘𝑏 (𝑙2 + 𝑎2 − 2𝑙2 𝑎 sin(𝜙)).
2
(19)
Combining these two energy terms gives the total energy of
the mechanism as a function of 𝜙. It can be seen that when
1
2
𝑀𝑔𝑙1 = 𝑘𝑏 𝑙2 𝑎
(20)
In this model, length 𝑙1 is not constant; so according to
Eq. (21) the distance 𝑎 should vary during the motion to
maintain a perfectly statically balanced system. However,
the balanced spring is combined with the bistable system
and since length 𝑙1 will not change a lot in the range of
motion, the behavior of the total system is the same when 𝑎
is calculated for the fixed length 𝑙10 . The total energy is
now given by
𝑃𝐸𝑡𝑜𝑡𝑎𝑙 = 𝑆𝐸𝑡 + 𝑆𝐸𝑟 + 𝑊𝑃𝑃1 + 𝑊𝑃𝑃2
+ 𝑃𝐸𝑀 + 𝑆𝐸𝑏 .
(22)
The plot of Fig. 16 is compared with the energy
diagram of the balanced mechanism in Fig. 20, for 𝑘𝑏 =
50 N/m and 𝑎 = 16.7 mm.
the total energy is independent of 𝜙 and thus is constant for
every position, as visualized in Fig. 18. This means that the
system can be statically balanced by choosing the
parameters properly. The behavior can only be achieved by
rotating the line 𝑎 by 𝛽 with respect to the vertical axis, as
shown in Fig. 17.
Fig. 20. Energy plot of balanced mechanism compared with the
unbalanced mechanism (with and without mass)
Fig. 18. Energy plot of balanced rigid mechanism
Since 𝑎 is the easiest parameter to adjust, when the
mass is changed after assembly, the requirement for the
system is given as
𝑎=
𝑀𝑔𝑙1
2𝑘𝑏 𝑙2
.
(21)
Next the models of Fig. 8 and Fig. 17 are combined to
obtain the model shown in Fig. 19.
The method to calculate the required port 1 and 2
actuation forces to switch to the other state was discussed
in Sections 2.2 and 2.3. The forces to switch between the
two equilibrium states of the balanced systems are 𝐹1 =
0.255 N and 𝐹2 = 0.138 N, so the error due to the variable
length 𝑙1 is negligible.
3.2 Tuning the energy behavior
The two-port bistable mechanism including load
compensation, shown in Section 3.1 results in a
configuration where it is required to add an additional force
(besides the mass) to switch from the first to the second
equilibrium state. It is also possible to design the
mechanism such that the only the weight of the person is
enough to switch to the second state. The following three
system behaviors are considered.
A. Additional force to switch from state 1 to state 2 and to
switch back.
B. No additional force required to switch from state 1 to
state 2, but there is a force required to come back to the
first state. The first state has become an unstable state,
so the bistability is lost when the mass in attached.
C. The system is so designed that the dynamic effect of
adding the mass will switch the system from state 1 to
state 2. The advantage is that the system maintains two
stable states, but there is now no additional force
required to switch to the second state. A force is
required to switch back, this force is lower than the
required force with configuration B and higher
compared to configuration A.
Fig. 19. Static balanced kinetoelastic model
The energy plots of all three configurations are shown
in Fig. 21 and can be analyzed using the ball-on-the-hill
analogy. As can be seen in the graphs, configuration A
needs a force to switch to the second state, configuration B
is only stable in its second state and configuration C uses
dynamic properties to switch to the second state, but is still
bistable. The only difference between the configurations is
the distance 𝑎, as a result of which the influence of the
balancing spring is different. Decreasing 𝑎 will result in
increased port 2 actuation force that is required to switch
back to state 1.
Fig. 21. Energy plots for three different configurations obtained
by varying parameter 𝑎 of the balancing spring.
advantage. On the other hand, increasing the thickness
while keeping the initial height constant, increases the
mechanical advantage. The same is seen in the analytical
model as increasing the thickness corresponds to an
increase in torsional spring stiffness (𝑘𝑟 ).
4.2 Experiments
The critical actuation forces for the mechanism were
experimentally measured using a BiSS (www.biss.in)
planar bi-axial test system as shown in Fig. 23. The bi-axial
test system is used so that port 2 actuation forces can be
applied on the two ends simultaneously while port 1
actuation is applied in the perpendicular direction. For both
ports only a position (push) force is recorded, up to the
moment the force becomes zero and the mechanism snaps
through. The experiment is repeated six times with three
identical copies of the prototype. An average critical port 1
force of 0.244 N was found while the average critical port
2 force was 0.154 N. The port 2 force is 63% of port 1
force. The discrepancy between the experimental forces
and analytical models are 4.3% and 11.6% respectively.
It is to be noted that any variation in the separation
distance between the pivot points from the stress-free
distance in the bistable mechanism will cause pre-load.
This has a large influence on the critical force values and
the ratio between the forces.
4 Validation
The analytical model is validated using finite element
analysis and experiments.
4.1 ABAQUS simulation
Nonlinear large displacement finite element analysis
(FEA) was performed using ABAQUS (www.simulia.com).
2D continuum quadratic elements were used in the
simulation. While in the analytical model, only one half of
the mechanism is considered due to symmetry, in the FE
simulations, the complete model was simulated. The two
dimensional profile was simulated by imposing a hinge
boundary condition (BC) at the pivot point and a
displacement at the center of the mechanism for port 1
actuation and displacement at the center of the flaps for
port 2 actuation, as shown in Fig. 22. A roller BC at the
center of the mechanism ensures it moves vertically
downwards.
Fig. 22. ABAQUS model with boundary conditions
The critical forces, 𝐹1 and 𝐹2 were respectively found
to be 0.254 N and 0.139 N, which are in close agreement
with the critical forces predicted by the analytical model
with less than 0.39% and 0.72% discrepancy respectively.
To arrive at optimum dimensions of the mechanism,
simulations were run varying the thickness and initial
height. Increasing the initial height of the mechanism while
keeping the thickness constant, reduces the mechanical
Fig. 23. Experimental setup of the used bi-axial test system, with:
(1) and (3) as the actuators to apply a constant rate displacement
(0.2 mm/s) at port 1 and 2 respectively, (2) and (4) as the sensors
to measure the force at both ports and (5) as the compliant
mechanism fixed on a base. The forces required to describe this
displacement are measured during the actuation. In the figure port
1 actuation is shown.
In Table 3, the results of the experiment and
simulations are compared with the analytical
approximation. It is to be noted that the experimental
results are close to the simulation values and the analytical
approximation, thus reinforcing the confidence in the
design and the prototype of the full-model. The measured
force for port 2 actuation is higher due to large influence of
the friction in the pivot joints.
Table 3. Comparison of critical (maximum) forces
Results
Analytical approximation
ABAQUS simulations
Experiments
𝐹1 (N)
0.255
0.254
0.244
𝐹2 (N)
0.138
0.139
0.154
5 Design of the easy chair
Having demonstrated the feasibility of the two-port
bistable mechanism to provide mechanical advantage, an
all-mechanical two-port bistable compliant easy-chair is
designed to gently propel and ease rising from it. The
addition of the balancing spring results in a chair that is
easily adjustable for people with different weights without
changing the stable positions of the mechanism. The
distance 𝑎 can also be used to change the energy behavior
of the system, as shown in Section 3 with three different
configurations. Depending on the final usage and the needs
of the user, the distance 𝑎 can be adjusted. A scale can be
added to guide the user (or the caregiver) to choose the
proper setting.
The port 2 force will be applied on the compliant
mechanism using the arm rests. A transmission can be used
to reduce the force even further. A bell-crank was chosen to
be the suitable actuation to transfer the force from the
armrest to the mechanism. The bell crank allows for
amplification of the force applied on the armrest, thus
enabling sufficient port 2 force to be applied on the
mechanism.
The seat of the chair can be designed as an extrusion of
the two-port bistable profile. By dividing the seat in several
compliant profiles with a different initial configuration and
mounting it on an inclined pin, as shown in Fig. 24, the
configuration of the stable positions can be further adjusted.
The shown configuration eases the rising from the chair
and maintains a comfortable inclined seat.
Fig. 24. Possible seat configurations
The complete chair frame was designed and prototyped
as a scaled model using 3D printers (Objet Connex and
Z250) to check the gross behavior of the chair. The
prototype is shown in Fig. 25. It was observed that the
combined effect of the mechanical advantage created by
the two-port bistable mechanism and the bell crank lever
results in lower actuation force at the armrest as compared
to the force required on the seat to switch to the second
state.
6 Conclusions
We analyzed a special bistable mechanism called a
two-port bistable mechanism, where the forces to switch
from one stable state to the other and vice-versa are
different as are the ports of actuation. An analytical
kinetoelastic model, which closely matches experimental
results, has been developed, which affirms the feasibility of
the mechanism for use in an easy-chair. The theory of static
balancing is used to make the chair easily adjustable for
Fig. 25. A scaled prototype of the easy-chair
different modality of use. The prototyped easy-chair is
inexpensive to manufacture and is fully mechanical. The
chair is therefore of particular interest for countries with
unreliable power supply in contrast to the existing
electrically actuated easy-chairs.
The prototype of this mechanism exhibits two-port
bistability with port 2 actuation force being lower than port
1 actuation force. To provide the force required for
actuation, a bell crank mechanism has been implemented.
Further research is needed to design the most
ergonomic and anthropometrically pleasant chair, to ensure
comfort while sitting and ease while rising from the chair,
which will depend on the capabilities of the users and the
different usage modalities.
The two-port mechanism proposed has been treated
using a single degree of freedom model, which inherently
ensures the same mode of deformation for both port 1 and
port 2 actuation. However, the click-clack tin-lid
mechanism appears to deform differently during the two
actuations. As such, a higher degree of freedom model
must be developed to capture the “bimodal” nature of the
mechanism. This is explored in our ongoing work.
Acknowledgment
The authors gratefully acknowledge the financial
support from the Department of Science and Technology
(DST) of the Government of India and Technology
Initiative for the Disabled and the Elderly (TIDE). We also
acknowledge Amit Kumar Singh, Akshay Varik, Rahul R
Kamath and H. R. Rangavittal from B.M.S. College of
Engineering, Bangalore and the members of the M2D2 lab,
specifically Dr. Santosh Bhargav and Anirudh N Katti, of
the Indian Institute of Science for their inputs in the design
of the chair.
References
[1] http://www.medicinenet.com/script/main/art.asp?articlekey=
23220, December 2014.
[2] http://www.la-z-boy.com/Collections/Lift-Chairs/,
December 2014.
[3]_ http://www.hermanmiller.com/products/healing-spaces/pat
ient-seating/cente-patient-chair.html, December 2014.
[4] Herder, J. L., “Energy-free Systems, Theory, conception and
design of statically balanced spring mechanisms”, PhD
Thesis, Delft University of Technology, Delft, The
Netherlands, November 2011.
[5] http://www.sprinq.eu/, December 2014.
[6] Jensen, B. D., and Howell, L. L., “Bistable Configurations of
Compliant Mechanisms Modeled Using Four Links and
Translational Joints”, Journal of Mechanical Design, Vol.
126, pp. 657–666, July 2012.
[7] Opdahl, P. G., Jensen, B. D., and Howell, L. L., “An
Investigation into Compliant Bistable Mechanisms”, Proc.
1998 ASME Design Engineering Technical Conf.,
DETC98/MECH-5914.
[8] B.D. Jensen, L.L. Howell, “Identification of compliant
pseudo-rigid-body four-link mechanism configurations
resulting in bistable behavior”, Journal of Mechanical
Design, Transactions of the ASME, Vol. 125, pp. 701–708,
2003
[9] Gallego, J. A. and Herder, J. L., “Criteria for the static
balancing of compliant mechanisms”, Proceedings of the
ASME Design Engineering Technical Conferences &
Computers and Information in Engineering Conference,
Montreal, Canada, 15–18 August 2010, DETC2010-28469,
2010.
[10] Deepak, S. R., “Static balancing of rigid-body linkages and
compliant mechanisms”, PhD Thesis, Indian Institute of
Science, India, May 2012.
[11] Deepak, S. R. and Ananthasuresh, G. K., “Static balancing
of a four-bar linkage and its cognates”, Mechanism and
Machine Theory, Vol. 48, pp 62–80, February 2012.
[12] Deepak, S.R. and Ananthasuresh, G. K., “Perfect Static
Balance of Linkages by Addition of Spring but not Auxiliary
Bodies,” ASME Journal of Mechanisms and Robotics, Vol.
4, May 2012, pp. 021014-1 to 12.
[13] Hoetmer, K., Woo, G., Kim, C., and Herder, J. L., “Negative
stiffness building blocks for statically balanced compliant
mechanisms: design and testing”, ASME Journal of
Mechanisms and Robotics, Vol. 2, November 2010.
[14] Zhang, S. and Guimin, C., “Design of compliant bistable
mechanism for rear trunk lid of cars”, Intelligent Robotics
and Applications. Springer Berlin Heidelberg, pp. 291-299,
2011.
[15] Howell, L. L., Magleby, S. P., and Olsen, B., “Handbook of
Compliant Mechanisms”, John Wiley & Sons, 2013.