Mechanical bearings with tunable compliance after biological role 
model of blood sinus hairs 
Bachelor:  Jorge Luis Ingaroca Quispe 
Thesis submitted for the degree of 
Master of Science in Mechatronics 
Advisors, Ilmenau: 
Dipl.-Ing. Thomas Helbig 
Advisors, Lima: 
Dr. Ing. Jorge Alencastre 
Supervising Professor, Ilmenau: 
Univ.-Prof. Dipl.-Ing. Dr. Med.(habil.) Hartmut Witte 
Ilmenau – Germany 
2016 
Hereby, 1 declare that I have elaborated the present work without any non-specified 
assistance. The people involved in the research, literature, as well as any another 
resource used in this thesis, has been completely specified throughout and the end of 
the text. 
26th April 2016, llmenau. 
\ 
Abstract
In follicle sinus complex of sinus hairs, two blood vessels seem to have a prominent
function. Up to date, their biological role is unknown, however, hypothesis suggest
that they are used as hydraulical bearing for the hair which is used to change the
stiffness and compliance of the fixation depending on the application. Because of
the size of the structures of interest experiments at the living object to clarify the
biological role are not possible so far. Therefore, mechatronic approaches can help to
investigate advantages and possibilities of a bearing with tunable compliance. Thus, in
the current thesis, it is develop a mathematical modeling, multi-body simulation and
a mechatronic demonstrator of a swinging rod in a bearing with tunable compliance.
The tunable compliance of the system was inspired of Jack Spring principle of work,
which change the compliance by decreasing the number of active coils.
Acknowledgment
I would like to express my gratitude to my advisor in TU-Ilmenau, Dipl.-Ing. Thomas
Helbig, for his immense support and advices during the develop of this thesis.
My sincere thanks to Professor Witte, for allowing me to work in this thesis theme of
the Department of Biomechatronic Engineering.
I appreciate the patience and support of Professors Dr. Ing. Jorge Alencastre and Dr.
Ing. Julio Tafur, from PUCP, during the development of this current thesis.
I would also like to express my gratitude to my family who supported me and always
believes in me, even before I do myself. A whole year far from them made me realize
I love them much more than I ever thought.
I would like to thank to my friends who accompanied me in this experience. Their
support was fundamental in this last section.
Finally, but not less important, I would like to express my gratitude to all the people
who always cared of me, with a call, a message, supporting words or being with me in
this stage of my life.
Contents i
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Biological paradigm 2
2.1 Vibrissa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Follicle-sinus Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 State of the art 7
3.1 Mechanical models of Follicle-sinus complex . . . . . . . . . . . . . . . 7
3.2 Actuator with adaptable compliance . . . . . . . . . . . . . . . . . . . 10
4 Theoretical foundation 14
4.1 Vibrations and damping . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1.1 Case 1: Under-damped system . . . . . . . . . . . . . . . . . . . 17
4.1.2 Case 2: Critically damped system . . . . . . . . . . . . . . . . . 17
4.1.3 Case 3: Overdamped system . . . . . . . . . . . . . . . . . . . . 18
4.2 Spring Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2.1 Stresses in Helical Spring . . . . . . . . . . . . . . . . . . . . . . 21
4.2.2 Curvature effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.2.3 Deflection of Helical Springs . . . . . . . . . . . . . . . . . . . . 23
4.2.4 Compression springs . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.6 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2.7 Design for static service . . . . . . . . . . . . . . . . . . . . . . 30
4.2.8 Critical frequency of Helical Springs . . . . . . . . . . . . . . . . 31
4.2.9 Fatigue loading of Helical Compression Springs . . . . . . . . . 33
Master Thesis of Jorge Luis Ingaroca Quispe
Contents ii
5 Mechanical modeling 35
5.1 Description of the 1DoF model . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Evaluation of spring as a compliant component . . . . . . . . . . . . . 37
5.3 Multi-body model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3.1 Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3.2 Couplings for limitation of active coils . . . . . . . . . . . . . . 43
5.3.3 Rigid body model . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Simulation of proposed system 49
6.1 Matlab simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Solidworks simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.3 Experiment results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.3.1 Comparison between Mathematical model and Multi-body model 53
6.3.2 Analysis of the compliance . . . . . . . . . . . . . . . . . . . . . 58
7 Conclusions 60
Bibliography 61
Master Thesis of Jorge Luis Ingaroca Quispe
1 Introduction 1
1 Introduction
1.1 Motivation
Nature, in its wise development, provided living beings with characteristic that made
them possible to survive and adapt to the conditions of their environment. There are
several differences among the characteristics of all the species of the world. Most of
those characteristics that living beings own, are still unknown for humans; however,
the understanding of the biological characteristics of the creatures in nature could
open multiple possibilities to be employ to solve problems update without solutions
or improve them with new advantages. In this context, many researches focused on
rats and their abilities such as nocturnal rodents to perform their locomotion with
high precision. These studies give some evidences that rats use a complex method of
detection surface and objects by using their vibrissa. The State of the are about vib-
rissa and Follicle-sinus complex provides information about how biological mechanism
inside the follicle-sinus complex might change viscoelastic properties in order to switch
operating modes. This variation of viscoelastic properties, hypothetically is achieved
by adjusting the flow of blood in the blood vessel inside the follicle-sinux complex.
Here, the concept of tunable compliance owns a great meaning. The idea of tunable
compliance has a prominent future in many fields of engineering. This ability to vary-
ing the viscoelastic properties of an actuator would mean a higher energy efficiency.
For example, in rehabilitation, prosthesis and walking biped robot, actuators with
tunable compliance improve their developments. There have been many researches
about actuators with tunable compliance, and had reached good results to be employ
in robotics. Besides, for control systems, it can be long employed to change parameter
that are not common to be tuned, providing robustness to the system. In summarize,
compliant actuators have a great potential for applications in mechatronics.
Master Thesis of Jorge Luis Ingaroca Quispe
2 Biological paradigm 2
2 Biological paradigm
2.1 Vibrissa
The vibrissa are sinus hairs that are commonly found in rats, cats, seals and rodents.
These sinus hairs provide them special skills not only for social interaction with their
own kind, but for grasping activities that involves the control of their locomotion
([KKL+10],[Bed00]). In the case of marine mammals, vibrissae is useful to detect
pressure changes [MAKL06]. Rats are known as nocturnal rodents, since their vibrissa
offer them detection of environment characteristics as a non-visual sensory system. It
is important to remark that vibrissa features, e.g. number, geometric arrangement,
size, morphology, shape and stiffness, vary among animals [GDFM12].
Figure 2.1: Vibrissae classification according of its location [Car10] (1)mystacial (2)rhi-
nal (3)supraorbital (4)infraorbital (5)postorbital (6)buccal (7)mystacial
(8)interramal (9)anconeal (10)ulnar carpal (11)calcaneal (12)medial ante-
brachial
In figure 2.1, it can be seen the different types of vibrissa that exists according to its
location on rats body. Also, according to its position has a position relationship and
Master Thesis of Jorge Luis Ingaroca Quispe
2 Biological paradigm 3
numbering, as it can be viewed in 2.2.
Figure 2.2: Position relation and numbering of carpal Vibrissa [SWZ+14]
Up to date, there are more knowledges about structure and function of mystacial sinus
hair than carpal sinus hair. However, lately, carpal sinus hairs have gained inter-
est because there are researches that evidence its influence in kinematics of rat legs
([HVN+14b], [BSWZ13], [Dör82]). Hypothesis, also suggest that carpal sinus hair is
used to vary the stiffness in the follicle-sinus complex of the legs in order to prepare
them for contact with substrate ([BSWZ13], [HVN+14b]). In the whole structure
of vibrissa, there is not any receptor to transmit the external information (external
forces) because it is composed by a dead material [BSWZ13]. Thus, the mechanore-
ceptors in the follicle-sinus complex, as it can be seen in figure 2.5, make possible
the detection when they suffer a pressure mechanically transmitted by the vibrissa.
The number of mechanoreceptors between mystacial and carpal sinus hairs are dif-
ferent, so it is understood that they have different functions ([FAR95], [EKM+02]).
FUNDIN ([FAR95]) explains that carpal sinus hairs have an importance in activities
such as eating and grasping or also in detection of vibrations of the ground. There
are also hypothesis about its influence in locomotion and kinematics of rat legs. Re-
searches made in [NWS15] and [HVN+14a] demonstrate that movements of rat vary
when they walk without carpal sinus hairs. Therefore, carpal sinus hairs might have
a function for locomotion. The mystacial sinus hair detects vertical obstacles and
can measure the spatial body position [SWZ+14]. Consequently, there should be an-
other sensor to guarantee the skill of recognizing obstacles and substrate composition
in order to stabilize the body position [SWZ+14]. This is why carpal sinus hairs is
Master Thesis of Jorge Luis Ingaroca Quispe
2 Biological paradigm 4
supposed to recognize irregular substrate when touchdown occurs [HVN+14a]. Carpal
sinus hairs has different characteristics than mystacial sinus hairs such as length and
curved shape. Because the position of forelimbs is important to stabilize locomotion,
carpal sinus hairs might have a prominent function to detect horizontal changes in
surface ([HVN+14a], [SWZ+14]). BERG and KLEINFELD distinguished two whisker
patterns in rats: exploratory whisking, when rats explore the environment sweeping
their whiskers with high amplitude and low frequency (5-15 Hz); and foveal whisking,
when rats contact surface of objects with small amplitude and high frequency (15-25
Hz) [BK03].
2.2 Follicle-sinus Complex
The tactile signals are transmitted by the vibrissae to the receptors in Follicle-sinus
Complex. The Follicle-sinus complex (FSC) is a sensory receptor located in the root
of vibrissae; which supports the sinus hair and contains many components that allow
the detection of force signals ([Car10], [SWZ+14], [HVN+14a], [BSWZ13]). The FSC
is composed by vibrissal shaft and a follicle which has an outer root sheath surrounded
by a glassy membrane ([KKL+10], [FAR95]). The blood sinus encloses the follicle. The
blood sinus is divided by two parts; a superior part, named Ring sinus, and a inferior
part, named Cavernous sinus ([Fli14], [BSWZ13], [Car10]). In the ring sinus, there is
a C-shaped Ringwulst. A collagenous capsule encloses the vibrissal shaft and blood
sinus, until the neck of the follicle creating an outer and inner conical body [KKL+10].
In the cavernous sinus, the superior part has less blood than the inferior one. The
Follicle-sinus complex has an important function because amplifies the movement of
the vibrissae, so the mechanoreceptors located at the base of vibrissa can detect small
stimuli [KKL+10].
Master Thesis of Jorge Luis Ingaroca Quispe
2 Biological paradigm 5
Figure 2.3: Histology of vibrissal Follicle-sinus complex [KKL+10]. (a) Sagittal section
of Follicle-sinus complex (b)Transverse section of ring sinus and ringwulst
(c) Transverse section of carvernous sinus (d) Transverse section of cav-
ernous sinus in the lower level. C=Collagenous capsule; CS=Cavernous
sinus; DVN=Deep vibrissal nerve; ICB=Inner conical body; OCB=Outer
conical body; ORS=Outer root sheath; RS=Ring sinus; RW=Ringwulst;
S=Vibrissal shaft; VCM=vibrissal capsular muscle
The FSC and the muscle system allow rodents to use their vibrissa in two modes of
operation: Passive mode, and Active mode. In passive mode, when the vibrissa is
deflected by external forces and return passively into its initial position. In active
mode, vibrissa is swung by contractions of intrinsic an extrinsic muscles ([BSWZ13],
behn2009finite). The figure 2.4 shows a schematic drawing of FSC where there are
intrinsic and extrinsic muscles. So, triggering oscillations with different frequencies
and amplitudes, rodents are able to investigate objects and its characteristics, such
Master Thesis of Jorge Luis Ingaroca Quispe
2 Biological paradigm 6
as shape and surface, in a quickly way. Hitherto, the way they can convert that
information of their surroundings is still unknown [BSWZ13].
Figure 2.4: Schematic drawing of Follicle-sinus-complex [BSWZ13]
Figure 2.5: FSC ofa vibrissa with different receptors (blue) [BSWZ13]
Theory suggest that probably rodent can adjust the stiffness of the support in the
vibrissa by varying the flow of blood in the ring sinus, so it changes the pressure in
the blood-sinus and, of course, its viscoelastic properties [Dör82].
Master Thesis of Jorge Luis Ingaroca Quispe
3 State of the art 7
3 State of the art
3.1 Mechanical models of Follicle-sinus complex
Because the mechanical behaviour of the FSC might be significant in the development
of autonomous robots, there have been researches trying to identify the mechanical pa-
rameters involve on it in order to study the dynamical behaviour of vibrissa [BSWZ13].
Usually, the models used are rigid body models and continuum models. In [MGR+04],
researches create an electro-mechanical model of rat mystacial Follicle-Sinus complex in
order to give a clearer idea about the transduction process to recognize the information
of force given by vibrissa to its Follicle-sinus complex.
Figure 3.1: (a)Simplify FSC anatomy (b)effect of deflection and (c)exploded FSC me-
chanical model [MGR+04]
Master Thesis of Jorge Luis Ingaroca Quispe
3 State of the art 8
The biological elements involve in the follicle are represented by dampers and springs.
There are also researches focussing on the relationship among the muscles associated
in vibrissae motion [HBZK08]. To achieve that, three follicles are linked by dampers
and springs among them in order to represent the influence of intrinsic muscle in the
other follicles.
Another approach is presented in [BK03], where the passive elements used in the
mechanical model of whisker represents the components in the FSC such as skin,
plate, and intrinsic and extrinsic muscles as it is shown in figure 2.4.
Figure 3.2: Model of mystacial vibrissae regarding components such skin, plate, and
intrinsic and extrinsic muscles [BK03]
Figure 3.3: Model of a rod-like vibrissa of 1 DoF. Where MU(t) represents the torque
produced by rodents in their vibrissae to use it in active mode [BSWZ13]
Master Thesis of Jorge Luis Ingaroca Quispe
3 State of the art 9
Figure 3.4: Model with multiple degrees of freedom [BSWZ13]
However, to achieve a better accuracy of the mechanical system involve in the FSC
of a vibrissa, the higher quantity of elements in the FSC should be represented. In
figure 3.5 a complex vibrissae model is shown with multiple springs and dampers. In
this model, the viscoelastic properties of the blood-sinus are represented by compliant
elements (spring and damper) [SSWB11].
Master Thesis of Jorge Luis Ingaroca Quispe
3 State of the art 10
Figure 3.5: Complex model of vibrissae in the mystacial pad [SSWB11]
3.2 Actuator with adaptable compliance
In the growing fields of wearable robotics, rehabilitation robotics, prosthetics, and
walking robots, actuators with adjustable compliance have gained a great importance
due to its ability to minimize large forces produced by shocks, to interact safely with
users, and to store and then release energy in passive elastic elements [HSV+09]. In
classic robotic, the actuators are desired to be as stiff as possible in order to reach
accurate positions or trajectories. However, mechanically, muscle has a great potential
because of its superior functional performance and control system with the advantage
of tunable parameters ([VHVVD+07], [HSV+09]). It is important to notice that sys-
tems with tunable compliance has less accuracy than stiff actuators. There are two
groups of applications that require system with adaptable compliance: Robot-human
interaction and for adjusting natural dynamics [HSV+09].
Robot-human interaction
In this first group, it is required to guarantee a safety and natural interaction between
robot and user. For instance, in industrial robots, the actuator are really stiff, and
do not relief the damage produced by shocks or collisions. It is not a big deal while
Master Thesis of Jorge Luis Ingaroca Quispe
3 State of the art 11
working in human-free environment. However, for application where exists interaction
with humans, it is needed to work with safer robots [ZKRS04]. In that case, the actu-
ator would act with a predefined stiffness when it is required to operate with precise
positioning, and compliant when it requires higher speeds. For rehabilitation, as well,
is required the dissipation of undesired movements. For example in the case of pa-
tients who suffer spasms, stiff actuators could damage their legs. Thus, actuators with
tunable compliance would reduce those errors preventing the damage to the system
and the user. Besides, at the beginning of rehabilitation process, it is desired a low
stiffness; however according to the progress of the patient by using the system, it will
be necessary a higher stiffness ([VHVVD+07], [HSV+09]).
Adjustment of natural dynamics In this group, it is required the adjustment of the
compliance of an actuator in order to change the natural motion into a desired motion.
It is applicable in fields of robotic prosthesis, where is required and adjustment of the
stiffness in actuators to enhance the comfort for user. Thus, by using stiff actuator,
the stiffness is fixed and would work just for certain applications. For instance, in
transtibial prosthesis [HSV+09], with a fixed stiffness will operate comfortably for user
but just for certain type of surface. Therefore, the ability of tuning the stiffness would
optimize the behaviour of the system. As well, for walking biped robots, tunable com-
pliance of actuator would upgrade the amount of energy because it could be stored
during the touchdown and then released during push-off [HSV+09]. Also, by varying
the stiffness, the frequency of the system is adjusted allowing the system to walk slower
or faster.
To achieve energy storage as well as adaptable compliance, it is required an elastic
element. Many researches has presented novel design to reach these objectives.
In figure 3.6, the principle of work of MACCEPA is shown. MACCEPA is a novel
compliant system. The compliance is achieved by pre-stretching the spring with a
servomotor, so it would change the response and frequency of the joint. This is a novel
compliant actuator that have won great importance because of its simplicity.
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3 State of the art 12
Figure 3.6: Working principle of MACCEPA [VHVVD+07]
In figure 3.7, the Jack spring actuator has an interesting manner to change the stiffness
by varying the number of active coils of the spring. It is because the stiffness of
the spring depends on some parameters such as wire coil, wire diameter, modulus
of rigidity, and number of active coils. Because modifying the values of the other
parameters is not that simple, it focuses on varying the number of active coils. Number
of active coils are inversely proportional to stiffness.
Figure 3.7: Concept of ’Jack Spring’ [HSH05]
In figure 3.8, it can be seen an actuator with two extensions springs that provide the
compliance, both springs are linked to an element controlled by a motor. In order to
Master Thesis of Jorge Luis Ingaroca Quispe
3 State of the art 13
safe space, a spring steel wire was also used as a replacement of extension springs.
Figure 3.8: Model proposed in [SCS05]. (a)Extension springs used to generate the
compliance in the actuator (b)Instead of extension springs, a single steel
wire is used to save space
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 14
4 Theoretical foundation
4.1 Vibrations and damping
The interest of the study of vibrations comes from ancient times where the first mu-
sical instruments were created. In their study they have involved great personages
of science as Pythagoras, Hooke, Galileo. The phenomenon of mechanical vibrations
has been part of the considerations on mechanical structures because it improves their
designs and prevent the collapse of these, as is in the case of the effects of resonance
frequency.
Vibrations or oscillations are movements that are repeated in a time interval around a
stable equilibrium position. A vibrating system contains three components. The first
one transfer potential energy. The second one stores the potential energy as kinetic
energy. And the third one is that dissipates the energy of the system. Therefore, in
a vibration exists an alternating transfer of kinetic energy to potential, and in some
cases contain an energy dissipater element. Mechanical vibrations can be classified in
multiple ways.
Free and forced vibrations
Free vibrations are those in which only inherent forces to the system elements act in
vibration. In other words, it does not use any external force on the system. In con-
trast, the forced vibrations are those where exists one or more external forces acting
on the vibration. At this point it is important to remember that if the frequency of
the external force is equal to the natural frequency of the system, the system goes into
resonance which is subjected to very large swings that could bring down the system.
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 15
Damped and undamped vibrations
If there is no loss of energy in the oscillations of a vibration, it would be a case of
undamped vibration. Then, in vibrations where their oscillations amplitude decreases
in each swing, they are called damped vibrations.
Linear and non-linear vibrations
If the components of a vibratory system have linear behavior, then we say that the
system results in a linear vibration. Conversely, if the components of a mechanic vi-
bration are non-linear, it implies that the vibration will be non-linear.
Then it may be the case where there is free or forced vibration damped and non-
damped. For the current thesis, theoretical behaviour of free damped vibrations will
be described.
The figure represents a mass-spring-damper system which is a system with simple
harmonic motion movement and damping.
Figure 4.1: Mass-spring-damper system [RY95]
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4 Theoretical foundation 16
In the equation 4.1, it is described the equation of forces.
mẍ+ cẋ+ kx = 0 (4.1)
Where:
m: represents the mass of the block
k: stiffness of the spring
c: damper of the system
To solve equation 4.1, we assume as a solution:
x(t) = Cest (4.2)
Where C and s are undetermined constants. So, if we replace in the equation 4.1 the
next equation is obtained:
ms2 + cs+ k = 0 (4.3)
Then, from equation 4.3 we obtain the following roots:
√ √
− ± 2 − 4 ( )
c c mk c c 2
s1,2 = 2 = −2 ± 2 − k (4.4)
m m m m
( √ ) ( √ )
2
− c + ( c − k t − c − c 2
− k t
( ) = 2m 2m) m 2m ( 2m) m
x t C1e + C2e (4.5)
The values of constants C1 and C2 are constants determined from initial conditions
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4 Theoretical foundation 17
4.1.1 Case 1: Under-damped system
This case occurs when it satisfies this situation:
√
c k
2 < (4.6)
m m
In this case the roots are complex and the response will result:
( √ ) ( √ )
c k c 2
− c k c 2
( ) = 2 +i −( 2 ) t − 2 −i −( 2 ) t
m m m m m m
x t C1e + C2e (4.7)
Then:
  √ ( )   √ 
2 ( )2 
x(t) = e−
c
2 t
m (C1 + C2) cos k c
t −  k c
2 + i (C1 − C2) sint − 
m m m 2m 
(4.8)
√ ( )  
2
x(t) = Xe−
c
2 t cos k − c
m 
2 t+ Φ (4.9)
m m
Where X and Φ can be obtained from initial conditions.
Thus, from equation 4.9, it can be predicted that its response will be periodic with a
tendency to decrease to zero. These vibrations are called under-damped vibrations.
4.1.2 Case 2: Critically damped system
It occurs when the following is true:
√
c = k
2 (4.10)
m m
Therefore, the roots are purely real.
√
s1 = k
s2 = (4.11)
m
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 18
Then, a solution for the response is given for the next equation:
( ) = ( + ) − c
x t C t
1 C2t e 2m (4.12)
Because of the effect of exponential function the response of this system will be non-
periodic with a tendency of decrease rapidly to zero.
4.1.3 Case 3: Overdamped system
This system occurs when:
√
c k
2 > (4.13)
m m
So, the roots are real but distinct and are given by:
√( )
c c 2
s1 = − + k
2 2 − (4.14)
m m m
√( )2
s2 = − c c k
2 − 2 − (4.15)
m m m
Where s1  s2, and the response of this system would be given by:
 (√ ) ( √ ) 
c 2 k c 2 k 
( ) = − c 
( 2 − t − − t
m) m ( 2m) m
x t e 2m C1e + C2e  (4.16)
The response of the system would be non-periodic. However, it decreases slowly to
zero.
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4 Theoretical foundation 19
Figure 4.2: Comparison of different types of damping [RY95]
In the graphic 4.2, it is shown the responses of the three cases of free vibrations with
damping.
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 20
4.2 Spring Design
Springs are mechanical elements with elastic properties which store energy when they
suffer longitudinal deformation and regain its initial length. They can be used for
multiple applications where it is required to apply a force and then being returned
as energy; e.g. vehicle suspension, connection cables, foldable chairs. Because of its
importance in the industry, they have been thoroughly studied for development and
manufacture taking into account the materials and the calculations necessary to com-
ply with their work properly. They can be classified in many ways e.g as spring form
(cylindrical helical, conical helical, spiral), shaped cross section (circular, square, rect-
angular) and the type of burden (compression, extension, torsion).
Spring’s main importance is the ability to store energy. So, for its construction there
should contains material with high Elastic Modulus (E). Depending of their applica-
tions, it could be also desired to store energy without excessive losses or to absorb
energy as much as possible to avoid rebound [Chu10].
Figure 4.3: Graphs of response of Force vs Deflection for a spring with linear (left) and
non-linear (right) behaviour[Chu10]
In figure (4.3), it can be seen that there are cases when springs have linear and non-
linear characteristic. In the first case, where loading and unloading lines are straight
(constant stiffness), the spring is composed of metal springs that have Low Hysteresis.
In contrast, in the case where there is non-linear behaviour of the spring, it can be
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4 Theoretical foundation 21
made of material such as rubber or polymer [Chu10].
4.2.1 Stresses in Helical Spring
The figure 4.4 shows a compression coil spring, made with round wire, supporting an
axial load. The parameter D indicates the mean diameter of the spring, while the
parameter d indicates the wire diameter of the spring.
Figure 4.4: Helical Spring loaded axial [ML+13]
The maximum shear stress becomes from superposition of direct shear stress and tor-
sional shear stress as it is seen in the next equation [SBM15]:
Tr F
τmax = + (4.17)
J A
Where τmax = τ , T = FD/2, r = d/2, J = πd4/32, and A = πd2/4. Finally, it is
obtain the following:
= 8FD + 4F
τ 3 (4.18)
πd πd2
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 22
Then is defined C as spring index:
D
C = (4.19)
d
According to Design Handbook [Spr81], the value of C must be between 4 and 12.
Low index could be difficult to manufacture and its process could trigger some small
damages that could reduce life time. Otherwise, high index is difficult to handle, could
tangle and may need a individual packaging [Spr81].
So, developing equation 4.18 and replacing spring index (C):
( )
= 8FD + 8FdD = 8FD 1 + d
τ
πd3 2πd3D πd3 2D
8 (2 + 1)= FD C = 8FD
τ 3 2 KS 3 (4.20)
πd C πd
(2C + 1)
KS = 2 (4.21)
C
Where KS is the shear stress-correction factor
4.2.2 Curvature effect
Because the wire in not straight, but curve, the curvature must be taken into consid-
eration. It will increase the stress in the inner surface of the coil ([Cor15], [SBM15],
[ML+13]) for that reason, it is an important factor to be considered in spring design.
This effect should be used for fatigue applications.
If the Curvature effect is added to shear stress-correction factor, the equation is ob-
tained:
τ = 8FD
K 3 (4.22)
πd
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 23
Where K can be replace for the next two forms:
4C + 2
K = KB = 4 3 (4.23)
C −
= = 4C − 1 + 0.615
K KW 4 4 (4.24)
C − C
KB and KW represents Bergsträsser factor and Wahl factor respectively. Literature
indicates that the difference between these two factor is around 1%; however, Wahl
factor is most used [SBM15].
4.2.3 Deflection of Helical Springs
When there is a load over compression springs, spring wire has a shear stress and
torsional stress because load tries to twist the coils in relation to the axis. To deter-
mine the relation between the deflection and force applied on the spring, Castigliano’s
theorem is commonly used. It describes the strain energy for a helical spring produced
by torsional and shear components [SBM15]. The strain energy in the spring is:
T 2
= l 2
U 2 + F l
GJ 2 (4.25)
AG
Where T = FD/2 (Torque), l = πDN (length of wire), J = πd4/32 (Moment of
inertia), A = πd4/4 (Transversal area of spring). Then it results in:
4F 2D3
= N + 2F 2DN
U 4 2 (4.26)
d G d G
N = Na is number of active coils.
To determine the deflection, Castigliano’s theorem is used:
= ∂U
δi (4.27)
∂Fi
Then:
8FD3 ( )
N 2 3 ( )
δ = 1 + d 8FD N 1
d4G 2D2 = 4 1 + 2 2 (4.28)
d G C
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 24
( )
The term 1 + 1
2 2 can be approximated to zero when 4 < C < 12
C
Then:
≈ 8FD3N
δ 4 (4.29)
d G
Finally, obtaining the spring rate:
F Gd4
k = ≈ 8 3 (4.30)
δ D N
4.2.4 Compression springs
Compression springs are used to support compressive stresses and shock. Therefore
they reduce their volume when a force exerted on them, making them energy storage
devices very efficient. According to its geometry, one can find multiple compression
springs of different shapes: cylindrical coil springs, coil stamping, conical helical, bi-
conical helical.
Compression springs have separate coils evenly. They are compressed when forces are
applied at its ends, when this happens, its coils are equally close since they receive the
same pressure resistance. Compression spring must never be deflected until its solid
length because it would trigger a permanent deformation. Actually, it is suggest to
do not deflect it too much because at the last 15% of its total deflection, the stiffness
becomes non-linear due to there would be some active coils at the end of the spring
which will close becoming them inactive.
Figure 4.5: Compression spring [Cor15]
As it can be seen in figure 4.6, there are four typical types of end.
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 25
Figure 4.6: Types of ends of compression springs [SBM15]
Figure 4.7: Formulas for Dimensional Characteristics of Compression springs [SBM15]
4.2.5 Stability
Spring must be studied for avoiding case of buckling when there is a large load. So, to
verify that spring will not suffer buckling, the critical deflection should be calculated,
and is given by:
 ( ) 
′ 1/2
ycr = ′ C 2
L0C 1 1− 1− 
2 (4.31)
λcff
ycr corresponds to the deflection where instability starts [,]. λcff corresponds to effec-
tive slenderness ratio and is given by:
λcff = αL0 (4.32)
D
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 26
The constants C ′1 and C ′2 are defined by:
C ′1 = E
2 (E −G)
2π2
′ = (E −G)
C 2 2G+ E
Absolute stability occurs when C′2
2 is greater than unity [SBM15]. So, the next con-
λ
cff
dition must be satisfied: [ ]
πD 2 ( − ) 1/2
E G
L0 < 2 + (4.33)
α G E
Figure 4.8: End-condition constant α [SBM15]
In the case of steels, the condition would be:
L0 < 2.63D (4.34)
α
4.2.6 Materials
Elasticity is a mechanical property of the objects to be deformed and regain its original
shape once upon forces that cause it disappear. The basis of operation of the springs
relies on the material properties, among them is the steel, which can undergo significant
elastic deformation. Among the materials used to manufacture springs, carbon steels
are used. To regain initial length of the spring, it is necessary that the material has a
high elastic limit. It is important to note that the yield depends on the diameter of
the wire, as long as the diameter is increased, the yield decreases. The springs in their
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 27
operation support alternative and repeated efforts, it is important that the steel to be
used have a high resistance to fatigue [Chu10].
It is also important that in manufacturing, spring do not suffer surface decarburization
since the peripheral portion of the spring is more fatigued, then its resistance would
be lower and may result in spring break [Chu10]. Another important factor is the
shear modulus. This factor characterizes the change in shape experimented by the
material when is subjected to shear forces. For linear elastic materials such as used
in compression springs, the shear modulus has the same value in all directions. It is
calculated as follows:
G = E
2 (1 + (4.35)
ν)
G: Shear modulus
E: Elasticity modulus
ν: Poisson’s ratio (For steel ν ≈ 0.3)
Spring material can be examined for their tensile strength, which vary according to
its wire diameter, so it is needed to be known. The processing of the material has
also an effect on tensile strength. The next equation is useful to estimate the tensile
strength by previously knowing the wire diameter to identify the constants involve in
the equation:
Sut = A (4.36)
dm
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 28
Figure 4.9: Constants A and m to determine Minimum Tensile Strength [SBM15]
In figure 4.36, the table is useful to identify the constants A and m according to wire
diameter.
Figure 4.10: Mechanical properties of some springs [SBM15]
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 29
Figure 4.11: Mechanical properties of some springs [SBM15]
Music Wire 0.80-0.95C (ASTM A228)
Music wire is material mostly used for small springs because of it is the toughest
material. This material has the higher tensile strength and also can withstand higher
stresses with repeated loading than any other material. It can be found in diameters
from 0.12 mm to 3mm[SBM15].
Oil-tempered wire 0.60-0.70C (ASTM A229)
This material is for general-purpose, and it is a good replacement when music wire
is more expensive or when it is required a large size of spring. However, it is not
recommended for uses of shock or impact loading. It is available in diameters from 3
mm to 12 mm [SBM15].
Hard-drawn wire 0.60-0.70C (ASTM A227-47)
It is the cheapest option for general-purpose spring steel. It should be only used when
it is not required a long life, accuracy or deflection. It is available in diameters from
0.8 mm to 12 mm [SBM15].
Chrome-vanadium (ASTM A231-41)
It is the most popular alloy for conditions with higher stresses and when it is needed
to work under fatigue and long endurance. It is also used for shock or impact loading.
Available in annealed or pre-tempered sizes 0.8 mm to 12 mm in diameter [SBM15].
Chrome-silicon (ASTM A402)
This material is ideal for highly stressed springs which require long life and are intended
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 30
for shock loading. Available wire from 0.8 mm to 12 mm [SBM15].
4.2.7 Design for static service
For a design, a recommended number of active coils is 3 ≤ Na ≤ 15. It is worth to
remind that to keep linearity, it is necessary that active coils do not gradually touch.
Although ideally, helical spring force-deflection is linear, in practice, it is nearly so.
However, not at each end of the curve of force-deflection [SBM15]. The spring is not
able to reproduce a force for small deflection. Also, when it is near to closure, it
lose its linearity because the pitch of spring is not accurately the same and produce
touching of active coils under higher deformations. For this reason, it is desired that
designer must confine the 75% of the curve between the spring without load (F = 0)
and loading spring to closure (F = Fs) as the maximum deformation of the spring.
Therefore, the maximum force endured for the spring should be F 7
max ≤ 8Fs. The ξ is
defined as fractional overrun to closure and it is defined by:
Fs = (1 + ξ)Fmax (4.37)
Then: (
= (1 + ) = (1 + ) 7)
Fs ξ Fmax ξ 8 Fs (4.38)
To solve the equation, ξ must be 1/7. So ξ = 1
7 = 0.143 ≈ 0.15. Finally, it is
recommended that ξ ≥ 0.15.
The design conditions the be considered are the followings:
4 ≤ C ≤ 12
3 ≤ Na ≤ 15
ξ ≥ 0.15
ns ≥ 1.2
The safety factor to avoid closure is ns For designing a helical coil compression spring,
a design strategy suggested by SHIGLEY [SBM15] advise to make a priori decision
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 31
regarding the material of steel and consequently select the diameter of wire d. With
decisions made before, it is possible to calculate the parameters: d, D, C, OD, ID, Na,
Ls, L0, (L0)cr, ns. It is required to use the material and wire size with the higher figure
of merit, i.e. the best performance or efficiency. The figure 4.12 follows a strategy for
designing a helical coil compression spring according to its application.
Figure 4.12: Design of helical coil compression spring for static loading [SBM15]
4.2.8 Critical frequency of Helical Springs
There are some applications that require helical coil compression spring for working
with fast motion loading. Therefore, designer must assure that physical dimensions of
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 32
the spring are not able to get close to natural vibratory frequency, because it might
produce resonance, triggering damaging stresses, given that internal damping in the
spring are quite low [SBM15]. The equation that describes the translational vibration
of a spring located between two parallel plate is given by:
∂2u = W ∂2u (4.39)
∂x2 kgl2 ∂t2
Where: k: spring rate
g: acceleration of gravity
l: length of spring
W : weight of spring
x: coordinate along length of spring u: motion of any particle at distance x
The natural frequencies of a spring that is placed between two parallel plates can be
obtained from the following equation:
√
= kg
ω mπ m = 1, 2, 3, .......... (4.40)
W
The fundamental frequency is obtained with m = 1. Commonly, the frequency is
interested in cycles per second, so the equation 4.41 determine it in hertz.
√
= 1 kg
f 2 (4.41)
W
The equation to determine the weight of the operating part of the spring is given by
the following:
πd2 2 2
W = ALγ = 4 (πDNa) ( ) = π d DNaγ
γ g 4 g (4.42)
In equation 4.42, γ corresponds to specific weight or specific density (steel = 7850 kg ).
m3
It is used when calculations are in metric system, otherwise γ = 0.284lb/in.
The fundamental critical frequency must be greater than frequency of force motion by
15 or 20 times. It is for avoidance of resonance with the harmonics. In case the design
does not satisfy that conditions, it should be redesign to increase k and W .
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 33
4.2.9 Fatigue loading of Helical Compression Springs
Mostly, springs must be designed for working in high number of cycles, which can go
from thousands to millions. For improve the mechanical properties of a spring, it can
be treated by a process of shot peening. This process can increase the fatigue strength
until 20% or more. According to SHIGLEY, the best data for about the torsional
endurance limits of spring steels are provided by Zimmerli. Zimmerli figured out that
size, material and tensile strength has no effect on endurance limits of springs steels
when their sizes are under 10 mm. SHIGLEY observed that endurance limits increases
with high tensile strengths[SBM15]. The endurance strength components for infinite
life for unpeened and peened spring are:
Unpeened:
Ssa = 241 MPa (35 kpsi) Ssm = 379 MPa (55 kpsi) (4.43)
Peened:
Ssa = 398 MPa (57.5 kpsi) Ssm = 534 MPa (77.5 kpsi) (4.44)
Sse = S(sa )2 (4.45)
1− Ssm
Ssu
Ssa
Sse =
1− S (4.46)
sm
Ssu
Joerres, a member of Associated Spring-Barnes Group, recommends the Goodman
relation for pulsation torsion [SBM15]. From Goodman diagram, Joerres uses:
Ssu = 0.67Sut (4.47)
Helical springs are never used as compression and extension, just have one of both
applications. Usually, spring withstand a pre-loading.
Master Thesis of Jorge Luis Ingaroca Quispe
4 Theoretical foundation 34
Figure 4.13: Sinusoidal fluctuating stress [SBM15]
Thus, from graph above, alternating and midrange component of forces are the follow-
ings:
= Fmax − Fmin
Fa 2 (4.48)
Fmax + Fmin
Fm = 2 (4.49)
Then, the shear stress amplitude and midrange shear stress are given by the equations:
τa = 8FaD
KB 3 (4.50)
πd
8FmD
τm = KB 3 (4.51)
πd
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 35
5 Mechanical modeling
5.1 Description of the 1DoF model
The aim of this research is not to represent exactly all the components involved in
the Follicle-sinus complex, but to exploit the internal behaviour for applications with
tunable compliance. The model used for this research is the represented in figure 6.1.
It contents a cylindrical rod with a pivot on a point of this. The lower end of the
rod is joined with a system damper-spring. The reason why the model used has the
system damper-spring in the lower end and not on other sides of the rod, it is because
it represents an equivalent of a distribution of dampers and spring in a single one
system.
This equivalent representation will change the values of proposed models of other
researches; however, the behaviour will be the same. Thus, it would allow us to study
the effects of a variation of the stiffness, which is our main objective in this current
thesis. There(fore, th)e model is des(cribed i)n the following equation:
= l − l l l l2 l2
Joθ̈(t) F(t) 2 cosθ(t) +mg 2 − sinθ(t)− k 2 sinθ(t)cosθ(t)− c 2 cos
2θ(t)θ̇(t)
a a a a
(5.1)
Equation 5.1 describes the relation among the angular position and the other parame-
ters, represented as variables. Where Jo is the inertia moment regarding the pivot, m
mass of rod, k stiffness of spring, c damper of the system, g acceleration due to gravity,
l length of the rod, a fraction of the rod regarding the pivot, θ(t) angular position of
the rod respect to vertical axis and f(t) the perpendicular force applied on the center
of the rod. Also, using Steiner Theorem the calculate inertia moment, it is obtained
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 36
the following: ( )2
ml2 l l
Jo = 12 +m 2 − (5.2)
a
In this equation 5.1, and with the objective of making it easier to solve, the friction in
the pivot is considered null.
Figure 5.1: Model of 1DoF proposed for current thesis. Modify from [BSWZ13]
In order to reduce complexity of the equation, it is used an approximation of trigono-
metric ratios into angles:
sinθ ≈ θ
cosθ ≈ 1
It is important to know that this approximation works properly until the angle of 15◦.
Therefore, it is expected to be some error when this angle increases more than 15◦.
However, this error will be also evaluated and compared with a multi-body simulation
in the next chapters.
Then, the reduced equation obtain with approximation is represented in equation 5.3.
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 37
( ) ( )
= l − l + l − l l2 l2
Joθ̈(t) f(t) 2 mg 2 θ(t) − k 2 θ(t) − c 2 θ̇(t) (5.3)
a a a a
In the process of solving of the equation 5.3, that can be seen in CD attached to this
document, it is obtained the new parameters n1;2:
√
2 √√( )2 [ ( )]
cl cl2 1 l2 l l
n1;2 = − ±√2 − k −mg − (5.4)
2Ja 2Ja2 J a2 2 a
And after applying LaPlace Transform to solve the equation, the response obtained is:
( ) ( )
= 1 t [ ]
l l ∫ 1 cl2
θ(t) 2 − g(t− σ)f(σ)dσ+θ(0) (n en1t − n en2t) + θ̇ + θ g(t)
J a n1 −
1 2 (0) (0)
n
0 2 Ja2
(5.5)
Where:
g = 1 (en1t − en2t
(t) ) (5.6)
n1 − n2
5.2 Evaluation of spring as a compliant component
In the model proposed, there are three options to provide compliance to the system.
The first one would be by tuning the stiffness of the spring. The second one, would
be by tuning the damping of the damper. The third one, would be by implementing
both before mentioned.
In this current thesis, the compliance of the actuator have been achieved for the first
option, with a spring with tunable stiffness. In order to accomplish the variation of
stiffness, the method used is the same of Jack Spring (seen in State of the art). The
equation of the stiffness of a spring is given by:
= Gd4
k 8 3 (5.7)
D Na
Where k is the stiffness of the spring, G is the Shear modulus or modulus of rigidity,
d is wire diameter, D is Diameter of coil, and Na is the number of active coils. Hence,
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 38
from equation 5.7, the stiffness could be change by varying the one or some of the other
parameters. Mechanically, from all the parameters involved in equation 5.7, the easiest
one to be changed is the number of active coils Na. The variation of number of active
coils can be done mechanically, it is required to have pre-defined the values of wire
diameter and diameter of coils in order to design the mechanical components of the
system with their dimensions. Thus, compression spring must be used because, unlike
extension spring, it has a pitch between its coils that make it ideal for implementing a
mechanical element to limit or extend the number of active coils. The modulus rigidity
G is a material parameter itself, given by books or by the manufacturer of springs.
Thus, the values of d and D must be analysed to select spring which could withstand
the operation proposed. As it was explained in state of the art, in foveal whisking, rats
contact surface of objects with a frequency until 25Hz. For that reason, the maximum
frequency of operation for this study would be 25Hz.
fop = 25Hz (5.8)
Then, the critical frequency must be analysed. From spring design section in this
document, and according to Shigley:
√
1 kg
f = 2 (5.9)
W
And for recommendation of Shigley’s book, the critical frequency should be greater
than 15 or 20 times the maximum frequency of operation. For this research, it is used
the mean between those:
fcr = 17.5fop (5.10)
Decomposing the stiffness k and weight W into basic parameters, it is obtained the
following: √
= 1 Gd2g
fcr 2 2 4 2 (5.11)
2π D Na γ
Here, it is worth to mention that equation 5.11 corresponds to an equation applying
values in English system, otherwise in metric system that acceleration due to gravity
g might be omitted.
The material selected for this research is Music wire, because it has strong properties
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 39
with would make robust for fatigue loading. Modulus of rigidity can be given by a
book or for previous research. However, manufacturer has its own value which is close
to the usually quoted in books. For this thesis, the characteristics of spring will be
selected from Century Spring Corporation. By working with calculations of springs
of Century Spring, it was determined that accurate calculations are obtained working
with English system, because the values given in metric system has only a decimal
point so it is rounded to its closer decimal producing loss of precision. Consequently,
all calculations will be made with English system, and posteriorly the mechanical
elements will be designed in metric system. Empirically, it was calculated that for
Century Spring Corporation, the music wire material has a modulus of rigidity about
11.5Mpsi which is closer to given by Shigley 12Mpsi. For calculation, the one used
will be G = 11.5Mpsi
Then, replacing the values to obtain the relationship among d, D and Na.√√√ 2
(17 5)(25) = 1√(11.5Mpsi)d (386)
. 2 2 4 2 (5.12)
2π D Na (0.284)
D4N 2
a
2 = 1038.7 (5.13)
d
From the catalogue of Century Spring Corporations, wire diameter and Coil diameter
can be selected to determined the number of active coils necessary for withstanding
25Hz. For this research, the values selected were d = 0.038in and D = 0.382in. Then,
the number of active coils should be:
Na = 8.39 ≈ 8.5 turns (5.14)
Usually, to select a spring, the total number of coils are presented, so it is needed to
know its value depending of the type of ends for the spring. For this purpose, it is
required a closed and ground end, which would mean that the total number of coils
should be:
Nt = Na + 2 = 10.5 (5.15)
The compression spring that satisfy the characteristic required is Century Stock num-
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 40
ber 71335 (page 108 from catalogue of Century Spring [Cor15]). It owns the following
dimensions and characteristic:
d = 0.038in (1mm)
D = 0.382in (9.7mm)
L0 = 1.75in (44.5mm)
k = 6.3 Lb/in (1.1N/mm)
Suggested Maximum Deflection = 1.1in (28mm)
Suggested Maximum Load = 7lbs (31N)
Ls = 0.40in (10.1mm)
Nt = 10.5 turns
To determine the pitch between the coils, we use the following equation:
= (L0 − 2d)
p
Na
p = 0.197in (5mm)
This value is important to know because it will determine the pitch of the actuator
which will vary the number of active coils.
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 41
5.3 Multi-body model
In order to have a clearer idea of how this system would operate in reality, a multi-
body simulation was implemented by designing mechanically the elements to achieve
the tuning of active coils of the compression spring. The software SolidWorks appro-
priate the tools necessary to observe the design proposed and a sight of its operation.
Besides, SolidWorks allow to analyse its responses with graphics. For construction
of element, the dimensions will be in metric system. SolidWorks 2016 version has
been employed since the computer where multi-body simulation was created has a
Microsoft’s Windows 10 operating system. Microsoft’s Windows 10 has compatibility
problems to apply SolidWorks Motion complement in SolidWorks’ version before 2016.
As it was mentioned before, the basement of the design is inspired in operation of Jack
Spring. Therefore, a motor is used to vary the number of active coils of the spring. The
mechanical elements must satisfy the dimensions of the compression spring selected in
the section before. It is worth to mention that the mechanical system must provide a
lineal and rotational displacement for the element which limits the number of active
coils. Also, because a spring can only operate as compression spring or extension
spring, no both, it is required to use two compression springs to compensate the case
of extension spring operation, as it can be seen in figure 6.2.
Figure 5.2: Frontal view of the multi-body model implemented in Solidworks 2016
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 42
5.3.1 Actuator
The actuator element used for generating motion is a motor, the velocity of changing
of stiffness would depend on the angular velocity of motor. It is a variation of a
coil per each turn (360◦). For the design of the current thesis, the type of motor
implemented in the multi-body simulation is a DC motor with encoder. It provides
spins in both directions, and also could reach high velocities, which might be useful in
applications that requires faster change of stiffness. The dimensions, in metric system,
of a commercial DC motor (in CD attached to this document) were employed for
building the body in SoliWorks. A DC motor has a cylindrical shape, so it required a
basement for supporting it to keep it static. In figure 5.3, it can be seen the basement
applied for supporting the DC motor. The motor is desired to have a fixed position,
then the rotation of its axis should provide both lineal translation and rotation to vary
the number of active coils of the spring.
Figure 5.3: Isometric view of the motor
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 43
5.3.2 Couplings for limitation of active coils
To vary the number of active coils, it is necessary that a coupling could displace lineally
with a pitch of 5mm. In order to achieve this, the idea of worm screw is exploited
to build elements that accomplish the required. For that reason, three elements were
built with characteristic of screws. The first one should allow a coupling with the shaft
of the DC motor, as can be seen in figure 5.4. Its external side has a shape of a bolt.
This coupling only rotates, but do not displace lineally.
Then, the coupling for the motor must be connected to another element that should
be displaced along of the spring (figure 5.5). This element has two sides. The first one
has a shape like a nut to couple with the elements described before. The second one
has a shape like a bolt which then will be couple to the support of the spring (figure
5.6) that has a nut shape and make possible the displacement.
Figure 5.4: Coupling element for DC motor
Figure 5.5: Mechanical element with catcher to limit the coils of the compression spring
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 44
Figure 5.6: Support of the compression spring
The element which make possible the limitation of active coils is the one proposed
in figure 5.5. It must catch a part of the active coils to convert the coils behind it
to inactive coils. In the model, it can be seen that it has two catchers which have
the enough space for the repose of wire coils. These catchers have a small angle of
inclination. For calculating that angle is necessary to have the value of Diameter coil
(D = 9.7mm) and the pitch (p = 5mm), as it can be observed in figure 5.7.
Figure 5.7: Geometry of a helical spring [HSH05]
α = tan−1 ( p ) (5.16)
πD
α = 9.3179◦ (5.17)
The reason why it possesses two catchers is because just one would not be sufficient
for a appropriate subjection of the coils. Since it might perform repeated loading, it
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 45
could induce vibrations in the spring if there is only one held point.
Thus, an mechanical element varies the number of active coils in the spring. However,
it can not use all the active coils because the catchers has dimension that will no be
possible to take the active coils from it starts, but a smaller number of coils. In figure
5.8, it is shown the dimension of the both catchers. The red point indicates the point
where the number of active coils starts. Then, by adding the dimensions of the right
side, we obtain:
Inactive_length = 2.50mm+ 1.74mm+ 0.5mm (5.18)
Inactive_length = 4.74mm (5.19)
Therefore, in reality, the active length of the compression spring would be:
Active_length = p(Na)− Inactive_length = (5mm)(8.5)− 4.74 (5.20)
Active_length = 37.76 (5.21)
Then the initial number of active coils would be:
= (42.5− Inactive_length)(8.5)
Na 42 5 (5.22)
.
Na = 7.552 turns (5.23)
Thus, the initial stiffness of the spring must be calculated with 7.552 turns (G =
11.55Mpsi = 76.924GPa):
Gd4 (76.934GPa)(0.00096524
k = )
8 =
D3Na 8(0.00973)(7 552) (5.24)
.
k = 1.252N/mm = 1252N/m (5.25)
According the number of active coils will be reducing, the stiffness will increase because
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 46
Number of active coils is inversely proportional spring stiffness.
Figure 5.8: Design dimensions of the mechanical element that catches the wire coil
The compression spring, then, is built with the dimensions provided for manufacturer.
The total number of coils is 10.5. However, it has closed and ground ends, so the
number of active coils is 8.5. Previously, it was determined that the pitch has 5mm.
With these dimensions, the compression spring can be built as it is shown in figure
5.9.
Figure 5.9: Compression spring built from Century Spring Corporation (Stock number
71335) with Closed and ground ends
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 47
5.3.3 Rigid body model
The system is inspired in the operation of a vibrissa, therefore to obtain a response
which could give a response related to its real behaviour, an element similar to it is
analysed. Vibrissa is a flexible sinus hairs; however, in this thesis it will be taken as a
rigid body for practicality, since it is a first attempt to analyse viscoelastic properties
of an actuator with tunable compliance. The aluminium cylindrical rod replace the
vibrissa, so it has dimensions defined as r = 2mm and L = mm, it has a pivot point
which will be the eighth of its total length (a = 8). In figure 5.10
Figure 5.10: Rigid body to be analysed in motion. Rod with a pivot point.
The density of the aluminium is Density = 2700kg/m3. Therefore, the mass of the
rod would be m = 0.0024kg.
To provide an appropriate subjection of the spring, an element similar to a flat is fixed
to the free end of the spring. Its importance endorse in transmitting correctly the force
produced for the force to the spring, otherwise, it would not applied a longitudinal
Master Thesis of Jorge Luis Ingaroca Quispe
5 Mechanical modeling 48
deflection. For this, the element seen in figure 5.11, is proposed since it limit the
movement in just one plane and provides the correct longitudinal deflection for the
springs.
Figure 5.11: Flat supports fixed to the each spring end
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 49
6 Simulation of proposed system
6.1 Matlab simulation
In Chapter 5, it was described a model proposed for the system. For developing
the equation that describes the system, an approximation of small angles were used,
obtaining a response depending on the force that triggers the swinging of the rod
and the initial conditions of the rod. It is important to observe how accurate this
approximation is, so its response should be analyse and compare with the response of
a Multi-body simulation, since the last owns more sources to simulate closer to reality.
For that reason, Matlab software is employed, for describing the equations obtained
and plot them. Once that dimensions of the elements of the systems are defined,
it is possible to obtain the plots of the response for different cases. It is worth to
remember that initial stiffness for the system is not the defined by the manufacturer
in the catalogue because of the mechanical element used for catching the wire coil. It
was previously calculated in Chapter 3 that initial stiffness is k = 12552N/m.
A system with vibrations needs a damper element to absorb the energy and stops the
oscillations, otherwise it would oscillate without stopping. This current thesis focuses
on the change of stiffness in a elastic element, in this case the spring. For that reason,
the value of the damper will be fixed to c = 5Ns/mm.
The response of the system changes by adjusting the stiffness of the spring. To observe
how it changes, the frequency of the system must be determined from the equation.
Thus, it is part of the interest of this thesis to calculate how it varies depending on
the value of the stiffness.
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 50
Figure 6.1: Frequency vs stiffness response with a damper of c = 0.5Ns/m
As it can be seen, the frequency increases regarding the increase of the stiffness. The
initial stiffness is k = 1255.2N/m, and for that value, the frequency corresponds to
25Hz approximately. This value was our initial point of start to select the spring, so
it satisfy the conditions of critical frequencies. Also, it is worth to mention that while
reducing the number of active coils of the spring, the critical frequency of operation
(view in Chapters before) will increase as well. Thus, the compression spring should
have no problem for operating. For experiments, the force will be a rectangular pulse
described by equation
f(t) = F [u(t)− u(t− t0)] (6.1)
Therefore,(the re)sponse of the angular position would be the following:
− l [ ( ) ( )]F l
a 1 u(t− t0)
θ n2t
(t) = n1e − n en1t + n en1(t−t0) − n en2(t−t0)
2 2 1 +
Jn1n2 n2 − n1 n2 − n1
( ) ( )
F l − l [ ] 2
a [1− u(t− t0)]+ 1
θ(0) ( cl
n1e
n1t − n en2t
2 ) + θ̇
− (0) +
Jn1n2 n1 n2 Ja2 θ(0) g(t) (6.2)
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 51
6.2 Solidworks simulation
Using the software SolidWorks, a mechanical demonstrator has been built for showing
the dimensions and clarify the principle of operation. The complement SolidWorks
Motion provides appropriate sources for a better Motion analysis of the system. Solid-
Works Motion allows to introduce gravity to the system, to create dampers and springs,
to measure parameters, to introduce lineal and rotational displacements, etc. The ob-
jective of the mechanical system is to limit the number of active coils of the compres-
sion spring. For this, the movement starts from the rotation of the shaft of the motor.
This movement rotates the mechanical coupling ahead, and go on. This generates a
transmission of movement until the mechanical coupling that catches the coils of the
compression spring. The catcher has the same inclination of the coils of the spring, in
order to couple precisely, i.e. the catcher and the wire coils are tangents. The mating
relationships in the multi-body system fit properly; however, empirically, and error is
observed when the mechanical couplings begin to rotate. It was found that catcher
of the coupling and the wire coil are not exactly tangent, producing an alteration of
the mating relationships. Then, another method must be applied in order to recreate
the behaviour of the spring. For that reason, it is implemented an alternative way to
recreate the behaviour of a the spring with a variation of their active coils.
As it is shown in figure 6.2, the main components which define the deflection and the
length of the active region of the spring are selected. For measuring parameters such
as distance and angular position, ’Results and Plots’ tool is useful since it measures
the parameter selected in all the simulation. Besides, these values are available to be
employed in equations. The parameters required for these simulation are two. The
first one, distance of the active length of the compression spring. And the second one,
the deflection of the compression spring. For that reason, D1 measures the distance
of the catcher from the axis Y. On another hand, D2 measures the deflection between
the touch point of the rod and the beginning of the active coils. For offering a dis-
placement of the coupling to simulate its rotation, Rotatory Motor is assigned to this
in mode of Lineal Motor. Since the pitch of the compression spring is 5mm, every
step of 5mm would mean a turn of the motor and a variation of an active coil. The
simulation incorporates a damper fixed to c = 0.5Ns/m and in the same direction of
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 52
the compression spring.
Figure 6.2: View of the alternative method for simulating the behaviour of the sys-
tem. F1 introduce external excitation and F2 the force due to compression
spring deflection
For analysing the response of the system, many experiments are performed. To ini-
tialize the displacement of the angular position, two ways will be implemented: A
perpendicular Force applied to the rod and variation of initial conditions of the rod,
i.e. angular position and angular velocity. To observe variation of angular position.
Results and Plots, from Solidworks motion, allow to measure in the process of simula-
tion. The parameter selected is projection angle respect to Axis Z. After introducing
deviation of angular position, the system stabilize in a short time. For that reason,
simulation time is defined until one second. For improving the view of the plot, pa-
rameters of simulation are set. Thus, the number of frames per second, resolution of
3D contact and Precision are increased. The interface that SolidWorks provides to
observe the response is not affordable for comparison with Matlab response. Then,
results obtained by SolidWorks are exported as a CSV file. Posteriorly, this file is read
by a Matlab command. and compared with its respective case.
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 53
6.3 Experiment results
6.3.1 Comparison between Mathematical model and Multi-body
model
For comparing the results obtained in both models, some experiments were performed
by changing the conditions for each one. In some cases, a perpendicular force is intro-
duce to the rod in its inertia center. In other cases, initial angular position is modified.
And also, there are cases were both, force and variation of angular position, are applied
to the rod. Because, in this section is desired to analyse the response of the system
described with approximation of small angles, the stiffness of the compression spring
will be fixed to its initial value k = 1255.2N/m
Experiment 1
For this experiment, the initial angular position is fixed in 10◦. There are not external
forces, so the rod tries to return to its stabilization point.
Figure 6.3: Comparison of responses of experiment 1 (c = 0.5Ns/m)
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 54
Graphs from figure 6.3, have a similar response. The initial angular position is 10◦, so
the approximation of small angles works properly regarding amplitude and frequency.
Experiment 2
For this experiment, the rod is introduce with a initial angular position of 45◦. No
external forces are introduced in the model.
Figure 6.4: Comparison of responses of experiment 2 (c = 0.5Ns/m)
As it can be observed in figure 6.4, the response are not quite similar. The amplitude
and frequency have almost the same value for both responses; however, there is an
error because of the bigger deflection obtained in SolidWorks simulation which means
more time for recovering its stabilization.
Experiment 3
For this experiment, the initial angular position is zero degrees, but a rectangular pulse
force is introduced. The force introduced has a value of F = 0.5N and t0 = 0.1s, as
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 55
seen in the equation 6.3
f(t) = 0.5 [u(t)− u(t− 0.1)] (6.3)
Figure 6.5: Comparison of responses of experiment 3 (c = 0.5Ns/m)
The responses obtained are quite similar. The main difference is the amplitude. The
SolidWorks simulation has less amplitude than the Matlab simulation. The difference
of deviation is not so high.
Experiment 4
For Experiment 4, the initial position is zero degrees. As well as experiment 3, rect-
angular pulse force is introduced, but it has a higher value F = 1.5N and duration
t0 = 0.25s as seen in equation 6.4
f(t) = 1.5 [u(t)− u(t− 0.25)] (6.4)
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 56
Figure 6.6: Comparison of responses of experiment 4 (c = 0.5Ns/m)
As it is expected, in figure 6.6, the difference of amplitudes increases respect to ex-
periment 3 because a greater deflection. Regarding to frequencies, they have a similar
response.
Experiment 5
For this experiment, the rod has a initial angular position of 45◦. Also, a force is
introduce with a value of F = 0.5N and a duration 0.2s (equation 6.5).
f(t) = 0.5 [u(t)− u(t− 0.2)] (6.5)
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 57
Figure 6.7: Comparison of responses of experiment 5 (c = 0.5Ns/m)
It is observed in figure 6.7 that rod oscillates while a force is applied on it. This
is because the angular deviation is so high that force of the compression spring is
increased enough to recover its stabilization despite of the force introduced. However,
that force will be reducing while angular deviation increases, so the force applied is
greater than force of the spring. Consequently, it increases the angular deviation and
go on, until rectangular pulse force disappears. In this experiment, the amplitudes have
not too much difference of values. Frequency is the same, but as seen in experiment
2, there is an error due to higher angular deviation in Solidworks simulation.
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 58
6.3.2 Analysis of the compliance
For this part of the simulation, the stiffness was tuned while the rod tries to return
to its stabilization point. The rod is rotated with a certain angle, depending on the
experiment. Experiment 1
For this experiment, the initial angular position is fixed in 10◦. There are not external
forces, so the rod tries to return to its stabilization point. At the time = 0.1s the
motor started to rotate at a speed of 10rsps or 600rpm until time = 0.2s.
Figure 6.8: SolidWorks’ response for experiment 1 (c = 0.5Ns/m)
As it can be seen in figure 6.8, the initial frequency is f = 25.88Hz until the rotation
starts. Later, after a transition state while motor stops, the frequency is increased
until f = 28.16Hz. Because the rotation speed is 10rsps, for a variation of time of
∆t = 0.1s the linear displacement would be 5mm. It means that a complete active
coil is decreased.
Experiment 2
For this experiment, the rod is introduce with a initial angular position of 45◦. No
Master Thesis of Jorge Luis Ingaroca Quispe
6 Simulation of proposed system 59
external forces are introduced in the model. In this experiment, the motor start to
rotate at time = 0.1s until time = 0.3s with the same speed of 10rps
Figure 6.9: SolidWorks’ response for experiment 1 (c = 0.5Ns/m)
From figure 6.9, it can be observed that frequency increases from time = 0.1s when
motor starts to rotate with a speed of 10rps until time = 0.3s. It means a displacement
of 10mm which would also mean a decrease of two active coils.
Master Thesis of Jorge Luis Ingaroca Quispe
7 Conclusions 60
7 Conclusions
Software for Multi-body simulation, as SolidWorks for this thesis, could have problems
in simulation due to its own limitation. For instance, the conflict of mates between
elements. Even when elements might have the same dimension to be mated, some
movements generate conflicts such as rotation. Nevertheless, it is possible to apply
sources for describing the behaviour as made in this work.
The comparison of the models developed in this current thesis indicates that they have
characteristics quite similar, even for angular deviation greater than 15◦. That is the
value limit where system is supposed to have greater errors. As it was observed, the
approximated response works properly for a maximum deviation of 45◦.
By tuning the compliance of an actuator, the response of the system changes not only
for frequencies, but for amplitudes as well because the value of stiffness is involved
in the calculations of both parameters. However, obtaining the response of both pa-
rameters with a tunable stiffness will increase its complexity, so it would be required
another strategy for calculations.
Master Thesis of Jorge Luis Ingaroca Quispe
Bibliography 61
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