Research Training Report
on
DESIGNING AND STRESS ANALYSIS OF VARIABLE PITCH
CONICAL SPRING
Submitted By
HARSH PANDEY
B. Tech (IV Year), Mechanical Engineering,
Sardar Vallabhbhai National Institute of Technology, Surat
Training Period
(14 June 2021 to 06 August 2021)
Under the Guidance of
Mr. AK Kashyap,
Scientist `F`
Aerial Delivery Research & Development Establishment(ADRDE)
Defense Research & Development Organization
Ministry of Defense, Government of India
Agra Cantt – 282 001
1
CERTIFICATE
This is to certify that the project report compiled by Mr. Harsh Pandey
which is entitled ''Designing And Stress Analysis of Variable Pitch
Conical Spring " is an authentic record of the effort carried out by him
during the period of his summer training from 14 June 2021 to 06
August 2021. The report is aimed towards the partial fulfillment of the
requirement of the training certificate duly accredited by Aerial Delivery
Research & Delivery Establishment (ADRDE), Agra under the guidance
of Mr. AK Kashyap, Scientist `F`.
Training In-charge
Mr. AK Kashyap,
Scientist `F`
Parachute Group
ADRDE
Agra, Uttar Pradesh
2
ACKNOWLEDGEMENT
Aerial Delivery Research & Development Establishment (ADRDE) is one of the
premier establishments of the Defense Research and Development organization
(DRDO). This establishment is committed to provide the users state-of-the-art
product & services to its customers in areas of the aerodynamic decelerators,
pneumatic structures, aircraft arrester barriers and allied systems through research
and development, innovation, team work and following up the same by continual
improvement based on user's perception.
I am highly obliged to Shri Arun Kumar Saxena, Director ADRDE, Agra for
allowing me to associate with this esteemed establishment as a summer trainee in
'Parachute Group’ for one month period from 14 June 2021 to 06 August 2021.
Further, I extend my heartfelt gratitude to the Scientist Mr. Gayasuddin Qureshi,
for entrusting me with such a substantial project on "Designing and Stress Analysis
of Variable Pitch Conical Springs'', and teaching the valuable Technical knowledge.
Working in this project has certainly been a good learning experience and has
reinforced my knowledge of Designing and Mechanical Finite Element Analysis to
a great deal.
Most importantly, I express my sincere thanks to Mr. AK Kashyap, Scientist `F`,
for his unflagging guidance throughout the progress of this project as well as
valuable contribution in the preparation and compilation of the text.
I am thankful to all those who have helped me for the successful completion of my
training at ADRDE. Agra.
Harsh Pandey
B. Tech (IV Year)
Mechanical Engineering
Sardar Vallabhbhai National Institute of Technology, Surat
3
ABOUT ADRDE
Aerial Delivery Research & Development Establishment (ADRDE) is one of
the premier establishments of the Defense Research and Development
organization (DRDO). ADRDE deals with high-level research in defense
related technologies. Various projects are under progress like parachutes for
different applications, aerostats for surveillance, platforms for heavy drop,
floatation for Space Recovery Experiment (SRE), aircraft arrester barrier
system for air force, etc. Defense Research and Development organization
(DRDO) is a premier organization of the Government of India, responsible
for development of technology for use by the three services of defense in
India. It was formed in 1958 by the merger of the Technical Development
Establishment and the directorate of Technical Development and Production
with the Defense Science Organization.
My training area for around 8 weeks was in the Department of Parachute
Group, Sub Division ABS, ADRDE Agra.
4
TABLE OF CONTENT
S NO.
1.
2.
3.
4.
i)
ii)
iii)
iv)
v)
vi)
vii)
viii)
5.
I.
i)
ii)
iv)
v)
II.
i)
ii)
iv)
v)
6.
7.
TOPIC
Certificate
Acknowledgment
About ADRDE
Literature Review
Introduction
Advantages of Variable pitch over Constant pitch
Application of Variable Pitch Conical Spring
Nomenclature
Elementary Calculation
Conical Spring With Constant Helix Angle.
Conical Spring With Constant Stress at Solid.
Conical Spring With Linear Behavior
Case Study
For Constant Spring With Constant Helix Angle
Theoretical Calculation
Modelling in SOLIDWORKS 2021
Stress Analysis in ANSYS R1 2021
Results
For Constant Spring With Constant Stress at Solid.
Theoretical Calculation
Modelling in Solidworks 2021
Stress Analysis in ANSYS R1 2021
Results
Conclusion
References
PAGE
NO.
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3
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5
INTRODUCTION
Springs are often used in mechanical devices for their ability to store and return
energy. The range of applications is very wide. In some applications, conical
springs can be preferred to cylindrical springs for their ability to have a smaller
solid length. Depending on their geometry, they may be able to fully telescope,
inducing a solid length equal to the wire diameter.
Conical springs with constant pitch are quite well known but many other types of
conical springs can be manufactured. For a given conical shape and a given
number of turns, the properties of the spring can evolve significantly depending on
the way the coils are distributed along the conical profile.
In fact, cylindrical springs can simultaneously possess a constant pitch, a constant
angle, a constant stress at solid and a linear behavior whereas, for a conical spring,
only one property can be achieved at a time.
Following properties of conical springs has Variable Pitch● Conical spring with a constant angle.
● Conical spring that leads to a constant stress when fully telescoped.
● Conical spring with a fully linear load-length behavior.
6
ADVANTAGES OF VARIABLE PITCH SPRINGS
OVER CONSTANT PITCH SPRINGS
Following are the advantages of variable pitch springs over constant pitch springs.
● Manufacturers find a number of advantages in the ability to vary spring pitch
For example, conical springs (and other types of variable diameter spring)
can be made to deflect at a linear rate.
● Variable pitch can also be used to prevent coils from fully closing in “soft”
springs, or springs with a low spring rate.
● Both extension and compression springs can sometimes benefit from the
inclusion of several rows of closed coils, which still allows flexibility but
prevents tangling.
● A more technical advantage is the prevention of spring surge. Spring surge
occurs when a spring’s natural frequency corresponds to the frequency of
applied force, causing the spring to oscillate back and forth (similar to the
way a slinky moves when held on both ends).
● Spring surge shortens the life of springs, and in some cases can cause
immediate failure.
● By varying the spring pitch, manufacturers can prevent spring surge in
high-frequency applications like valves and automatic weapons.
7
APPLICATION OF VARIABLE PITCH CONICAL
SPRING IN ADRDE
These Spring are used in various parachute system like ●
●
●
●
●
●
Ram Air 9 Cell Parachute​.
Aircrew Parachute..
Controlled Aerial Delivery System (CADS).
Space Payload Recovery Parachute.
AN-32 Heavy Drop System
Brake Parachute.
Conical springs are also found in●
●
●
●
●
●
High-Performance Vehicle Suspension
Commercial Vehicle Suspension
High-End Mattresses
Gravity equilibrators
Laser light applications
Non-linear joints of robots.
8
NOMENCLATURE
→
α - current helix angle
→
θ - angle that defines the position on the conical helix
→
δ - deflection
→
δ1- deflection of the part of the spring that is freely deflect (from D1 to DL)
→
δ2 - deflection of the part that is at solid (from DL to D2)
→
τ - uncorrected stress
→
τi - constant uncorrected stress at solid.
→
d - wire diameter
→
dn - elementary coil
→
dr - elementary radial displacement on the conical shape
→
dl - elementary orthoradial displacement on the conical shape
→
dz - elementary axial displacement on the conical shape
→
D1 - minimum mean diameter
→
D2 - maximum mean diameter
→
DL - diameter that defines the limit between coils that are free to deflect and coils at solid for a given load P
→
F - initial spring flexibility
→
Fe - elementary flexibility
→
G - torsional modulus
→
La - active length
→
Lf - free length
→
Na - number of active coils
→
NL - number of active coils that defines the limit between coils that are free to deflect and coils at solid for a given load
P
→
pe - elementary axial pitch
→
P - axial load
→
PM - load when all the coils come to solid (when DL=D1).
→
PT - load from which the spring starts to come to solid (when DL=D2)
→
r - running helix radius (varies from D1/2 to R)
→
R - current helix radius (varies between D1/2 and D2/2)
9
⮚ The geometry of the conical helix can be defined as follows:
⮚ Global equation of the conical spring:
....................(1)
In this equation, the helix angle α is a function of the current radius r and depends
on the type of conical spring considered.
⮚ If the spring can telescope fully, the next equation must be satisfied for any
value of θ :
.........................(2)
10
ELEMENTARY CALCULATIONS
✔ The stress in an elementary coil defined by its radius r and subjected to a
load P can be calculated using the formula proposed by Wahl (uncorrected
stress of a coil that is free to deflect):
........................(3)
✔ The elementary flexibility is:
….....................(4)
✔ The elementary axial pitch is:
........................(5)
Now analyzing following properties of conical springs that has Variable Pitch one by
one● Conical spring with a constant angle.
● Conical spring that leads to a constant stress when fully telescoped.
● Conical spring with a fully linear load-length behavior.
11
CONICAL SPRING WITH A CONSTANT HELIX ANGLE
With a constant helix angle (and a variable axial pitch) which induces a
logarithmic spiral (the radial pitch increases with the radius)It can be noted that a conical spring can't have a constant pitch and a constant helix
angle at the same time (only cylindrical springs have this property):
...................(i)
Also, we know that
So, we get
..........................(ii)
Reversing the equation, we get -
...........................(iii)
LOAD CALCULATION
For a given load P, the limit between the coils that are free to deflect and the coils
that are at solid is obtained for Pe = Fe*P. Thus, combining Eq. (4) and (5), the
associated diameter can be defined:
12
The load from which the spring starts to come to solid (when DL=D2).
The load when all the coils come to solid (when DL=D1 … Full Compression
For a load lower than PT, the deflection curve is linear
For a load between PT and PM, the deflection of the spring can be expressed as the
addition of two deflections: the deflection of the part of the spring that is free to
deflect (from D1 to DL) and the deflection of the part that is at solid (from DL to
D2).
13
CONICAL SPRING WITH CONSTANT STRESS AT
SOLID
An elementary coil situated at a radius r and subjected to a load P induces an
elementary deflection (when it is free to deflect) of:
.............(I)
When the elementary coil comes to solid, the following equation is obtained:
.............(II)
Combining Eqs. (I), (II) and (3), the value of the helix angle
Eq (1) becomes-
...............(III)
But we know that
,
...............(IV)
Eq. (III) and (IV) can be combined:
14
And Above Equation can be reversed:
LOAD CALCULATION
For a given load P, the diameter that separates the part of the spring that is free to
deflect from the part that is already at solid:
The load from which the spring starts to come to solid (when DL=D2).
The load when all the coils come to solid (when DL=D1).
CONICAL SPRING WITH LINEAR BEHAVIOR
To obtain fully linear behavior, any elementary coil situated at a radius r should
come to solid for the same load PM. The elementary deflection thus has to be equal
to the elementary axial pitch.
Thus Equations (4) and (5) give:
15
Eq. (1) then becomes:
..............(a)
LOAD CALCULATION
But we still know that
Thus, the constant value of PM can be calculated:
..............(b)
There is no transition load for this kind of conical spring. Eq. (a) can be simplified
by exploiting Eq. (b):
Above equation can be reversed:
Being fully linear, this kind of spring is the only one to offer a conical shape at any
compression state. The other kinds of springs have a conical shape only when
uncompressed and their shape is non-conical when they are compressed, even for
conical springs with constant pitch.
16
CASE STUDY-
Given thatTotal Number of Coil = 6
Total Number of Active Coil, N a = 4
Free length, Lf =368 mm
Diameter of wire, d= 5 mm
Active Length, La = 368-6*5 = 338 mm
Minimum mean diameter, D1= 56 mm
Maximum mean diameter, D2 = 128 mm
Conical Spring with a Constant Helix Angle,
and calculating theoretically,
➢
Putting above values, we get
tanα = 0.30
17
→ Calculating the diameter of each coil of Spring
R(π/2) = 29.48;
D(π/2) = 58.97;
R(5π/2) = 36.25;
D(5π/2) = 72.50;
R(9π/2) = 44.57;
D(9π/2) = 89.15;
R(13π/2) = 54.81;
D(13π/2) = 109.62;
→
PT = 346.04 N = 35.28 kgf
→
PM = 1807.9 N = 184.35 kgf
18
19
20
MODELLING
In the present work Conical spring with constant helix angle is taken for modeling
and analysis, to compute and compare results. The dimensions for the same are
obtained theoretically as calculated above. The 3D modeling of Conical spring is
carried out using SOLIDWORKS 2021 and it is shown in Figure.
The dimensions of the Variable Pitch Conical spring are shown below:
21
MATERIAL SELECTION
Structural steel is the conventional material used for Conical spring. The
properties of the materials used in the present work are as mentioned in Table
Material Property
Density ρ
Unit weight γ
Modulus of elasticity E
(Young's modulus)
Value
≈ 7850 kg/m3
≈ 78.5 kN/m3
Shear modulus G
G = E / [2 ⋅ (1 + ν)] ≈ 76923 MPa
Poisson's ratio in elastic range ν
Coefficient of linear thermal expansion α
0.30
12 ×10-6 °K-1
210000 MPa
22
FINITE ELEMENT ANALYSIS OF A CONICAL SPRING
Finite Element Method (FEM) in ANSYS 2021 R1 software.
● Capable of solving large, complex problems with general
geometry, loading, and boundary conditions.
● Increasingly becoming the primary analysis tool for designers and
analysts.
● The Finite Element Method is also known as the Matrix Method of
Structural Analysis in the literature because it uses matrix algebra
to solve the system of simultaneous equations
MESHING
23
DETAILS OF MESHING-
24
LOADS AND BOUNDARY CONDITIONS
For this static structural analysis, the following boundary conditions are applied
Applying Fixed Support at Coil of Diameter 128mm as shown below-
Applying Load along Y axis in smaller coil of diameter 5 mm as shown below-
25
STATIC STRUCTURAL ANALYSIS
1. Applied load = 100 N
Static structural analysis was carried out on Conical springs with constant helix angle and their
corresponding deflections and von mises' stresses were determined. Total deflection and Von misses
stress results are shown in, respectively.
26
2. Applied load = 500 N
Static structural analysis was carried out on Conical spring with constant helix angle and their
corresponding deflections and von misses' stresses were determined. Total deflection and Von mises
stress results are shown in, respectively.
27
2. Applied load = 1500 N
Static structural analysis was carried out on Conical spring with constant helix angle and their
corresponding deflections and von misses' stresses were determined. Total deflection and Von misses
stress results are shown in, respectively.
28
RESULTS
OBSERVATION TABLE
● FOR CONICAL SPRING WITH CONSTANT HELIX ANGLE
APPLIE
D LOAD
FEM ANALYSIS
EQUIVALENT
STRESS
(In MPa)
100N
200N
500N
1000N
1500N
157.26
314.51
786.28
1572.6
2358.8
THEORETICAL ANALYSIS
TOTAL
EQUIVALENT
DEFORMATIO
STRESS
N
(In MPa)
(In mm)
30.56
144
61.13
288
152.83
720
305.67
1440
386.85
2160
TOTAL
DEFLECTION
(In mm)
51.51
103.03
239.31
314.75
335.09
29
30
Conical Spring with a Constant Stress at Solid,
and calculating theoretically
R(π/2) = 29.02;
D(π/2) = 58.04;
R(5π/2) = 33.97;
D(5π/2) = 67.94;
R(9π/2) = 40.96;
D(9π/2) = 81.92;
R(13π/2) = 51.56;
D(13π/2) = 103.13;
= 554.02 N = 56.49 kg-f
= 1266.34 N = 129.13 kg-f
= 1444.67 N
31
32
33
MODELLING
Modelling Conical Spring considering above calculated information in
SOLIDWORK 2021 Software, we get-
Analysis done in ANSYS 2021 R, same as done earlier.
34
RESULTS
OBSERVATION TABLE
● FOR CONICAL SPRING WITH CONSTANT STRESS AT SOLID
APPLIED LOAD
FEM ANALYSIS
THEORETICAL ANALYSIS
TOTAL DEFORMATION
TOTAL DEFLECTION
100 N
26.728
43.84
200 N
53.456
87.69
500 N
133.64
219.24
1000 N
267.28
330.83
1500 N
323.64
338
35
CONCLUSION
● Most research papers that exploit conical springs focus only on conical
springs with a constant pitch. In order to increase the range of possibilities,
this paper has studied conical springs with Constant helix angle and Conical
Spring with Constant stress at Solid.
● The analytical study enabled us to define the theoretical geometry of the
spiral in order to obtain a conical spring with a constant angle and with a
constant maximum stress, load -length relation for fully telescoping springs.
Based on the spirals proposed, the corresponding compression load has been
calculated using the common assumptions for springs.
● The formulae can be used for any kind of conical shape (whether the spring
is able to telescope or not). Tests on conical springs made using Fused
Deposition Modeling showed that all the analytical formulae proposed
enable the initial rates to be determined with accuracy.
● On the other hand, the theoretical formulae tend to under-estimate the load
required to reach a given length. It would be of great interest to increase the
accuracy of the predictions. Thus advanced finite element studies could be
used to evaluate the effect of end coils, which is suspected to be the main
cause of the gap.
● To reach the required accuracy, the studies would have to manage large
displacements, contact between coils and contact between coils and the
ground. Such finite element studies would be able to test several options
for the end coil geometry and may help to find the most suitable ones.
● Another source of improvement could be to precisely identify the loads
(forces and moments) induced by the end coils as they are, and to perform
another analytical study to determine the associated load -length relations.
36
REFERENCES
Analytical and Experimental Study of Conical Telescoping Springs With
Nonconstant Pitch.
By Manuel Paredes ,Université de Toulouse; INSA, UPS, Mines Albi, ISAE; ICA
(Institut Clément Ader) 135, avenue de Rangueil, 31077 Toulouse, France
Available from:
https://www.researchgate.net/publication/330410037_Analytical_and_Experimenta
l_Study_of_Conical_Telescoping_Springs_With_Nonconstant_Pitch.
Constraint Analysis and Optimization of NES System Based on Variable Pitch
Spring
Zhenhang Wu, Manuel Paredes (B), and Sébastien Seguy, Institut Clément Ader
UMR CNRS 5312, Université de Toulouse, UPS, INSA, ISAE-SUPAERO,
MINES-ALBI, CNRS, 3 rue Caroline Aigle, Toulouse, France
Available from:
https://www.researchgate.net/publication/351040236_Constraint_Analysis_and_O
ptimization_of_NES_System_Based_on_Variable_Pitch_Spring
Optimal design of conical springs
MANUEL PAREDES, EMMANUEL RODRIGUEZ, Université de Toulouse;
INSA, UPS; LGMT (Laboratoire de Génie Mécanique de Toulouse); 135,avenue
de Rangueil, F-31077 Toulouse, France
Available from:
https://www.researchgate.net/publication/220677684_Optimal_design_of_conical_
Springs
37