Unit 5- balancing of reciprocating masses, Dynamics of machines of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Static force analysis, Unit-1 of Dynamics of machines of VTU Syllabus compiled by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
ME010 801 Design of Transmission Elements
(Common with AU010 801)
Teaching scheme Credits: 4
2 hours lecture, 2 hour tutorial and 1 hour drawing per week
Objectives
To provide basic design skill with regard to various transmission elements like clutches, brakes, bearings and
gears.
Module I (20 Hrs)
Clutches - friction clutches- design considerations-multiple disc clutches-cone clutch- centrifugal clutch -
Brakes- Block brake- band brake- band and block brake-internal expanding shoe brake.
Module II (17 Hrs)
Design of bearings - Types - Selection of a bearing type - bearing life - Rolling contact bearings - static
and dynamic load capacity - axial and radial loads - selection of bearings - dynamic equivalent load -
lubrication and lubricants - viscosity - Journal bearings - hydrodynamic theory - design considerations -
heat balance - bearing characteristic number - hydrostatic bearings.
Module III (19 Hrs)
Gears- classification- Gear nomenclature - Tooth profiles - Materials of gears - design of spur, helical,
bevel gears and worm & worm wheel - Law of gearing - virtual or formative number of teeth- gear tooth
failures- Beam strength - Lewis equation- Buckingham’s equation for dynamic load- wear loadendurance strength of tooth- surface durability- heat dissipation - lubrication of gears - Merits and
demerits of each type of gears.
Module IV (16 Hrs)
Design of Internal Combustion Engine parts- Piston, Cylinder, Connecting rod, Flywheel
Design recommendations for Forgings- castings and welded products- rolled sections- turned parts,
screw machined products- Parts produced on milling machines. Design for manufacturing - preparation
of working drawings - working drawings for manufacture of parts with complete specifications including
manufacturing details.
Note: Any one of the following data book is permitted for reference in the final University examination:
1. Machine Design Data hand book by K. Lingaiah, Suma Publishers, Bangalore/ Tata Mc Graw Hill
2. PSG Design Data, DPV Printers, Coimbatore.
Text Books
1. C.S,Sarma, Kamlesh Purohit, Design of Machine Elements Prentice Hall of India Ltd NewDelhi
2. V.B.Bhandari, Design of Machine Elements McGraw Hill Book Company
3. M. F. Spotts, T. E. Shoup, Design of Machine Elements, Pearson Education.
Reference Books
1. J. E. Shigley, Mechanical Engineering Design, McGraw Hill Book Company.
2. Juvinall R.C & Marshek K.M., Fundamentals of Machine Component Design, John Wiley
3. Doughtie V.L., & Vallance A.V., Design of Machine Elements, McGraw Hill Book Company.
4. Siegel, Maleev & Hartman, Mechanical Design of Machines, International Book Company
This book is intended to cover the basic Strength of Materials of the first
two years of an engineering degree or diploma course ; it does not attempt
to deal with the more specialized topics which usually comprise the final
year of such courses.
The work has been confined to the mathematical aspect of the subject
and no descriptive matter relating to design or materials testing has been
included.
The various forces acts on the reciprocating parts of an engine.
The resultant of all the forces acting on the body of the engine due to inertia forces only is known as unbalanced force or shaking force.
Unit 5- balancing of reciprocating masses, Dynamics of machines of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Static force analysis, Unit-1 of Dynamics of machines of VTU Syllabus compiled by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
ME010 801 Design of Transmission Elements
(Common with AU010 801)
Teaching scheme Credits: 4
2 hours lecture, 2 hour tutorial and 1 hour drawing per week
Objectives
To provide basic design skill with regard to various transmission elements like clutches, brakes, bearings and
gears.
Module I (20 Hrs)
Clutches - friction clutches- design considerations-multiple disc clutches-cone clutch- centrifugal clutch -
Brakes- Block brake- band brake- band and block brake-internal expanding shoe brake.
Module II (17 Hrs)
Design of bearings - Types - Selection of a bearing type - bearing life - Rolling contact bearings - static
and dynamic load capacity - axial and radial loads - selection of bearings - dynamic equivalent load -
lubrication and lubricants - viscosity - Journal bearings - hydrodynamic theory - design considerations -
heat balance - bearing characteristic number - hydrostatic bearings.
Module III (19 Hrs)
Gears- classification- Gear nomenclature - Tooth profiles - Materials of gears - design of spur, helical,
bevel gears and worm & worm wheel - Law of gearing - virtual or formative number of teeth- gear tooth
failures- Beam strength - Lewis equation- Buckingham’s equation for dynamic load- wear loadendurance strength of tooth- surface durability- heat dissipation - lubrication of gears - Merits and
demerits of each type of gears.
Module IV (16 Hrs)
Design of Internal Combustion Engine parts- Piston, Cylinder, Connecting rod, Flywheel
Design recommendations for Forgings- castings and welded products- rolled sections- turned parts,
screw machined products- Parts produced on milling machines. Design for manufacturing - preparation
of working drawings - working drawings for manufacture of parts with complete specifications including
manufacturing details.
Note: Any one of the following data book is permitted for reference in the final University examination:
1. Machine Design Data hand book by K. Lingaiah, Suma Publishers, Bangalore/ Tata Mc Graw Hill
2. PSG Design Data, DPV Printers, Coimbatore.
Text Books
1. C.S,Sarma, Kamlesh Purohit, Design of Machine Elements Prentice Hall of India Ltd NewDelhi
2. V.B.Bhandari, Design of Machine Elements McGraw Hill Book Company
3. M. F. Spotts, T. E. Shoup, Design of Machine Elements, Pearson Education.
Reference Books
1. J. E. Shigley, Mechanical Engineering Design, McGraw Hill Book Company.
2. Juvinall R.C & Marshek K.M., Fundamentals of Machine Component Design, John Wiley
3. Doughtie V.L., & Vallance A.V., Design of Machine Elements, McGraw Hill Book Company.
4. Siegel, Maleev & Hartman, Mechanical Design of Machines, International Book Company
This book is intended to cover the basic Strength of Materials of the first
two years of an engineering degree or diploma course ; it does not attempt
to deal with the more specialized topics which usually comprise the final
year of such courses.
The work has been confined to the mathematical aspect of the subject
and no descriptive matter relating to design or materials testing has been
included.
The various forces acts on the reciprocating parts of an engine.
The resultant of all the forces acting on the body of the engine due to inertia forces only is known as unbalanced force or shaking force.
Method of Moment analysis of a printed Archimedian Spiral antenna Piyush Kashyap
A single arm Archimedean spiral printed on a grounded dielectric substrate is analyzed using the method of moments. Piecewise sinusoidal subdomain basis and test functions are used over curved segments that exactly follow the spiral curvature. Results for the input impedance obtained using the curved segmentation approach on MATLAB are compared with those obtained after simulating the model on FEKO. A comparison with published results shows that the curved segment model requires fewer segments and is therefore significantly more computationally efficient than the linear segmentation model.
IS800:2007 GENERAL CONSTRUCTION IN
STEEL — CODE OF PRACTICE with latest amendments and bookmarks so as to facilitate the navigation through the document, to get onto a particular clause or table directly.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
AIRCRAFT GENERAL
The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
The A318, A319, A320 and A321 are twin-engine subsonic medium range aircraft.
The family offers a choice of engines
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSE
Chapter 17(leaf springs)
1. Beam shape of uniform strength.•
What is leaf spring and where it is used?•
Stress and deformation analyses of leaf spring.•
Designing of a leaf spring in terms of deter-•
mination of the leaf’s requisite cross-sectional
dimension.
Leaf Springs
Leaf springs (sometimes also known as laminated springs) are made up of beams of uniform strength
and are normally used in automobiles to absorb the shock and vibration produced by road undula-
tions, thereby providing comfort to passengers. Figure 17.1 shows a general arrangement of a typical
semi-elliptical leaf spring.
The figure also shows the different components of a typical leaf spring1
. Figure 17.2 shows a typical
application of the spring in an automobile.
Span 2L
Camber
Eye
Master leaf Central clamp
Graduated leaves
Rebound clip
Figure 17.1 Laminated semi-elliptical spring.
1
An interested reader is further directed to refer to Reference (24) in the Bibliography section of the book to know
further details of a leaf spring.
Learning goaLS
After completing this chapter, you will be able to understand the following:
17
Chapter
SOM_Chapter 17.indd 741 4/24/2012 8:51:36 PM
2. 742 • Chapter 17—Leaf SpringS
17.1 Beams of Uniform Strength
Let us consider a cantilever beam of a rectangular cross-section loaded at its free-end as shown in Figure 17.3.
Now, we know that for such beams, the bending stress is same throughout the length of the beam.
If we consider a section at a distance of x from the free-end of the beam as shown in Figure 17.3(a)
and show its free-body diagram in Figure 17.3(b), we can find out the magnitude of the bending
moment Mx prevailing at that section according to the equation:
Mx = −Px (17.1)
Obviously, the magnitude of the maximum bending stress is given by:
σ = =
MC
I
Px
t
bt
( )
2
1
12
3
Figure 17.2 Leaf spring in an automobile.
Figure 17.3 (a) Cantilever beam, (b) free-body diagram of any section.
(a)
(b)
P
x
Vx
Mx
x
P
b
t
L
SOM_Chapter 17.indd 742 4/24/2012 8:51:37 PM
3. 17.2 DefLeCtion of Beam of Uniform Strength • 743
or σ =
6
2
Px
bt
(17.2)
If we keep the thickness t of the beam section constant and vary the section width b along the beam
length in such a way that everywhere the stress is same, say equal to so, then Eq. (17.2) can be used to
define b as:
b
P
t
x=
6
2
σo
or b = lx (17.3)
For a given value of P, the parameter λ σ= 6 2
P t/ o is constant and evidently the above Eq. (17.3)
suggests that b must then vary linearly with x as shown in the Figure 17.4(a).
However, practically, absence of any material at the cantilever tip as shown in Figure 17.4(a) is
unacceptable as load P (at the tip of the beam) then cannot be borne by it. This above design of beam
is the basic element of a leaf spring.
17.2 Deflection of Beam of Uniform Strength
As is evident from the previous Figure 17.4(a), the width of the section at a distance x from the free-end
of the beam is given by:
b x
b
L
x( ) =
o
(a)
L
b(x)
x
bo
(b)
A
P
b(x)
t
L
A
y
x O
A–A section
Figure 17.4 Beam of uniform strength: (a) Top view of beam, (b) front view of beam.
SOM_Chapter 17.indd 743 4/24/2012 8:51:38 PM
4. 744 • Chapter 17—Leaf SpringS
and centroidal area moment of inertia is given by:
I x b x t( ) ( )=
1
12
3
=
1
12
3
b t
L
x0
or I x
I
L
x( ) =
0
(17.4)
To compute the bending deflection y, we use the flexure equation, Eq. (7.6) of Chapter 7 as:
EI
y
x
Mx
d
d
2
2
= −
Although in Chapter 7, we mostly used the above equation for a prismatic beam (i.e., beam with
constant EI), now we use it for a non-prismatic beam as well. From Eq. (17.1), as Mx = −Px, we get
EI
L
x
y
x
Pxo d
d
2
2
=
or
EI
L
y
x
Po d
d
=
2
2
Integrating successively with respect to x:
EI
L
y
x
Px Co d
d
= + 1 (17.5)
and
EI
L
y
Px
C x Co
= + +
2
1 2
2
(17.6)
Putting the boundary conditions of y = 0 and dy/dx = 0 at x = L in the above equations, we obtain:
C PL C
PL
1 2
2
2
= − =and
Thus, putting x = 0 in Eq. (17.6) we get the maximum deflection, which is the deflection of the beam
at its free end as:
EI
L
Px
PLx
PL
x
o
= − +
=
δmax
2 2
0
2 2
SOM_Chapter 17.indd 744 4/24/2012 8:51:39 PM
5. 17.3 Leaf Spring • 745
or
EI
L
PLo
δmax =
2
2
or δmax =
PL
EI
3
2 o
(17.7)
Comparing the free-end deflection of a prismatic cantilever beam with area moment of inertia equal
to Io, we note that the above deflection is 1.5 times more. This large deflection under load P is used to
make the beam to act as a spring. If instead of a cantilever beam had we considered a simply supported
beam, the uniform strength would have led to the geometry known as Lozenge-shaped geometry as
shown in Figure 17.5.
As each half of the above beam can be modelled as a cantilever beam of uniform strength, we get
from Eqs. (17.2) and (17.5) by putting L/2 in place of L and P/2 in place of P, the following relations
for the lozenge-shaped beam:
σo
o
=
3
2
PL
b t
(17.8)
δmax =
PL
LEI
3
3 o
17.3 Leaf Spring
A leaf spring or a laminated spring is made up from a beam of uniform strength by cutting, say ng
equal strips from the original beam of uniform strength and stacking them one on top of the other.
The construction is shown in Figure 17.6.
This arrangement of stacked plates produces cantilever-type leaf spring as shown in Figure 17.6(b).
In Figure 17.6(a), a triangular plate of width bo (which is a cantilever of uniform strength) is cut into ng
equal strips of width b of the central strip (1) where bo = ngb and stacked on as shown in Figure 17.6(b)
t t
b(x)
P
bo
P/2 P/2
L/2L/2
Figure 17.5 Lozenge-shaped beam.
SOM_Chapter 17.indd 745 4/24/2012 8:51:40 PM
6. 746 • Chapter 17—Leaf SpringS
to form a cantilever-type leaf spring. In practice, one or more number of extra full-length plates of uniform
width are used to place on top of the graduated plate (1). This is done because to carry the load at the tip
of plate (1) we need sufficient material to provide the necessary shear force. However, as plate (1) has a
pointed tip, it is not possible to do so if no extra top plate(s) is/are used. This extra plate is known as the
master leaf.
Yet another variation of the laminated spring, known as the semi-elliptic spring as shown in
Figure 17.1. It is used in practice, where a favourable curvature known as cambering is provided to
the entire assembly of plates to carry more loads with uniform stress distribution using favourable
conditions of residual stresses. The load on the beam due to which this initial curvature can be reduced
to zero is called proof load.
In the following sections, we provide the analysis of the cantilever-type leaf springs, which is appli-
cable also for the semi-elliptic leaf springs.
Stress Deformation Analysis for Leaf Springs
Let us assume that to the top of the assembly of the ng number of graduated plates of width b, we
use nm number of extra full-length master leaf each of uniform width b [i.e., width of the topmost
graduated plate (1)]. Thus, essentially, we have a cantilever beam of width nmb and length L connected
parallel to the other cantilever beam of uniform strength of width ngb and length L as shown in the
Figure 17.7. Clearly, through this arrangement, the applied load P is shared by both the beams. The
parallel connectivity of the two beams is modelled as if the beams are connected by a massless rigid link
connected at the free ends of the beams. Let Pm and Pg be the loads shared by the master leaf and the
graduated leaf, respectively. Clearly,
P = Pm + Pg (17.9)
Also, we note that the free-end deflections of the beams are equal as they are connected by a rigid link.
Hence,
P L
EI
P L
EI
m
m
g
g
3 3
3 2
=
L
(a)
b1
2
3
(b)
3
2
1
bo = ngb
bg/2
Figure 17.6 Cantilever leaf spring.
SOM_Chapter 17.indd 746 4/24/2012 8:51:41 PM
7. 17.3 Leaf Spring • 747
or
P
P
I
I
m
g
m
g
= ⋅
3
2
where Im and Ig are second moments of area of the master leaf and graduated leaf, respectively.
Therefore,
P
P
n bt
n bt
m
g
m
g
=
3
2
1
12
1
12
3
3
or
P
P
n
n
m
g
m
g
=
3
2
(17.10)
Solving Eqs. (17.9) and (17.10), we get
P
n
n n
Pm
m
m g
Load carried by master leaf=
+
=
3
3 2
(17.11)
Beam of uniform strength
Beam of uniform width
Massless rigid link
Master leaf
P
t
t
L
nmb
ngb
LGraduated leaf
Figure 17.7 Parallel assembly of cantilever leaf springs.
SOM_Chapter 17.indd 747 4/24/2012 8:51:41 PM
8. 748 • Chapter 17—Leaf SpringS
and P
n
n n
Pg
g
m g
Load carried by the graduated leaf=
+
=
2
3 2
(17.12)
Now, maximum stress developed in the beams’ master and graduated leaves are as follows:
σm
m
m
=
6
2
P L
n bt
in the master leaf as it is a rectangular section. Putting Pm from Eq. (17.11), we get
σm
m g
=
+
18
3 2 2
( )n n
PL
bt
(17.13)
The above equation gives us the maximum stress developed in the master leaf. Similarly, by putting the
expression of Pg in the stress equation, we get the maximum stress in the graduated leaf as
σg
m g
=
+
12
3 2 2
n n
PL
bt
(17.14)
Comparing Eqs. (17.13) and (17.14), we observe that the stress developed in the master leaf is 50%
more than that of the laminated plates. Obviously, this limits the load-carrying capacity of the entire
assembly. Hence, in practice, the master leaf is given a different radius of curvature, thereby putting
favourable residual stress in it in such a way that when the load is applied to the assembly, an equal
stress distribution exists and the system can carry more load. Let us now focus on the deflection of the
entire assembly. If dmax is the maximum free-end deflection of the assembly, then from Eq. (17.7):
δmax =
P L
EI
g
g
3
2
Now from Eq. (17.12) and noting I n btg g= 3
12/ , we get
δmax =
+
6 2
3 2
3
3
L
n bt E
n
n n
P
g
g
m g
or δmax
( )
=
+
12
3 2
3
3
PL
Ebt n nm g
(17.15)
Equations (17.13) and (17.14) indicate the strength of the spring and Eq. (17.15) gives the rigidity of the
assembly. We have to note that the above modelling is approximate, and hence the foregoing equations
SOM_Chapter 17.indd 748 4/24/2012 8:51:42 PM
9. 17.3 Leaf Spring • 749
approximately depict the strength and deformation analysis of the cantilever-type of laminated springs.
Recall that these equations are also applicable to the semi-elliptic springs.
exampLe 17.1
A cantilever leaf spring is designed to meet the following specifications:
Load on the spring = 2 kN
Total number of leaves = 8
Number of extra full-length leaves = 2
Width of each leaf = 50 mm
Length of spring = 500 mm
Design stress in tension = 350 MPa
What is the thickness of leaf required to meet the above requirements?
Solution
Assuming no pre-stressing, we observe that stress in the master leaf is the deciding factor as its stress
is 50% more than that in the graduated leaves. From Eq. (17.13), we get
σm
m g
=
+
18 1
3 22
PL
bt n n
So, the thickness of the leaf is given by
or t
PL
b n n
=
+
18 1
3 2σm m g
Putting the necessary values, we get
t =
+
×
=
⋅ −( )( )( )( . )
( )( )( . ) ( ) ( )
.
18 2 10 0 5
350 10 0 05
1
3 2 2 8
10
6 84
3
6
3
mm
mm
Thus, the required thickness of the plates is 6.84 mm. [Answer]
exampLe 17.2
A laminated semi-elliptic spring under a central load of 12 kN is to have an effective length of 1 m and
is not allowed to deflect more than 75 mm. The spring has 10 leaves, 2 of which are of full length and are
pre-stressed so that all leaves have the same stress after the full load is applied. All leaves have the same width
and thickness. The maximum stress in the leaves is not to exceed 350 MPa. Find the width and thickness of
the plates. Assume that for the spring material, E = 200 GPa.
SOM_Chapter 17.indd 749 4/24/2012 8:51:43 PM
10. 750 • Chapter 17—Leaf SpringS
Solution
In the given condition, the stresses are all equal in the leaves. We thus, consider the semi-elliptic
spring as two cantilever-type leaf springs connected as shown in Figure 17.8.
A C
B P/2 = 6 × 103
NP/2 = 6 × 103
N
P = 12 × 103
N
500 mm = L/2
1000 mm = L
Figure 17.8 Arrangement of beams in Example 17.2.
As shown in the figure, we consider the portion BC of the beam as the cantilever-type leaf spring with
load P/2 and length L/2. Applying the stress and deflection equations, and by noting that stresses are
all equal in the leaves, we get
σmax = =
=
6
6
4 3
22 2 2
M
nbt
PL
nbt
PL
nbt
where n is the total number of leaves, b denotes plate width and t is plate thickness. (Note that the
above equation is applied as the stresses in the leaves are all equal because of pre-stressing.)
3
2
12 10 10
10
350
3 12 10
2 10 350
3 3
2
2
6
( )( )
( )
( )( )
( )( )bt
bt= ⇒ =
or bt2 3
5142 86= . mm (1)
Again, in Eq. (17.15) by putting P/2 and L/2 in place of P and L, respectively, we get
δmax
( )
=
+
12
2 2
3 2
3
3
P L
Ebt n nm g
=
+
3
4 3 2
3
3
PL
Ebt n nm g( )
or bt
PL
E n nm g
3
3
3
4 3 2
=
+δmax ( )
=
× + ×
3
4
12 10 1000
200 10 75 3 2 2 8
3 3
3
( )( )
( )( )( )
SOM_Chapter 17.indd 750 4/24/2012 8:51:44 PM
11. 17.3 Leaf Spring • 751
or bt3 4
27272 73= . mm (2)
Now from Eqs. (1) and (2), we get
b = 183.1 mm and t = 5.30 mm
Thus, the required width and thickness of the plates are 183.1 mm and 5.30 mm, respectively.
[Answer]
exampLe 17.3
A 100 mm outer diameter steel coil spring having 10 active coils of 2.5 mm diameter wire is in contact
with a 750 mm long steel cantilever spring having 6 graduated leaves, 100 mm wide and 6.5 mm thick as
shown in Figure 17.9. Assume for steel, E = 200 GPa.
(a) Calculate force F, which when gradually applied to the top of the coil spring will cause the cantilever
spring to deflect by 25 mm.
(b) What will be the maximum shear stress in the coil spring?
750 mm
F
Figure 17.9 Problem 17.3.
Solution
We observe from the figure that the coiled spring and the graduated leaf spring are in series arrange-
ment and carry the same axial load F.
(a) By applying deflection equation, Eq. (17.15) for the leaf spring, we get
δmax
( )
=
+
12
3 2
3
3
FL
Ebt n nm g
or F
Ebt n n
L
=
+( ) ( )max
3
3
3 2
12
δ m g
SOM_Chapter 17.indd 751 4/24/2012 8:51:44 PM
12. 752 • Chapter 17—Leaf SpringS
Now putting the given values in the above expression, we get
F =
× + ×200 10 100 6 5 25 3 0 2 6
12 750
3 3
3
( )( )( . ) ( )( )
( )
N
or F = 325.5 N [Answer]
(b) Recalling our Eqs. (2.15) and (2.16) from Section 2.5, we get the following results. The maxi-
mum shear stress in the coiled spring considering Wahl’s correction factor is
τ
π
max
.
=
−
−
+
8 4 1
4 4
0 615
3
FD
d
C
C C
Putting C D d= = =/ / .100 12 5 8
τ
π
max
( )( . )( )
( . )
( )
( )
.
=
−
−
+
8 325 5 100
12 5
4 8 1
4 8 4
0 615
83 2
N
mm
or τmax .= 50 25 MPa [Answer]
and considering theoretical stress equation, we get
τ
π
max = +
8
1
23
FD
d
d
D
= +
8
1
0 5
3
FD
d Cπ
.
= +
( )( . )( )
( . )
.8 325 5 100
12 5
1
0 5
83 2
π
N
mm
or τmax .= 45 1 MPa [Answer]
Summary
In this chapter, we have carefully introduced the
concept of beam of uniform strength. From that
concept, we went on discussing the use of such
beams with constant thickness as spring elements,
as they suffer more deflection than their ordi-
nary counterparts. These designs (two of which
SOM_Chapter 17.indd 752 4/24/2012 8:51:45 PM
13. nUmeriCaL proBLemS • 753
have been considered – cantilever type, simply
supported type) find their useful practical applica-
tions especially in the case of automobiles.
A preliminary approximate stress analy-
sis of these springs, known as leaf springs, have
been presented along with their deformation
characteristics also. This chapter will surely be
able to introduce the students to a complex design
analysis of leaf springs, which they will take up in
their Machine Design course.
Key terms
Leaf spring
Laminated spring
Uniform strength beam
Semi-elliptical leaf spring
Rectangular cross-section
Cantilever beam
Prismatic beam
Lozenge-shaped beam
Cantilever-type leaf spring
Master leaf
Graduated leaf
Coiled spring
Wahl’s correction factor
review Questions
1. What is a leaf spring?
2. Explain the function of a leaf spring.
3. Why is a leaf spring used?
4. What is proof load?
5. What do you mean by beam of uniform
strength?
6. Discuss the construction of a leaf spring.
7. Derive the stress deformation theory of leaf
springs.
numerical problems
1. A 1 m long cantilever spring is composed of
8 graduated leaves and 1 additional full-
length leaf. The leaves are 45 mm wide.
A load of 2000 N at the free-end of the
spring causes a deflection of 75 mm.
Assuming no pre-stressing, calculate the
maximum bending stress developed in the
spring. Assume that for the spring material,
E = 200 GPa.
2. Repeat the above problem assuming pre-
stressing of the additional full-length leaf.
3. For Problem 1, calculate the thickness of the
leaf required.
SOM_Chapter 17.indd 753 4/24/2012 8:51:46 PM