VARDHAMAN COLLEGE OF ENGINEERING
(Autonomous)
Shamshabad, Hyderabad

501218
.
DESIGN OF MACHINE MEMBERS

I
LECTURE NOTES
by
D.V.RAMANAREDDY
ASSISTANT PROFESSOR
MECHANICAL ENGINEERING DEPARTMENT
UNIT

VII
SYALLABUS:
Mechanical Springs :
Stresses and deflections of helical springs
–
Extension

compression springs
–
Springs for
fatigue loading
–
natural frequency of helical
springs
–
Energy storage capacity
–
helical
torsion springs
–
Co

axial springs,leaf springs.
Instructional Objectives
:
The focus is on blending fundamental development of
concepts with practical specification of components
The objectives of the text are to:
1.
Cover the basics of machine design, including the design
process,
Engineering
mechanics and materials, failure
prevention under static and variable loading, and
characteristics of the principal types of mechanical elements.
(PEO 2)
2.
Offer a practical approach to the subject through a wide
range of real world applications and exampl
es.
(PEO 1,2)
3.
Encourage readers to link design and analysis.
(PEO 2)
4.
Encourage readers to link fundamental concepts with
practical component specification
.
(PEO 2)
At the end of this lesson, the students should be able to understand:
1. Ability to
design the different type of springs
2. Ability to understand the
natural frequency of torsion springs.
MECHANICAL SPRINGS
Spring is a type of mechanical device on which if we apply some load, it
deforms but when we remove this load, it returns to its o
riginal shape.
Springs can be used for many purposes which are following
Springs are used to cushion, control or absorb energy in shock absorbers
Springs are used to apply some force in brakes, clutches
Springs are used to limit the motion of two contact
elements like cams and
followers
Springs are used to measure force like in spring balance
Springs are used to store energy like in watches
There are many types of springs which are following
1.
Helical springs
2.
Conical and volute springs
3.
Torsion springs
4.
Leaf s
prings
5.
Bellevile or disk springs
6.
Special purpose springs
Helical springs
Springs are made from wire and this wire can be circular, rectangular or
square but we use circular wire commonly. This spring has helix shape and
there are two types of helical sprin
gs depending upon the load applied which
are compression helical springs and tension helical springs.
In
compression helical springs
load in applied in such a way that it tends
to deform spring. Due to this load applied the length of spring decreases and
diameter increases.
In
tension helical springs
load in applied in such a way that it tends to
extend spring. Due to this load applied the length of spring increases and
diameter decreases.
Now consider the geometry of helical springs, there are also types
which are
close coil springs and open coil springs.
Close coil springs
are those in which the numbers of turns are
perpendicular to helix axis and the gap between each turn is very small. We
can define close coil springs in terms of helix angle also, if h
elix angle is less
than 10° it is called close coil springs.
Open coil springs
are those in which the numbers of turns are not
perpendicular to helix axis and there is a gap between each turn. We can
define open coil springs in terms of helix angle also, i
f helix angle is greater
than 10° it is called open coil springs.
Major stresses produced in this spring are shear stresses due to twisting. We
recommend these springs to use in our daily because of advantages over
other springs which are following
They
are more reliable
Easy to manufacture
Not so much costly
Available in many shape
Change in property after changing the shape
Have constant spring rate
Properties can be predict more accurately
Conical and volute springs
These types of springs are used in s
pecial applications and these are the
types of springs in which spring rate increases as number of active coils
decreases. Conical springs are made in triangular shape and have uniform
pitch whereas volute springs are made in paraboloid shape with constant
pitch.
These springs are made in partially or completely telescoping shape. In both
types trend of spring rate remains same. We can use these springs where
we have varying loading and sometimes we uses these springs in vibration
problems.
–
Major
stress that produced in these springs is shear
stresses due to twisting
Torsion springs
In reality these springs are helical or spiral type. In case of
helical spring
type
, we extended the ends and on these ends we apply load. Due to
loading on extended en
ds, springs winds up. In the result its length will
increase and diameter will decrease.
In case of
spiral type
, we have one extended end and the other is attached
to a fixed point. When we apply load on this type, number of turns of spring
increases. T
his type of spring is used to store energy and is made of
rectangular shape.
In both types compressive and tensile stress are the major stresses due to
bending.
Leaf springs
This type of springs has number of flat plates and these plates are called
leaf. F
lat plates have varying length and all are placed in such a position that
they form triangle. These plates are fixed at their position by means of
clamps and bolts. These springs are mostly used in automobile sectors to
absorb shocks. Major stresses produc
ed in this spring are compressive or
tensile due to bending.
Disk or bellevile springs
This spring has number of conical discs that are held at one place by using
central bolts. We can use these springs where we need large spring rate and
compact shape.
The major stresses that are produced in this spring are
compressive and tensile.
Special purpose springs
These springs are air or liquid spring, ring springs, rubber springs etc and
are used for special purposes.
HELICAL SPRINGS SUBJECTED TO
FATIGUE LOAD
ING
If we want to design a helical spring under fatigue loading we use Soderberg
line method. In this method we test spring for torsional edurance strength
for repeated loading whose value varies from 0 to maximum.
But there is a problem in helical spring
that they can be load in only one
direction so tackle this problem we use modified Soderberg diagram in which
for reverse loading endurance limit is shown at point A and τ
e
/2 is equal to
mean shear stress and variable shear stress.
A
line drawn from A to B shows Soderberg failure stress line and if we add
factor of safety in yield strength τ
y
, we will get a safe value for design. The
line which will obtain is CD and is parallel to line AB.
To get a specific relation let us take a point
P on line CD. Draw two
perpendiculars to x and y axes we will get two triangles PQD and AOB. From
these triangles we have
PQ / QD = OA / OB
PQ / (O
1
D
–
O
1
Q) = OA / (O
1
B
–
O
1
O)
τ
e
. τ
y
/ F.S = 2 τ
v
.τ
y
–
τ
v
.τ
e
+ τ
m
.τ
e
After rearranging the above expression
we will get the following expression
1 / F.S = (τ
m
–
τ
v
) / τ
y
+ 2 τ
v
/τ
e
In the above expression the value of τ
m
can be calculated by using following
formula
τ
m
= K
s
x (8W
m
D)/πd
3
where K
s
is the shear stress factor and its value is 1 + 1/2C and W
m
is the
mean value of minimum and maximum value.
Similarly just like we find out the value of τ
m
, we can also find out variable
shear stress value
τ
v
= K
s
x (8W
v
D)/πd
3
where in the above expression K
s
is the Wahl’s factor and its value is (4C
–
1)/(4C
–
4
) + 0.615/C and the value of W
v
is obtained by dividing the
difference of maximum and minimum value by 2.
Helical Torsion Springs
Helical torsion springs are an important component in for example seat belt
locks, boot lid springs or backrest springs
in cars.
Helical torsion springs in a wide range of variations
Torsion springs can either have a left

hand or right

hand helix. They are
made to give a force and torque when one leg is turned relative the other.
Our helical torsion springs are made in
dimensions from 0.03

26 millimeter
and we select wire diameter, spring body diameter, length and pitch to suit
your requirements.
For closed coiled helical torsion springs there will be a friction between the
coils giving a hysteresis in the torque

angle
performance.
There are generally two common types, single and double coil torsion
springs. Normally is the coil controlled inside with a mandrel but also an
outer control with a cylindrical housing is possible. With fixed assembled legs
the spring type ca
n work without a coil body control. Tangential legs give the
lowest stresses but also radial and axial legs of different types are possible.
In many cases the strain in the legs must be considered in the design.
Tension Springs
Tension springs have a wide
range of applications. They are for example
used in industrial robots and are also important components in
perambulators and door locks.
Tension springs in a wide range of variations
Tension springs are designed to give a specific pull force when extende
d to a
a specific length. Our tension springs have a material dimensional range
from 0.03

26 millimeter.
Cold coiled extension springs can be given an initial tension which needs to
be overcome before any elongation of the spring takes place. The level o
f
initial tension can be controlled.
Most common ends are different types of end loops and hooks, there are
also a wide range of other mechanical attachment possibilities such as coiled
in and screwed in attachments. Examples are, half hooks, German loops,
English loops, extended hooks, side hooks, side loops, screw plugs and
screw shackles.
Unique competence in manufacturing tension springs
Lesjöfors have long experience of tension springs and constantly give
priority to technological developments in th
e field. Our main competitive
advantages are our wide product range, technology support, high quality
and service of tension springs.
Our experience of springs for a large number of different branches and
applications give you the possibility to receive
the best recommendations
and springs optimized for your actual application.
Environment evaluation and material recommendations, possible stress
levels, load and deflection, initial tension and fatigue strength are factors our
spring experts have unique kn
owledge about. We can support you from the
idea and understanding of your expectations and application demands to the
validation of the design.
Our spring operations also include manufacturing with highest flexibility and
delivery precision. For immediate
needs or small orders we also provide a
wide extension spring stock program range with both normal spring steel
and stainless steel.
Irrespective of where you need to use a standard spring or a specially made
tension spring, we can help you to find an opti
mum solution.
We constantly improve, through meeting new requirements. Our spring
experts are experienced problem solvers and we have succeeded in reaching
the position we have today, through deep technical competence, a close
relationship with you, effici
ent production and delivery reliability.
Irrespective of the challenges we face, there is a good chance that we will be
able to deliver the tension springs that exactly meets your company's needs.
Helical Compression Springs
Helical compression springs ar
e our most common spring type. They have a wide
range of applications and can be found in almost all mechanical products, for
example as important components in door locks, in compressors, in valves, in
electric switches and more.
Compression Springs
A
compression spring is an open

coil helical spring that offers resistance to a
compressive force applied axially. Compression Springs are the most
common metal spring configuration and are in fact one of the most efficient
energy storage devices available.
Other than the common cylindrical shape,
many shapes are utilized, including conical, barrel and hourglass. Generally,
these coil springs are either placed over a rod or fitted inside a hole. When
you put a load on a compression coil spring, making it shor
ter, it pushes
back against the load and tries to get back to its original length
Common Applications
Compression springs are found in a wide variety of applications ranging from
automotive engines and large stamping presses to major appliances and
lawn
mowers to medical devices, cell phones, electronics and sensitive
instrumentation devices. Cone shape metal springs are generally used in
applications requiring low solid height and increased resistance to surging.
Compression spring design cases:
Design
to Physical Dimensions
Required Information: Wire Size, Outside Diameter or Hole Size OR Inside
Diameter or Rod Size, Free Length, Number of Coils (Active / Inactive), Ends
Configuration, Solid Height Requirements.
In addition to the general considerations
(in the previous sections) the
following information must be calculated:
Spring Rate
Spring rate is generally defined between 20% and 80% of the available
deflection where it is linear.
Design to Spring Rate
Required Information: Wire Size, Outside Diam
eter or Hole Size OR Inside
Diameter or Rod Size, Free Length, Spring Rate, Number of Inactive Coils,
Ends Configuration, Solid Height Requirements.
In addition to the general considerations (in the previous sections) the
following information must be calc
ulated:
Number of Active Coils
Design to Two Loads
Required Information: Wire Size, Outside Diameter or Hole Size OR Inside
Diameter or Rod Size, Load 1 @ Length 1, Load 2 @ Length 2, Number of
Inactive Coils, Ends Configuration, Solid Height Requiremen
ts.
In addition to the general considerations (in the previous sections) the
following information must be calculated:
Spring Rate
Number of Active Coils
Free Length
Torsion Spring Design Theory
Torsion springs are stressed in bending. Recta
ngular wire is more efficient in
bending than round wire, but due to the premium cost of rectangular wire,
round wire is preferred. Torsion springs, whose ends are rotated in angular
deflection, offer resistance to externally applied torque. The wire itsel
f is
subjected to bending stresses rather than torsional stresses, as might be
expected from the name. Springs of this type are usually close wound. The
coil diameter reduces and body length increases as they are deflected. The
designer must consider the e
ffects of friction and arm deflection on the
torque.
The number of active turns in a helical torsion spring is equal to the number
of body turns, plus a contribution from the ends. For straight torsion ends,
this contribution is equal to one

third of the
moment arms and is usually
expressed as an equivalent number of turns:
Diameter Reduction
When the direction of loading tends to reduce the body diameter, the mean
diameter changes with deflection according to:
where D1 is initial mean diameter and D?
is deflection in revolutions.
Body Length Increase
Most torsion springs are close

wound, with body length equal to the wire
diameter multiplied by the number of turns plus one. When a spring is
deflected in the direction that will reduce the coil diameter,
body length
increases according to:
Spring Rate for Torsion Springs
Spring rate (per turn) for helical round wire torsion springs is given by:
The 10.8 (or 3888) factor is greater than the theoretical factor of 10.2 (or 3670) to
allow for friction bet
ween adjacent spring coils, and between the spring body and
the arbor. This factor is based on experience and has been found to be satisfactory.
Stress
Stress in torsion springs is due to bending, and for round wire is given by:
During elastic deflectio
n of a curved beam, the neutral axis shifts toward the center
of curvature, causing higher stress at the inner surface than the outer. Wahl has
calculated the bending stress correction factor at the ID of a round wire torsion
spring as:
The following for
mulas may also be used as an approximate bending stress
correction factor at the ID or OD of the spring. For low index springs, the stress
concentration will be much higher at the ID than at the OD.
Torsion Spring Design General Considerations
1.
Loads for torsion springs should be specified at a fixed angular position
and not at a fixed deflection from the free position.
2.
Torque testing is not easily performed. While torque measurements
can be taken, they can be inconsistent.
3.
Presently, there is n
o standard way to test loads for torsion springs.
4.
Diameter reduction and potential binding should always be considered.
5.
For applications that require minimum hysteresis (load loss) springs
should be designed with space between adjacent coils to reduce
fric
tional losses.
6.
A torsion spring should always be loaded in a direction that causes its
body diameter to decrease. The residual forming stresses are
favorable in this direction, but unfavorable when the spring is loaded in
a direction that increases body di
ameter.
7.
Clearance must be maintained between the mandrel and spring at all
times to prevent binding. The ideal mandrel size is equal to, or slightly
less than, 90% of the I.D. when the spring is fully deflected (minimum
diameter). Mandrels significantly s
maller than 90% should be avoided
to prevent buckling during large deflections.
8.
Most torsion springs are close wound. Hence, their body length will
increase when a spring is deflected in the direction that will reduce the
coil diameter. In tight housing si
tuations, this increase should be
considered.
9.
Direction of wind must always be specified for a torsion spring.
INTRODUCTION
NATURAL FREQUENCY OF HELICAL SPRING
If a spring which is subject to a vibratory motion which is close to its natural
frequency the spring can start to surge.
This situation is very undesireable
because the life of the spring can be reduced as excessive internal stresses
can result.
Th
e operating characteristics of the spring are also seriously
affected.
For most springs subject to low frequency vibrations surging is
not a problem.
However for high frequency vibrating applications it is
necessary to to ensure, in the design stage
, that the spring natural
frequency is 15 to 20 or more times the maximum operating vibration
frequency of the spring.
Natural Frequency of a Loaded spring System
Consider a mass M supported on a weightless spring with a spring rate k is
illustrated belo
w
Overview
o
Helical Compression Spring

Estimate Natural Frequency
Equations
o
n= [(1.12 (10
3
) d)/(D
2
Na)] (G g/Den)
0.5
o

o
n

Natural Frequency (hz)
o
D

Mean Diameter (in)
o
Na

Number of active coils
o
G

Modulus of Rigidity (psi)
o
g

Acceleration due to gravity (386.4 in/sec^2)
o
Den

Density (lb/in^3)
Assumptions
o
Spring with both ends fixed, no damper
This equation derives the natural frequency of a simple mass

spring oscillator, of
mass
m
and wit
h a stiffness of
k
. The frequency is proportional to the square root of
the ratio of stiffness to mass.
FUNDAMENTAL FREQUENCY
NATURAL FREQUENCY
The natural frequency is the rate at which an object vibrates when it is not
disturbed by an outside force. Each degree of freedom of an object has its own
natural frequency, expressed as ω
n
(omega subscript n). Frequency (omega) is
equal to the speed of v
ibration divided by the wavelength (lambda),
Other equations to calculate the natural frequency depend upon the vibration
system. Natural frequency can be either undamped
or damped, depending on
whether the system has significant damping. The damped natural frequency is
equal to the square root of the collective of one minus the damping ratio squared
multiplied by the natural frequency,
Equation of Motion : Natural Frequ
ency
Figure 2 shows a simple undamped spring

mass system, which is
assumed to move only along the vertical direction. It has one
degree of freedom (DOF), because its motion is described by a
single coordinate
x
.
When placed into motion, oscillation will ta
ke place at the natural
frequency f
n
which is a property of the system. We now examine
some of the basic concepts associated with the free vibration of
systems with one degree of freedom.
Newton's second law is the first basis for examining the motion of
the system. As shown in Fig. 2 the deformation of the spring in the
static equilibrium position is D , and the spring force
kD
is equal to
the gravitational force
w
acting on mass m
By measuring the displacement
x
from the static equilibrium
position, t
he forces acting on
m
are
and
w
. With
x
chosen to be positive in the downward direction, all
quantities

force, velocity, and acceleration are also positive in the
downward direction.
We now apply Newton's second law of motion to the mass
m
:
and beca
use
kD = w
, we obtain :
It is evident that the choice of the static equilibrium position as
reference for
x
has eliminated
w
, the force due to gravity, and the
static spring force
kD
from the equation of motion, and the resultant
force on
m
is simply the spring force due to the displacement
x.
By defining the circular frequency w
n
by the equation
Eq. 6 can be written as
and
we conclude that the motion is harmonic. Equation (8), a
homogeneous second order linear differential equation, has the
following general solution
where A and
B
are the two necessary constants. These constants
are evaluated from initial conditions
and
Eq. (9) can be shown to reduce to
The natural period of the oscillation is established from
OR
and the natural frequency is
What is the potential energy
stored in the spring when the
spring is compressed by 20 cm?
A block of mass 500 grams is
attached to a spring with a spring constant
k=100 N/m. What is the potential energy stored in the spring when the
spring is compressed in the spring when the spring is compressed by 20 cm?
If the spring is released after it is compressed, with what veloci
ty will the
block leave the spring?
he
potential energy
stored in a spring is given by this formula:
U = 1/2 • k • x^2
Where U is the potential energy, k is the spring constant, and x is the distance that
the spring is compressed or stretched. Thus, in
this case:
U = 1/2 • k • x^2
U = 1/2 • 100 N/m • (0.2 m)^2
U = 50 N/m • 0.04 m^2
U = 2
Joules
The
kinetic energy
of the block will be equal to the potential energy of the spring.
This happens because all of the energy stored as elastic potential energy i
n the
spring is converted into kinetic energy when the spring is released. Since KE = 1/2
• m • v^2, we know that:
1/2 • m • v^2 = U
m • v^2 = 2U
v^2 = 2U/m
v = SQRT (2U/m)
By substituting m = 0.5 kg and U = 2 Joules, you get:
v = SQRT (4 J / 0.5 kg)
v
= SQRT (8 m^2/s^2)
v = 2.828 m/s
So, our final answer is that the potential energy in the spring is 2 Joules, and the
velocity of the block is 2.828 m/s. Hope this helps!
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