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Modeling and Analysis of a Vibratory Bowl Feeder
Conference Paper · December 2021
DOI: 10.1109/ICASE54940.2021.9904038
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Modeling and Analysis of a Vibratory Bowl Feeder
Sadia Azhar, Dr. Syed Irtiza Ali Shah
Department of Mechanical and
Aerospace Engineering
Air University, Islamabad, Pakistan.
Islamabad, Pakistan
201610@students.au.edu.pk
Abstract— Vibratory bowl feeders are required in industrial
automation for the correct orientation of feed parts. These are
considered to be efficient machines that are cheap and effective.
The biggest disadvantage of the traditional vibratory bowl feeder
is its inability to adapt to new designs, shapes, and orientations of
the feed parts. Traditional vibratory bowl feeders are very
inflexible and can work with only one orientation of the part.
Although vibratory bowl feeders are considered to be traditional
devices, these are now very much modernized and have adapted
to diversified industrial needs. Robotic arms are also often used
along with the cameras to orient the parts. However, with the
employment of robots comes a high cost, extensive maintenance
requirements, and complexity in the assembly line. So a viable
solution is to use the best features of both machines, i.e. while
keeping the cheap effective and efficient qualities of a traditional
vibratory bowl feeder, alter it to adapt to the changing part
shapes and assembly lines. By doing so we can use the same
vibratory bowl feeder for various orientations of components and
in various assembly lines. This would eliminate the need for
redesigning the vibratory bowl feeder with the orientation of each
component, with which assembly line is not only expensive but
also time-consuming. Thus we propose a resourceful, cheap,
time-saving, and also flexible alternative. This is done by
mounting cameras on the feeder, where the feed parts are fed.
This camera can analyze the orientation of the part being fed and
then compare it with the three-dimensional model fed to the
system. It can then decide in which direction the part must be
rotated to have the desired orientation. By using this technique,
we can orient almost any part, which is being fed to the feeder
with the traditional vibratory feeder, without the use of fancy
equipment or robots, just by altering and modernizing the
traditional vibratory feeder. This would make the traditional
vibratory bowl feeder, a device that is very flexible and can adapt
to the changing feed, solving many problems. We can thus alter
an already available system, to become versatile and adaptable, in
numerous assembly lines for different parts, with very little
alteration to the already existing systems.
Keywords—Vibratory bowl feeders, Feed, Two degrees of
freedom system, Response graphs.
BFT: Tangential VBF
BFV: Vertical VBF
: Damping coefficient of rubber mount
: Damping constant of leaf springs
: Force acting on base
: Natural frequency
: input frequency
: Stiffness of rubber mount
: stiffness constant of leaf springs
: Mass of the base
: Mass of bowl
: Moment of vibratory motor
XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE
: Weight of structure
: Displacement of the base
: Displacement of bowl
I.
INTRODUCTION
Vibratory feeders are used to orient randomly oriented parts.
Small parts are aligned and fed to the system assembly.
They are generally used in product assembly. Automation
consistency is key and vibratory feeders provide that so each
segment of the process can work predictably, minimizing
downtime and operator error. The most common type of
vibratory bowl feeders are straight wall feeders, which are
also the most versatile. It can take on the wok or cascade
style vibratory bowl feeder styles to suit specific needs.
To satisfy even the most difficult or delicate part conditions
feeder bowls can be coated or covered in a variety of
materials to reduce the noise level, excess wear or to
provide extra grip or traction for the parts that are fed.
Coatings and linings include plastic compounds, bristled
matting, rubber mating, metal plating, and many more.
Vibratory feeders impact the production of many of the
products we interact with daily and with so many custom
variations it can sometimes be difficult to interpret what
kind of feeder is best for each application or why a certain
feeder was selected. The springs allow the bowl to only
translate in one dimension. The tooling mounted along this
inclined path rejects the misaligned parts down to the center
of the bowl and also properly orients the parts in the desired
orientation. The rejected parts again travel on the inclined
track to be properly oriented. The tooling uses mechanical
movement, pressurized air, magnets, etc. for the orientation
of parts. Feed is controlled through these vibratory feeders.
Hence the feed is put through in a controlled manner to
down the process line.
We know that the vibrations produced in the body of
equipment at the operating frequency and have a lot of
importance in the design because of resonance, which
produces large uncontrolled amplitudes of vibrations. So,
for design safety, it is taken into consideration that the
operating frequency (fo) and the natural frequency (fn) do
not match. There is a transient period in which fn = fo [1].
To overcome this a tuned feeder can be used. As the head
load increases the natural frequency is decreased because of
the increase of pan mass hence decreasing the natural
frequency and increasing the frequency ratio fo/fn [2]. The
support structure can face fatigue if it operates at resonance
for long periods [3]. If the operating frequency and the
natural frequency are made sure to not match, then we can
neglect the elastic forces of suspension [4]. At the support,
high accelerations should be avoided because of its high
consumption of energy and because the vibrations
transmitted to the foundation are increased through them
which reduces the performance [5]. The resultant motion of
vibration is almost circular [6]. The dynamic forces are
transmitted to the foundation [7]. According to a research
[8], if more than 99.28% of the total power of the motor is
being used, a 10% larger motor should be used instead. For
feed to travel in the vibratory feeder, frictional force must be
overcome, which is not possible if the frequency is low with
a small amplitude of vibrations. However, it is also observed
that if the vibration frequency of the conveyor belt is too
large the feed loses contact with it and the acceleration of
the feed surpasses the gravity acting on it and hence the rate
of hopping is increased and exceeds the desired value [9].
The flow of particles depends upon the material
characteristics of the feed such as the grain size and the
shape of grain, the moisture content in the feed, electrostatic
forces (generated between feed and surface or between
particles), the surface texture of the particles or structures of
feed, etc. [10].
The damping varies from 0.1 to 0.01 [11]. Feed rate is the
velocity with which the particles accelerate along a straight
line which is the resultant of the asymmetric vibrations and
the friction force. The asymmetric vibrations are because of
the longer forward (positive) motion [12].
Whether the feed is at rest or in the flight phase is
determined by the feed and the vibratory feeder conveyor
motion [13] [9]. External simple harmonic motion affects
the velocity of the feed material [14]. For vibratory
equipment, the effects of speed are very important and
cannot be neglected. In different elements, the stress
distribution is also analyzed which varies from part to part
and is very essential for the safety calculations of the modal
of any vibratory equipment. As the equipment is vibrating,
the applied static and dynamic forces are affected by this
vibratory motion which depends upon the operation speed
[15]. A model is analyzed for dynamic analysis and among
many benefits, the power output can also be optimized [16].
For the rotary equipment whose speed is up to 10Hz, the
vibration is measured in the form of displacement. The
vibration level of equipment is measured in terms of
acceleration or velocity [17]. An increase in speed of feed
particles is not achieved by maximum deflection of the
feeder but rather a maximum acceleration of the feeder
which depends on the operating frequency of the system [7].
Figure 1. FBD of the vibratory feeder assuming it as SDOF system
[7]
The traditional system of vibratory feeders is specific and
tailored to a particular part or component. This is a
hindrance to maximum productivity in assembly which can
be achieved by having a device that is rather flexible
towards the variations in components. In a production line,
the parts and components are to be fed in a known location
and orientation. For this purpose, robots and humans both
work to increase the efficiency of the manufacturing line. In
today’s growing industrial world it is suitable for the
feeding system to be able to handle variation and adapt to
the parts’ shape and design and also location or orientation
to have flexibility in the production. In recent years,
advancement, there has been an effort to increase the
flexibility in the feed system. One of the major
developments is a vision-guided flexible system. The robots
of the first generation were not sensitive to any kind of
variation and performed the tasks repetitively with parts of
known orientation and shapes. They were incapable of
working in an environment where Parts were fed in
unknown positions and were different in shape and size to
each other. For such an environment a vision system was
required to help cope with all the variations of parts, their
orientation, and position in the assembly line. This was
achieved by placing camera-equipped feeders. Many
methods are available for the use of vision this sensitive
method is given in [18].
A. Camera Equipped Feeders
1) 2D mounted cameras:
The first method uses two-dimensional cameras for feeding.
Stationary two-dimensional cameras are mounted. Parts are
fed randomly with random orientation. For more accuracy,
two cameras are used. One is stationary and the other is
mounted on the robot arm. The limitation of this technique
is that the parts are to be fed at the same height because twodimensional cameras are incapable of detecting depth. This
is a cheap and accurate method other than this one
limitation. It is a very flexible and adaptable solution
because of the ease of two-dimensional camera coding.
2) 3D structured light cameras:
The parts are located with the help of three-dimensional
structured light cameras. The pose of the parts is identified
by a depth sensor. The camera is kept stationary for
assessing the next part while the first part is being reoriented
and re-positioned.
3) CAD support:
A single two-dimensional camera is used with the CAD
geometry data which helps to identify the correct orientation
for the part or component
4) Vibratory surface:
Parts are fed in bulk on a surface that is reoriented after the
detection of their location and their original orientation.
Although this system can adapt to different types of parts
without any rebuilding, this process takes a longer time than
the traditional dedicated system [19].
5) 3D printed bowl feeders:
This type is an add-on to the traditional vibratory bowl
feeder. The top part which deals with the parts or
components can be changed. This part of the machine is
made by additive manufacturing. It reduces the rebuilding
process and has a shorter change over time.
2
Figure 2. (a) Motion of the upper part with respect to the base in
operational conditions for the feeder with vertically-oriented
electromagnets, and (b) feeder with tangentially-oriented
electromagnets without bowl [5].
In an electrodynamic analysis of a modal of vibratory bowl
feeder [5] different important parameters such as inertia of
bowl, the inertia of base, stiffness of the leaf spring, forces
acted upon by the electromagnets, and the effects of
damping are taken into account for the analysis. The key
points which can be extracted from the results are: the force
which is transmitted must be reduced, the motion of bowl
which causes the transfer of the parts must be improved for
efficient traveling of the feed, the frequency of
electromagnetic force at which the feeder is excited must be
estimated for the feeder to work at a frequency near
resonance. These results are satisfactory when we take into
account the modeling done to replicate the rubber mounts,
leaf springs, and the base, but the modal can be improved by
using the finite element method for the dynamic analysis of
the bowl. This will help to capture the behavior of
components with higher frequency and bowl flexibility. This
analysis is satisfactory because the data for this analysis is
collected by the experimentation done on two different
vibratory bowl feeders which are assembled of an assembly
line.
F
igure -3 Vibratory bowl feeders: (a) with tangentially-oriented
electromagnets, and (b) with vertically-oriented electromagnets
Two vibratory bowl feeders in Fig. 4 are tested [5]. The
vibratory bowl feeder with three tangentially oriented
electromagnets (BFT) is shown in (a), and the second
vibratory bowl feeder with two vertically oriented
electromagnets (BFV) is shown in (b). The common
features are a base and a bowl but with different sizes and
geometry of the structures. The vibratory bowl feeder with
three tangentially oriented electromagnets is larger than and
the vibratory bowl feeder with two vertically oriented
electromagnets. Also, there are three leaf springs in the BFT
and four-leaf springs in the BFV. The leaf springs are all
equally spaced hence the angle between the leaf springs of
BFT is 120 degrees and the angle between the equally
spaced leaf springs of BFV is 90 degrees. The vibratory
feeders, the BFT and the BTV both are mounted to the floor
with three mounts, the purpose of which is to isolate them
from the floor and create a distance between the floor and
the vibratory equipment so that the vibrations traveling to
the floor is limited with the help of these mounts. With the
sinusoidal current, these feeders are acted upon by the quasi
sinusoidal forces. From this analysis, we get information
about the following parameters we can look into: the modal
analysis through experimentation, how mounts respond to
the frequency which is acted upon them, the dynamic forces
which are exerted by the electromagnets, during the
operation of the feeder the measurement of the acceleration.
The feed material is conveyed at a specific velocity which is
determined by several parameters. It also depends upon the
material properties of feed material e.g. density of particles
or bulk density [20] [21].
In a vibratory feeder, the conveying speed of components is
affected by the inclination of the track the fed travels upon,
the frequency of excitation for the vibratory bowl feeder, the
amplitude of vibration, and the friction coefficient of the
components and between the components and the track.
These are based on a theoretical analysis of feeding a part
on a vibratory feeder with simple harmonic motion [9].
So far the majority of the research is based on a twodimensional approximation of the three-dimensional part
motion of the dynamic analysis of the vibratory bowl feeder.
In the analysis of the new dynamic modal for the motion of
the part under which is under the effect of excitation from
the base which is moving on or separating from a threedimensional spiral track, different scenarios such as hopping
of the part or sliding of the part on the track can be predicted
which occur due to the interaction between the part and the
track. For each one of these scenarios, a solution algorithm
must be presented consisting of equations of the motion of
the parts. These systems of equations are a function of time
and are used to get information about the location of the part
by their numerical solution. From these results the following
observations can be concluded [22]: the motion of the part
on the vibratory bowl feeder is greatly affected by the
frequency of excitation which is applied. Sliding motion is
observed to be the only mechanism involved in the motion
of the components at a low frequency of excitation, at
higher frequencies of excitation the dominant action of
motion is the hopping of parts on the vibratory feeder.
In the research of a 2D numerical model [23] which was
based on the concept of discrete element method, a modal
was developed to study the behavior of a feeding part by
reciprocating its motion in a vibratory bowl feeder. In an
automated assembly, the vibratory bowl feeders are
considered a major component. The components of a
vibratory bowl feeder which deal with the feeding of parts
are repeatedly experiencing impacts with friction involved.
This is a typical example of a nonlinear dynamic problem.
In previous studies the assumption of point mass was made
however in this study instead of a point mass, a rectangular
shape of the part was assumed. The model demonstrated
both chaotic and periodic behavior which shows that the
model is capable of exhibiting both behaviors of the feeding
part. It can also be seen that by using this model output is
generated in the forms of diagrams which explain the
3
dynamic behavior of the parts in motion very accurately.
When the modal is analyzed at various amplitudes of
vibrations, it was observed that the velocity at which
maximum parts are conveyed is the velocity in between the
velocities at which the parts exhibit the periodic and chaotic
behavior i.e. the velocity at the transition point between the
periodic and chaotic behavior of parts. If the amplitude of
velocity is increased beyond this velocity which is a
transition point, the conveying velocity was observed to be
reduced. Also, it was observed that the conveying velocity is
also affected by the shape of the part. If the length of the
rectangle (of the part which is assumed to be a rectangle, not
a point mass) is increased, the chaotic behavior of the part
began to take place at higher amplitudes, hence a shift was
observed, and subsequently, the value of the maximum
conveying velocity was also increased. If the angle of the
vibration was increased the transition point shifted at a
lower amplitude hence the chaotic behavior also started at a
lower amplitude and the maximum conveying velocity was
also decreased. This analysis was conducted with many
materials with different coefficients of frictions. It was
observed that the coefficients of frictions affected the sliding
regime of the parts. However, it was observed that at higher
frequencies after a certain point the conveying velocity
showed independent behavior as it was not affected by the
coefficient of friction in the hopping regime.
Graph 1 Mean conveying velocity as a function of vibration
amplitude [23].
In the analysis [23] of the vibratory bowl feeder, a twodimensional modal of discrete elements feeding part is
developed. It investigates a more accurate dynamic behavior
of a feeding part. Because in contrast to previous research it
takes the shape of the feeding part to be a rectangle instead
of a point mass which is assumed normally in researches.
Due to this assumption, it was also assumed that this
rectangular shape can rotate and hence has three degrees of
freedom. Simulations were performed at different
amplitudes of vibrations. The following deduction was
made: the conveying velocity depends on the amplitudes of
vibrations. Graph 1 shows the behavior of conveying
velocity vs the amplitude of vibrations. This graph has three
distinct regions. This is the plot of the hopping regime. The
analysis was done for the amplitude of vibration ranging
from the value of 1 to 4 maximum amplitude. According to
the graph when the amplitude is increased from A1 to A2
we can see that the mean conveying velocity also increases
linearly with a slope which is the same as the slope of the
sliding regime i.e. maximum amplitude < 1. After A2 the
slope increases linearly but the slope is greater and hence
the value of conveying velocity also increases greatly. At
A3 the value of conveying velocity is almost maximum.
From A3 to A4 the conveying velocity remains almost
constant when the amplitude is increased. After A4 the as
the amplitude is increased the conveying velocity decreases.
So, from graph 1 It can be deduced that the conveying
velocity increases as we increase the amplitude until a point
i.e. 3.0 maximum amplitude value. After this point, we see a
decrease in the conveying velocity with some oscillations.
The vibratory bowl feeder is vibrating equipment. It is often
dealt with hand by workers on the floor. In many industrial
sectors, there is a risk of exposure of workers to machineproduced mechanical vibrations [24]. The reduction and
assessment of these vibrations are considered a priority in
today’s world where strict safety rules are applied. The
distinction between hand-arm vibrations and whole-body
vibrations is made with the European and Italian regulations
involving the risk connected with exposure of workers with
the vibrations they come in contact with [25].
B. New Technologies
Following are a few new areas of research in the field of
vibratory bowl feeders:
The drop test is another technique used in the vibratory bowl
feeder. A trap is designed for the correct orientation. It is
used to determine the most favorable rest position of the part
being fed which is then used as a reference to orient the
parts. The drop test technique is verified in [26] [27].
The coefficient of friction has effects on the flow. At a
lower frequency, the amplitude of excitation can control the
flow characteristic of granular material but the angle of
inclination hinders it [28].
Self-learning aerodynamic feeding systems are advanced
techniques that can be used for other part feeding devices as
well [29]. The collaborative robots (cobots) have flexibility
in working routines which fill the gap of traditional robot
techniques [30]. Another technique for stable equilibrium
position is using three dimensional model of a work piece
and then compare the top view of it with the camera placed
on the assembly [31]. A configuration system is also used
where the knowledge about each part is acquired in dynamic
simulation which aids the feeder for correct orientation [32].
The design process of the feeder is very time-consuming.
Approaches for automatic design of the feeders taking into
account the parameters of the feed parts are also available
[33] [34]. Traditional bowl feeders cannot keep up with
modern times. One of the new alterations is the Flexi bowl
which comes under the category of flexible feeders [35].
Above are some promising areas of research regarding the
part feeding industry
II. RESEARCH METHODOLOGY & MODELLING
A. COMMONLY USED MODELLING TECHNIQUES
Vibratory bowl feeders can be categorized into two types
with respect to the type of spring attachments. These two
architecture of vibratory bowl feeders are BFT vibratory
bowl feeder and BFV vibratory bowl feeder. The BFT
vibratory bowl feeder has three tangentially oriented
electromagnets and the BFV vibratory bowl feeder has two
vertically oriented electromagnets. The BFT vibratory bowl
feeder has equally spaced three leaf springs i.e. 120 degrees
apart and the BFV vibratory bowl feeder has four leaf
springs also equally spaced i.e. 90 degrees apart. Most work
4
has been done on BFT vibratory bowl feeder as it is most
commonly used in the industry. Due to the tangentially
oriented leaf springs of the of the BFT vibratory bowl
feeder, the mathematical modelling includes the torques and
moments. Most common and recent method is the use of
LaGrange’s approach to solve which can easily include the
resolution in the Cartesian or polar coordinates. It uses the
kinetic and strain energy terms to solve the model.
SPRING MASS DAMPER SYSTEM
Figure 6 Equivalent 2 DoF system [37]
EQUATIONS OF MOTION
Equation 1 Equations of motion.
Figure 4 Schematic of VBT vibratory bowl feeder with three
tangential leaf springs [36]
STATE SPACE FORM
A most common and recent method of modeling is used by
many researchers for analytical modeling is the use of
LaGrange’s approach to solving which can easily include
the resolution in the Cartesian or polar coordinates. It uses
the kinetic and strains energy terms to solve the model [5].
B. MODELLING APPROACH
The approach used is the simplification of the model to
minimize the complex terms such as moments and torques
and devious calculations in the mathematical modeling. It
will also make the mathematical modeling simpler and more
generalized. The BFV vibratory bowl feeder is considered
which has four leaf springs that make them 90 degrees apart.
For simplicity, the springs are assumed to be vertically
oriented.
The process and approach steps are given below
Equation 2 State space equation.
MATRIX FORM
SCHEMATIC DIAGRAM
Equation 3 M, C and K Matrices.
TRANSFER FUNCTION FORM
Equation 4 Transfer function form of EoM's
Figure 5 Equivalent spring-damper system
C. ADVANTAGES AND LIMITATIONS
This modal is a simplified model which can be applied to a
variety of systems. This modal helps to understand the
behavior of the vibratory bowl feeder, its analysis, and the
response of the system.
5
However, there are some of the limitations of this design
approach due to the simplification of the modal it excludes
forces such as torque and moments. Hence some variation in
results can be seen, but the general behavior of the system
can be understood from this modal.
D. MATHEMATICAL MODELLING
The system is a 2 dof system with two masses i.e. the base
and the bowl of the vibratory feeder. EoMs are written for
both the masses. Three different approaches to solve the
modal further are, namely; the matrix method, the state
space conversion, and the transfer function method.
1.
For
Determinant to find roots of the equation
Equation 8 Determinant of Equation of motion
EQUATIONS OF MOTION
3. STATE SPACE REPRESENTATION
These equations in matrix form can also be written in a
state-space form which converts the equations into firstorder equations. These equations are then easy to
manipulate and use in software like Matlab.
(base):
For
,
Equation 5 Equation of motion for mass 1.
Where,
=stiffness of rubber mount
Equation 9 State space equation for m1.
For
,
= damping constant of the rubber mount
0
= displacement of base
=force on base for base excitation
For
(bowl):
Equation 10 State space equation for m2.
If we assume,
Equation 6 Equation of motion for mass 2.
Where,
= equivalent stiffness of leaf springs
= damping constant of the leaf springs
The equation can be re written as:
= displacement of bowl
2.
MATRIX CONVERSION
Equation 11 Matrix form of state space equation.
4.
TRANSFER FUNCTION
Equation 7 Equations of motion in matrix form.
The following transfer functions can be derived from the
equations of motion:
Where,
Equation 12 Equations in the form of transfer function for both
masses m1 and m2
III. IMPLEMENTATION
The software is used to solve these equations and get the
response of the system in Matlab. Matlab has many inbuilt
functions to solve the modes and mode shapes etc. It also
6
has first-order solving algorithms as well as transfer
function solving algorithms which makes it easy and a very
convenient choice to solve long and tedious calculations.
Matrices can also be solved in Matlab.
Matlab also has a plotting option that gives the response of
the system in the form of graphs. The graph plotting in
Matlab has many editing options. By using these various
options, informative and clear graphs can be obtained.
Matlab can also generate graphs with various characteristics
of the system like time constant, rise time, settling time,
peak time, and percent overshoot shown on the graphs.
Matlab produces accurate results and has short codes and
inbuilt programs and algorithms which help to solve the
equations of the system efficiently and fast. Hence it is a
reliable method that is accurate, efficient, and time-saving
which makes analysis and modelling very easy.
Matlab can be used in many fields due to its diverse abilities
some of these are control systems, image processing and
robot control.
Matlab also has Simulink which is very user-friendly. It
simplifies the problem and also makes it easy to visualize
the problem which helps in better understanding of the
system or modal.
Other soft wares used for the analysis of the model like
ANSYS use the geometry and material used in the system to
produce results that are even more diverse. Hence, they can
give very accurate results. This accuracy has made it
possible to research and development without wasting
money and time on actual prototypes which is a major
contributing factor in the research and development
technology.
INPUT CONSTANTS
M1 (mass of base)
100kg
M2 (mass of bowl)
30kg
C1 (damping of rubber
mounts)
C2 (damping of leaf
springs)
K1 (stiffness of rubber
mounts)
K2(stiffness
of
leaf
springs)
F (step input)
50000 N-s/m
F(sinusoidal input)
10cos5t; f=10 N; w=5 Hz
500 N-s/m
100000 N/m
1800000 N/m
1N
Table IV-1 Input constants
IV. RESULTS
The 2 DoF mathematical model, with force acting on the
system from base i.e. base excitation, has been analyzed
with no force, step input, and sinusoidal input. The
characteristics of the system like time constant, rise time,
settling time, natural frequency, damping ratio, peak time,
and percent overshoot and mode shapes have been
determined. These results are then compared and discussed
and an analysis of these results is deduced.
7
A. NO APPLIED FORCE
RISE TIME
RESPONSE GRAPH
=
SETTLING TIME
=
PERCENTAGE OVERSHOOT
TIME CONSTANT
1/a is defined as the time constant of system response. In
this time the natural response decays to 37% of its value and
the step response rises to 63% of its final value.
Also ‘a’ is the exponential frequency and initial slope of the
system.
This graph does not depict exponential behaviour.
DAMPING RATIO
= 2.8
NATURAL FREQUENCY
Rad/sec
MODE SHAPES
Graph 2 Homogeneous Response graph showing displacement,
velocity and acceleration w.r.t. time.
=
m
=
m
ANALYSIS OF RESULTS
PEAK TIME
Time of first peak overshoot.
The two natural frequencies are due to the two degrees of
freedom i.e. one for each mass.
In mode 1 both masses move in the same direction with the
same magnitude. In mode 2 the direction of motion of both
masses is opposite.
This can also be seen by plotting both modes.
8
a) MODE 1
The damping ratio is calculated using the formula and the
natural frequencies of the system and its mode shapes are
calculated using Mat lab built-in functions.
B. STEP INPUT
RESPONSE GRAPH
Graph 4 Step response showing amplitude vs time.
Graph 3 Displacement, velocity and acceleration w.r.t time for
mode 1 (no applied force).
PEAK TIME
b) MODE 2
= 0.046 seconds
RISE TIME
=
SETTLING TIME
= 0.098 seconds
PERCENTAGE OVERSHOOT
= 56 %
Percentage overshoot is given by poles of the pole plot.
TIME CONSTANT
Graph 4 Displacement, velocity and acceleration w.r.t time for
mode 2 (no applied force).
1/a is defined as the time constant of system response. In
this time the natural response decays to 37% of its value and
the step response rises to 63% of its final value.
Also ‘a’ is the exponential frequency and initial slope of the
system. From graph:
9
Seconds
PEAK TIME
DAMPING RATIO
=2.6
seconds
= 0.47
NATURAL FREQUENCY
SETTLING TIME
=
= 0.23 seconds
Hz
=
Hz
ANALYSIS OF RESULTS
Natural frequencies are determined by the pole plot. As the
damping ratio is less than one, system is underdamped. And
it is also verified by the fact that the natural frequencies are
distinct and complex.
The settling time, rise time, and peak time are determined
using these values and their respective formulas. It can be
seen that these are also verified by the graph of the response
of the system
The time constant is determined using the response graph of
the step input
PERCENTAGE OVERSHOOT
= 0.18
DAMPING RATIO
=0.3
NATURAL FREQUENCY
C. SINUSIDAL INPUT
RESPONSE GRAPH
Hz
Hz
1) MODE SHAPES
m
m
The two complex modes show again that it is an
underdamped system. The mode shapes also show that the
bowl and the base vibrate in the same directions i.e. they are
not out of phase in both modes.
2) ANALYSIS OF RESULTS
Graph 5 Response graph for sinusoid input showing the
displacement of both masses w.r.t time.
For a forcing function of magnitude f=10N and forcing
frequency of 5Hz. The damping constant calculated shows
the system to be an underdamped system.
The two complex modes show again that it is an
underdamped system, hence verified. Both modes are
almost identical.
10
V. CONCLUSION
For the analysis of the vibratory bowl feeder, it was
modeled into a two-degree of freedom system. The
mathematical modeling was done using the approach of
summation of forces. The characteristics of the system like
time constant, rise time, settling time, natural frequency,
damping ratio, peak time, and percent overshoot and mode
shapes have been determined with response graphs and
formulas.
The system was subjected to three types of inputs. First, no
input was provided to the system to check the homogeneous
response. Then a unit step input was given to the system to
check stability. The last input was a sinusoid input.
Response graphs were plotted for each input to analyze the
behavior of the system.
To get the homogenous solution no force was applied to the
system. The response graph was plotted using Matlab. An
initial displacement of 1 unit was given to mass 1 i.e. the
bowl of the vibratory bowl feeder as the initial condition.
The response graph shows the starting displacement of mass
1 to be 1 unit and mass 2 to be 0 units as it was at rest at its
initial position. The graph shows that the system is stable
because after the initial disturbance the graphs tend to
approach zero i.e. the equilibrium position of the system.
From mode shapes, it can be seen that in the first mode both
masses vibrate in the same direction but with different
frequencies. In mode 2 however, it can be seen that the
vibration of both masses is out of phase which can be seen
in the graph for mode 2.
The step response is usually analyzed to check the stability
of a system. If the system is stable it should change its initial
equilibrium position according to the step input. We can see
from the step response graph that the system is stable. The
equilibrium position of the system has changed from zero to
1 unit, which was the input unit step function.
The response plot for sinusoid input with initial conditions
set to zero shows a homogeneous behavior at first, which
dies out quickly and the system shows sinusoidal behavior.
This shows the system is stable under sinusoidal input. For
the system to be stable we want the system to act according
to the input, so a sinusoidal response to a sinusoidal input
shows the stability of the system. Also, the frequency of
both the masse is the same which means the bowl and feeder
are perfectly in phase. Which is desirable for the smooth
operation of the vibratory bowl feeder.
As seen from the bode plot at r
the is almost unity at
every value of damping which is also verified by
calculations. To achieve isolation
must be less than .
But we can see that
.
B. BODE PLOT
From bode plot we can see that
vibration isolation.
to achieve
• We can change the forcing frequency as it was just
assumed and set it to be greater than 7.9.
• If the forcing frequency is set, say in an industry
assembly line we can alter the natural frequency by
changing M, C and k by reducing or increasing the mass or
by changing the materials used.
VI. RECOMMENDATIONS FOR FUTURE WORK
•
•
•
•
•
A. TRANSMISSIBILITY FACTOR
•
The transmissibility factor comes out to be almost 1. This
shows that almost the same amount of force is transmitted to
the bowl as applied on the base.
>
By assuming the leaf springs to be vertically
aligned simplified the modal but it excluded major
torques and moments acting on the modal.
Assumption of input variables has made the system
an underdamped system which is favorable in the
case of vibratory bowl feeders but it should be
optimized so that it does not damage the system
with uncontrolled vibrations.
The vibration isolation method should be used so
that force transmission to the foundation can be
reduced. It is of utmost importance because VBF is
a vibratory device and vibrations can cause great
damage over a long period.
Mass vibration reduction should also be applied to
optimize the vibrations. This optimization is very
important because excessive vibrations damage the
system and excessive reduction in vibrations may
affect the efficiency of the system to transport feed
parts.
The perfect hopping motion of parts should be
maintained which can be done by the dynamic
analysis including the feed. Slipping or sliding
back of parts should be avoided. Also, the chaotic
behavior of parts must be avoided. This is why
optimization of the dynamic model is needed.
Vibration isolation design chart and vibration
identification Nomo graph can be used for this
optimization
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