by
NGONGO KATEMBO MA GALA
Thesis submitted in partial fulfillment of the requirements for the degree of
MASTER IN MECHANICAL AND MANUFACTURING
ENGINEERING
in the
FACULTY OF ENGINEERING
at the
RAND AFRIKAANS UNIVERSITY
SUPERVISOR : PROF. Z. KATZ
ABSTRACT
Parts feeding devices or feeders are used in automated assembly systems to deliver correctly oriented parts to the assembly station workhead. These devices play an important operational role since feeding is one of the major operations involved in an automated assembly process. However they account for much of the cost of an automated assembly system because most of the engineering time spent to develop such a system is used to devise a means of feeding the components in the correct orientation for the assembly process.
This thesis describes the implementation of an integrated and computer controlled feeding and transfer system. The system consists of a vibratory bowl feeder for selecting, orienting and feeding parts and a flat conveyor belt for transferring parts to a prescribed location. The work focusses mainly on the design and analysis of the bowl feeder, on the mechanical and information interfacing aspects of the integration problem and on the control of the system. Sensing and electronic control circuits were also built to complete the system.
The system implemented is to be integrated at a later stage with an industrial robot for handling purposes. Therefore, some issues related to the handling of parts from the conveyor belt by the robot are also discussed.
Experimental results show that the recommendable operating frequency for the vibratory feeder is close to the value predicted by theoretical analysis.
Several concurrent activities with critical time constraints and different periods were involved in the system, making the control more difficult due to the limited control capabilities of Visual basic, an easy to learn programming language used to implement the control program and the relatively slow speed of the computer used Nevertheless, it was observed that for feed rates close to 3 parts/min, the program developed performs well regarding the random control of the flow rate of parts on the conveyor, parts position and speed profiles obtained compare satisfactorily with the corresponding theoretical profiles.
Affectionately dedicated to my wife Rose and my daughter Diane in appreciation for their enduring patience.
Acknowledgements
I would like to express my sincere appreciation to my supervisor Professor Katz for is guidance and support. I would also like to thank the following peoples in assisting me to complete this thesis:
My brother Alexis Mangala Mr Karl OhIm from the energy lab
Abstract ii
Dedication iii
Acknowledgements iv
Contents
List of abbreviations vii
List of figures viii
List of symbols
List of tables xiii
CHAPTER 1 INTRODUCTION- SYSTEM DESCRIPTION
1.1 Introduction 1
1.2 System description and requirements 2
CHAPTER 2 HARDWARE DESIGN AND IMPLEMENTATION
2.1 Vibratory bowl feeder design 5
2.1.1 Vibratory bowl feeder design parameters. 5
2.1.2 Description of the part used in the feeder 10
2.1.3 Design of selecting and orienting devices 10
2.1.4 Bowl feeder drive system 19
2.2 Conveyor belt speed control 27
2.2.1 Speed control of DC motors 27
2.2.2 Speed control implementation 29
CHAPTER 3 SYSTEM INTEGRATION
3.1 Information and mechanical interfacing 39
CHAPTER 4 SYSTEM ANALYSIS
4.1 Vibratory bowl feeder analysis 45
4.1.1 Natural frequency of vibration of the bowl feeder 46 .4.1.2 Response of the system to the periodic forcing function 49 4.2 Kinematic of the relative motion of an object on the conveyor 64
4.2.1 Introduction- projective geometry 64
4.2.2 Object relative motion 65
CHAPTER 5 CONTROL AND APPLICATIONS
5.1 Pick-and-place operation 73
5.1.1 Objects removal feed rate control 73
5.1.2 Speed control software implementation 80
5.1.3 Experimental results 89
5.1.4 Objects removal 92
5.2 Interception of an moving object on the conveyor 93
5.2.1 Problem statement 93
5.2.2 Catching time estimation 94
5.3 CONCLUSION AND RECOMMENDATIONS 96
5.3.1 Conclusion 96
5.3.2 Recommendations 96
REFERENCES
APPENDIX A : Vibratory bowl feeder analysis APPENDIX B : PC 30 Specifications
APPENDIX C : Bowl feeder drawing APPENDIX D : Control software (on disc)
vi'
LIST OF AB
ADC : Analog to digital converter DAC : Digital to analog converter DC : Direct current
CPU : Central process unit DMA : Direct memory access FVC : Frequency to volt converter I.0 : Integrated circuit
PWM : Pulse width modulation PC : Personal computer
TTL : Transistor-transistor logic
REVIATIONS
LIST OF FIGURES
Figure 1.1 A vibratory bowl feeder
Figure 1.1 A block diagram of the planned system Figure 2.1 Equilibrium of a part on the feed track Figure 2.2 Static deflection of the leaf spring Figure 2.3 Part fed
Figure 2.4 Probable feeding orientations: side view Figure 2.5 Probable feeding orientations: plan view Figure 2.6 Selecting device: plan view
Figure 2.7 Selecting device: cross section Figure 2.8 Selecting device: front view Figure 2.9 Determination of wmi n and wmax Figure 2.10 Narrowed track design parameters Figure 2.11 Railed shelf
Figure 2.12 Feed track- reorienting device Figure 2.13 Feed track cross section
Figure 2.14 Vibratory bowl feeder and associated devices Figure 2.15 Bowl feeder drive circuit
Figure 2.16 Bowl feeder drive system Figure 2.17 Pulse width modulation
Figure 2.18 Block diagram of the control system Figure 2.20 Conveyor's motor drive system Figure 2.21 Speed feed back circuit
Figure 2.22 Emitter Figure 2.23 Receiver
Figure 2.24 Speed feedback system Figure 2.25 Bi-directional motor drive Figure 2.26 Position sensing circuit Figure 3.1 PC30 Architecture Figure 3.2 Components interfacing Figure 3.3 Completed system Figure 4.1 Bowl feeder model
Figure 4.2 Geometric constraints diagram Figure 4.3 Forcing function
Figure 4.4 Electromagnet attachment Figure 4.5 Electromagnetic circuit Figure 4.6 Equivalent magnetic circuit Figure 4.7 The system's response
Figure 4.8 Point transformation in projective geometry Figure 4.9 System object conveyor
Figure 4.10 System object-conveyor-robot Figure 4.11 Relative motion of a conveyed part Figure 5.1 Diagram for two conveyed objects Figure 5.2 An example of a speed profile Figure 5.3a Constant velocity profile Figure 5.3b Displacement profile • Figure 5.4a Acceleration profile Figure 5.4b Velocity profile Figure 5.5 Displacement profile Figure 5.6 Overflow speed profile
Figure 5.7 Speed selection flowchart
Figure 5.8 Open loop motor control diagram Figure 5.9 Calibration set-up
Figure 5.10 Calibration function Figure 5.11 Program flowchart Figure 5.12 Program user interface
Figure 5.13 Real and theoretical position profiles Figure 5.14 Actual and theoretical steps of motion Figure 5.15 Actual and theoretical speed profiles Figure 5.16 A block diagram for the catching process Figure 5.17 Catching time estimation
UST OF SYMBOLS
Symbol Description Units
A Linear acceleration / deceleration , Normal track acceleration, area in/s 2, - ,m2 A Rotation matrix in projective co-ordinates system - A' Rotation matrix in a rectangular co-ordinates system
a Vector component, motor armature, actual, Amplitude of vibration -, -, -, m
Damping matrix, magnetic field density W/m2
b Vector component , track width m
b Robot base co-ordinates system
C Linear speed, constant m/s,
c Vector component
c Conveyor co-ordinates system
Distance, dimension m,-
e End-effector co-ordinates system eg Air gap
F Frequency, force Hz, N
F. Amplitude of force
f Frequency Hz
Center of gravity
Gn Normal acceleration due to gravity in/s2
g Part co-ordinates system , gravity Boolean variable
I Inertia, current Kgm2, A
i input, step
Spring constant, Boolean variable N/m,-
Length, lagrangian, Boolean variable m, -
1 Vector component
M Masse, point, motor Kg, -, -
M Inertia matrix
Symbol Description Units
m Mass Kg
N Number of steps, speed , number of turns _, rpm,-
Vector component, normal Dimension
0 Vector component, output, origin, initial -3 - 7 - 3
P Point
p Vector component, part co-ordinate system
q Vector component
R Radius, resistance, resistor, magnetic reluctance m, SI, -, At/W R Position vector in projective geometry
R * Position vector in a rectangular co-ordinates system
r Radius, reference m,-
Position vector
T Time, cycle time, kinetic energy s, s, J
T Homogeneous transformation matrix _
t Time s
V Voltage, potential energy V, J
X Linear displacement, position m,m
X Position vector in projective geometry
Position vector in a rectangular co-ordinates system
w Position of the center of masse
Greek symbols
a Spring angle deg
(3 Relative torsion angle rad
A Displacement, interval, variation, amplitude of vibration m, - - m
8 Error, deflection m
Frequency ratio
Symbol Description Units
(i) Torsion angle, diameter rad, m
(P Vibration angle deg
F Resonance factor
X Vector, generalized co-ordinate
a Eigenvalue
8 Track angle, robot joint position deg, deg
IA Eigenvector, coefficient of friction, magnetic permeativity -, -, Him
110 Magnetic permeativity of air H/m
11, Relative permeativity
S.2 Speed, amplitude rpm, m
Table 2.1 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 5.1
Leaf spring deflection measurements Amplitudes of vibration for e g = 4mm Amplitudes of vibration for e g = 10mm Amplitude of vibration and dimensionless
for eg = 4mm
Amplitude of vibration and dimensionless for eg = 10mm
Conveyor's linear speed measurements
normal track acceleration normal track acceleration
1.1
IntroductionAssembly forms an important link in the whole manufacturing process, because this operational activity is responsible for an important part of the total production cost and the throughput time. It is one of the most labour-intensive sectors in manufacturing[ 1]. On the other hand rationalization and automation offer good opportunities to minimize the production cost and the throughput time. Automated assembly systems can vary widely in configuration, depending on the needs of the individual manufacturing setting. However, regardless of the setting, the four components of an automated assembly system are[5] : robots, parts feeders, transfer devices and controllers. From an operational point of view, an automated assembly process comprises a cycle of operations. These operations can be divided into feeding, handling, transferring, composing, checking, adjusting and special process. The feeding and handling operations can also be divided into sub-operations. In that manner, feeding comprises the storage, separating, positioning and orienting of parts. The handling process can be divided into picking-up, moving and putting down of parts. Transferring involves the displacement of parts between different components of the assembly system[5]. Automated assembly are integrated system. Integration in this context means that information and mechanical interfaces have been implemented between all the
system's components for a smooth and harmonious operation.
The present work deals with the implementation of an integrated and computer controlled system for parts feeding, handling and transferring applications. The system comprises a conveyor belt, a vibratory bowl feeder, a robot and a personal computer. The conveyor belt and the vibratory bowl feeder have already been built but needed to be completed with either electronic control circuits or some mechanical devices or both so that they could operate properly. Although the robot is not integrated to the system at this stage, some provisions have been made for its future integration to the system. These provisions are mainly concerned with information interface with the computer for data exchange on the system state.
This work is divided in five chapters : the first chapter is an introduction and a description of the system hardware. The second chapter deals with the design and completion of the hardware. Chapter three describes how interfacing s implemented between the system's components. In the fourth chapter, an analysis of the feeder motion is presented.
The aim is to determine the suitable operating frequency for the feeder. The relative motion of a component part in respect off the conveyor belt and the robot is also studied for pick-up or tracking applications. Chapter five presents two applications in connection with the handling operations. The work also comprises a conclusion and an appendix section.
1.2 System description and requirements a) System requirements
From the start component parts are randomly dropped onto the flat bottom of the bowl feeder. These parts are oriented inside the bowl and feed out in the correct orientation. They then drop onto a feed track that connects the feeder and the conveyor. The feed track keeps them in the correct orientation before dropping them onto the conveyor belt. The belt then transfer them to a certain point where they are picked-up by the robot and put in another place for subsequent operations. The whole process is controlled from a computer.
The requirements for the system are: The system should:
Deliver correctly oriented parts at a certain location.
- Detect the presence of parts at specified points on the conveyor belt. Provide a means of regulating the feed rate of parts on the conveyor. - Provide a means of data acquisition
- Ensure communication between its components - Be computer controllable.
b) Components description Vibratory bowl feeder
Vibratory bowl feeder( fig.1.1) feed parts into an assembly station. It consists of a helical track welded to the inside wall of a bowl. The bowl is supported by inclined leaf springs attached to the bowl. An electromagnet mounted on the base periodically pulls against a flat plate attached to the bowl. The result is a torsional vibration about its vertical axis coupled with a linear vertical vibration. This cyclic " spiral up/ spiral down " oscillating screw motion causes any component part randomly deposited on the bottom of the bowl to climb up the track to the outlet at the top of the bowl. Vibratory bowl feeders are fitted with some devices generally placed at the top of the bowl. These devices ensure that only correctly oriented part are fed to the workhead, whereas improperly oriented parts are rejected back onto the bowl. These devices are referred to as in-bowl tooling. If such device 'is not incorporated in the bowl but is fitted between the feeder and the workhead it is called out-of- bowl tooling. The basic electronic drive uses either half
wave or full wave rectified as electrical input and a rheostat to limit the current to the magnet. An alternative type of drives uses a silicon controlled rectified (SCR) to limit the duration of the excitation signal. Since the workhead operates at a fixed cycle time, the vibratory bowl feeder should also function at the same cycle time, this means the feeder's feed rate should comply with the fixed cycle time of the workhead. The feed rate is controlled by the rheostat or by changing the conduction angle of a SCR. The feeder to be used in this work has already been built but needs to be completed with a bowl and its selecting devices. A drive is also needed to complete the devices.
Fig. 1.1 Components of a vibratory bowl feeder.( With acknowledgement to G.Boothroyd, C.Poli, L.E.Murch, Automatic assembly, Marcel Dekker pp 28)
Conveyor bellit
Robot
Sensors
owli feeder
Computer
Data acquisition and control interface a Material flow Data Control signal Parts Conveyor belt
A conveyor system is used when materials must be moved in relatively large quantities between specific locations over a fixed path. The conveyor to be used in this work is a small flat conveyor. The belt is supported on a metal base( slider bed) and is powered by a small geared dc motor ( car wiper motor). A toothed belt is used to transmit the power to the driver roll. This conveyor needs a drive and a speed feedback system so that it can run properly. Position sensors will be fitted to the conveyor to detect component parts being carried.
Computer
The P.C. is a Proline 60Mhz pentium processor. Robot
The robot is an ABB IRB 2400, 6 axis industrial robot with external communication capabilities.
Fig. 1.2 A block diagram of the planned system.( The robot is shown here just to give the reader a preview of the system when it is fully completed but one shouldnot forget that the robot is not part of the present system)
Chapter 2
HARDWARE DESIGN AND IMPLEMENTATION
2.1 Vibratory bowl feeder design
The functioning of a bowl feeder is mainly governed by the following parameters: bowl diameter, base weight, track angle, spring angle, coefficient of friction between the track and the conveyed part, the drive frequency etc... Vibratory bowl feeders are often designed with parameters based on past experience. Many hours of effort are usually spent experimentally and in some cases analytically developing designs to built a bowl feeder for a given part. However, progress is being done to minimize the necessity of developing vibratory bowl feeder systems by trial-and-error methods. The literature provides more detailed information about the operating conditions of a vibratory feeder and some relationships between the above mentioned parameters [1][2]. General design data for various orienting devices for vibratory bowl feeder are also available [2].
A mathematical model of the feeder's suspension system can be found in the appendix section. The model gives the natural frequencies of vibration and may help to optimize the selection of some elements of the suspension system once the design of a bowl feeder for a given part is completed. Vibratory bowl feeders are designed to feed parts at a certain feed rate since the workheads on an assembly machine are designed to work at a fixed cycle time. To achieve this a certain mean conveying velocity must be achieved. This is done by a proper combination, of the above mentioned parameters. In the present work we have to complete an existing vibratory bowl feeder system by designing and implementing its upper part i.e. the bowl. So the design will not cover the suspension system and because we are not directly concerned with an assembly process, achieving a fixed feed rate is for us of little importance. Our work will focus on the following points: design and implementation of the bowl and related devices, design and implementation of the feeder's drive system. A theoretical analysis of the bowl movement under the exciting force will be done in chapter 4 to establish if the condition for forward conveying is satisfied given the system's actual parameters.
2.1.1 Vibratory bowl feeder design parameters
By studying analytically and experimentally the motion of a part on a feeder's track (figure 2.1), G. Boothroyd [1] and co-authors have established some important relationships between the feeder's parameters. In order to achieve good operating conditions, they suggest the use of the following values for the following parameters:
The track angle 0
The track angle is the angle of the helix formed by the track. Experimental results[1] show that highest velocities are always achieved when the track angle is zero and that forward conveying is obtained only with small track angles because when the track angle increases the part tends to slide down the track. Experimental studies mentioned have established that a positive value of 3 or 4 degrees for the track angle is necessary in order to raise the parts to the bowl outlet.
mpg
cp is the vibration angle 0 is the track angle
a„ is the amplitude of vibration,
co is the angular frequency of vibration( f = 27cco ) g is the acceleration due to gravity ( g = 9.81rn/s 2). m is the mass of the part
F represents the frictional resistance between the part and the track.
Fig. 2.1 Equilibrium of a part on the feed track.
Coefficient of friction 11
The practical range of the coefficient of friction in vibratory feeding is from 0.2 to 0.1. The lower limit is representative of a steel part conveyed on a steel track . The coefficient of friction may be raised to approximately 0.8 by lining the track with rubber, this leads to an increase in the conveying velocity. Coating can also reduce the noise level due to the parts motion.
Vibration angle 9
The vibration angle is the angle between the track and its line of vibration. Theoretical results [I] show that an optimum vibration angle exits for some given conditions.
The most efficient feeding condition is obtained when operating with values of the dimensionless normal acceleration A. / G. greater than unity but below the values that will produce unstable conditions. Thus
A. 1
G ,, (2.1)
Where A.=oan 2= aoco2s. n, is the normal track acceleration,
G. = gcosa is the normal acceleration due to gravity ( g = 9.8Im/s 2).
The operating frequency f
It's has been shown[1] that for constant track acceleration the mean conveying velocity is inversely proportional to the vibration frequency. This means high conveying velocities and hence high feed rate are obtained by using as low a frequency as practicable, but a lower limit is imposed on the frequency due to the mechanical problem of connecting the feeder to a stationary machine( feed track, transfer mechanisms...). The lower limit is 25 Hz, usually in practice 60 or 50 Hz is used for the drive system. In the present case some of the feeder's physical parameters have already been fixed, therefore we need to analyse the system to determine the most suitable frequency for forward sliding to occur. For this reason the operating frequency is not chosen at the design stage.
Spring stiffness k
Generally, vibratory bowl feeders are tuned to a natural frequency just slightly higher than the frequency of the drive to minimize the power needs by using the ease of transmitting vibration at or near the natural frequency. This is a good indication for the computation of the spring constant k a , since the natural frequency and the spring constant are related. In the present case, the suspension system exists already but the value of the spring constant of the springs used was unknown to us.
8
The suspension system uses three identical leaf springs of rectangular section weighing approximately 1 kg each. Given the importance of this parameter in bowl's motion, we tried to find it by setting-up a simple experiment described hereafter.
Determination of the tangential spring constant value.
5
Fig. 2.2 Determination of the static deflection
To determine the spring constant, the spring is clamped in one end so that it remains horizontal and is loaded with a known mass M at the free end (fig. 2.2). The static deflection 5 is noted for each mass and the spring constant is found by using the following equation
K
= (M + 0.23m)g5 (2.2)
Where m = lkg is the mass of the spring and g = 9.81m/s 2 The results found are indicated in the table bellow.
hSU
Bsg
4.8'875 (10e-3m) 4
Table 2.1 Leaf spring deflection measurements
As stated before, the feeder was designed and partially built in a previous work, therefore some of the parameters have already been fixed. These are:
bowl's base maximum diameter cl) = 580 mm spring angle a = 60° at a radial position r = 140mm spring constant k a ( see page 8)
support base weight
The following values were adopted for the following parameters track angle 9 = 4 °
bowl external diameter .13,, = 540mm track width L = 40mm
The bowl was built using 1 mm steel sheet. The sheet was cut to dimensions and spot welded. A drawing of the bowl is shown in appendix c.
Others physical parameters to be used later in the bowl analysis have also been determined. These include :
The bowl mass(including the wood base support and the spring attachments but excluding the component parts to be used).
Mb = 9kg
The bowl inertia was approximated by decomposing it in a hollow right circular cylinder and a circular base and applying the formulas for each case. The total inertia thus calculated is
2.1.2 Description of the part to be used
The geometry of the part to be fed plays an important role in the design of the selecting and orienting devices. The part used in this work is a PVC conduit adaptator. It's a hollow cylindrical stepped part shown in the following figure. Solid parts with the same shape, right solid or hollow cylindrical objects and rectangular parts can also be fed using the same feeder .
22 32
Fig. 2.3. Part fed
2.1.4 Design of selecting and orienting devices
During the conveying process inside the bowl, the part adopts different orientations , but only one orientation is acceptable at the outlet of the bowl. The parts in the unwanted orientations should therefore be rejected back into the bowl. This is accomplished by the selecting devices. Before designing the selecting devices, all orientations occurring during the conveying process are examined in order to select the suitable devices to accomplish the rejection process. It is assumed that all the orientations have the same probability of occurrence.
Bowl wall 11 Feeding direction 4 a e Track
Fitz. 2.4 Probable orientations Side view
Bowl wail
4 Feeding direction
0,
a b c d e
Track Fig.. 2.5 Probable orientations : Plan view
Normally the orientation chosen when a part is to be conveyed on the conveyor belt is orientation e because its offers treater stability during conveying. This help to avoid using guiding devices on the belt. But if component parts are allowed to feed out of the bowl in orientation e only, we will have to reject five other orientations. These are a, b. c, d and f In order to increase the efficiency of the bowl feeder, parts travelling in incorrect orientations should be reoriented in the right orientation instead of rejecting them back into the bowl. However the design of orienting devices requires among others information some details concerning the motion of the part on the track which could not be obtained.
A
Direction of parts flow
A
On the other hand to reorient all incorrectly oriented parts will make the design more complicated. To reduce the number of rejections, will let the parts come out in either orientation e or f. An out- of-bowl device will then be used to reorient part feeding with orientation f In orientation e. To simplify the design, we used only simple passive orienting device inside the bowl. So parts travelling in orientation a, b, c, or d should be rejected back into the bowl.
a) Design of passive orienting devices
The device used to achieve the rejection process is a combination of a narrowed track and a railed shelf (fig. 2.6). This single device carries only those parts that remain upright those travelling in horizontal orientation are ejected trough the cut-out.
c a e w= 12mm w = 23.06mm w = 8.96mm w = 12mm w Feeding direction bowl wall track Fig ..2.7 Cross section A-A Fig. 2.8 Front view
b) Narrowed track design
The narrowed track is designed in such a way that all the parts fall back onto the bowl but the railed shelf will prevent parts travelling in the taller orientation to be rejected back into the bowl. Thus only parts in orientation e and f are fed out of the bowl.
The key parameter to determine in the design of a narrowed track is the dimensionless track width b/w which is a function of the mean conveying velocity . Given the ratio b/w, the track width b is then calculated by determining the maximum and minimum value of w which is the distance from the bowl wall to the centre of the mass of the part. The value of w change according to the part orientation, to determine w max and wmia, different orientations are considered.
bowl wall
From figure 2.9, wmae and w min are obtained when the part is in orientation c and a or e respectively. These values determine the range for the track width b. To obtain this range, the value of the ratio b/w corresponding to the feeder's mean conveying velocity should be determined from the graph in figure 2.10. But because the mean conveying velocity is unknown and since all parts should be rejected, we used the value 1.0 for the ratio b/w, this value is likely to comply with a wide range of values for the mean conveying velocity and the operating range of the device. For the reason stated above, the cut-out width should correspond to w n,in . Thus :
b = 1. Oxwnth, = 1. Ox12 = 12mm
The length of the narrowed L track is one and half time the part length, thus
narrowed track
wIdthwfse ports rejeclfr•d while
Only One row of lengl hwiSe parts
to delivery crass chute Narrowed track. Part lenc.,,n 1I 1 i I • cylinOe''S 1___ -=-reciarig'eS
all parts ac:ceoted,,_ I \ ...-- . 1 H.---J__ i _ - _- _r— r
-r
_______, j,____,2
.7_..
1 /., 1i
/.. _.:_
1__:__
--I
-7.-- ,
; l• !--
---;
al! ;arts rf..jected
!
. 0 r'5 100 cc%nve ,.•ing now! ,a I center .l mass (r) (a) 1.5 10Fig. 2.10 Narrowed track design parameters. ( With acknowledgement to
G.Boothroyd,
Automation and product design,Marcel Dekker, pp 90-91)
c) Railed shelf design
The railed shelf is designed to prevent parts travelling in the taller orientation to be rejected when passing trough the narrowed track. Its dimensions are given below .
a = 42mm b = 26 c = 26
length L = 100mm
Fig. 2.11 Railed shelf.
The values of parameters for the two devices thus determined are approximates, some trials were required to get these devices function correctly, so the real values may differ slightly from those mentioned here.
d) Out-of-bowl tooling
The out-of-bowl tooling is a device situated outside the bowl which reorients the parts in the desired orientation for conveying on the conveyor belt. Since the device is located outside of the bowl, no rejection is permitted. .The device used for this purpose is a simple rectangular inclined groove. It acts as feed track and also as orienting device. It uses gravity and part geometry to reorient parts ( fig. 2.12).
When the parts leave the bowl feeder, they are discharged onto the groove. Because of the inclination of the groove, gravity forces acting on the part create a moment which tends to stabilize parts with smaller diameter up on the groove, meanwhile parts with smaller diameter down are forced to fall onto the groove with the bigger diameter leading. A rolling pin placed on the groove forces parts with smaller diameter up to fall onto the groove with big head leading. At the end of the groove the part is fed into a bent tube chute and exits vertically to be discharged onto the conveyor belt (fig.2.12).
From the bowl feeder
Conveyor Fig. 2.12 Feed track- reorienting device.
Pin
Feed track Electromagnet Leaf spring Steel plate Support base Part Railed shelf Track
2.1.3 Bowl feeder drive system Drive requirements
The drive is basically an on-off switch that control the power to the electromagnet coil.
In the present work, the drive must satisfy the following requirements:
- Provide a variable frequency power control system. This requirement is dictated by the fact that the feeder's operating frequency was not fixed at the design stage but has to be determined by experiments where the operating frequency .is to be varied over a certain range.
- Allow the amplitude of vibration to be changed. The amplitude of vibration along with the operating frequency are two important parameters governing the functioning of the feeder. The amplitude of vibration depends on the tractive force developed by the electromagnet, which in turn depends on the current in the coil and the width of the air gap existing between the coil armature and the plate. To augment the amplitude of vibration, the air gap should be augmented. However an increase in the air gap produces a decrease of the tractive force if the current is kept constant. At a certain air gap width the decrease in the pulling force is such that the coil is no more able to pull-up the plate. The current should therefore be increased to increase the pulling force when the air gap is increased. However the increase in the air gap width has be to such that the current drawn by the coil doesn't overcomes the maximum allowed.
Coil electrical characteristics
The electromagnet has already been built, its support system was initially designed to pull the bowl verticaliy. In this design much of the tractive force developed is used to compress the spring along its axis producing less desirable effect. The electromagnet was
later placed aside to pull the bowl tangentially, creating a torque along the bowl's
vertical axis. The advantage of this design is that for the same amplitude of vibration, the current drawn is less than in the previous design.
Same of the coil's electrical parameters were measured for the purpose of sizing the drive components. These are:
Inductance L = 7.25 mH at 50 Hz Serie resistor Rs = 3.5 SI
Number of turns N = 950 Current max Imax = 5A
+50 Vdc T 4:1 1KVA R From computer AC 220V c) Drive description
The drive is a transistorised switch in a full wave bridge configuration (fig 2.15). This circuit accomplishes dc power control of the electromagnet. The mains voltage is stepped down to 50V by transformer T and then rectified trough the bridge B. The step-down operation is done for safety reasons and also because the use of an higher supply voltage would have forced us to use big serie limiting resistors for the coil and other components making the drive more expensive and bulky. After rectification a capacitor C is used as filter to reduce the ripple voltage. The current is then fed trough the coil L via a limiting resistor formed by a rheostat Rh and a fixed value resistor Ro . The current trough the coil is controlled by transistor T i which is driven from the computer via another transistor T2 and an optocoupler. Transistor T 1 is protected against transient voltage by diode D. Resistor Rd speeds-up the turn-off process of the inductor by dissipating the energy stored in the coil faster. The drive is optocoupled to the computer via an optoisolator to prevent any inductive glitches caused by the coil turning on and off from affecting the digital circuit. The optoisolator P is connected to a DAC channel of the PC30 data acquisition board. A compter program was written in Visual basic which produce the driving signal at the frequency set (see appendix section). The computer may be replaced by a function generator with the same result.
d) Components sizing
Inductor serie resistor 11,
For a fixed inductance value L, the time constant of the oscillator depends only on the total serie resistor R t . This means the charge and discharge of the coil is determined only by Rt . The current in the coil will reach its maximum value and decay to zero after the same laps of time, this time is equivalent to five times the time constant of the R t-L oscillator. This implies that the duty cycle of the chopper should be 50% or T..= Tar. The circuit is designed to handle frequencies up to 50Hz. The on and off time can be calculated from the following relation
1 1 1
f= =
T T„„ +Toff 2T,„ With f = 50 Hz ,
T.. = Tar = 10ms
The current in the coil should reach it maximum value and decay to zero after 10ms. Thus,
T. = —5L = 10ms
With L = 7.25mH, it yields
R, = 3.63 52 The coil current will be
50V
– = 13.8A
3.6352
Which is far more than the maximum allowed. We have to increase R, in order to reduce the current to a safe value. Let fix the minimum value for the current to 3A, the total serie resistor needed will be
50
= —3 = 16.7f2
And the on time becomes
To.= -LL = 2.17ms <10ms R,
This acceptable because the inductor will charge and discharge in less time at 50 Hz. On the other hand R t = RL + Rh +
If we use a rheostat of 652 and 6A max, the fixed value resistor R o will be Ro = R, - Rh - RI, = 16.7 - 3.5 - 6 = 7.20
We will use Ro = sn and Rh will be set at 5.252
The current reaches its maximum value in the coil when R h = 0. Thus 50
Imax
+ R„ – 4.35A < 5A The power dissipated trough R. at I ma, is
PRo = Ro x 1 2max = 8 x 19= 151W
Ro is a portion of a heating coil rated 852 and 170W Capacitor C
The capacitor C was sized to allow an output ripple of 1% .For a full wave rectification using a filter capacitor, the ripple r is given by the relation
I r
Vft4J
For r = 0.01, A, V = 50V, and f= 50 Hz we found C = 2886811F
The nearest standard value is C = 30000g , 63V
1:, ridge rectifier B
The bridge was chosen to handle a minimum current of 5A under a reverse voltage of 220V. The component used is KBPC2509 which can handle 25A with a peak reverse voltage of 1000V.
Transistor T1
The transistor is selected based upon the minimum collector current it must handle and the ability to block a voltage slightly higher than the supply dc voltage. It must also operates in the safe operating zone regarding the second breakdown.
We selected an MJ 15024 because it meets the above requirements. Its maximum ratings are:
Ic = 16A
VcE = 250 V hFE = 30
PD = 250W
Let's consider a value of 15 for h r. the maximum base current I BI needed will be = 0) IBS
_
ic
e R t h m 50 — 295mA 11.5 x 15 Transistor T2This transistor amplifies the output from the optoisolator to drive transistor T 1 , so its must provide enough base current to T i .
IC2 = IBI = 290 mA
Thus the minimum value of the current limiting resistor for T2 will be
RT2 — 50 — 17052 290 x 10 -3
The power dissipated in RT2 is
In practice we used a rheostat rated 50052, 0.632A max and set the proper value for RT2 We selected the transistor TIP 29 for the job. Its maximum rating are:
= 4A VcE = 120
hFE = 75 PD = 40 W Optoisolator P
The optoisolator selected is a TLP 521 wich has the following maximum ratings Phototransistor ratings Led ratings
VCE = 55V IF = 60mA
= 50mA VR = 5V
The output from the phototransistor drives transistor T2. If we consider a value of 35 for the current gain hFE of transistor T2, its base current will be
I c , _ 290 x 10 _
= 8.3mA
14.E 35
Since Icp = 1B2 , the value of the current limiting resistor Rp for the phototransistor will be
RR — 50 — 60242 8.3 x 10 -3
The nearest standard value is Rp = 6k2. And the power dissipated in Rp is
PRp= Rp Xic2p = 0.43W We selected Rp = 6k2, 5W
Resistor R is selected so that a current of no more than 5mA is drawn from the computer Thus
5
- ikn and PR = 0.025 W
Diode Clamp D
Diode D is chosen based upon the average amount of current it must handle during turn-off Ian is given by the following relation:
L x V I n avg — dc f R; 7.25 x 10 -3 x 50 x 50 la vg — 0.14A 11.5 2
The diode selected is 6A1100 which has a forward current of 6A and a peak reverse voltage of 800V.
Resistor RD was added to make the turn-off process faster. For transistor T 1 , the maximum value of VcE occurs at the instant the transistor id turned off and is given by:
VcE( max) = Vdc( 1 ±) VD
R
Neglecting the voltage drop across the diode VD, the selection of RD must satisfy the following criterion:
V
RD CE(max) 1)
Nick
RD S 6852
The power dissipated in RD is given by:
PRD = RD X I a2vg = 1.33W
The component selected is RD = 68Q, 5W R=
Transformer Rheostat Transistor switch
Rheostat
Fig. 2.16 Bowl feeder drive system. Computer connection
2.2 Conveyor belt speed control
2.2.1 Speed control of DC motors : Theoretical background
Component parts leaving the bowl are discharged onto the belt trough appropriate devices with an undetermined feed rate whereas the retrieval process is done according to a certain cycle with a fixed time period. Therefore there is a need to control the components feed rate on the conveyor. This implies the control of the conveyor speed. The conveyor belt is driven by a small DC motor ( car wiper motor) fitted with a worm gear. The power is transmitted to the drum by a toothed belt. The generalised equation for a DC motor is:
N— V — R "1
KD ( 2.3 )
Where N is the speed (rpm), V. is the armature voltage, R. is the armature resistance , 1a is the armature current, K is a constant and (I) is the field flux.
The value of Bala is usually less than 5% of the terminal voltage V., so that
V
N a ( 2.4)
One very important method of speed control of DC motor drives evident from relation 2.4, is to vary V. . This is known as voltage control . A high efficient method of varying the armature voltage when the available source is dc, is the use of a chopper.
A chopper is basically an on-off control which supplies pulses of source voltage whose average value is controlled by varying the duty cycle of the pulses. This method is known as pulse width modulation or PWM.
T
Fig. 2.17 Pulse width modulation
When applying PWM method, the source voltage V i and the output voltage V. of the chopper are related by the following equation:
V. — T„. Vi T'on -I- Toil
(2.5)
Where T. t Toff = T is the chopping period (fig. 2.17) . T. and T off are respectively the on and off time of the chopping period. T. is also known as pulse width. The chopping frequency is F = 1/ T and the ratio T. / T is called the duty cycle of the chopper. Thus both the pulse width Ton and the frequency I/ T may be varied to obtain a particular
value of the duty cycle therefore a certain output voltage.
Methods of generating PWM signals can be classified in to two categories: software and hardware method. Software methods are based on the timing loops, while hardware methods use sophisticated peripheral IC that contain appropriate internal programming to produce complex waveforms. In general, software methods require dedicated operation of the CPU and are sensitive to variations in the CPU clock speed , wait states, interrupts and code changes, while hardware methods are more expensive, relatively insensitive to the CPU timing and less early modified.
2.2.2 Speed control implementation
The speed control scheme implemented uses an hardware method to generate PWM signals because of the accuracy required in the determination of the conveyor speed. The hardware used is a dc to pulse width modulator circuit. This circuit converts a dc control voltage into a series of pulses, such that the pulse duration is directly proportional to the value of the dc voltage. The great advantage of the circuit lies in the fact that the PWM generator and the motor driver are combined into one single unit. making the control system simple . A speed feedback is obtained from an optical sensing circuit. Full details of these circuits are given in section (2.2.3 b).
Overview of the control system
Figure 2.18 illustrates the configuration of the control system , which accomplish a proportional control scheme. The motor speed is converted into pulses by the optical sensor circuit, a frequency to volt converter( FVC) then converts the pulses into a dc voltage v which is a measure of the motor speed. The actual speed SI a is compared to the reference speed S2 d( setting of desired speed). through a comparator which generates an error signal e . The error signal is used to set a new control voltage v' corresponding to a new duty cycle of the pulses applied to the motor
The order of events in a (full ) control cycle is as follows:
1 Measure the motor speed and compare it with the desired speed to produce the error
2 Change the compensator in proportion of the error
3 Output a new control voltage to vary the duty cycle of the motor 4 Go to the first step.
The cycle is implemented over and over to keep the motor speed at ( or very close to ) the desired speed. A dead time could be included between step 3 and 4 to allow the motor to respond to any change in duty cycle made at the end of step 3.
St
d
Vcomparator
compensator
DCto
PWMV'
Oa
converter
FVCspeed sensor
computer
Fig . 2.18 Block diagram of the control system
2.2.3 Circuits hardware
a) DC to
PWMunit
The DC
toPWM
unit is designed by VELLEMAN and is sold as a kit to be assembled. This circuit uses the IC SG 3525 to converts a dc control voltage to a series of pulses such that the pulse duration is directly proportional to the value of the dc voltage. The duty cycle range form 0 to 100% . The minimum and maximum output voltage can be adjusted by using respectively pre-set RV1 and RV2. The output frequency range is 1001-1z to 5khz and can be adjusted by using pre-set RV3. The input voltage varies from 0 to 30 vdc and the operating voltage from8
to 35 vdc. The maximum output current is 6.5A.VREF VCC 16 15 C2 CUTS Dl 14 VREF R4
I
MIN R7 RV1G
o' DC LOAD R5 [ c=" C6 COMP T2 9 'Cl 11
SG3525 IN kr
RV2 GNE R3 RV3 FR Pr' RI DISCHARGE = C3 CNDI
I SHUTDOWN 5 SYNC 12 10 3 NC SOFT START S Si I + /. P2 C5 .4i> 1 ,---0o---, I T1 8 C4 --- -4i .. R9•1R10 Ii
. ,
R 4 CSC CUT NC T* 0 L. 0 T -.a, OG ND MAX R6DC to PWM circuit . -. .. _ L.. ■ ..: 1b 00000.13.2.00000oo 000C.0900000.40000 7 00000000000000000 ppocroocoaoacaomo...o o.ib00000000m000000s oo 1,2...2.000000.00.00.0.000.0.
Computer connection CaC,0 00....o0000M000eco
00000000000.0 0 00 0 0000
000.510000 .000.00.0 000.00000000000000000
,0,<Z).13 000000000000 o 00 0
Dual power supply , •- , ....it.,i, ',1, •":-,- ,
v
, ),,,,•-,.•'.:, :-•'',, , • 1 ' ' i• ••:"' ' -•:, • '1 3 1 4
5
-
1 g +12 V 1k 0.5 I IH 480 12 111 10 LM 2917 EVC 35k 100k 10kTO
ADC receiver emitter from DAC Db) Speed sensing circuit [37]
This circuit comprises a disc encoder D connected to the motor and an emitter- detector circuit providing an infrared thrubeam detection system. When the motor rotates, the receiver generates square waves whose frequency vary with the speed of the motor. These signals are amplified by the IC LM741 and converted into dc voltage by the frequency to volt converter IC LM2917. The output of the LM2917 is then translated into rpm by the computer ( fig 2.21). The optical encoder uses a infrared emitting diode LD271 to produce the infrared beam (fig 2.21). The receiver comprises a phototransistor FPT 100 which is kept on by the infrared beam. This produces a high logic level voltage at pin 1 of the schmitt-trigger inverter ( IC 7414) and a TTL compatible low logic at pin 5 of the monostable ( IC 74121). When the infrared beam is blocked, the phototransistor turns off the schmitt trigger inverter to trigger the one shot ( fig. 2.23).
LM 741 -1
g
2k2 2k2 D= LD271 T = 2N3904 +5V 15k r> TO LM741 +5V Fig. 2.22 Emitter 14 14 11 FPT100 IC IC 10 7414 74121 10k 6 Fig. 2.23 Receiver
Infrared sensor Drum
to*
Motor
,c.; 0
OO
Disc encoder FVC ircuit
c) Bi-directional motor control
In order to control the direction of the conveyor, a separate circuit that provide a bi-directional motor speed control using pulse width modulation has been implemented. The circuit has two inputs : speed(1) and direction(2). The last one qualifies the +15 V or
-15V supply. Isolation between the motor power supply and the digital supply is provided by two optical isolators preventing any inductive glitches caused by the motor turning on and off from affecting the digital circuit.
G1 = G2 = G3 = G4 = 1/4 IC 743 8 ( nand buffer gate ) P = IRF 9530
N = IRF 520 M = motor
= PWM input = direction input
2k2 (a) 2k2 Dl = D2 = LD271 T = 2N3904 +10 V +5V (b) FPT100 TO ADC d) Position detection
To determine the position of a part travelling on the belt, we need to fix a reference position on the belt from which the distance travelled by the part is calculated. For the retrieval process it is also important to know whether a part is present at the retrieval point or not. Two optical sensors were used for detection purposes : one provide a reference line from which positions are computed, and the other detect the part presence at the retrieval position. These sensors can be moved to different positions on the belt. The sensors are constructed in such a way that they are movable along the conveyor belt. These sensors are identical to those used in the speed sensing circuit previously described (section 2.2.3.b). The only difference is that the emitter uses two diodes in series to produce more power radiant and increase the detection range. The receiver does not comprise a one shot, the output from the schmitt trigger is directly fed to the computer. A drawback with this arrangement is when a part blocks the infrared beam, a train of pulses is sent to the computer over a period of several seconds. Thus it difficult to determine whether it is one or many parts that have crossed the detector's beam. This could be misleading especially when it comes to count the number of parts conveyed. To combat this problem, a software hold-off method was used. The computer automatically switches-off the sensor after the first pulse is detected. The sensor is switched-on after a certain amount of time corresponding to the passage of the part trough the detection system.
Part Receiver
Receiver
Infrared sensor
Infrared sensor I Emitter
Emitter
Chapter 3
SYSTEM INTEGRATION
3.1 Information and mechanical interfacing
To implement an integrated system in automatic assembly applications, one needs to ensure that appropriate interfacing devices are used between the system's components. These devices should provide information and mechanical interfacing between the hardware. The information interface is concerned with the flow of information that must accompany the movement of the parts. It encompasses the problem of part identification and tracking, also data communication required to co-ordinate and control the various components of the system involved in the process. The mechanical interface deals with the problems of transferring parts between the storage and feeding system, material handling system and production system. The present work is concerned with both material and information flow .The first occurs between the bowl feeder and the conveyor belt, and also from the conveyor to an hypothetical production machine, after transfer by the robot. Information flow occurs when the system's components are communicating to each others for an harmonious operation. Majors devices used to implement this integration are described in this chapter. Figure 3.2
shows
a block diagram of the integrated system. The bold arrows represent material flow direction whereas normal arrows represent information flow. The robot is represented in dotted lines to highlight the fact that it is not integrated to the system at the present stage. Components are connected to the computer via an efficient and versatile data acquisition board (fig 3.1). Beside data acquisition the board also comprises some digital-to-analog channels that allows the ouput of control signal and some digital input-output ports for communication with an external device. The board can be programmed using several programming languages. Material flow between the feeder and the conveyor is implemented by using a feed track. The handling is to be performed by the robot. The complete system is shown in the picture in figure 3.3a) Information interface - PC30 Architecture and operation
PC30 Architecture
This section describes the architecture of the PC30 board. The block diagram in figure 3.1 highlights the major elements contained on the 'board and their interrelationship. There are four major subsections. These are the following:
D/A subsystem
The D/A subsystem contains two 12-bit D/A converters, two 8-bit D/A converters as well as their associated circuitry, including buffer registers .
A/D subsystem
The A/D subsystem contains several components:
The input multiplexer. The multiplexer selects one of the sixteen single ended input channels. The channel is selected by a channel address, obtained from the channel list. The channel list contains a list of channels to be converted.
The sample and hold unit. This unit holds the selected input channel steady for the duration of the A/D conversion process.
- The timing and control unit described later, generates strobe to start AID conversion. Data may be transferred from A/D either by polled I/O or DMA.
Bus interface
The bus interface is responsible for three functions:
The decoding of the board's base address. The board's base address is set by a DIP switch.
The generation of interrupts. - The generation of DMA signals.
Timing and control
The timing and control subsection is responsible for the generation of A/D strobes, and also contains an uncommitted counter / timer which can be used for signal generation, or as a frequency or pulse period counter. A/D strobes cause the A/D converter to began
conversion. The timing and control unit contains four major components: a crystal oscillator, a clock divider, a clock selection multiplexer and an counter/timer.
Counter / timer
The counter timer, can be configured by jumpers to count either pulses from the master oscillator, pulses from the output of the clock divider, or pulses from an external source. This allows the counter to be used for measuring frequency, or for generating a frequency. Counting can also be enabled from an external source, so allowing pulse width to be measured.
3
1t-D1 D/A Camvcrt=. 12-.STS 3/A Cerxvortzr 0-011 II/A Occasart=. D-Cn 3/A Colva-sa- PC-30D ant savrL 0 3Ts"
A:1 0 Block Scan Counter 16-D3 Cascrtairt;=.0
Part A Part 13
3
3
Peat CChannels 8 ,3, and 7 of the A/D subsystem are used to monitor respectively position sensorl, position sensor2 and the conveyor's motor speed. Channels 0 and 1 of the D/A subsystem are respectively used to drive the conveyor and the feeder. Communication with the robot can be implemented trough digital input-output ports operating in bi-directional bus mode.
PC30 Operation
The PC 30 comes with a set of device drivers for use with a wide variety of software. These device drivers are written in C, and are callable from several programming languages( C,C ++, Fortran, Pascal, Visual basic ....). Using these device driver's functions, a user can write is own software control application. But first of all the board has to be configured to suit the user's individual requirements. This configuration is set by the position of various mini jumps on the board. Further information are provided in the user's manual. In this work we used Visual basic to implement a control program for the system. Unlike C -", Visual basic for Windows is easy to learn in a very short time, the only setback when using this language is to implement applications requiring many time constraints because the full control of the computer's CPU cannot be obtained.
b) Mechanical interfacing : Feed track
The elements of a feeding system are usually located some distance from the assembly workhead. A feed track is used to transfer the components from the feeder to the location of the assembly workhead, maintaining proper orientation of the parts during the transfer . Feed tracks are also used to create a buffer stock in order to maintain a certain feed rate at the workhead. There are two general categories of feed tracks: gravity and powered. The gravity feed track is most common . In this type the feeder is located at an elevation that is above the elevation of the workhead. The force of gravity is used to deliver the components to the workhead. The powered feed track uses vibratory action , air pressure, or other means to force the parts to travel along the feed track toward the assembly workhead. In the present work, a gravity feed track was used to ensure material flow from the bowl feeder to the conveyor belt (see fig. 2.10). This feed track also reorients some parts into a desired orientation.
MOTOR DRIVER SPEED SENSOR CONVEYOR COMPUTER
DIGITAL I/O BLOCK ADC BLOCK PC-30 DAC BLOCK BOWL FEEDER,,,, DRIVE FEED TRACK POSITION SENSOR CIRCUITS ROBOT ROBOT CONTROLLER Information flow Material flow Control signals -
U
Vibratory bowl feeder
Feed track
Power supplies
Computer
Conveyor belt
Feeder's drive
Chapter 4
SYSTEM ANALYSIS AND EXPERIMENTAL RESULTS
4.1 Vibratory feeder analyses
The purpose of this analysis is to determine the most suitable operating frequency for the feeder. A general analysis of the feeder's suspension system is presented in appendix.. The same methodology used is applied here to determine the natural frequency of the feeder built and to study its motion under the influence of a periodic forcing force produced by the electromagnet. The amplitude of vibration of this motion will then be computed for a set of operating frequencies to determine the frequency satisfying the most the condition for forward conveying i.e. A ilGt,>1. The result will be validated by an experiment.
It is assumed that the vibrations of the bowl's base are negligible. This means X2 = 0 and 4)2 = 0 (see appendix A). The system is reduced to a single degree of freedom system We will first study the feeder's motion in order to establish its natural frequency of vibration and then use the result obtained to study the motion under a periodic pulling force.
Base (x = 0, = 0)
4.1.1 Natural frequency of vibration of the bowl feeder a) Differential equation of motion for the bowl feeder
Because the vibrations of the base are negligible, the vibration angle cp at the radius L where the leaf spring is attached to the bowl is the complement of the spring angle a . The vibration angle at this radius will be :
= 90 - a
Thus the following geometric constraints can be established from figure 4.2
x = Acosa (4. 1 )
41, = Asina (4.2)
Combining the two relations above, we may write
x = 434 Lcota (4.3)
and
x = ■Lcota (4.4)
Using eq. 4.1 and these constraints, only one degree of freedom will remain for the system(A, x or 4)). Let's use 4) as primary degree of freedom, the kinetic energy of the system will be
1 1 1
4)2 2 2 1 142
T = —MX 2 +—kir = — M L cot a + —14) /
2 2 2
and the potential energy is
2 1 2 •
V = — KiA = — Kuc) (L/ sina)2
The partial derivatives of the Lagrangian L = T - V are
a
= mL24)cot 2 a + 14) = iviL2cot2a + I )40
a
•
—L = - 1c4)(L/ sina) 2
The equation of motion is given by —d (— 0 L = 0 dt 84) -a4)
Using the partial derivatives, the differential equation of the bowl's torsional motion will be
•• K a L2
—
(1) 0
(ML2 cot 2 + I) sin' a
The natural frequency of vibration is
L Ka sina ML2 cot 2 + I (4.5) (4.6)
(oh =
(4.7) (4.8)The three leaf springs are attached in parallel, therefore the value of the spring constant for the whole system is the sum of spring constants of the three spring. That is given by:
Kq = 37648.33Nml
With M = 9 Kg
I = 1.286 Kgm2 L = 0.14m a = 600
We found for the natural frequency:
= 27 rad/s
And the frequency fi, is given by
fh = h = 4.3 Hz 2 it
The torsional vibration is an harmonic motion which can be completely determined given certain initial conditions.
4.1.2 Response of the system to the periodic forcing function a) Tractive force
A constant force created by the electromagnet is periodically applied to the bowl to make it vibrates. If the duty cycle is 50%, the time representation of this periodic pulling force is shown in figure 4.3. For computational convenience the origin of the t-axis is taken at the beginning of one pulse.
F(t)
Fo
T*/2 -T*12
Fig 4.3. Forcing function
The forcing function can be defined over the interval (-T */2 , T */2) by
0 for -T */2 < t < 0
(4.9) F(t) =
Fo for 0 t T*/2
Where T + is the forcing function period and F o is the amplitude of the pulling force produced by the electromagnet. In reality the pulling force is not constant but increases with the current grow in the coil to reach a maximum value when the current reaches its maximum value. Fo is the maximum force developed by the electromagnet. Because we are interested with the torsional vibration of the bowl, it is convenient to consider instead a pulling moment.
The pulling force Fo creates a pulling moment
Mo = Fo Lm (4. 1 0)
coil armature
steel plate
eg = air gap bowl base
Fig 4.4 Electromagnet attachment
b) Differential equation of motion and system's response
The mathematical expression of the periodic forcing moment M(t) can be obtained by replacing F0 by M o in eq 4.9. The equation of motion of the system under the influence of the forcing moment M(t) can be written as
..L
(ML2cot2cc + + 1( (I) — M(t)
sin ' a (4.11)
Where coh is the natural frequency of vibration of the system (eq.8).
To find the system ' s response under M(t), we need to express M(t) in terms of harmonic functions using Fourier expansion. For a given periodic forcing function F(t) with a period T, the Fourier expansion of this function can be expressed in terms of harmonics functions as
a
F(t) = —o + a o cos not + b. sin ncot (4.12)
Where co = 2n — is the fundamental frequency and ao, a n and bn are known as Fourier coefficients and are determined as follow
2 T ao = F(t)dt T 0 = —2 11F( t) cos ncotdt 0 bn F(t) si n notdt T 0 (4.13) (4.14) (4.15)
Since a,, = 0 and bn = 2M0 n = 1,3,5,... It can be shown that the Fourier expansion of nn
M(t) can be expressed in terms of harmonics functions as
, M o ,`" 2 1\4 0 = + sin m e t = -+ 1 —sinw o t 2 rut co o = nco 2 M (— 1 co t ) n (4.16) (4.17) M(t) M o 2 nit
Where ‘..o. = 7 is the fundamental frequency and T *
Subtituting eq 4.16 into eq 4.11 yields:
2
+ + KL —
(ML2cot2a — sin
sin 2 a 2 + n=1 . 3 , 5 nit
The solution of eq. 4.17 consists of two parts, the homogeneous function 4)11 and the particular solution Op, that is:
52
(1)h is the solution of eq. 4.7. Since eq. 4.17 is a linear, second-order ordinary differential equation with constant coefficients, the principle of superposition can be applied in order to obtain the particular solution 4Ip . (Op consists of two parts, the particular solution .po
due to the constant term ° only and the response due to each of the terms 2M 0
sin co o t noted Itspn. nn
Therefore the complete solution of eq. 4.17 can be written as
(I) = (1)h 10p0 (1)pn (4.19)
Response of the system to the constant moment Mo
Since Mo is constant, 440 is also a constant, that is
4:00 = C (4.20)
Where C is a constant. It follows that
4;p0 = (iopo = 0
Substituting into eq. 4.18, gives
(ML2cot2a + 1)01,0 + K —
sin' a 2
one obtains
M o sin' a
53
Response of the system to the moment t ° sin co t nit
This response is the solution of the following equation
(ML2cot2a. + + K x
I-2 (I) — 2M° sin (0 t
sin 2 a nn (4.22)
The steady state solution of eq. 4.22 is
2M 0 sin 2 a
Ypn
IcL2 nic sin co n s
I - ( ML2 cot 2 a + I )sin 2 a.o) K x L
Rearranging different terms and using eq. 4.8 yields
(1)pn 2M 0 sin 2 1 sinco n t (4.23) K a. L2 nic 2 I 2h n Provided con # (oh
The steady state response of the system under M(t) is:
(Op = (1)pl) ()pn
Substituting eq. 4.21 and eq. 4.23 in eq. 4.24 yields
M o sin 2 a 2M 0 sin' a 1 sine) ° t (4.24) (4.25) (4.26) (4.27) (4.28) 4)p 2K a L2 K a. L2 rin
Eq. 4.25 can be written as
—Mosin2a + F( 1„ ca 1 -21 2 0.) 4)p 11< ix L2 n 1— n's - 2M ° sin- a Where F = and s = )sino o t sinco n t lc,L2 7c h
co is the fundamental frequency of vibration.
F , 1 Let 12 = 7 , ) n I — Eq. 4.26 becomes (11 sin2 , a + p
EQ.
n=1,3,5From eq. 4.27, it can be shown that , C2 n tends to become zero when n increases. and when c becomes big. Therefore a good approximation of eq. 4.28 can be obtained by considering the Fourier serie truncated at the first harmonic only. Thus eq. 4.28 can be written as
Where co l = co
M o sin 2 a
()p + S2 I sinco l t
2K,,L2 (4.29)
The amplitude of torsional vibration can be determined if the value of M o and the operating frequency o; are known. An estimation of M. is done in the following section. The approach used here is to determine the values of the amplitude of vibration for a set of operating frequency and verify the condition for forward sliding for each of these frequencies. The operating frequency for the feeder will be the one that satisfy the most the criterion for forward sliding.
c) Tractive force estimation
To determine the amplitude of vibration from eq. 4.29, the moment M o should be known. This moment is generated by the tractive force F. produced by the electromagnet. F. is given by the following equation
Fo — B2AT — 4:12
21.1 0 21..i o A T (4.30)
Where B is the magnetic field density, (1) = BA T is the magnetic flux and AT is the total area covered by the air gaps. po is the permeativity of air (4n10-7H/m).
