Ahmed Yousry Winding Machine Mathematical Model
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Transcript of Ahmed Yousry Winding Machine Mathematical Model
Winding machine Mathematical Model
Firstly Motor Drive
Motor Dynamics is separated into 3 Stages:
1. Electrical Dynamics equation.
Representing the dynamics of the motor Electrical
Circuit
2. Electromechanical Linkage equation.
Representing the Electromagnetic Torque Developed
from the electric circuit.
3. Mechanical Dynamics equations.
Representing the Dynamics of the motor mechanical
Drive (Mechanical System coupled to motor ).
Bipolar Stepper Motor Mathematical Model
Bipolar Stepper motor Construction:
1. A multi pole permanent magnet Rotor.
2. Multiple Stator Winding creating Stator phases.
Theory of operation:
When energizing a certain stator phase a magnetic field in direction of the stator
phase is induced, this field produces a torque over the rotor causing the rotor to
rotate until is aligns with the field.
So when energizing the stator phases in sequence a rotating magnetic field is
produced and the rotor rotates trying to align with the rotating magnetic field.
Mathematical Model
Electrical Equations:
Assume a 2 phase (a and b) Stepper motor
Each phase can be represented by this equation
ππ = π πππ + πΏπ
πππ
ππ‘+ ππ
ππ = βπΎποΏ½ΜοΏ½ sin(ππ π)
Where
ππ: Phase a applied Voltage [Volt].
π π: Phase a winding Resistance [Ohm].
ππ: Phase a Current [ampere].
πΏπ: Phase a winding inductance [H].
ππ : Back emf (electromagnetic force) induced in phase a [Volt].
πΎπ: Electromotive Force Constant, (Motor Torque constant) [ππππ‘. π ππ πππβ ]
Same equations represents phase b.
Electromechanical Torque Equation: In the Stepper Motor case the Torque developed over the rotor is the sum of the torque induced from each phase.
The Torque developed by each stator phase is dependent on the position of the
rotor in reference to that phase (as mentioned before) maximum torque when the phase induced magnetic field is perpendicular on the rotor (rotor field line
which connects the rotor poles) and minimum torque is when the rotor is aligned with the phase magnetic field.
this relation is represented by a sinusoidal wave (sine for first phase and cosine
for next phase as it varies with 90 mechanical degree )
So:
ππ = βπΎπ (ππ βππ
π π) sin(πππ) + πΎπ (ππ β
ππ
π π) cos(πππ)
Where:
π: Rotor Position
πΎπ: Motor Torque Constant [π. π π΄ππ]β
ππ : phase a current [π΄ππ ]
ππ : phase b current [π΄ππ ]
ππ : Back emf (electromagnetic force) induced in phase a [volt]
ππ : Back emf (electromagnetic force) induced in phase b [volt].
ππ: Is the number of teeth on each of the two rotor poles. The Full step
size parameter is (Ο/2)/Nr.
π π: Magnetizing Resistance in case of neglecting iron losses itβs assumed to
be infinite which causes the term ( ππ
π π= 0 ).
As Shown the system equations is nonlinear.
Dc Motor Model
Dc Motors Control Techniques:
1. Armature Control.
2. Field Control.
Armature Controlled Dc Motor Mathematical Model:
Fig 1 Armature Controlled Dc Motor Schematic
Electrical Equations:
ππ = π π ππ + πΏπ
πππ
ππ‘+ πππππ
πππππ = πΎππππ οΏ½ΜοΏ½
Where:
ππ: Voltage applied over Motor Armature [Volt].
π π: Armature Resistance [Ohm].
ππ: Armature Current [ampere].
πΏπ: Armature Inductance [H].
πππππ : Back volt induced from rotor into armature [Volt].
πΎππππ:Electromotive Force Constant [ππππ‘. π ππ πππβ ]
Taking Laplace Transform
ππ(π ) = ππ(π )(π π + π πΏπ) + π πΎππππ π
Current volt Transfer Function ππ(π )
ππ(π )βπΎππππ οΏ½ΜοΏ½(π )=
1
π π+π πΏπ
Electromechanical Torque Equation: In general, the torque generated by a DC motor is proportional to the armature current and the strength of the magnetic field.
ππ = πΎππππ
The Strength of the magnetic field is constant as its armature controlled so:
ππ = πΎπππ
Where:
πΎπ: Motor Torque Constant [π. π π΄ππ]β
ππ : Armature current [π΄ππ ]
Current to Torque Transfer function
ππ (π )
ππ (π )= πΎπ
So Volt Torque Transfer Function: ππ(π )
ππ(π )βπΎπππποΏ½ΜοΏ½(π )=
ππ(π )
ππ(π )βπΎπππποΏ½ΜοΏ½(π ) .
ππ(π )
ππ(π )=
πΎπ
π π+π πΏπ
Field Controlled Dc Motor Model:
In Field Controlled Dc motor the armature current is kept constant, the torque is
controlled by modulating the magnetic flux.
The magnetic field is controlled by controlling the voltage applied on the field
winding (ππ) so in this case the manipulated variable is field winding Voltage
(ππ).
Electrical equation:
ππ = π π ππ + πΏπ
πππ
ππ‘
ππ: Voltage applied over Motor field winding [Volt].
π π: Field winding Resistance [Ohm].
ππ: Field winding Current [ampere].
πΏπ: Field winding Inductance [H].
Taking Laplace Transform
ππ (π ) = π π ππ(π ) + π πΏπ ππ(π )
ππ(π )
ππ(π )=
1
π π + π πΏπ
Electromechanical Torque:
As armature current is constant
ππ = πΎπππ
πΎπ: Motor Torque Constant [π. π π΄ππ]β
ππ : Field winding current [π΄ππ ]
Current to Torque Transfer function:
ππ (π )
ππ(π )= πΎπ
So Volt Torque Transfer Function: ππ(π )
ππ(π )=
ππ(π )
ππ(π ).
ππ(π )
ππ(π )=
πΎπ
π π+π πΏπ
ππ = πΉπ‘1ππ€πππππ
ππ :Load Torque over the winder Driver
πΉπ‘1: Web Tension force at Winder Region (Control Variable).
πΉπ‘0: Web Tension force at unWinder Region (assumed constant).
ππ : Winder Tangential Velocity.
π1 : unwinder Tangential Velocity.
πππππππ : Displacement of Dancer.
ππ€πππππ :Raduis of winder cylinder.
ππ : Dancer Velocityππ =ππππππππ
ππ‘.
ππ€πππππ
πππππππ
πΉπ‘1
πΉπ‘1 πΉπ‘π
Model Assumptions:
1. The paper velocity from the unwinder is constant
2. The cross section area of the web is uniform
3. The definition of strain is normal and only small deformation is
expected.
4. The deformation of the web material is elastic this assumption is used
because plastic deformation is unwanted during the winding process and
quite difficult to model.
5. The density of the web is unchanged
6. The dancer movement is negligible compared to the length of the web between the unwinder and the winder.
7. The speed of the dancer is negligible compared to the speed of the web
ππ<<π1
8. The web material is very stiff, hence ππβπ1
If assumption 6 is correct and the material is stiff the unwinder paper
speed and the winder paper speed is approximately the same.
9. The tension in the unwinder section is constant.
10. The change of roll radius does not change the web length between the
winders:
as one radius is increasing the other is decreasing therefore the changing
radius is estimated to only having little influence on the web length and
is therefore neglected.
Mechanical Equations:
In our case we have two torques opposing the torque from the motor:
1. The tension in the web is acting as the load on the winder motor
ππ = πΉπ‘1ππ€πππππ
2. Friction Torque Consisting of :
a. Coulomb Friction (Static Friction) throughout the Drive
system (as in bearing , Gears β¦etc.)
ππππ’π
b. Viscous Friction (Dynamic Friction) throughout the Drive
system
πππ = ποΏ½ΜοΏ½
So Mechanical Differential Equation:
β π = π½οΏ½ΜοΏ½
ππ β ππ β ππππ’ β ποΏ½ΜοΏ½ = π½οΏ½ΜοΏ½
Taking Laplace Transform:
ππ(π ) β ππ β ππππ’ β ππ π(π ) = π½π 2π(π )
Angular Position Transfer Function:
π
ππ β ππππ’ β πΉπ‘1ππ€πππππ
=1
π½π 2 + ππ
π½: is variable as winding Cylinder mass increases as winding goes on
its calculations is at the end of paper
Web Material Model
The purpose in modelling the web material is to find an expression for the tension force
development in the web material located between the winders. This requires a physical
interpretation on how stress arises in the web material and how the stress is related to the
winders tangential velocities ππ and π1 .
In the following the Voigt model is used to explain arising stress and with the before
mentioned assumptions, control volume analysis and continuum mechanics it is shown how
the stresses are related to ππ and π1 .
Voigt Model:
The Voigt model consists of a viscous damper and an elastic spring in parallel as shown
With this model the Stress on the material π =πΉπ‘
π΄ is expressed as follows
π =πΉπ‘
π΄= πΈπ + πΆπΜ
Where
π:Stress on web material
πΈ: material youngβs modulus of elasticity
π΄:Material Cross Section area A=material width *material thickness
π:Strain Due to Tension force π =βπΏ
πΏ (Deformed length over normal length)
πΉπ‘:Tension Force over material
Taking Laplace Transform we get π =πΉπ‘
π΄πΈ+π΄πΆπ (1)
Mass Continuity Definition:
Mass of material doesnβt change as the material is Stretched
ππ΄πΏ = ππ΄π πΏπ
Where
A: Normal Area of material
L: Normal length of material
πΏπ :Stretched Length of material
π΄π :Stretched Cross sectional area
As Density is assumed constant β΄ π΄πΏ = π΄π πΏπ (2)
From Strain Definition π =βπΏ
πΏ=
πΏπ βπΏ
πΏ=
πΏπ
πΏβ 1 (3)
From (2) and (1) π΄π =π΄
(π+1) (3)
Since π βͺ 1 equation (3) can be expressed as
π΄π = π΄(1 β π) (4)
Mass Conservation Law:
The definition of mass conservation states that the change in mass of the control
volume equals the difference between the mass entering and exiting the control
volume.
π
ππ‘ππ΄πΏ = ππ΄πππ β ππ΄1π1
In our case and since density constant π
ππ‘π΄πΏ = π΄π πππ β π΄π 1π1
From equation (4) we get
π
ππ‘π΄(1 β π1)πΏ = π΄π(1 β ππ)ππ β π΄1(1 β π1)π1
As area is assumed uniform over all machine
β΄π
ππ‘(1 β π1)πΏ = (1 β ππ)ππ β (1 β π1)π1 (5)
Since the Length of web is influenced only by the Dancer
Displacement (assumption 10)
Dancer Displacement affects web length from both sides
β΄ πΏ = πΏπ β 2π
Where: πΏπ:Constant Length of web.
π : Dancer Displacement.
Then equation (5)
π
ππ‘(1 β π1)(πΏπ β 2π) = (1 β ππ)ππ β (1 β π1)π1
By differentiating and simplifying we get
(πΏπ β 2π). π1Μ = π1βππ β 2ππ + ππππ β (π1 β 2ππ)π1
By taking Laplace Transform
(πΏπ β 2π). π π1 = π1βππ β 2ππ + ππππ β (π1 β 2ππ)π1 (6)
From transfer function (1) into (6) we get equation (7)
πΉπ‘1 (π +π1 β 2ππ
πΏπ + 2π) =
π΄1 πΈ + π΄1 πΆπ
πΏπ β 2π(βππ + π1 β 2ππ ) +
π1πΉπ‘π
πΏπ β 2π.π΄1
π΄2
From assumption 6, 7 and 8
Dancer displacement is negligible to total web length between winder and
unwinder
Dancer speed is negligible relative to winder and unwinder relative velocities
Material is stiff therefore ππβπ1
Therefor πΏπβπΏπ β 2π and ππβπ1 β 2ππ (8)
πΏπ: Approximate web length between winder and unwinder
Substituting in equation (7)
πΉπ‘1 (π +ππ
πΏπ
) =π΄1 πΈ + π΄1 πΆπ
πΏπ
(βππ + π1 β 2ππ ) +π1 πΉπ‘π
πΏπ
.π΄1
π΄2
The Term π1πΉπ‘π
πΏπ.
π΄1
π΄2 is constant due to assumptions 1, 2 and 9 and it
represents the initial Tension force.
Finally we get the transfer function of tension force from inputs (paper Linear
velocities and dancer velocity)
πΉπ‘1
π1 β ππ β 2ππ=
π΄πΏπ
(πΆπ + πΈ)
π +πππΏπ
Dancer Mathematical Model
From Newtonβs Second law of motion
β πΉππ₯π‘πππππ = ππ
By deriving equation and taking Laplace Transform
πΉπ‘π + πΉπ‘1 β ππ . π = (πππ 2 + πΆππ + πΎπ)π
Where
ππ: Dancer mass
πΆπ: Dancer damping coefficient
πΎπ:Spring Stiffness
π:Dancer Dsiplacment
Dancer position Transfer function π
πΉπ‘1+πΉπ‘πβππ .π=
1
ππ π 2+πΆππ +πΎπ
πΉπ‘1 πΉπ‘π
ππ . π
Complete model Summary:
Stepper Motor:
ππ = π πππ + πΏπ
πππ
ππ‘+ ππ
ππ = βπΎποΏ½ΜοΏ½ sin(ππ π)
ππ = βπΎπ (ππ βππ
π π) sin(πππ) + πΎπ (ππ β
ππ
π π) cos(πππ)
Dc motor armature controlled:
ππ(π )
ππ(π ) β πΎπππποΏ½ΜοΏ½(π )=
πΎπ
π π + π πΏπ
Dc motor Field controlled:
ππ(π )
ππ (π )=
πΎπ
π π + π πΏπ
Mechanical Transfer Function: π
ππβππππ’βπΉπ‘1ππ€πππππ=
1
π½π 2+ππ
Material Transfer Function: πΉπ‘1
π1βππβ2ππ=
π΄
πΏπ(πΆπ +πΈ)
π +πππΏπ
Dancer Transfer function:π
πΉπ‘1+πΉπ‘πβππ.π=
1
ππ π 2+πΆπ π +πΎπ
Calculation of varying moment of inertia:
Length of winded material:
πΏπ€πππππ = β« π1ππ‘
Radius of Winder Cylinder:
ππ€πππππ = βπΏπ€πππππ . π‘
π+ πππππ
2
Material Mass:
π = ππ£. π. π€(ππ€πππππ2 β πππππ
2)
ππ£: Material mass per unit volume
πππππ: Winder Cylinder Core radius
Variable material Moment of inertia:
π½π€ =1
2π(ππ€πππππ
2 + πππππ2)
Total Drive moment of inertia:
π½ = π½π€ + π½π
π½π : Winder Cylinder Core Moment of inertia