it shows on how the slot windings of machine motor
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This paper presents a mathematical model of the proportional valve main stage flow characteristics. Model is verified with static measurements. Valve spool dynamics are modelled but not verified. ...
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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 π ππ + πΏπ
πππ
+ ππ ππ‘ ππ = βπΎπ πΜ sin(ππ π) Where ππ : Phase a applied Voltage [Volt]. π π : Phase a winding Resistance [Ohm]. ππ : Phase a Current [ampere]. πΏ π: Phase a winding inductance [H]. ππ : Back emf (electromagnetic force) induced in phase a [Volt]. πΎπ : Electromotive Force Constant, (Motor Torque constant) [ππππ‘. π ππβπππ ]
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:
Where: π: Rotor Position πΎπ : Motor Torque Constant [π. πβπ΄ππ] 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 π be infinite which causes the term ( π = 0 ). π π
As Shown the system equations is nonlinear.
Dc Motor Model Dc Motors Control Techniques: 1. Armature Control. 2. Field Control.
Where: ππ : Voltage applied over Motor Armature [Volt]. π π : Armature Resistance [Ohm]. ππ : Armature Current [ampere]. πΏ π: Armature Inductance [H]. πππππ : Back volt induced from rotor into armature [Volt]. πΎππππ :Electromotive Force Constant [ππππ‘. π ππβπππ ]
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 = πΎππ ππ The Strength of the magnetic field is constant as its armature controlled so:
ππ : Armature current [π΄ππ ] Current to Torque Transfer function ππ (π ) = πΎπ ππ (π )
So Volt Torque Transfer Function: ππ (π ) ππ (π ) = ππ (π )βπΎπππππΜ (π ) ππ (π )βπΎππππ πΜ (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 ).
ππ : Voltage applied over Motor field winding [Volt]. π π : Field winding Resistance [Ohm]. ππ : Field winding Current [ampere]. πΏπ: Field winding Inductance [H].
ππ : Field winding current [π΄ππ ] Current to Torque Transfer function: ππ (π ) = πΎπ ππ (π )
So Volt Torque Transfer Function:
ππ (π ) ππ (π )
=
ππ (π ) ππ (π ) ππ (π )
.
ππ (π )
=
πΎπ π π +π πΏπ
πΉπ‘π
πΉπ‘1 ππ€πππππ
πππππππ
πΉπ‘1
ππ = πΉπ‘1 ππ€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 =
ππ ππππππ ππ‘
.
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‘1 ππ€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 = ππΜ So Mechanical Differential Equation: β π = π½πΜ ππ β ππ β ππππ’ β ππΜ = π½πΜ Taking Laplace Transform: ππ (π ) β ππ β ππππ’ β ππ π(π ) = π½π 2 π(π ) Angular Position Transfer Function:
π 1 = 2 ππ β ππππ’ β πΉπ‘1 ππ€πππππ π½π + 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 =
π=
πΉπ‘ π΄
is expressed as follows
πΉπ‘ = πΈπ + πΆπΜ π΄
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
πΉ
π‘ π = 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 πΏ π :Stretched Length of material π΄π :Stretched Cross sectional area
As Density is assumed constant
From Strain Definition
From (2) and (1)
π= π΄π =
β΄ π΄πΏ = π΄π πΏπ βπΏ πΏ
=
π΄ (π+1)
πΏπ βπΏ πΏ
=
πΏπ πΏ
(2)
β 1 (3)
(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‘
From equation (4) we get π π΄(1 β π1 )πΏ = π΄π (1 β ππ )ππ β π΄1 (1 β π1 )π1 ππ‘ As area is assumed uniform over all machine β΄
(πΏπ β 2π ). π π1 = π1 βππ β 2ππ + ππ ππ 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 ) +
π1 πΉπ‘π
.
π΄1
πΏπ β 2π 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
πΏπ βπΏπ β 2π
and
ππ βπ1 β 2ππ
(8)
πΏ π: Approximate web length between winder and unwinder Substituting in equation (7)
πΉπ‘1 (π +
The Term
ππ πΏπ
)=
π1 πΉπ‘π π΄1 . πΏπ π΄2
π΄1 πΈ + π΄1 πΆπ πΏπ
(βππ + π1 β 2ππ ) +
π1 πΉπ‘π π΄1 . πΏπ 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)
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 = (ππ π 2 + πΆπ π + πΎπ )π Where ππ : Dancer mass πΆπ : Dancer damping coefficient πΎπ :Spring Stiffness π:Dancer Dsiplacment Dancer position Transfer function
Calculation of varying moment of inertia: Length of winded material: πΏπ€πππππ = β« π1 ππ‘ Radius of Winder Cylinder: ππ€πππππ
Material Mass: π = ππ£ . π. π€(ππ€πππππ 2 β πππππ 2 ) ππ£ : Material mass per unit volume πππππ : Winder Cylinder Core radius Variable material Moment of inertia: π½π€ =