PH132
Section 17
Team “ACME”
Silas Magee
Chris Ouellette
Josh Kehs
Joshua Smith
1
Physics Team Design II
Modeling the Velocity of an Electric Train
April 20, 2004
Introduction:
An important aspect of the field of physics is developing models that describe physical phenomena. Models can be constructed which describe the strength of an electromagnet,the most efficient design of an electrical circuit, or the forces on a current carrying wire in a magnetic field. This report modeled the velocity of a HO-scale electric train on a straight track as a function of voltage. For the development of any model the underlying process is the same.
In order to scientifically model a system, a scientist’s first task is to identify the system that he or she wishes to study, and define what it is that they want to model about that system. Next, the system is studied, and relevant information is described using schematics. Scientists then decide which physical laws govern the system, based on hypotheses and background research. The physical laws are then applied to the system of interest. In this report laws are applied to an electric train, by defining variables and mathematical relationships that describe its various components. The relationships are then combined and used to create a theoretical model for the entire system. Sometimes this process is best achieved by looking at the system as a whole and modeling it generally and then looking at each individual piece of the system and repeating the modeling process as necessary.
The goal of the modeling process is two-fold: primarily it is to develop a mathematical relationship which theoretically describes any model electric train, but is also apply this general model to the train used in this project so that the model can be experimentally verified. Experimentation is done throughout the modeling process to isolate and calculate parameters for the system and most importantly at the end when the theoretical model is compared with experimentally collected data in complex cases. The theory is then revised and further developed until a model which satisfactorily predicts experimental observations, and thus accurately models the physical system, is created.
Objectives:
The purpose of this project is to accurately model the velocity of a train on a flat, straight track. Ultimately, the researchers wish to know the velocity of the train as a function voltage. This model will allow the researchers to predict the location of the train at any time, given a voltage function. Following the construction of this model, the theoretical calculations will be tested in two ways: first, by comparing experimental velocity data with an arbitrary voltage to theoretical predictions based on the same voltage profile, and second, by the success of the predictive model during a series of challenge sessions.
Development of the Theoretical Model:
There are two major sections of an electric train which need to be modeled before a function relating velocity and voltage can be made. On first observation, it is evident that the desired model describes a relationship between a mechanical quantity, velocity, and an electrical quantity, voltage. These two quantities directly relate to the two major components of any electric train: the mechanical parts such as the gears and wheels; and the electrical parts such as coils of wire in the motor, and train track and voltage source that can be modeled as a circuit. By first modeling the train as a whole, and then modeling the electrical and mechanical parts independently, the mechanical and electrical descriptions of the train’s components can be combined back into the overall description, allowing the researchers to predict a mechanical quantity based on an electrical one.
This paper will present the development of the theory in a similar fashion: first a mechanical schematic of the train will be presented, identifying relevant variables. The mechanical components will then be modeled using concepts of rotational and translational motion and forces learned in Physics I. The electrical portions of the train and track will also be simplified in a schematic and relevant variables will be identified. The electrical portion will be modeled using concepts of circuits, and magnetic field theory. These two models will then be combined and manipulated into a form so that needed physical parameters (such as constants specific to the train used) can be determined through experimentation. By calculating these constants, a predictive model specific to the train used in the research can be made and verified through comparison with experimental data.
Part I: Mechanical Train Description and Model
A schematic of our train’s mechanical components is shown in Figure 1 with relevant variables and parameters labeled.
Figure 1
Note: This diagram accounts for electrically induced torque by the electric motor (τe), but this will not be developed and included in our model until it is derived in Part II and combined in Part III.
Observing Figure 1, we first note that there are two types of motion involved: translational and rotational motion. The internal components of the train, including all parts from the torque associated with the motor to the wheels in the wheel assembly, are rotational. The resulting motion of the train down the track is translational. These two major sections will be addressed separately and then combined through substitutions.
Using Newton’s second law, applied for translation and rotation, we can relate the variables and parameters shown in Figure 1 together as shown below:
1
Newton’s Second Law for Rotation:
Newton’s Second Law for Translation:
1
Using Newton’s law for rotation with the knowledge that a torque can be described as a force applied at a distance or as a “pure” torque (such as the electrically induced torque in Figure 1), there are two equations that describe the rotational motion of mechanical parts inside the train:
Drive Shaft Components:
Eqn. 1
Wheel Assembly Components:
Eqn. 2
Newton’s second law for translation governs the interaction between the rotating wheels and the track. As seen in the figure, there are two translational forces on the train: the static frictional force (F) pulling the train ahead, and the kinetic frictional force resisting its motion (fk) Or:
Eqn. 3
Where M is the mass of the train, a is the train’s linear acceleration, and . Since the train being modeled is always traveling on a horizontal-flat track, the normal force (N) is equal to the trains weight force Mg by Newton’s second law.
In order to combine Equations 1, 2 and 3 together into one equation, we need to express the translational variable acceleration in terms of rotational ones.
By definition, , so applied to the train’s wheel assembly:
This yields a description of the translational forces in Equation 3 in terms of rotational variables:
Eqn. 4
We now want to eliminate the variables r1, r2, F, P1, P2, in Equations 1, 2, and 4 developed above and combine them all into one relationship so we can see how electrical torque () relates to rotational acceleration of the train’s wheels. This will allow us to ultimately determine how an electrical quantity relates to a mechanical one, after developing a model for the electrical portion of our train in Part II.
We will use the following substitutions to combine these equations:
, Since by Newton’s 3rd Law, forces come in pairs which are equal in magnitude and opposite in direction, as seen in Figure 1.
1
Reference Figure 2 to the right for the following substitutions:
By definition , so: ,
, or , where
Also:
Figure 2
1
Substituting into the translation equation (Equation 4) into the rotational equation for the wheels (Equation 2):
Eqn. 5
Substituting for r1, P1 and in Equation 1:
Rearranging to solve for P we get:
Substituting P(as written above) intoEquation 5:
Substituting in and distributing:
Reorganizing:
Reorganizing more, the final equation then becomes:
, Eqn. 6
Equation 6 completes the mechanical description of our train. The electrical description of the train will now be developed to find the electrically induced torque for substitution back into this equation.
Part II: Electrical Train Description and Model
In this section we consider the electrical motor of the train and how it creates torque, which drives the mechanical portion of the train. This is done in two steps, first by modeling the electrical torque created by an electric motor, and then combining this model with what we know about the circuit which the train runs on.
Consider Figure 3, which diagrams a single loop of wire residing in a magnetic field. This can be considered a very simple electric motor.
Figure 3
We know that a charge moving in an electric field experiences a force. In this case, a current i flows around the loop. The force that a charge feels moving in a magnetic field can be modeled as:
For a wire, a more specific model can be written as: .
Using the definition of a cross product and the right hand rule, we can determine the magnitude and direction of the forces on each of the four segments of straight wire around the loop. This can be written as:
, Eqn. 7
Where theta is the angle between the wire segment vector (in the direction of the current) and the magnetic field.
Using the right hand rule, the force on the wires that are perpendicular to the axis of rotation are both zero since the sine of the angle between the wire segment vector L and the magnetic field B iszero.These portions of wire therefore do not effect the rotation of the loop.
The direction of force on the wires that are parallel to the axis of rotation can be calculated using the right hand rule and Equation 7; this is shown in the side-view schematic in Figure 3. The forces acting on the wire that are parallel to the axis of rotation are equal in magnitude and opposite and direction. While these forces cancel in a linear sense, since the forces act at a distance r from the axis of rotation, they create a torque on the drive train: the electrically induced torque.
This torque can be modeled as follows:
, where beta is the angle between the position vector from the axis of rotation to the position of the applied force and force vector.
Since two forces act on the loop of wire, creating torque in the same direction, the electrically induced torque can be written as:
Substituting (by Equation 7, where between the length and magnetic field vectors) we obtain:
This result is the basic model fortorque induced by an electric motor. This model can be simplified further however with a couple observations. First, 2rL represents the area of the loop which the current i flows through. This can be represented as A. Also, since beta will vary between 0 and 180 degrees as the loop rotates, this would imply that torque acts in one direction for a period, and then switches direction. However, when the loop passes the perpendicular orientation to the magnetic field, where beta is zero, the current switches direction around the loop. So instead of torque induced in the opposite direction, the direction of the forces due to the magnetic field changes direction when the current switches and the motion continues. Consider Figure 4 below:
Figure 4
Since this shows that there is always a pair of forces which aid in creating electrical torque in the same direction, varying with the magnitude of the current and the sine of the angle between the force and position vector we can write:
.
We assume that the magnitude of the torque is proportional to the number of coils N, so:
, where N is the number of loops of wire.
Observing this model, it is evident that a vehicle would not run well with only a single (or even multiple) loops of wire with only one orientation. This is because the sine of the angle will vary between 0 and 1 periodically creating spurts of torque, followed by nearly none. In other words, the torque is reliant on both the angle between the magnetic field and area vector and the magnitude of the current. This can be overcome by using multiple orientations of loops. That way the torque becomes less sensitive to the angle between the loop and the magnetic field. With multiple orientations we can average the values of sine becoming:
, where theta has been substituted for beta—the interpretation of the angle remains the angle between the radius vector and the force on the wire.
Since the number of loops, magnetic field, area of the loops and average sine value are all constant for each motor design, we can further simplify the electrically induced torque to become a constant multiplied by the relevant variable i:
, where Eqn. 8
This equation describes the electrical torque created by the train’s motor as a function of current (i).
The primary goal of this project was to find the velocity of the train as a function of voltage. This means that the electrical torque needs to be modeled as a function of voltage instead of a function of current, as seen in Equation 8. This can be done by considering the circuit that that train operates on, and applying Ohm’s law. First consider the basic layout of the train traveling on a track in Figure 5:
Figure 5
Simplifying this into a circuit schematic, we can apply Kirchoff’s Voltage Law. Consider schematic circuit design in Figure 6:
Figure 6
We can write the following loop equation for this circuit in order to find the relationship between voltage and current:
or Eqn. 9
ε in this equation represents the “back” electromotive force created by the coils of wire rotating in a magnetic field, in accordance to Faraday’s Law and Lenz’s Law. This occurs because of induction of the closed loop of current carrying wire. Consider the magnetic field, set-up by the permanent magnets, flowing through the current loop in the electric motor, shown in Figure 7 below:
Figure 7
To find an expression for ε the definition of magnetic flux and Faraday’s law need to be applied.
1
Definition of Magnetic Flux:
Faraday’s Law:
1
In the case of the train, the magnetic field is constant since it is created by permanent magnets, and can be pulled out of the integral. The area of the loop is also constant so the flux can be written as: .
Consider Figure 8 below to see the relationship between the direction of the area vector and magnetic field:
Figure 8
Since the angle between the magnetic field and the area vector varies with time, when Faraday’s Law is applied the chain rule will be needed.
.
Note that when the chain rule was applied to cosθ, this yielded its derivative, or that rate at which theta is changing. Since theta is related to the spinning of the coil loops that make up the electric motor, this rate is the same as the rate at which the drive shaft is spinning, or:
, where (reference Figure 1 and 2 and related Equations for further explanation of these substitutions).
Combining the model of back electromotive force with the developed relationship for current(Equation 9) and substituting it back into Equation 8:
Eqn. 10
The model for electrical torque is now complete. Beginning by considering the force exerted on a current carrying wire in a magnetic field and the torque that this induces on a closed loop, thereafter adapting this result to represent the train’s electrical motor setup, and then modeling the circuit on which the train operates to determine torque in terms of a voltage, a final result was obtained.
In the next section the electrical description of the train’s operation will be combined with the mechanical developed in Part I, allowing a final result between voltage and velocity to be derived.
Part III: Combination of Mechanical and Electrical Descriptions
In this section the results of Part I and II will be combined together into one equation. After combining these equations, the results will be simplified so that needed constants can be measured and experimentally determined in Part IV.
Combining Equations 6 and 10:
The last remaining portion of the model that has not been incorporated is the frictional torque which opposes the rotation of the drive shaft. This report assumes that the torque is proportional to angular velocity of the drive shaft so that: .
The final combined model is:
This relationship can be greatly simplified through a series substitutions and algebraic manipulation. The goal of this simplification is to relate the independent variable V (voltage) to the dependent variables of velocity (v) and acceleration. A simplified expression allows isolating constant “lumps” that can be determined experimentally.
Making the substitutions: and , and separating dependent variables from independent:
.
Needing an equation for v (velocity), we divide through by its constant coefficient to isolate it:
Eqn. 11
Observing this relationship one can see that there is a group of constants in front of dv/dt and the right hand side of the equation can be written as a function of V.Simplifying in this fashion we obtain: