DET 2113 Chapter 5 Introduction to Machinery Principles

DET 211/3 Chapter 5 Introduction to Machinery Principles

Rotational Motion Angular Position, θ The angular position θ of an object is the angle at which it oriented, measured from some arbitrary reference point. Unit: Radians or Degree (rad or deg) Rotation of a rigid object P about a fixed object about a fixed axis O.

Angular velocity, ω Angular velocity (or speed) is the rate of change in angular position with respect to time. It is assume positive if the rotation is in a counter clockwise direction. Unit: Radians per second (rads-1) The following symbols are used in this course: ωm = angular velocity expressed in radians per second Fm = angular velocity expressed in revolutions per second nm = angular velocity expressed in revolutions per minute

Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. The direction of the angular velocity vector will be along the axis of rotation; in this case (counter-clockwise rotation).

Angular acceleration, α is the rate of change in angular velocity with respect to time. Unit: Radians per second (rads-2) Torque, τ (tau) The torque on the object is defined as the product of the force applied to the object and the smallest distance between the line action of the force and the object’s axis of rotation.

where θ is the angle between vector r and the vector F. The direction of the torque is clockwise if it tends to cause a clockwise rotation and counter clockwise if it would tend to cause a counter clockwise rotation

In physics, a torque (τ) is a vector that measures the tendency of a force to rotate an object about some axis. The magnitude of a torque is defined as force times its lever arm. Just as a force is a push or a pull, a torque can be thought of as a twist.

The Magnetic Field Four basic principles describe how magnetic fields are used in electrical machines (transformer, motor, and generator): • A current carrying produces a magnetic field in the area around it. • A time changing magnetic field induces a voltage in a coil of wire if it passes through that coil. (This is the basis of transformer action) • A current carrying wire in the presence of a magnetic field has a force induced on it. (This is the basis of motor action) • A moving wire in the presence of the magnetic fields has a voltage induced in it. (This is the basis of generator action)

Faraday’s Law – Induced voltage from a time changing magnetic field Faraday’s law states that if a flux passes through a turn of a coil, a voltage will induced in the turn of wire that is directly proportional to the rate of change in the flux with respect to time. where eind is the voltage induced in the turn of the coil and Ф is the flux passing through the turn. If a coil has N turns and if the same flux passes through all of them, then the voltage induced across the whole coil is given by where N = number of turns of wire in coil The minus sign in the equation is an expression of Lenz’s Law.

Lenz’s Law Lenz’s law states that the direction of the voltage buildup in the coil is such that if the coils ends were short circuited, it would be produce current that would cause a flux opposing the original flux change. Since the induced voltage opposes the change that causes it, a minus sign is included in equation above.

the flux shown in the figure a is increasing in strength, than he voltage built up in the coil will oppose the increase. A urrent flowing as shown in Figure b would produce a flux pposing the increase, so the voltage on the coil must be uilt up with the polarity required to drive that current hrough the external circuit. Therefore, the voltage must be uilt up with polarity shown in Figure b. Since the polarity of he resulting voltage can be determine from physical onsiderations, the minus sign in equation above is often left ut.

If leakage is quite high or if extreme accuracy is required, a different expression that does not make that assumption will be needed. The magnitude of the voltage in the ith turn of the coil always given by If there are N turns in the coil of wire, the total voltage on the coil is Faraday’s law can be rewritten in terms of the flux linkage as (unit: weber turns)

Faraday’s law is the fundamental property of the magnetic fields involved in transformer operation. The effect of Lenz’s in transformer is to predict the polarity of the voltages induced in the transformer windings.

Production of induced force on a wire The basic concept involved is illustrated in Figure below. The figure shows a conductor present in a uniform magnetic field of flux density B, pointing into the page. The conductor itself is l meters long and contains a current of i amperes. The force induced on the conductor is given by

The direction of the force is given by the right hand rule. If the index finger of the right hand points in the direction of the vector l and the middle finger points in the direction of flux density vector B, then the thumb points in the direction of the resultant force on wire. The magnitude of the force is given by the equation F= il. Bsin θ where θ is the angle between the wire and the flux density vector The induction of the force in a wire by a current in the presence of a magnetic field is the basis of motor action.

Induced voltage on a conductor moving in a magnetic field If the wire with the proper orientation moves a through a magnetic field, a voltage is induced in it. This idea is shown in figure below. The voltage induced in a wire is given by

Where v = velocity of the wire B = magnetic flux density vector l = length of the conductor in the magnetic field Vector l points along the direction of the wire toward the end making the smallest angle with respect to the vector v x B. The voltage in the wire will be built up so that the positive end is in the direction of the vector v x B.

Example Figure above shows a conductor moving with a velocity of 5. 0 m/s to the right in the presence of a magnetic field. The flux density is 0. 8 T into the page and the wire is 1. 0 m in length. What are the magnitude and polarity of the resulting induced voltage?

Solution Example Direction of the quantity v x B in this example is up. Therefore, the voltage on the conductor will be built up positive at the top with respect to the bottom of the wire. The direction of vector I is up. So that makes the smallest angle with respect to the vector v X B.

Since v is perpendicular to B and since v X B is parallel to I, the magnitude of the induced voltage reduces to: Thus, the induced voltage is 4. 0 V, positive at the top of the wire