Angular Kinematic Equations

3 Key excerpts on "Angular Kinematic Equations"

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  eBook - ePub

  Higher Engineering Science

  • William Bolton(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  4 Linear and angular motion 4.1 Introduction This chapter is concerned with the behaviour of dynamic mechanical systems when there is uniform acceleration. The terms and basic equations associated with linear motion with uniform acceleration and angular motion with uniform angular acceleration, Newton’s laws of motion, moment of inertia and the effects of friction are revised and applied to the solution of mechanical system problems. The terms scalar quantity and vector quantity are used in this chapter, so as a point of revision: Scalar quantities are those that only need to have their size to be given in order for their effects to be determined, e.g. mass. Vector quantities are those that need to have both their size and direction to be given in order for their effects to be determined, e.g. force where we need to know the direction as well as the size to determine its effect. 4.2 Linear motion The following are basic terms used in the description of linear motion, i.e. motion that occurs in a straight line path rather than rotation which we will consider later in this chapter: 1 Distance and displacement The term distance tends to be used for distances measured along the path of an object, whatever form the path takes; the term displacement, however, tends to be used for the distance travelled in a particular straight line direction (Figure 4.1). For example, if an object moves in a circular path the distance travelled is the circumference of the path whereas the displacement might be zero if it ends up at the same point it started from

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  eBook - ePub

  Advanced Flight Dynamics with Elements of Flight Control

  • Nandan K. Sinha, N. Ananthkrishnan(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  b ” refers to components along body-fixed axes. Thus, the rates of change of the aerodynamic angles are related to the difference between the angular velocities of the body- and wind-fixed axes.

  At this point we can pause to write out the airplane kinematic equations—these relate the airplane position to its inertial velocity and the orientation angles to the angular velocity. And, as we have seen, these may be written in terms of either body- or wind-axis variables. They are summarized in Table 1.3 .

  Table 1.3 Summary of Airplane Kinematic Equations

  Now we have all the machinery required to derive the equations for the translational and rotational dynamics of an airplane in flight.

  1.5Translational Equations of Motion

  The translational equations of motion relate the rate of change of the inertial velocity V to the forces acting on the airplane. As discussed previously, we assume the airplane in flight to be a rigid body with six degrees of freedom with fixed mass m and fixed CG and MI. Additional effects, as in the examples in Section 1.1 , may be taken into account as and when needed. A flat Earth is chosen as the inertial frame of reference.

  Figure 1.14 shows the representation that we shall use of the aircraft for the purpose of deriving the equations of motion. The body-fixed axes X B Y B Z B are fixed to the CG labeled O B . The Earth-fixed axes X E Y E Z E provide an inertial reference frame. The airplane position is given by the position vector R from the origin O E of the X E Y E Z E axes to the aircraft CG (O B ). The inertial velocity of the airplane is the velocity V of its CG relative to the Earth-fixed axes. Likewise its angular velocity relative to the inertial X E Y E Z E axes is denoted by ω . We shall assume the airplane geometry to be symmetric about the X B Z B plane, which is called the longitudinal plane of the aircraft.

  Figure 1.14 Aircraft representation for the purpose of deriving equations of motion.

  Consider an element of mass δ m at a distance r C from the CG O B as marked in Figure 1.14 . The velocity of this elemental mass as seen by an observer in the inertial frame of reference (X E Y E Z E

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  eBook - ePub

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Rotational Kinematics

  Not all rotational motion is uniform; in non-uniform circular motion, the angular speed changes. The average angular acceleration is the rate of change of the angular speed

  (7.7)

  where Δω = ω - ω0 is the change in angular speed. The usual unit for angular acceleration is the rad/s2 , but we’ll also use rev/min2 .

  From the definitions of angular displacement, speed, and acceleration, we can derive four basic equations which describe the angular position and speed of objects rotating with a constant acceleration.

  (7.8a)

  (7.8b)

  (7.8c)

  (7.8d)

  For constant angular acceleration, the average angular speed is .

  Do these equations look familiar? These are the same as Eqs. (2.5) from Chapter 2. The connection between these angular quantities and their linear counterparts is the radius R of the circular path.

  The radius is perpendicular to the displacement, velocity, and the component of the acceleration that changes the object’s speed.

  Example 7.5

  Turn that crap off

  After the LP record in our first two examples is finished, you turn it off and it slows to a stop after 10 seconds. Calculate the record’s angular acceleration, assuming it to be constant. How many times does the record turn while it is slowing down?

  Solution.

  Since we want Δθ in revolutions, let’s use Eq. (7.7) to Evaluate the acceleration in revolutions and minutes.

  Now that we know the time and angular acceleration, we can use Eq. (7.8a) and Evaluate Numerically to find the number of times the record turns while it’s slowing down.

  The record turns about 2.8 times while it’s slowing down.

  When the angular acceleration is not constant, you must use calculus to describe the object’s rotational motion. The angular velocity

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