Technology & Engineering
Equation of Motion
The equation of motion is a mathematical expression that describes the relationship between an object's motion and the forces acting upon it. It is commonly used in physics and engineering to predict the behavior of objects in motion, such as projectiles, vehicles, and machinery. The equation of motion is derived from Newton's laws of motion and can be used to solve for various parameters like velocity, acceleration, and displacement.
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3 Key excerpts on "Equation of Motion"
eBook - PDF- Dominic J. Diston, Peter Belobaba, Jonathan Cooper, Allan Seabridge, Peter Belobaba, Jonathan Cooper, Allan Seabridge(Authors)
- 2024(Publication Date)
- Wiley(Publisher)
2 Equations of Motion 2.1 Introduction 2.1.1 The Problem with Equations of Motion Everybody understands equations of motion … or they should! At school, everybody learns Newtons’ Laws of Motion and, at higher levels of education, these are generalised and applied in different contexts. It is not entirely clear how much of this subject is understood by students or engineers although, in fairness, most individuals do not need expert insight. However, flight dynamicists do need a better-than-average grasp of the applicable science and mathematics and they need to understand the various formulations and foundations for representing the motion of a vehicle through space. One televised documentary include commentary from a physics teacher who ‘explained’ flight wholly in relation to Newton’s Third Law, which is wrong. A prominent researcher proclaimed that ‘nobody uses quaternions’, which is also wrong. A lecturer once claimed that aerospace students do not need to study this subject because companies always employ physics graduates to do this kind of work. Wrong again! While experts will be acutely aware of mis-interpretations and confusions, others may not be. So, in the interest of educating the next generation of experts, this author believes that textbooks should present a complete explanation of Equations of Motion. This is because simulation has to predict aircraft motion and, to that end, it is important to know what formulation is being applied and what computational is being performed. 2.1.2 What Chapter 2 Includes This chapter includes: • Spatial Reference Model (covering reference frames for earth, aircraft, and flight path) • Aircraft Dynamics (developing equations of motion in terms of force, moment, and velocity components) • Aircraft Kinematics (developing equations of motion for position and orientation) 25 Computational Modelling and Simulation of Aircraft and the Environment: Aircraft Dynamics, First Edition, Volume II.
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
25 © 2010 Taylor & Francis Group, LLC Motion in One Dimension This chapter addresses deriving and using the equations of motion that describe the time depen-dence of an object’s displacement, velocity, and acceleration. Relationships between displacement, velocity, and acceleration are also of importance and will be derived. This chapter starts with the basic definitions of displacement and average velocity of an object moving in one dimension. Such a simplified start will help to lead a complete set of equations of motion for an object moving along one dimension, east–west, north–south, or up and down. In the context of coordinate systems that were treated in the previous chapter, the one-dimensional motion will reduce the time and effort needed on the study of motion in two dimensions, which is the subject of the next chapter. 2.1 DISPLACEMENT Motion can be defined as a continuous change in position and that change could occur in one, two, or three dimensions. As the treatment here will be limited to motions only in one dimension, one axis of the coordinate system described in Chapter 1 will suffice. Choosing this axis as the x-axis, we depict on it a point, O, which will be considered as an origin of zero coordinate. Any point on the right of the origin O will have a positive coordinate and any point on the left of O will have a negative coordinate (Figure 2.1). The displacement, denoted by Δ x, of an object as it moves from an initial position x i to a final position x f along the x-axis can be defined as the change in the object’s position along the x-axis. That is Δ x = x f – x i . (2.1) 2.1.1 S PECIAL R EMARKS The quantities in Equation 2.1 are treated as vectors. x i is the initial position vector, x f the final posi-tion vector, and Δ x the displacement vector. Accordingly, all these quantities have a direction and magnitude.
eBook - PDF- Jitendra R. Raol, Jatinder Singh(Authors)
- 2008(Publication Date)
- CRC Press(Publisher)
3 Equations of Motion 3.1 INTRODUCTION In Chapter 2, a system was de fi ned as a set of components or activities that interact to achieve a desired task. The primary requirement to study a system, be it a chemical plant or a fl ight vehicle, is a model that closely resembles the system. In other words, a model is a simpli fi ed representation of a system that helps to understand, predict, and possibly control its behavior. It presents a workable knowledge of the system and simulates its behavior. Some authors also refer to a model as a less complex form of reality. A fl ight vehicle is a complex system that requires a number of models for its various components (Figure 3.1). In this chapter, we describe the model form that represents aircraft dynamics. When an aircraft fl ies, it experiences gravitational, propulsive, and inertial forces arising from the air fl ow over the aircraft frame. To achieve a steady, unaccelerated fl ight, these forces must balance out one another. The upward force, due to the lift, should be in equilibrium with the downward force, due to the weight of the aircraft, so that it does not experience unsteadiness (unintentional one). Similarly, the forward thrust force should be in equilibrium with the opposing drag force so that the aircraft does not accelerate and, hence, is steady. An aircraft satisfying these requirements is said to be in a state of equilibrium or fl ying at trim condition. Normally at trim, the translational and angular accelerations are zero. The notations and terms used in describing aircraft dynamics are given in Appendix A. Both the equations of motion (EOM) and the models of the aerodynamic force and moment coef fi cients have to be postulated to describe the aircraft motion in fl ight (Figure 3.2).
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