Chapter 2
Equations of Motion
The equations of motion form the basic building blocks for any system under consideration. These equations should be formulated as accurately as possible to model the desired system. The δinaccuracies in formulating these equations could result in faulty behaviour of the system which could be very difficult to understand. However, modern control systems are designed to accommodate model inaccuracies to a certain degree. It is very important to ensure that our model is modelled within this range. Errors could also enter the system during the calculation stage due to the precision and number of digits used to represent the values.
The equations of motion for any given airplane are non-linear in nature. It is linearised by making certain assumptions. The assumptions are as follows:
The OX and OZ axes are the planes of symmetry.
The polar moment of Inertia J is zero in XY and YZ is equal to zero.
The mass of the airplane remains constant during the whole analysis.
The aircraft is a rigid body.
Earth is an inertial reference.
Formulation of Equations of Motion:
According to Newton’s Second Law of Motion, the vector sum of the total forces in a system is equal to the product of the mass (m) and acceleration ( a ) of the system.
Therefore, in this case, we have :
∑ F= m (d Vt / dt ) , where Vt = Velocity of Aircraft (1.1)
Since mass m is constant.
Therefore we have,
F=m*(dVt/dt)+(d/dt) ∑( Vt+(dr/dt))* δm (1.2)
F=m*(dVt/dt)+d^2/dt^2(∑r*δm ) (1.3) r*δm = 0 since r is measured from center of mass.
Similarly the moment can be found can be formulated as follows: δm=d/dt(δH)=d/dt(r x v ) δm (1.4)
The velocity of this small element can be expressed as ...
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...= (A – B*K)x + Bv.
We formulate a Performance Index J as there can be n number of optimal solutions to a problem. Once we formulate a performance Index J, we can call the solution which minimises this performance index as Optimal. This can be visualised as an energy function, as our ultimate objective is to reduce the energy consumption of the system.
Let us define our Performance Index J as:
J = ∫_0^∞▒xTQx + uTRu dt
Substituting the State Variable Feedback control into this yields,
J = ∫_0^∞▒xT(Q + KT R K ) x dt
Here we have assumed that input v(t) is equal to zero. This assumption is valid as we are only concentrating on the internal stability of the system.
In this performance index, both the states x(t) and the control input u(t) are weighted, hence if J is small, neither x(t) nor u(t) would be too large. If we can minimise J, then it is definitely finite.
Ever since I was little I was amazed at the ability for a machine to fly. I have always wanted to explore ideas of flight and be able to actually fly. I think I may have found my childhood fantasy in the world of aeronautical engineering. The object of my paper is to give me more insight on my future career as an aeronautical engineer. This paper was also to give me ideas of the physics of flight and be to apply those physics of flight to compete in a high school competition.
Many people are amazed with the flight of an object, especially one the size of an airplane, but they do not realize how much physics plays a role in this amazing incident. There are many different ways in which physics aids the flight of an aircraft. In the following few paragraphs some of the many ways will be described so that you, the reader, will realize physics at work in the world of flight.
Since the time of the World Wars and the Wright brothers, aviation has become a huge part of global society. The Orville and Wilbur Wright’s names will forever be remembered into United States history as the first men who were the first to fully realized human flight. Their successful invention of a working, powered airplane brought about whole new ways of wars, including new strategies for both offense and defense. Many technological advances might not have occurred without the need for new weapons and systems for airplanes. Travel and commerce would be much slower without the usage of airplanes. Orville and Wilbur have made a lasting impact on the world with their invention of a working, human-controlled, powered airplane; who knows what the world would be without it.
Flight is one of the most important achievements of mankind. We owe this achievement to the invention of the airfoil and understanding the physics that allow it to lift enormous weights into the sky.
Kinematics unlike Newton’s three laws is the study of the motion of objects. The “Kinematic Equations” all have four variables.These equations can help us understand and predict an object’s motion. The four equations use the following variables; displacement of the object, the time the object was moving, the acceleration of the object, the initial velocity of the object and the final velocity of the object. While Newton’s three laws have co-operated to help create and improve the study of
The Volume Library, vol. I, Physics: Newton's Law of Motion. Pg. 436. The Southwestern Company, Nashville, Tennessee, 1988.
Throughout every angle used, each of the trials were between 41 and 36 swings per minute. When releasing the pendulum from 15 degrees, the results of the trials were 39 swings per minute, then 40 swings per minute, 38 swings per minute ,39 swings per minute, then finally 39 swings per minute. When releasing the pendulum from 30 degrees the results of the trials were 39 swings per minute , then 41 swings per minute, 40 swings per minute, 36 swings per minute, and finally 37 swings per minute. When the pendulum was released from 45 degrees, the results of the trials were 39 swings per minute, then 38 swings per minute,38 swings per minute,38 swings per minute, then 41 swings per minute. When the pendulum was released from 90 degrees, the results of the trials were 38 swings per minute, then 39 swings per minute, 38 swings per minute,38 swings per minute, and finally 41 swings per
F = ma : where F is force; m is the mass of the body; and a is the acceleration due to that particular force
Here θ is the angle between the “tails” of the force and the differential displacement. The integration must account for the variation of the force’s direction and magnitude.
In Newtonian physics, free fall is any motion of a body where its weight is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it and it moves along a geodesic. The present article only concerns itself with free fall in the Newtonian domain.
VII). This definition can be explained through Figure 2. In the figure, there are two bodies of the same size both with a gravitational pull, body A with center B, and body C with center D. A third body, E, is at unequal distances from centers B and D, with D being closer. Because the bodies are of the same size, but the center D is closer to E, the body E will be drawn towards body C. Accelerative quantity of centripetal force is proportional to the velocity that the force generates in a given time, and because the body E is closest to the center that is D, the accelerative quantity of centripetal force will be greater for the body C, and it will have a greater pull on body
Henderson, T. n.d. The physics classroom tutorial. Lesson 2: Force and Its Representation [Online]. Illinois. Available at: http://gbhsweb.glenbrook225.org/gbs/science/phys/class/newtlaws/u2l2b.html [Accessed: 28th March 2014].
The second law is, “the relationship between an objects mass (m), its acceleration (a), and the applied force (f) is F= ma.” The heavier object requires more force to move an object, the same distance as light object. The equation gives us an exact relationship between Force, mass, and acceleration.
...e following subsections, only a few, which are used in introductory fluid mechanics, are mentioned.
... resultant speed and, by the definition of the tangent, to determine the angle of which the object is launched into the air.