Aircraft vectoring is a navigation service provided to aircraft by air traffic control. The controller decides on a particular airfield traffic pattern for the aircraft to fly, composed of specific legs or vectors. The aircraft then follows this pattern when the controller instructs the pilot to fly specific headings at appropriate times.
To better understand the science of propulsionit is necessary to use some mathematical ideas fromtag_hash_105_______________.Most people are introduced to vectors in high school or college,but for the elementary and middle school students, or the mathematically-challenged:
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There are many complex parts to vector analysis and we aren't going there.We are going to limit ourselves to the very basics.Vectors allow us to look at complex, multi-dimensional problemsas a simpler group of one-dimensional problems.We will be concerned mostly with definitionsThe words are a bit strange, but the ideas arevery powerful as you will see.If you want to find out a lot more about vectors you can downloadthis report onvector analysis.
Math and science were invented by humans to describe andunderstand the world around us.We live in a (at least) four-dimensional world governed bythe passing of time and three space dimensions; up and down,left and right, and back and forth.We observe that there are some quantities and processes inour world that depend on the direction in whichthey occur, and there are some quantities that do not dependon direction. For example, thevolumeof an object, the three-dimensional space that an object occupies,does not depend on direction.If we have a 5 cubic foot block of iron and we move it up and down andthen left and right, we still have a 5 cubic foot block of iron.On the other hand, thelocation,of an object does depend on direction.If we move the 5 cubic foot block 5 miles to thenorth, the resulting location is very different thanif we moved it 5 miles to the east.Mathematicians and scientists call a quantitywhich depends on direction a vector quantity. A quantitywhich does not depend on direction is called a scalar quantity.
Vector quantities have two characteristics, a magnitude and a direction.Scalar quantities have only a magnitude. Whencomparingtwo vector quantities of the same type, you have to compare boththe magnitude and the direction. For scalars, you only have tocomparethe magnitude. When doing any mathematical operation on a vector quantity(like adding, subtracting, multiplying ..) you have toconsiderboth the magnitude and the direction. This makes dealing with vectorquantities a little more complicated than scalars.
On the slide we list some of the physical quantities discussedin theBeginner's Guide to Aeronauticsand group them into either vector or scalar quantities. Of particularinterest, theforceswhich operate on a flying aircraft, theweight,thrust, andaerodynmaic forces, are allvector quantities. The resultingmotionof the aircraft in terms of displacement, velocity, andacceleration are also vector quantities.These quantities can be determined by application ofNewton's lawsfor vectors.The scalar quantities include most of thethermodynamic statevariables involved with the propulsion system, such as thedensity,pressure, andtemperature of the propellants.Theenergy,work,andentropyassociated with the engines are also scalar quantities.
Vectors have magnitude and direction, scalars only have magnitude. The fact that magnitude occurs for both scalars and vectors canlead to some confusion.There are some quantities, like speed, which havevery special definitions for scientists. By definition,speed is the scalar magnitude of a velocity vector. A cargoing down the road has a speed of 50 mph. Its velocityis 50 mph in the northeast direction. It can get veryconfusing when the terms are used interchangeably! Another exampleis mass and weight. Weight is a force which is a vectorand has a magnitude and direction. Mass is a scalar. Weight and mass are related to one another, but they are not the same quantity.`
While Newton's laws describe the resulting motion of asolid, there are special equations which describe the motionof fluids,gases and liquids.For any physical system, themass,momentum, andenergyof the system must be conserved. Mass and energyare scalar quantities, while momentum is a vectorquantity. This results in a coupled set of equations,called theNavier-Stokes equations,which describe how fluids behave when subjected to external forces.These equations are the fluid equivalent of Newton's laws of motionand are very difficult to solve and understand.A simplified version of the equations called theEuler equationscan be solved for some fluids problems.
A force may be thought of as a push or pull in a specific direction. A force is a vector quantity so a force has both a magnitude and a direction. When describing forces, we have to specify both the magnitude and the direction. This slide shows the forces that act on an airplane in flight.
Weight is a force that is always directed toward the center of the earth. The magnitude of the weight depends on the mass of all the airplane parts, plus the amount of fuel, plus any payload on board (people, baggage, freight, etc.). The weight is distributed throughout the airplane. But we can often think of it as collected and acting through a single point called the center of gravity. In flight, the airplane rotates about the center of gravity.
To overcome drag, airplanes use a propulsion system to generate a force called thrust. The direction of the thrust force depends on how the engines are attached to the aircraft. In the figure shown above, two turbine engines are located under the wings, parallel to the body, with thrust acting along the body centerline. On some aircraft, such as the Harrier, the thrust direction can be varied to help the airplane take off in a very short distance. The magnitude of the thrust depends on many factors associated with the propulsion system including the type of engine, the number of engines, and the throttle setting.
The motion of the airplane through the air depends on the relative strength and direction of the forces shown above. If the forces are balanced, the aircraft cruises at constant velocity. If the forces are unbalanced, the aircraft accelerates in the direction of the largest force.
Note that the job of the engine is just to overcome the drag of the airplane, not to lift the airplane. A 1 million pound airliner has 4 engines that produce a grand total of 200,000 of thrust. The wings are doing the lifting, not the engines. In fact, there are some aircraft, called gliders that have no engines at all, but fly just fine. Some external source of power has to be applied to initiate the motion necessary for the wings to produce lift. But during flight, the weight is opposed by both lift and drag. Paper airplanes are the most obvious example, but there are many kinds of gliders. Some gliders are piloted and are towed aloft by a powered aircraft, then cut free to glide for long distances before landing. During reentry and landing, the Space Shuttle is a glider; the rocket engines are used only to loft the Shuttle into space.
A vector airplane can be significantly affected by a side wind, as it can cause the aircraft to drift off course and make it more difficult for the pilot to maintain control. This is because the wind hitting the side of the aircraft creates an imbalance in the forces acting on the plane, making it harder to maintain a stable flight path.
Pilots of vector airplanes use various techniques to compensate for a side wind, such as adjusting the angle of the aircraft's wings, using rudder controls to steer the plane, and making adjustments to the throttle to maintain a steady airspeed.
Yes, flying a vector airplane in a side wind is generally more challenging for pilots. It requires a higher level of skill and experience to maintain control and keep the aircraft on course when faced with strong winds.
Pilots prepare for flying a vector airplane in windy conditions by checking weather reports and wind forecasts before the flight, ensuring that the aircraft is properly maintained and balanced, and practicing techniques for compensating for a side wind during training flights.
A recent question about the resultant velocity of an airplane illustrates different ways to make a diagram showing the bearings of air velocity and wind velocity, and to work out angles without getting too dizzy.
Numerically, I can figure this out, but I am having difficulty imagining what this looks like as a picture. The only examples I've done thus far are ones where the wind was pointing either due north or at an angle smaller than $90^{\circ}$, and so I'm having trouble drawing a picture of the scenario - I'm not sure how the wind should displace the path of the airplane. Could somebody please draw me a picture? That's really all I need here to figure out the rest on my own.
The vector with length $130~\text{mph}$ at an angle of $45^\circ$ north of east represents the trajectory of the airplane in the absence of wind. The vector with magnitude $25~\text{mph}$ in the easterly direction represents the velocity of the wind. The vector $v$ represents the resultant velocity of the airplane.
Edit with picture:I've modified your picture a bit. Red is v. dist is the length of blue and green, equal to v dot normal. Blue is normal*dist. Green is the same vector as blue, they're just plotted in different places. To find planar_xyz, start from point and subtract the green vector.
A complete answer would need an extra parameter. Say, you supply also the vector that denotes the x-axis on your plane.So we have vectors n and x. Assume they're normalized.
The problem of finding a \(H_{\infty } -\) observer of the state vector of a linear continuous non-stationary dynamic system with finite time of functioning is considered. It is assumed that a mathematical model of a closed-loop linear continuous deterministic dynamic system with an optimal linear regulator, found as a result of minimization of the quadratic quality criterion, is known. We find a solution to the problem of state vector coordinates estimation in the presence of limited external influences and disturbances in a linear model of the measuring system. As an example, the equations of motion of an L_1011-type airplane are used. 0852c4b9a8
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