Digital Computer Simulation of Period Oscillation - Term Paper Example

Name: Tutor: Task: Date: Digital Computer Simulation of Period Oscillation Introduction It is the responsibility of Aeronautical engineers to control and design stability in aircrafts. An airplane is said to be steady if the level at which it flies is in a straight course and whose control by the pilot does not require any attention. Such a plane, when a gust of wind disturbs it and is being control from a neutral point will try and stabilize itself without much influence by the pilot. Scientists explain that the complete dynamic response of a plane so as to stabilize itself is dependent on some factors which can be described by four differential equations. One of these equations can be described by the longitudinal response which often results into two dynamic modes. Vertical velocity and pitch rate which often shortens the oscillation and the forward height or speed which often causes the long period oscillation that is also known as phugoid can easily be presented by four sets of equations that are differential in nature. In the experiment that will be presented here, a simulation of a digital computer or an appropriate mathematical model will be analysed by observing short period oscillations of a plane. Whether a plane is stable or unstable is an aspect which can be explained in engineering terms in that when a pilot loses control of the plane, then he will consequently cause the plane to crash and as result cause accidents. Stability is therefore a very important aspect when handling issues concerning the designing and controlling of planes. Many accidents and incidences which involve planes has been caused as a result of planes becoming unstable in their course. Such accidents have even caused the lives of innocent people who might be straying innocently before the plane crashes on them. Objective This experiment is aimed at coming up and understanding how the features of short and long longitudinal oscillations of a model of a plane can be said to be in a position of instability, stability or neutrality. Theory Attitude and motion play a very vital role in determining the moments acting on a plane and these moments can either be propulsive or aerodynamic. The corresponding accelerations and the angular and linear velocity components are used to describe the motion which takes place when an aircraft moves. The dynamic aircraft variables with derivatives with respect to time determine the moments and forces which make the plane move. The different groups of primary variables are subject to the sensitive nature of the moments and forces. It is therefore true to say that low sensitivity of longitudinal forces which is responsible for the motion of lateral variables and it is also true that when this is reversed the high sensitivity of longitudinal forces can apply. When this is the case, an aspect known as decoupling occurs causing an accurate or approximate response that is dynamic and is responsible for reducing the dependence of some moments and forces. The following equation represents the short period of oscillation (SPO); This indicates that for any given aircraft, the parameters and that representing motion will vary from one plane to the other. This equation can alternatively be represented in the forms; and Where D represents differentiation with respect to time i.e., And, The amplitude of the aircraft during the SPO is given by the equation; For these equations, the following values will hold; For an oscillation that is damped, For an oscillation that is undamped, The Ratio of attenuation, The double amplitude time, The number of oscillations to half amplitude, Decrement Logarithmic notation, The Time to reach half amplitude, Number of oscillations to double amplitude, The lateral stability is defined as the stability which occurs around the longitudinal axis of a plane. Both wings of an aeroplane often give the same level of lift if the plane is moving in a straight flight where there is no slip at all. This can also be used to explain why velocity is an important aspect in determining the stability of planes. If instability occurred within the wings, the plane is said to roll over causing one of the wings to produce more lift than the other. If this condition causes the plane to roll over, one of the wings of the plane is said to increase its angle of attack while the other is decreasing it. After a short while, the difference in the lift between the two wings will be balanced. This will again bring stability in the plane. This condition can be explained in a situation when a plane tilts at a given angle. While one of the wings creates an angle of attack to increase, the other causes it to reduce and within a given time, this balance will be stabilized because the plane moves at a fast rate the two motions cancel one another. Planes which are light often display a greater form of stability in a lateral manner than other planes. As opposed to lateral stability, longitudinal stability is that form of stability which is centered on the lateral axis. This form of stability, which is also known as the pitch stability is dependent on the center of gravity of the plane. Pilots ought to understand this form of stability since it keeps the stability in the plane intact and thus a safer flight. When flying, the pilot needs to check the longitudinal stability because a negative stability would mean that it is impossible for the pilot to fly for more than four hours without concentrating so much on the control of the plane. Weather is one of the major determinants of longitudinal stability as opposed to lateral stability which is dependent on the position of the wings. Most planes however, often have a stabilizer which makes the wings to have stability in them which is often referred to as directional stability. The pictures in appendix 1 a, b and c indicate the different types of stability which airplanes can gain; ( a) This type of stability is obtained when there is a damp of oscillations over time. (b) This type occurs when oscillations remain constant over some time (c) This type occurs when oscillations grow smaller or larger with time without any speculation Modern planes especially those which are built by Airbus Company, are made in such a manner that they have a neutral longitudinal stability in its pitch. This means that when there is a change in the altitude, the pilot has to release the controls so that the plane will automatically change to the new altitudes. This effect is often created by the computers which are on-board and the pilot should be aware not to assume the fact that there is an actual instability within the system of the plane. Such instability is only temporary and the plane will return to the original angle of attack and speed after it has encountered a slight interference. The MATLAB software is of great use in computation of signal processing, matrix computation and generation of graphics as will be used in this experiment to come up with the simulations taking place within the plane. Apparatus description So as to come up with positive results during the experiment, a digital computer simulation was carried out. This involved the use of the SIMULIN model, which is represented in the appendix 2; With the aid of the MATLAB software, the parameters which were obtained were loaded into the simulation programme. The following table shows the figures which were computed by the MATLAB software; Flight condition1 Flight condition 2 Flight condition 3 m/s) 177 177 177 (kgm2) (Nm per rad/s) 0 (Nm per m/s) (Nm per rad/s) 0 So as to make the results to be more accurate and relevant, most of the data of the experiment were obtained from a computer printout. Data was printed out three times by using this manner, depending on the conditions in which the flights were exposed to. Test procedure The procedure followed during the experiment involved opening the computer and making sure that MATLAB software was installed and fully functional, entering the parameters from the results of the aeroplanes into the system, tabulating the figures and drawing a comparison in form of a graph out of it and finally coming up with a print out which would be act as an evidence for the procedures followed. Results Experimental results (see appendix 3a, b, c) Normal calculated results At flight condition 1-stable case; The values of chart and simulation will be given by; The logarithmic decrement denoted as , the period of damping oscillation represented as and the ratio of attenuation indicated as . It is therefore evident from the figure 2 given that; ., while = In this case; while, At flight condition 2-at a critically stable condition; The value of chart and simulation is given by; where and Further calculations yield that where At flight condition 3-when the plane is at unstable condition; The values for chart and simulation are given by; Damping oscillation period, , decrement of logarithms, and attenuation ratio to be Analysis After the experiment had been carried out, the results were summarised as follows; For the stable case; Parameter Td(secs) A δ N1/2 ζ t1/2 wd wn σ (/s) Value from Simulation 1.37 0.7209 0.67 1.1 0.1 1.507 4.5863 4.609 0.4609 and chart                   Value from Calculation 1.363 0.729 0.631 1.103 0.105 1.504 4.581 4.606 0.4816 For the critical stable case; Parameter Td(secs) A δ N1/2 ζ t1/2 wd wn σ (/s) Value from Simulation 1.36 1 0 infinity 0 infinity 4.62 4.62 0 and chart                   Value from Calculation 1.364 1 0 infinity 0 infinity 4.606 4.606 0 For the unstable case Parameter Td(secs) A δ N1/2 ζ t1/2 wd wn σ (/s) Value from Simulation 1.37 1.47 -0.78 1.09 -0.103 1.493 4.586 4.609 -0.475 and chart                   Value from Calculation 1.3699 1.38 -0.65 1.107 -0.124 1.459 4.641 4.606 -0.573 Accuracy level The results obtained from this experiment were not so much different from that in theory and in calculation. The accuracy factor can therefore be taken as +/-0.5mm. Conclusion The experiment stated was very important in understanding how flights are controlled and directed. By comparing the data presented, the values from the theoretical point of view could be compared with those from the practical view. This gave many pilots and military combats a chance to understand more about the stability of flight. Work cited: Pamadi, B. N. Performance, stability, dynamics, and control of airplanes. American Institute of Aeronautics and Astronautics, 2004 (1) 218-328 Appendix: 1 (a) 1 (b) 1(c) Appendix 2 Appendix 3a, A stable case; 3b For a critically stable case; 3c For an unstable case; Read More