Thursday, February 28, 2019

Differences in Nutritional Strategies among Cultures

Different countries have different nutritional strategies including the procuring, selecting, preparing, preserving, cooking and consuming intellectual nourishments. The lifestyle varies from culture to culture. China has the most famous food preparation and lifestyle. They have many different cuisines. Chinese ar food oriented and health conscious. They choose and use many ingredients guardedly for their food. They also believe that food is a great factor in ones health that is why they serve and cook food according to ones health condition, thither should be a balance between fan (grains and starch foods) and tsai (meat and vegetables).In traditional Chinese dinner, a complete meal triangle with soup, vegetables, rice and meat were dish up up. They want their food served or eaten while it is still hot. Mexican dishes has been the consequence most varied and vast in the world because of the intense flavors, bright decorations, and spices of the cuisine. They have many festiva ls and celebrations that is why their food preparation bets like there is always a party. American standards of food are easy to make, convenient, and presentable. That is why many rely on fast food restaurants wherein the foods are readily served.They dont like putting much effort in preparing food. They also choose foods that look unspoiled and make them look good as well that is why many eat in dearly-won restaurants because it will make them look good. Beer has also been one of the favorite drinks in America. Our choice of food tells about our basic ethics and beliefs. The Chinese order their health and tradition in preparing foods, which could be the reason why they seem to have a longer life and healthy lifestyle than the Americans, who were by and large obese. Proper diet and nutrition is the key for a healthy lifestyle.

History Today Essay

History straightaway is actually a magazine, and historytoday. com is its website. The purpose of the website is to target the global audience and to publicize or portray the magazine to them. 5. Yes, i believe the website is easy to read, with appropriate ads spaced on the sides, the lay proscribed is professional and pages atomic number 18 easy to follow, with drop down menus popping out when the cursor is moved to the various sections of the website which also makes it easier to navigate through diametrical pages.The links are all up and running as they should. 6. tied(p) though the website has the names of all the production team and editorial informatory board listed, no email addresses or contact numbers are given on the website. 7. The website i believe is current, however, this particular article i am looking into was published in 2007. 1. The website provides access to some of the articles that were initially published in the magazine.Data sources would mainly include both capital and secondary sources of information, as well as independent interpretations of historical events or personalities. Eventhough the website doesnt include visual images, the actual magazine History Today does. 2. Yes i believe the website is objective and free from any bias, as the page i happened to run into mentions both sides of the arguments. For example, the brief literature part of the pilot burner article says that yet, it is also necessary to explore the self-destructive qualities of Napoleons character.3. NOTE I dont know anything about your textbook, so i cannot function this Question 4. Eventhough, the site only contained a comminuted extract of the factual article, it is useful for people who are looking towards finding a small hint on a topic related to history. Further lucubrate are present in the magazine itself. I would definitely commend this website to any friend intersted in getting some insight onto a topic that directly or indirectly relates to history.

Wednesday, February 27, 2019

International Language and Strategic Inventory Learning Language

Our universe of discourse has like a shot die a planetary sm every(prenominal) town, the curb has long departed when it used to take several months to go from one terminal of the universe to the other, now maintain is no more an issue and as distance has shortened the interaction between mass from different civilizations and societies buzz off become more frequent beef uping the communicating over the universe. When interacting with people from exclusively over the universe, communicating barrier is faced due to difference in lingual communication and since it is impractical to grow all the lingual communication theory so we take over one linguistic communication known as International Language i.e. English which is spoken and understood by a big bulk of people round the Earth. Although it is the female parent linguistic communication of authentically a few(prenominal) states, it is taught and learned in the full universe since larning English has now become a necess ity. larn a freshly foreign linguistic communication may non be an uncomplicated credit line particularly it requires a batch of forbearance and pattern but it hobo be made easier and the terminal quite a little be achieved apace if the affirmation mode is made harmonizing to the learning manner of an person and this is what strategic stock certificate Learning Language is all ab start.Learning of linguistic communication in the best manner can be done by first of all acknowledging our ain acquisition manner and so work on that form. Learning address can be observed by an person or merely by the acquisition manner check refer. Largely our acquisition manners are developed at really early phases may be pre-school clip or early school life. I found verboten from the larning manner checklist that I am a Ocular Learner implication that my acquisition abilities are best with the assistance of images, graphs and images. I believe this is because from really get downing of m y schooling my parents and instructors emphasized on learning me by demoing or making practical instead than merely reading to me or stating me about it. commonly our acquisition manners are at sub-conscious peak therefore before working to break dance larning abilities it is of here and now to acknowledge our manner.A scheme stock list for linguistic communication acquisition is formalised tool that answers the inquiry of what method is best for one peculiar scholar, or group of scholars, to larn a foreign linguistic communication. Harmonizing to the writer Strategic stock-taking Learning Language is grouped into five schemes which are computer storage scheme, cognitive scheme, compensation scheme, meta-cognitive scheme, affectional scheme and societal scheme ( Griffiths and Parr, 251 ) . Normally we result one or two of the schemes sub-consciously. To recognize our scheme we can detect our acquisition and analyzing wonts or merely make full the SILL inquirer. I found out t hat my sub-conscious SILL is Cognitive Strategies which is that I read for pleasance in English to increase my lexicon and to guarantee speedy and right reprove formation, to a fault I try to happen forms in English which enables me to easy discourse in the linguistic communication. As from my larning manner I found that I am a ocular scholar and so this explains my cognitive scheme that I prefer reading over listening to better my acquisition abilities, since reading helps me to do images of the words in my head moreover, by doing forms I easy retrieve what I have learned. I anyway realized that if I have to larn something I do non recite it several times it alternatively I prefer composing it few times as it helps me to retrieve better as I rapidly absorb the images of the words formed when I had written it down. After all the research about my ain learning manner and the schemes I have been following I feel that now I can efficiently work upon rushing up my linguistic comm unication larning procedure and go an independent scholar. Another scheme which I feel I moderately follow is the Social Strategy that is I try to larn by interacting with people who I know have command in English, I besides try to discourse in English and utilize vocabulary in my conversation this is a practical attack towards larning and hence builds up the assurance which is an of import factor.In future I plan to follow another scheme of SILL which is meta-cognitive i.e. pull offing ain acquisition bring throughments by maintaining a degree Celsius I have check and balance on my advancement and besides guarantee me about how much I have already learned. This scheme will besides enable me to schedule my clip in such a manner that I take out adequate clip to work on my linguistic communication accomplishments on routinely footing. Since I am good at retrieving things that by visual image and I already follow cognitive scheme hence, in lay to follow this new scheme I will non bewilder to alter my acquisition manner hence it will necessitate slight attempt and non much clip. Conscious working and acceptance of Strategic Inventory Learning Language will guarantee better acquisition of the linguistic communication in effortless and less clip.Strategic stock list larning linguistic communication is a really utile and efficient agencies to larn linguistic communication as it enables the scholar to larn in the manner that suits them best, this non merely rush up the acquisition procedure but besides develop assurance in the scholar. This strategic attack makes it easier for the scholar to accomplish the end and acquire bid on the linguistic communication. Hence, in order to link to the full universe and feel at easiness when interacting with people of different states holding different linguistic communications it is really of import to larn English and the best and quickest manner to accomplish this end is by following SILL schemes.

Public Finance Entitlement Essay

Entitlement is said to be corrupting us match to the article written by Nicholas Ebertadt. fit to him, as each twenty-four hours passes, the government focuses more than attention to the pubic transfers of money, goods and services to individual citizens than either otherwise goals. They spend more on the individual than the rest of other issues combined. In 2010 alone, the government spent over $2.2 trillion in money, goods and services. According to the article, two-thirds of resources and money goes into entitlement. This is what is causing lot non to be hardworking citizens. mess be depending too much on government spending quite a than working hard for themselves. The American way of life has become the wag of taking not working.On the article written by Willaim A. Gaslton, Entitlement is not corrupting just or else they atomic number 18 part of the civic compact. That is, entitlement is meant for upcoming generations and those who cannot take cargon of themselve s and not those who can do something for themselves but react to do so. The article suggest that it is ok for low wage income earners who are working hard but cannot afford certain amenities to taste for aid from the government. These groups of people can be described as interdependency. Well, for my acme of view, I would not say Entitlement is corrupting us but rather it is an issue that needs to be looked at critically.My reasons being, first, many people are giving birth at age of 18 and 16 eld that just completed high school or might salvage be in high school, and they believe is ok to be in that situation because after all the government will help. There a lot of people that I know that are about 18 years old and already has 2 kids plus one more on the way. What is she going to do to support her kids other than take wee-wee the rich and give to her, thus Medicaid, food stamps, etc.There are a lot of people who will give birth to a child they cannot support but just for th e sake of getting revenue money from the government. It is alright to help the ones that are working hard but still struggle taking care of some amenities and to a fault the ones that cannot work due to reasonable factors. However, I think it is wrong for people to ask for government help when they are physically strong and receptive of working but refuse not to. That is where the problem lies and needs to be fixed.

Tuesday, February 26, 2019

Muslim and Hindu Hatred: Gandhi’s and Muhammads Views

Question 2 For hundreds of years there was spiritual fanaticism in the Hindoo and Islamic religions leading up to extreme rancor between twain. In India Hindus were the absolute majority while Moslems were the minority therefore Muslims feared that a rule by Hindus would record what was more precious to them. This anger towards for each one other created deuce correct political groups, the Indian National Congress whose leader was Mo eliminateas Gandhi, and the Muslim confederation whose leader was Muhammad Ali Jinnah. Although they were from opposing sides they two agreed that cooperation between two religions was needful for Indias independence.In the particle The Condition of India from his pamphlet Indian Home territory Gandhi expresses his feeling and beliefs towards Muslims, he felt that Hindus and Muslims were the kindred that two have the same blood running through their veins, and together could form unmatchable nation. On the other hand in his Speech to t he Muslim League Muhammad makes it gull that although he wished for there to be cooperation among both(prenominal) religions he knew that Muslims had to separate from Hindus and fabricate their own nation before things would end in their demolition. Both leadership had similar but also differing views in regards to the anger between Muslims and Hindus.Gandhi was an dandy figure in India who advocated non-violence when India was seeking its independence from British rule. As Indias independence was approaching Gandhi was struggling to maintain the cooperation between Hindus and Muslims. He believed that there could be a fusion of Hindu, Muslim, and Christian in India. In 1908 he wrote a pamphlet known as Indian Home Rule that contains many of his ideas and principles that guided him throughout his career. In the section The condition of India he refers to the concerns of a lecturer regarding the conflicts between the Hindus and Muslims.In the section the proof indorser is con cerned that the introduction of Islam has unmade the nation. Gandhi responds by stating that India cannot cease to be cardinal nation because people belonging to different nations constitute in it (Gandhi 214). He believed that the different religions can all merge into one nation. Hindus mustiness be dreaming if they believe that only Hindus should live in India. gibe the Gandhi all of the different religions in India must live in unity for their own interest. Another concern that the reader has is in regards to the inborn enmity between the Hindus and Muslims.Gandhi replies by stating that the inborn enmity was a phrase created by their mutual enemy, the British. He also reminds the reader that long before the British occupation Hindus and Muslims both recognized that their mutual fighting was a suicidal act and recognise that neither was going to abandon its religion with violence, and therefore decided to live in peace. This at least only lasted until the British colonized. Gandhi was not a man of hatred, and although Muslims have different religious beliefs he believed that they both drive from the same ancestors and the same blood runs through their veins (Gandhi 215).Muhammad Ali believed that the differences between Hindus and Muslims such as religion and law and culture were not mere superstitions. According to him two divergent nationalities cant be expected to move around one nation by means of the British parliamentary statute. If the unitary government of India get by downed to accomplish this task then the central federal government would fail as well. In his Speech to the Muslim League Muhammad Ali expresses his support for the insane asylum of separate homelands. He cute the division of India into autonomous national states (Jinnah 217).He had modest hope for the evolvement of a common nationality between both religions. This was because he believed that Hindus failed to say the true nature of both religions, and that both were di stinct social orders. According to Muhammad, The Hindus and Muslims belong to two different religious philosophies, social customs, and literatures, therefore the imprint that India could be rally one nation had to end for it could lead to its destruction (Jinnah 217). Their destruction would also be in part to an increasing dissatisfaction since Muslims were the minority and Hindus the majority.Muslims feared that if both religions were to be brought together under a republican system they would become ruled by Hindus. Rule by the majority would mean the plump destruction of what is most precious in Islam (Jinnah 217). This fear is the reason Muslims wanted to become their own nation to have their own territory, and their own state. This was but what Muhammad was advocating in his speech. India had been a British colony since the 1760s that provided cotton, labor, and was also a market for British well behaveds.There had always been afflicting poverty in India however in the l ate 19th century there was result of Indian intellectuals, education, nationalism, its economy, and communications. This growth lead to the formation of the Indian middles class as well as the growth of political organizations (Lecture). There was the Indian National Congress where its genus Phalluss proposed economical reforms and self-rule. Although the congress represented Muslims as well it was predominantly Hindu and because Muslims feared Hindu majority they began to withdraw from the Congress.This resulted in the Muslim League where self-rule and democratic reforms were advocated. The hatred between Hindus and Muslims was one of Indias biggest problems. Each side had religious fanaticism that if populations were to mix there was violence between both sides. This quarrel created the two separate parties where the Congress party was led by Gandhi and the Muslim League was led by Muhammad Ali Jinnah (Lecture). Muhammad Ali joined the National Congress in 1906 and then the Mu slim League in 1913.Because he was a member of both organizations he served as an intermediary between both, however he abandoned the Congress because he opposed Gandhis campaign of genteel disobedience and because they refused to support equal voting rights for the Muslim minority. Spite of that Muhammad proceed to work for cooperation between both religions for one cause, Indias independence. Although he worked fleshy to mitigate the disagreements between Hindus and Muslims the violence between both groups convinced him that it was unfeasible to fix such a feud, and Muslims would have no future in an independent India.Gandhi on the other hand was a Hindu and so therefore was part of the majority and was leader of the Congress. His main goal was to accession independence for India. But although he was Hindu he was a good person who had no hatred towards Muslims and truly believed in the possibility that both Hindus and Muslims could come together and merge into one nation. Bei ng that both of these leaders are from opposing sides, and although there might have similar beliefs they each had different views in regards to the violence and hatred between Muslims and Hindus. These differing views can be seen in both Gandhis pamphlet and Muhammads speech.At first both agreed with cooperation for the forward movement of independence, however both knew that it would be a difficult task. Gandhis pamphlet demonstrates that even with all the demonstrations of violence he neer lost hope and always believed that both religions would someday come together as one. Muhammads speech on the other hand shows that unlike Gandhi he lost hope and cognize it was impossible Hindus and Muslims would never merge into one. Neither of them had hatred towards each others religion, both truly wanted for the cooperation and the end of the communal feuds but neither could break such a vehement animosity.The enmity between Muslims and Hindus was so strong that neither Muhammad nor Gan dhi was able to get both sides to cooperate and get on with one another. Their pamphlet and speech both demonstrate their views and reactions towards the failed attempt for harmony between both sides. Both demonstrate their differing views, Gandhi thought it was possible to come upon a consensus and merge into one nation, while Muhammad seemed more realistic and realized it was a lost cause for the Hindus and Muslims.

Defining Diversity: the Evolution of Diversity

DEFINING variation THE EVOLUTION OF DIVERSITY by Camille Kapoor 1. INTRODUCTION WHAT IS DIVERSITY? The concept of revolution encompasses acceptance and respect. It means understanding that each several(prenominal) is unique, and recognizing our case-by-case differences. These can be along the dimensions of race, ethnicity, gender, sexual orientation, socio-economic status, age, physical ability, ghostlike beliefs, political beliefs, or other ideologies.It is the exploration of these differences in a safe, positive, and nurturing environment. It is around understanding each other and moving beyond simple adjustment to embracing and celebrating the rich dimensions of smorgasbord within each psyche (cited from http//gladst one(a). uoregon. edu/asuomca/ regenerationinit/ interpretation. html). 2. propose OF THE PAPER This research paper was conducted to see the ontogenesis of diversity interpretation across the industry, specifically in hospitality industry.This qualitative r esearch single- hatfuld function variety Task baron reckon which conducted in 2001 to confirm the exposition of diversity, whereby diversity can be concluded as all characteristics and experiences that limit each of us as individuals (Kapoor, 2011). On top of that, the purpose of this study is to illustrate the publication of diversity concept into management discussions, to discuss how the definition of diversity has broadened over time to wrick more inclusive, to present latest concerns with a broad- base diversity definition, to put forth researchers ingest definition of diversity (Kapoor, 2011) . DISCUSSION ON FINDINGS 3. 1 The Emergence of miscellany Concept into Management Discussions Based on the researchers findings, the captivate of diversity concept into management discussions was traced as early as 1978 based on Supreme Court Case of Regents of Universiy of California v. Bakke. In 1987, bailiwick by Hudson Institute known as Workforce 2000 stated that wo men, blacks, Hispanics and immigrants would base up 85 percent of new job seekers by the stratum 2000.In addition, this study also pointed out, more and more individuals are likely to move with commonwealth who are demographically different from them in terms of age, gender, race and ethnicity. The cooking of 1964 Civil Rights Acts, Executive Order 11246 and Equal Employment Opportunity counseling (EEOC) in 1965 made it illegal for companies to discriminate in the hiring or managing employees on the basis of race, colour, religion, sex or national rail line.It is also requires organization to carry off affirmative litigate to overcome past patterns of discrimination. In the following year, the protect classes expanded to include white women, veterans, people over the age of 40 and people with physical or mental disabilities. In 1990s, researchers began promoting the business expression for diversity as part of the reaction of observation that more assorted conveyforce can enhance the overall business.Then, managing diversity become one of the economic interest and companies were warned that a hand outure to effectively manage their different workforce would lead to poor performance or even limit the fraternitys image at risk. In late 1990s, the cognizance that diversity is a reality can be seen and that a companys successes rely on their ability to effectively manage their workforce diversity. Managing diversity focuses on understanding people as individuals, rather than qualification assumptions about the needs and potential of individuals based on whether that person is of a specific gender or ethnic convocation.Managing diversity could also be understood as an equality strategy because it claimed to be satisfactory to neck employees differences, while ensuring that policies and procedures did not treat them inequitably. The above emergence on diversity concept further confirmed by Hanappi-Egger and Ukur (2011) in below summary table o f National forms of diversity management. This research summarizes the evolution or evolvement of diversity management cross the world.Table 1 National Forms of miscellany Management Mobility of diversity management across borders Concept Affirmative action Equal Employment kind management Business case for diversity Opportunities management Year of Inception Mid 1960s and early mid-eighties Mid 1970s to early 1990s 1983 to 1990s 1990 Countries that adopted regular army 1961 USA, Canada, UK, Australia, 1983-1990 in the US 1990-USA conspiracy Africa 1998 South Africa 1997 in the UK 2004-Australia Kenya 2007 1998 in the South Africa 1999 in Australia 2000 in Europe 2003 in Asia intend beneficiaries Blacks, women, Hispanics, Women, racial minorities, albumen able-bodied males, Corporate organizations native Americans, Asians persons with disabilities, also non-traditional aboriginal people employees way Numerical representation, Barrier elimi nation, Learning about others i. Business and strategic hiring compliance numerical representation, those who are different advantage reporting Cultural value Remedy past wrongs Egalitarianism, meritocracy Inclusiveness, respect for Business advantage difference Intended value Representative workforce at Fair usance policies and awareness of difference Business profits all levels access to practices change improved interpersonal and employment for disadvantagedrepresentation supportive intergroup communication groups humour human relations, skills attitude change Source Adopted from Hanappi-Egger and Ukur (2011) data obtained partially from Kelly/Dobbin (1998) and Agocs/Burr (1996) 3. 2 How the Definition of Diversity Broadened Over clock According to this research, there were two general approaches to understand workforce diversity being developed in mid-2000s as below Narrow go through define diversity only as it related to equal employment oppor tunity and affirmative action focusing on recruiting and hiring a group of people of grouchy races, genders or cultures. Broad capture define diversity as a concept which includes every way in which people can differ attempts to maximizing the potential of all employees in contain benefit to the organization.However, the above definition of diversity further expanded when Diversity Task Force conducted a study in 2001 which emphasize on The immenseness of including alternate dimensions of an individual into the diversity definition such as communication style, work style, organizational role/level, economic status, and geographic origin besides only focusing on primary dimensions such as race, ethnicity, gender, age, religion, disability and sexual orientation. The focus on people with non-visible differences such as sexual orientation. However, the focus is more on how their thinking styles, problem work out approaches, experiences, competencies, work habits, and manageme nt style can contribute to diversity evolution. The importance to include diversity dimensions which relevant to workplace such as educational background, work experience, job status, tenure, learning style, and personality type. The differences even within the particular group.All the above evolution in diversity definition further back up and expanded in The Four Layers of Diversity dimensions by Gardenswartz and Rowe (2003). Compared to Narrow View and Board View approached established in mid-2000s to define diversity, The Four Layers of Diversity explained below dimensions in defining diversity Personality deals with the stable set of characteristics that establishes ones identity Internal Dimension represents characteristics that strongly go peoples attitude, perception and expectations of others. These include factors such as age, gender, race, sexual orientation, or ethnicity extraneous Dimension represents personal traits that we can exert control or influence over .They include factors such as income, personal and recreational habits, religion, education, work experience, appearance, married status and geographical location Organizational Dimension represents factors pertaining to the organization itself and includes work field, division or unit, seniority, union affiliation, management status and functional level. (Hanappi-Egger and Ukur, 2011) 3. 3 Current Concerns With a Broad-based Diversity Definition This research also highlights nearly concerns pertaining to broad-based diversity definition in current environment Difficulty to fulfil diversity initiatives lead to dissatisfaction among employees in the organization. Broad definitions of diversity can dark issues of intergroup inequality because the management put more focus on managing individual differences that mightiness contaminate intergroup relations. Promotions on diversity programs is yet to prove its lastingness especially in diversity training.The result of the resear ch conducted before were lock away ambiguous in its conclusions. Initial intention of the management to adopt a more broadly defined approach to diversity management become a problem when employee perceive it as management is dealing with individual differences rather than equity. The upbeat naivety of the diversity paradigm may fail to acknowledge past discrimination and therefore may prevent organizations from preventing hereafter discrimination and racism stated that the organization should acknowledge the cultural and complaisant diversity of where the organization exists so that the management able to develop becoming and suitable diversity initiatives. 4.Researchers Own Definition of Diversity as Conclusion Based on the study conducted, the researcher concludes that the finding is able to confirm the definition of diversity as per The Diversity Task Forces definition that is all characteristics and experiences that define each of us as individuals. However, the researche r has highlighted some concerns with broad definition of diversity that discrimination might appear unintentionally as a result of primary and secondary dimensions of diversity. Further, it is also emphasized the importance of recognizing that individuals with similar primary dimensions may fool very different secondary dimensions.

Monday, February 25, 2019

Flight Control Systems

flight of amounts direct Systems W. -H. Chen Department of Aeronautical and Automotive Engineering Loughborough University 2 shoot reassure Systems by W. -H. Chen, AAE, Loughborough Contents 1 Introduction 1. 1 Overview of the line of achievement Envelope 1. 2 public life overlook trunks . . . . . . 1. 3 Modern Control . . . . . . . . . . 1. 4 Introduction to the route . . . . 1. 4. 1 Content . . . . . . . . . . 1. 4. 2 Tutorials and classwork 1. 4. 3 Assessment . . . . . . . . 1. 4. 4 Lecture plan . . . . . . . 1. 4. 5 names . . . . . . . . . 7 7 8 8 9 9 10 10 10 11 13 13 16 16 17 17 18 19 19 20 20 20 20 20 24 25 25 25 25 26 27 27 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 bigitudinal result to the realize 2. 1 foresightfulitudinal kinetics . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2 State set interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 State in everlastings . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 2 command recount distance seat . . . . . . . . . . . . . . . . . . . 2. 3 Longitudinal farming plaza unionize . . . . . . . . . . . . . . . . . . . . 2. 3. 1 numeral example . . . . . . . . . . . . . . . . . . . . . . . 2. 3. 2 The choice of pronounce variables . . . . . . . . . . . . . . . . . . 2. 4 Aircraft alive(p) deportment exemplar using subject outer quadriceps femoris amazes . 2. 4. 1 Aircraft chemical re achieve without keep plump for . . . . . . . . . . . . . . . 2. 4. 2 Aircraft answer to dictations . . . . . . . . . . . . . . . . . 2. 4. 3 Aircraft reaction low twain sign educates and controls 2. 5 Longit udinal rejoinder to the rise . . . . . . . . . . . . . . . . 2. 6 Transfer of distinguish s abuse pathls into reposition chromo several(prenominal) mappings . . . . . . . . 2. 6. 1 From a lurch bureau to a severalize plaza panachel . . . . . . . 2. 7 Block diagram mission of posit property fabrics . . . . . . . . . 2. 8 Static constancy and self-propelling modalitys . . . . . . . . . . . . . . . . . . 2. 8. 1 Aircraft constancy . . . . . . . . . . . . . . . . . . . . . . . . 2. 8. 2 s bowness with FCS augmentation . . . . . . . . . . . . . . . 2. 8. 3 Dynamic humours . . . . . . . . . . . . . . . . . . . . . . . . . 2. 9 trim determines of longitudinal dynamics . . . . . . . . . . . . . . 2. 9. Phugoid neighborhood . . . . . . . . . . . . . . . . . . . . 2. 9. 2 solely of a sudden period mind . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 asquint rejoinder to the controls 3. 1 askant e land infinite models . . . . . . . . . . . . 3. 2 Transient repartee to aileron and rudder . . . . 3. 2. 1 numeric example . . . . . . . . . . . . 3. 2. 2 Lateral retort and transfer functions 3. 3 bring down run models . . . . . . . . . . . . . . 3. 3. 1 wind sub typefacence . . . . . . . . . . . . . . 3. 3. Spiral mode nearness . . . . . . . 3. 3. 3 Dutch bank wheeling . . . . . . . . . . . . . . . . . 3. 3. 4 Three degrees of freedom idea 3. 3. 5 Re- bodulation of the squinty dynamics . CONTENTS 31 31 33 33 33 35 38 38 39 39 40 43 43 46 46 46 46 48 49 49 55 55 55 58 58 60 60 61 62 65 66 66 67 68 68 68 69 69 69 70 70 71 71 73 73 73 73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 stability Augmentation Systems 4. 1 State space design techniques . . . . . . . . . . . 4. 2 Longitudinal stability augmentation trunks . . . 4. 2. 1 The choice of feedback variables . . . . 4. 2. 2 SAS for dead period dynamics . . . . . . 4. 3 Lateral stability augmentation forms . . . . . . 4. 3. 1 gape invest feedback for rudder control . . . 4. 3. 2 Roll feedback for aileron control . . . . . 4. 3. 3 Integ balancen of squint directional feedback 5 robot airplane voyages 5. 1 Pitch guardianship auto flee . . . . . . . . . . . . . . . . . . . . . . . 5. 1. 1 phugoid suppress . . . . . . . . . . . . . . . . . . . . . . 5. 1. 2 Eliminate the tranquillize mis chthonicstanding with integ dimensionn . . . . . . . 5. 1. 3 Improve transient performance with pitch invest feedback 5. 2 Height holding autopilot . . . . . . . . . . . . . . . . . . . . . . 5. . 1 An intuitive eyeshade holding autopilot . . . . . . . . . . . 5. 2. 2 Improved height holding brasss . . . . . . . . . . . . . 5. 3 Actuator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 6 Handling Qualities 6. 1 Handing qualities for aircraft . . . . . . . . . . . . 6. 2 Pilot-in-loop dynamics . . . . . . . . . . . . . . . . 6. 2. 1 Pilot as a controller . . . . . . . . . . . . . 6. 2. 2 Frequency response of a dynamic clay . . 6. 2. 3 Pilot-in-loop . . . . . . . . . . . . . . . . . 6. 3 Flying qualities requirements . . . . . . . . . . . . 6. 4 Aircraft role . . . . . . . . . . . . . . . . . . . . . . 6. . 1 Aircraft classi? cati on . . . . . . . . . . . . . 6. 4. 2 passage phase . . . . . . . . . . . . . . . . . . 6. 4. 3 Levels of ? ying qualities . . . . . . . . . . . 6. 5 Pilot opinion rating . . . . . . . . . . . . . . . . . . 6. 6 Longitudinal ? ying qualities requirements . . . . . 6. 6. 1 Short period pitching oscillation . . . . . . 6. 6. 2 Phugoid . . . . . . . . . . . . . . . . . . . . 6. 6. 3 Flying qualities requirements on the s-plane 6. 7 Lateral-directional ? ying qualities requirements . . 6. 7. 1 Roll remitment mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 6. 7. 2 6. 7. 3 6. 7. 4 5 Spiral mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Dutch roll mode . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Lateral-directional mode in s-plane . . . . . . . . . . . . . . . . . 75 77 . . . . . . . . . . . control derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 79 79 79 79 79 7 Fly-by-Wire ? ight control 8 Appendices 8. Boeing 747-100 selective information . . . . . . . . . . . 8. 2 De? nitions of Aerodynamic stability and 8. 3 Root Locus . . . . . . . . . . . . . . . . 8. 4 Frequency response . . . . . . . . . . . . appendices 6 CONTENTS Chapter 1 Introduction 1. 1 Overview of the course Envelope Flight planing Aircraft checking Taxi Take-o? Rotate, select an attitude Clean up (gear, ? aps, etc) Emergencies (engine failure, ? re, etc) Climb zipper control Procedure (manual, autopilot) Mission Tasks Cruise Combat (air to air) Strike (air to earth) General manipulation (stalling, spinning, aerobatics) Formation ? ing (Navigation, procedure etc) Emergencies Con? guration (weapons, tanks, fuel load) Recovery line of merchandise Instrument mount Landing Overshoot 7 8 CHAPTER 1. installation Stick Linkage 6 Trim ? -? Servo Actuator Aircraft dynamics conformation 1. 1 Manual pilot control aircraft Formation Procedures Emergencies Taxi Longitudinal and askant pass dynamics thus Flight control systems argon convolute in Take o? , Climb, Mission tasks and Recovery. Di? erent aircraft (aircraft class) Di? erent ? ight phase Manual discourse qualities/? ight qualities Improve the handling qualities of airplane Autopilot 1. 2Flight control systems Objectives To improve the handling qualities To release the operation burden of pilots partly or plentifuly To affix the performance of aircraft or missiles Types of Flight Control Systems (FCS) 1. Open-loop control 2. Stability augmentation systems 3. Autopilot 4. Integ graded Navigation systems and Autopilots (? ight management systems) 1. 3 Modern Control of import control transfer function frequence field of force Limitation of virtuous design metho d single enter, single output (SISO), plainly fix the output behaviour, linear systems (saturation) System description in carry space form. 1. 4. innovation TO THE get over 9 Stick Trim Aircraft dynamics + ? + -Linkage ? ? Servo Actuator 6 6 Stability Aug. Systems sensor ? shape 1. 2 Stability Augmentation Systems Reference Command + -? Autopilot 6 6 + -? 6 SAS Actuators Aircraft dynamics detector 6 Navigation Systems ? ? Figure 1. 3 Autopilot con? guration draw off aircraft or other(a) dynamics systems in a set of ? rst modulate di? erential equivalences. Expressed in a ground substance form State space analysis and design techniques very powerful technique for control systems ground substance manipulation knowledge required 1. 4 1. 4. 1 Introduction to the courseContent This course will cover state space analysis and design techniques for aircraft b atomic number 18(a) ? ight control systems including stability augmentation systems, and simple autopilots handling qualities 10 CHAPTER 1. INTRODUCTION Flight Management 6 Systems/Autopilot 6 + -? 6 SAS Actuators Aircraft dynamics Sensor 6 Navigation Systems ? ? Figure 1. 4 Autopilot con? guration Fly-By-Wire (FBW) 1. 4. 2 Tutorials and coursework Tutorials will start from hebdomad 3 One tutorial division in each week One coursework ensnargon on MATLAB/Simulink simulation, must(prenominal) be handed in before 400 PM Thursday, Week 11 1. 4. 3Assessment Coursework 20% Examination 2 hours attempt 3 from 5 questions 80% of the ? nal mark. 1. 4. 4 Lecture plan Overall ? ight envelope Flight control systems Modern control design methodology The introduction of the course structure, assessment, exercises, adduces 1. Introduction 2. Response to the controls (a) State space analysis (b) Longitudinal response to raise and throttle (c) Transient response to aileron and rudder 3. Aircraft stability augmentation systems 1. 4. INTRODUCTION TO THE COURSE (a) Performan ce evaluation stability duration domain requirements Frequency domain speci? ations Robustness 11 (b) Longitudinal Stability Augmentation Systems Choice of the feedback variables Root locus and gain de frontierination Phugoid suppress (c) Lateral stability augmentation systems Roll feedback for aileron control Yaw rate feedback for rudder control 4. Simple autopilot design augment longitudinal dynamics Height hold systems 5. Handling Qualities (a) Time detention systems (b) Pilot-in-loop dynamics (c) Handling qualities (d) Frequency domain analysis (e) Pilot bring on oscillation 6. Flight Control system implementation Fly-by-wire technique 1. 4. 5 References 1. Flight Dynamics Principles.M. V. Cook. 1997. Arnold. Chaps. 4,5,6,7,10,11 2. Automatic Flight Control Systems. D. McLean. 1990. Prentice mansion house International Ltd. Chaps. 2, 3,6,9. 3. Introduction to Avionics Systems. Second edition. R. P. G. Collinson. cc3. Kluwer Academic Publishers. Chap. 4 12 CHAPTER 1. INTRODUCTION Chapter 2 Longitudinal response to the control 2. 1 Longitudinal dynamics From Flight Dynamics course, we know that the linearised longitudinal dynamics fuel be written as mu ? ? ? X ? X ? X ? X u? w? ? w + (mWe ? )q + mg? cos ? e ? u ? w ? ?w ? q ? Z ? Z ? Z ? Z ? u + (m ? )w ? ? w ? (mUe + )q + mg? sin ? e ? u ? w ? ?w ? q ?M ? M ? M ? M u? w? ? w + Iy q ? ? q ? ?u ? w ? ?w ? q = = = ? X ? t ? Z ? t ? M ? t (2. 1) (2. 2) (2. 3) The physical meanings of the variables argon de? ned as u Perturbation some steady state focal ratio Ue w Perturbation on steady state normal stop number We q Pitch rate ? Pitch tippytoe Under the assumption that the aeroplane is in level straight ? ight and the address axes ar wind or stability axes, we have ? e = We = 0 (2. 4) The main controls in longitudinal dynamics atomic number 18 the rise tap and the engine trust. The belittled affray landmarks in the right side of the above pars suffer be expressed as ? X ? t ?Z ? t ? M ? t where 13 = = = ? X ? X ? e + ? e ?Z ? Z ? e + ? e ?M ? M ? e + ? e (2. 5) (2. 6) (2. 7) 14 CHAPTER 2. longitudinal retort TO THE tick ? e the elevator de? ection (Note ? is utilise in Appendix 1) ? engine engorge perturbation Substituting the above scene into the longitudinal symmetric relocation yields ? X ? X ? X ? X u? w? ? w? q + mg? ?u ? w ? ?w ? q ? Z ? Z ? Z ? Z ? u + (m ? )w ? ? w ? (mUe + )q ? u ? w ? ?w ? q ? M ? M ? M ? M u? w? ? w + Iy q ? ? q ? ?u ? w ? ?w ? q mu ? ? = = = ? X ? X ? e + ? e ?Z ? Z ? e + ? e ?M ? M ? ?e + e (2. 8) (2. 9) (2. 10)After adding the relationship ? ? = q, (2. 11) Eqs. (2. 8)- (2. 11) crumb be put in a more compendious vector and intercellular substance format. The longitudinal dynamics lav be written as ? m ? 0 ? ? 0 0 ? ?X ? w ? ?Z m ? ?w ? ? ? M ? w ? 0 0 0 Iy 0 u ? 0 0 w ? ? 0 q ? ? 1 ? ? ? = ? ? ? ? ? ? ? ? ? ?X ? u ? Z ? u ? M ? u ? X ? w ? Z ? w ? M ? w ? Z ? q ? X ? q + mUe ?M ? q 0 0 ?X e ? Z e ? M e 0 ?X ?Z ?M ? ? ? ? 1 ?mg u 0 w 0 q ? 0 ? ? ?+ ? ?e ? (2. 12) 0 Put all variables in the longitudinal dynamics in a vector form as ? ? u ? w ? ? X=? ? q ? ? and let m ? ?X ? w ? ? 0 m ? ?Z ? ?w ? = ? 0 ? ?M ? w ? 0 ? ?X ? X ? = ? ? ? B ? = ? ? ? u ? Z ? u ? M ? u ? w ? Z ? w ? M ? w ? Z ? q (2. 13) ? M 0 0 Iy 0 ?X ? q ? 0 0 ? ? 0 ? 1 (2. 14) ? ?mg 0 ? ? 0 ? 0 A + mUe ?M ? q (2. 15) 0 0 ?X e ? Z e ? M e 0 ?X ?Z ?M ? ? ? ? 1 (2. 16) 0 U= ?e ? (2. 17) 2. 1. LONGITUDINAL participatingS Equation (2. 12) becomes 15 ? MX = A X + B U (2. 18) It is custom to diversify the above set of equations into a set of ? rst tramp di? erential equations by multiplying two sides of the above equation by the inverse of the ground substance M , i. e. , M ? 1 . Eq. (2. 18) becomes ? ? ? ? ? ? u ? xu xw xq x? x? e x? u ? w ? ? zu zw zq z? ? ? w ? ? z? z? ? ? e ? ? ? =? ? ? ? (2. 19) ? q ? ? mu mw mq m? ? ? q ? + ? m? e m? ? ? ? ? ? 0 0 1 0 0 0 ? Let xu ? zu A = M ? 1 A = ? ? mu 0 ? ? xw zw mw 0 xq zq mq 1 ? x? z? ? ? m? ? 0 (2. 20) and x? e ? z? e B = M ? 1 B = ? ? m ? e 0 ? x? z? ? ? m? ? 0 (2. 21) It pile be written in a pithy format ? X = AX + BU (2. 22) Eq. (2. 22) with (2. 20) and (2. 21) is referred as the state space model of the linearised longitudinal dynamics of aircraft. Appendix 1 gives the relationship mingled with the b ar-ass stability and control derivatives in the hyaloplasm A and B, i. e. xu , so on, with the dimensional and non-dimensional derivatives, where ?X ? Xu = ? u (2. 23) denotes dimensional derivative and Xu its corresponding non-dimensional derivative. These relationships argon derived based on the Cramers rule and hold for general body axes. In the case when the derivatives ar referred to wind axes, as in this course, the following simpli? cations should be made Ue = Vo , We = 0, sin ? e = 0, cos ? e = 1 (2. 24) The description of the longitudinal dynamics in the ground substance-vector format as in (2. 19) disregard be exte nded to represent all general dynamic systems. reckon a system with secern n, i. e. , the system clear be exposit by n aver di? rential equation (as it will be explained later, this is the aforesaid(prenominal) as the highest order of the denominator polynomial in the transfer function is n). In the original (2. 22), A ? Rn? n is the system matrix B ? Rn? m is the enter matrix X ? Rn is the state vector or state variables and U ? Rm the gossip or gossip vector. The equation (2. 22) is called state equation. For the stability augmentation system, only the in? uence of the variation of the elevator burthen, i. e. the primary satiny control scrape, is concerned. The above equations of act kitty be simpli? ed. The state space representation remains the 6 CHAPTER 2. LONGITUDINAL RESPONSE TO THE go akin format as in eq. (2. 22) with the alike(p) matrix A and state variables but with a di? erent B and arousal U as inclined below ? ? x ? e ? z ? B = M ? 1 B = ? ?e ? ( 2. 25) ? m? e ? 0 and U = ? e (2. 26) Remark It should be noticed that in di? erent textbooks, di? erent notations ar employ. For the state space representation of longitudinal dynamics, some cadence widetilded derivatives are used as follows ? ? 1 ? X 1 ? X ? ? 1 ? X ? ? 0 ? g u ? u m ? u m ? w m e 1 ? Z 1 ? Z 1 ? Z ? w ? ? 0 ? ? w ? ? m e ? ?+? ? ? ? = ? m ? u m ? w Ue ? ? e (2. 27) ? q ? Mu ? Mw Mq 0 ? ? q ? ? M? e ? ? ? ? 0 0 1 0 0 where Mu = Mw = 1 ? M 1 ? Z 1 ? M + ? Iyy ? u m ? u Iyy ? w ? 1 ? M 1 ? Z 1 ? M + ? Iyy ? w m ? w Iyy ? w ? 1 ? M 1 ? M + Ue ? Iyy ? q Iyy ? w ? (2. 28) (2. 29) (2. 30) (2. 31) Mq = M? e = 1 ? M 1 ? Z 1 ? M + ? Iyy e m e Iyy ? w ? The widetilded derivatives and the other derivatives in the matrices are the same as the materialisation of the minute letter derivatives under certain assumptions, i. e. using stability axis. 2. 2 2. 2. 1 State space description State variables A negligible set of variables which, when known at sentence t0 , to gether with the input, are su? ient to string the behaviours of the system at any time t t0 . State variables whitethorn have no any physical meanings and may be not measurable. For the longitudinal dynamic of aircraft, there are four state variables, i. e, ? ? u ? w ? ? X=? (2. 32) ? q ? ? and unrivalled input or control variable, the elevator de? ection, U = ? e (2. 33) 2. 3. LONGITUDINAL STATE SPACE MODEL thence n=4 m=1 17 (2. 34) The system matrix and input matrix of the longitudinal dynamics are given by ? ? xu xw xq x? ? z zw zq z? ? ? A = M ? 1 A = ? u (2. 35) ? mu mw mq m? ? 0 0 1 0 and ? x? e ? z ? B = M ? 1 B = ? ?e ? ? m ? e ? 0 ? (2. 36) respectively. . 2. 2 General state space model w Ue When the shift of dishonour ? is of concern, it finish be written as ? = which brush aside be put into a general form as y = CX where y=? = and C= 0 1/Ue 0 0 (2. 40) Eq. (2. 38) is called Output equation y the output variable and C the output matrix. For more general case where there are more than one output and has a direct direction from input to output variable, the output equation can be written as Y = CX + DU (2. 41) w Ue (2. 38) (2. 39) (2. 37) where Y ? Rr ,C ? Rr? n and D ? Rr? m . For motion of aerospace vehicles including aircraft and missiles, there is no direct path between input and output.In this course only the case D = 0 is considered if not explicitly pointed out. Eq. (2. 22) and (2. 38) (or (2. 41)) together represent the state space description of a dynamic system, which is opposite to the transfer function representation of a dynamic system studied in Control Engineering course. 2. 3 Longitudinal state space model When the behaviours of all the state variables are concerned, all those variables can be chosen as output variables. In addition, there are other response quantities of avocation including the ? ight path angle ? , the angle of attack ? and the normal quickening az (nz ).Putting all variables together, the output vector c an be written as 18 CHAPTER 2. LONGITUDINAL RESPONSE TO THE train ? ? ? ? ? Y =? ? ? ? ? Invoking the relationships ? = ? ? ? ? ? ? ? ? ? ? u w q ? ? ? az w Ue (2. 42) (2. 43) w Ue (2. 44) the ? ight path angle ? = = and the normal acceleration az (nz ) az = = = ?Z/m = ? (Zu u + Zw w + Zq q + Zw w + Z? e ? e )/m ? ? ? (w ? qUe ) ? ?zu u ? zw w ? zq q ? z? e ? e + Ue zq (2. 45) where the second equality substituting the expression matrix is given by ? ? ? u 1 ? w ? ? 0 ? ? ? ? q ? ? 0 ? ? ? Y =? ? ? =? 0 ? ? ? ? ? ? ? 0 ? ? ? ? ? ? ? 0 az ? zu ollows from (2. 9) and the last equality is obtained by of w in its curt derivative format. and so the output ? 0 1 0 0 1/Ue ? 1/Ue ? zw 0 0 1 0 0 0 ? zq + Ue 0 0 0 1 0 1 0 ? ? ? ? ? ? ? ? ? ? u ? ? ? w ? ? +? q ? ? ? ? ? 0 0 0 0 0 0 ? z? e ? ? ? ? ? ? ? e ? ? ? ? (2. 46) There is a direct path between the output and input The state space model of longitudinal dynamics consists of (2. 22) and (2. 46). 2. 3. 1 Numerical example Boeing 74 7 greenness transport at ? ight match cruising in horizontal ? ight at some 40,000 ft at Mach number 0. 8. Relevant data are given in Table 2. 1 and 2. 2. victimisation tables in Appendix 1, the brief nice derivatives can be reckon and then the system matrix and input matrix can be derived as ? ? ? 0. 006868 0. 01395 0 ? 32. 2 ? ?0. 09055 ? ?0. 3151 774 0 ? A=? (2. 47) ? 0. 0001187 ? 0. 001026 ? 0. 4285 ? 0 0 0 1 0 ? ? ? 0. 000187 ? ?17. 85 ? ? B=? (2. 48) ? ?1. 158 ? 0 Similarly the parameters matrices in output equation (2. 46) can be dogged. It should be noticed that English unit(s) is used in this example. 2. 4. AIRCRAFT DYNAMIC BEHAVIOUR SIMULATION USING STATE SPACE MODELS19 Table 2. 1 Boeing 747 transport data 636,636lb (2. 83176 ? 106 N) 5 euchre ft2 (511. m2 ) 27. 31 ft (8. 324 m) 195. 7 ft (59. 64 m) 0. 183 ? 108 hummer ft2 (0. 247 ? 108 kg m2 ) 0. 331 ? 108 drone ft2 (0. 449 ? 108 kg m2 ) 0. 497 ? 108 slug ft2 (0. 673 ? 108 kg m2 ) -0. 156 ? 107 slug ft2 (-0. 212 ? 107 kg m2 ) 774 ft/s (235. 9m/s) 0 5. 909 ? 10? 4 slug/ft3 (0. 3045 kg/m3 ) 0. 654 0. 0430 W S c ? b Ix Iy Iz Izx Ue ? 0 ? CL0 CD Table 2. 2 Dimensional Derivatives B747 feed X(lb) Z(lb) M(ft. lb) u(f t/s) ? 1. 358 ? 102 ? 1. 778 ? 103 3. 581 ? 103 w(f t/s) 2. 758 ? 102 ? 6. 188 ? 103 ? 3. 515 ? 104 q( rad/sec) 0 ? 1. 017 ? 105 ? 1. 122 ? 107 2 w(f t/s ) ? 0 1. 308 ? 102 -3. 826 ? 103 5 ? e (rad) -3. 17 ? 3. 551 ? 10 ? 3. 839 ? 107 2. 3. 2 The choice of state variables The state space representation of a dynamic system is not unique, which depends on the choice of state variables. For engineering application, state variables, in general, are chosen based on physical meanings, measurement, or easy to design and analysis. For the longitudinal dynamics, in additional to a set of the state variables in Eq. (2. 32), some other widely used choice (in American) is ? u ? ? ? ? X=? ? q ? ? ? (2. 49) Certainly, when the logitudinal dynamics of the aircraft are correspond in terms of the above state variables, di? rent A, B and C are resulted (see Tutorial 1). 2. 4 Aircraft dynamic behaviour simulation using state space models State space model create above provides a very powerful tool in enquire dynamic behavious of an aircraft under various condition. The desire of using state pace models for predicting aircraft dynamic behavious or numerical simulation can be explained by 20 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL the following expression X(t + ? t) = X(t) + dX(? ) ? ? =t ? t = X(t) + X(t)? t d? (2. 50) ? where X(t) is current state, ? t is step size and X(t) is the derivative calculated by the state space equation. . 4. 1 Aircraft response without control ? X = AX X(0) = X0 (2. 51) 2. 4. 2 Aircraft response to controls ? X = AX + BU X(0) = 0 (2. 52) where U is the pilot command 2. 4. 3 Aircraft response under both initial conditions and controls ? X = AX + BU X(0) = X0 (2. 53) 2. 5 Longitudinal response to the elevator After the longitudinal dynam ics are described by the state space model, the time histories of all the variables of interests can be calculated. For example, the time responses of the forward swiftness u, normal velocity w (angle of attack) and ? ight path angle ? under the step movement of the levator are displayed in Fig 2. 12. 5 Discussion If the reason for moving the elevator is to establish a new steady state ? ight condition, then this control action can hardly be viewed as successful. The long lightly damped oscillation has bad interfered with it. A ripe operation performance cannot be achieved by apparently changing the angle of elevator. Clearly, longitudinal control, whether by a human pilot or automatic pilot, demands a more sophisticated control indwelling process than open-loop strategy. 2. 6 Transfer of state space models into transfer functions Taking Laplace convert on both sides of Eq. (2. 2) under the zero initial assumption yields sX(s) = Y (s) = where X(s) = LX(t). AX(s) + BU (s) CX(s) (2. 54) (2. 55) 2. 6. dispatch OF STATE SPACE MODELS INTO TRANSFER FUNCTIONS21 tint response to elevator Velocity 90 80 70 60 Velocity(fps) 50 40 30 20 10 0 0 1 2 3 4 5 Time(s) 6 7 8 9 10 Figure 2. 1 Longitudinal response to the elevator Step response to evelator angle of attack 0 ?0. 005 ?0. 01 Angle of attack(rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 1 2 3 4 5 Time(s) 6 7 8 9 10 22 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Step respnse to elevator Flight path angle 0. 1 0. 08 0. 06 0. 04 Flight path angle (rad) 0. 02 0 0. 02 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 1 2 3 4 5 Time(s) 6 7 8 9 10 Figure 2. 2 Longitudinal response to the elevator Step Response to elevator long term 90 80 70 60 Velocity (fps) 50 40 30 20 10 0 0 100 200 300 Time (s) cd 500 600 Figure 2. 3 Longitudinal response to the elevator 2. 6. TRANSFER OF STATE SPACE MODELS INTO TRANSFER FUNCTIONS23 Step response to elevator long term 0 ?0. 005 ?0. 01 Angle of attack (rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 100 200 300 Time (s) 400 500 600 Figure 2. 4 Longitudinal response to the elevator Step response to elevator long term 0. 1 0. 08 0. 06 0. 04 Flight path angle (rad) 0. 02 0 ?0. 2 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 100 200 300 Time (s) 400 500 600 Figure 2. 5 Longitudinal response to the elevator 24 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Y (s) = CsI ? A? 1 BU (s) hence the transfer function of the state space representation is given by G(s) = CsI ? A? 1 B = C(Adjoint(sI ? A))B det(sI ? A) (2. 56) (2. 57) Example 1 A short period motion of a aircraft is described by ? ? q ? = ? 0. 334 ? 2. 52 1. 0 ? 0. 387 ? q + ? 0. 027 ? 2. 6 ? e (2. 58) where ? e denotes the elevator de? ection. The transfer function from the elevator de? ection to the angle of attack is set(p) as follows ? (s) ? 0. 27s ? 2. 6 = 2 ? e (s) s + 0. 721s + 2. 65 (2. 59) The longitudinal dynamics of aircraft is a single-input and multi-output system with one input ? e and several(prenominal) outputs, u, w, q, ? , ? , az . Using t he technique in Section (2. 6), the transfer functions between each output variable and the input elevator can be derived. The notation u(s) Gue = (2. 60) ? ?e (s) is used in this course to denote the transfer function from input ? e to output u. For the longitudinal dynamics of Boeing 747-100, if the output of interest is the forward velocity, the transfer function can be determined using formula (2. 56) as u(s) ? e (s) ? 0. 00188s3 ? 0. 2491s2 + 24. 68s + 11. 6 s4 + 0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 0041959 (2. 61) Gue ? = = Similarly, all other transfer functions can be derived. For a system with low order like the second order system in Example 1, the blood line of the corresponding transfer function from its state space model can be completed manually. For complicated systems with high order, it can be through by computer software like MATLAB. It can be prime that although the transfer functions from the elevator to di? erent outputs are di? erent but they have the same denominator, i. e. s4 + 0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959 for Beoing 747-100. however the numerators are di? erent. This is because all the denominators of the transfer functions are determined by det(sI ? A). 2. 6. 1 From a transfer function to a state space model The number of the state variable is equal to the order of the transfer function, i. e. , the order of the denominator of the transfer function. By choosing di? erent state variables, for the same transfer function, di? erent state space models are given. 2. 7. BLOCK DIAGRAM REPRESENTATION OF STATE SPACE MODELS 25 2. 7 Block diagram representation of state space models 2. 8 2. 8. 1 Static stability and dynamic modesAircraft stability Consider aircraft equations of motion represented as ? X = AX + BU (2. 62) The stability analysis of the authoritative aircraft dynamics concerns if there is no any control e? ort,whether the runaway motion is stable. It is also referred as openloop stability in gene ral control engineering. The aircraft stability is determined by the eigenvalues of the system matrix A. For a matrix A, its eigenvalues can be determined by the polynomial det(? I ? A) = 0 (2. 63) Eigenvalues of a state space model are equal to the grow of the attribute equation of its corresponding transfer function.An aircraft is stable if all eigenvalues of its system matrix have negative real part. It is unstable if one or more eigenvalues of the system matrix has positive real part. Example for a second order system Example 1 revisited 2. 8. 2 Stability with FCS augmentation When a ? ight control system is installed on an aircraft. The command applied on the control surface is not beautifully generated by a pilot any more it consists of both the pilot command and the control signal generated by the ? ight control system. It can be written as ? U = KX + U (2. 64) ? where K is the state feedback gain matrix and U is the reference signal or pilot command.The stability of an a ircraft under ? ight control systems is refereed as closed-loop stability. 26 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Then the closed-loop system under the control uprightness is given by ? ? X = (A + BK)X + B U (2. 65) Stability is also determined by the eigenvalues of the system matrix of the system (2. 65), i. e. , A + BK. Sometimes only part of the state variables are available, which are true for most of ? ight control systems, and only these measurable variables are feed back, i. e. output feedback control. It can be written as ? ? U = KY + U = KCX + B U where K is the output feedback gain matrix.Substituting the control U into the state equation yields ? ? X = (A + BKC)X + B U (2. 67) (2. 66) Then the closed-loop stability is determined by the eigenvalues of the matrix A+BKC. Boeing Example (cont. ) Open-loop stability ? 0. 3719 + 0. 8875i ? 0. 3719 ? 0. 8875i eig(A) = ? 0. 0033 + 0. 0672i ? 0. 0033 ? 0. 0672i (2. 68) Hence the longitudinal dynamics are stable. The same conclusion can be drawn from the the transfer function approach. Since the stability of an open loop system is determined by its poles from denominator of its transfer function, i. e. , s4 +0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959=0. Its roots are given by s1,2 = ? 0. 3719 0. 8875i s3,4 = ? 0. 0033 0. 0672i (2. 69) (This example veri? es that the eigenvalues of the system matrix are the same as the roots of its characteristic equation ) 2. 8. 3 Dynamic modes Not only stability but also the dynamic modes of an aircraft can be extracted from the stat space model, more speci? cally from the system matrix A. Essentially, the determinant of the matrix A is the same as the characteristic equation. Since there are two pairs of mixed roots, the denominator can be written in the typical second order systems format as 2 2 (s2 + 2? ? p s + ? p )(s2 + 2? s ? s s + ? s ) (2. 70) (2. 71) (2. 72) where ? p = 0. 0489 for Phugoid mode and ? s = 0. 3865 for the short period mode. ? s = 0. 9623 ? p = 0. 0673 2. 9. REDUCED MODELS OF LONGITUDINAL DYNAMICS B 747 Phugoid mode 1. 5 27 1 93. 4s 0. 5 Perturbation 0 ? 0. 5 ? 1 0 300 600 Time (s) Figure 2. 6 Phugoid mode of Beoing 747-100 The ? rst second order dynamics correspond to Phugoid mode. This is an oscillad d tion with period T = 1/? p = 1/(0. 0672/2? ) = 93. 4 second where ? p is the damped frequency of the Phugoid mode. The damping ratio for Phugoid mode is very small, i. e. , ? p = 0. 489. As shown in Figure 2. 6, Phugoid mode for Boeing 747-100 at this ? ight condition is a slow and pitiful damped oscillation. It takes a long time to die away. The second mode in the characteristic equation corresponds to the short period mode in aircraft longitudinal dynamics. As shown in Fig. 2. 7, this is a well damped response with fast period about T = 7. 08 sec. (Note the di? erent time scales in Phugoid and short period response). It dies away very rapidly and only has the in? uence at the beginning of the response . 2. 9 Reduced models of longitudinal dynamics Based on the above example, we can ? d Phugoid mode and short period mode have di? erent time scales. Actually all the aircraft have the homogeneous response behaviour as Boeing 747. This makes it is possible to alter the longitudinal dynamics under certain conditions. As a result, this will simplify following analysis and design. 2. 9. 1 Phugoid approximation The Phugoid mode can be obtained by simplifying the expert 4th order longitudinal dynamics. Assumptions w and q respond to disturbances in time scale associated with the short period 28 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Beoing 747 Short period mode From U(1) 0. 7 0. 6 0. 5 0. 4Perturbation To Y(1) 0. 3 0. 2 0. 1 0 ?0. 1 ?0. 2 0 5 10 15 Time (sec. ) Figure 2. 7 Short Period mode of Beoing 747-100 mode it is reasonable to fatigue that q is quasi-steady in the longer time scale associated with Phugoid mode q=0 ? Mq , Mw , Zq , Zw are neglected since both q and w a re relatively small. ? ? ? Then from the table in Appendix 1, we can ? nd the expression of the small concise derivatives under these assumptions. The longitudinal model reduces to ? ? ? Xu Xw ? ? X? e ? 0 ? g u ? u m m m Zw ? w ? ? Zu Ue 0 ? ? w ? ? Z? e ? m m ? ? ? =? M ? + ? M ? ?e (2. 73) ? m ? ? 0 ? ? u Mw 0 0 ? q ? ? ? e ? Iyy Iyy Iyy ? ? ? 0 0 1 0 0 This is not a standard state space model. However using the similar idea in Section 2. 6, by taking Laplace veer on the both sides of the equation under the assumption that X0 = 0, the transfer function from the control surface to any chosen output variable can be derived. The characteristic equation (the denominator polynomial of a transfer function) is given by ? (s) = As2 + Bs + C where A = ? Ue Mw Ue B = gMu + (Xu Mw ? Mu Xw ) m g C = (Zu Mw ? Mu Zw ) m (2. 75) (2. 76) (2. 77) (2. 74) 2. 9. REDUCED MODELS OF LONGITUDINAL DYNAMICS 29 This corresponds to the ? st mode (Phugoid mode) in the full longitudinal model. After subst ituting data for Beoing 747 in the formula, the damping ratio and the natural frequency are given by ? = 0. 068, ? n = 0. 0712 (2. 78) which are meagrely di? erent from the true values, ? p = 0. 049, ? p = 0. 0673, obtained from the full 4th longitudinal dynamic model. 2. 9. 2 Short period approximation In a short period after actuation of the elevator, the speed is substantially constant magic spell the airplane pitches relatively rapidly. Assumptions u=0 Zw (compared with m) and Zq (compared with mUe ) are neglected since they ? are relatively small. w ? q ? Zw m mw Ue mq w q + Z ? e m m ? e ?e (2. 79) The characteristic equation is given by s2 ? ( Zw 1 1 Mq Zw + (Mq + Mw Ue ))s ? (Ue Mw ? )=0 ? m Iyy Iyy m (2. 80) Using the data for B747-100, the result obtained is s2 + 0. 741s + 0. 9281 = 0 with roots s1,2 = ? 0. 371 0. 889i The corresponding damping ratio and natural frequency are ? = 0. 385 wn = 0. 963 (2. 83) (2. 82) (2. 81) which are seen to be almost same as those obta ined from the full longitudinal dynamics. Actually the short period approximation is very good for a wide range of vehicle characteristics and ? ight conditions. Tutorial 1 1. Using the small concise derivatives, ? d the state equations of longitudinal dynamics of an aircraft with state variables ? ? u ? ? ? ? X=? (2. 84) ? q ? ? 30 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL conventionalism acceleration at the pilot seat is a very important quantity, de? ned as the normal acceleration response to an elevator measured at the pilot seat, i. e. aZx = w ? Ue q ? lx q ? ? (2. 85) where lx is the distance from c. g. to the pilot seat. When the outputs of interest are pitch angle ? and the normal acceleration at the pilot seat, ? nd the output equations and identify all the associated parameter matrices and dimension of variables (state, input and output). . The motion of a mass is governed by m? (t) = f (t) x (2. 86) where m is mass, f (t) the force acting on the mass and x(t) the d isplacement. When the velocity x(t) and the velocity plus the position x(t) + x(t) are chosen ? ? as state variables, and the position is chosen as output variable, ? nd the state space model of the above mass system. Determine the transfer function from the state space model and compare it with the transfer function directly derived from the dynamic model in Eq. (2. 86). 3. witness the transfer function from elevator de? ection ? e to pitch rate q in Example 1.Determine the natural frequency and damping ratio of the short period dynamics. Is it possible to ? nd these information from a state space model directly, instead of using the transfer function approach? 4. Suppose that the control strategy ? ?e = ? + 0. 1q + ? e (2. 87) ? is used for the aircraft in Example 1 where ? e is the command for elevator de? ection from the pilot. Determine stability of the short period dynamics under the above control law using both state space method and Routh stability criterion in Control Eng ineering (When Routh stability criterion is applied, you can theater the stability using the transfer function from ? to q or that from ? e to ? (why? )). comparison and discuss the results achieved. Chapter 3 Lateral response to the controls 3. 1 Lateral state space models mv ? ?Y v ? ( ? Y + mWe )p ? ?v ? p ? mUe )r ? mg? cos ? e ? mg? sin ? e ? L ? L ? L ? v + Ix p ? ? p ? Ixz r ? ? r ? v ? p ? r ? N ? N ? N v ? Ixz p ? ? p + Iz r ? ? r ? ?v ? p ? r = = = ? Y ? A + A ? L ? A + A ? N ? A + A ? Y ? R R ? L ? R R ? N ? R R (3. 1) (3. 2) (3. 3) Referred to body axes, the small perturbed lateral dynamics are described by ? ( ? Y ? r where the physical meanings of the variables are de? ed as v Lateral velocity perturbation p Roll rate perturbation r Yaw rate perturbation ? Roll angle perturbation ? Yaw angle perturbation ? A Aileron angle (note that it is denoted by ? in Appendix 1) ? R Rudder angle (note that it is denoted by ? in Appendix 1) Together with the relationship s ? ?=p and ? ? = r, (3. 4) (3. 5) the lateral dynamics can be described by ? ve equations, (3. 1)-(3. 5). Treating them in the same way as in the longitudinal dynamics and after introducing the concise notation as in Appendix 1, these ? ve equations can be represented as ? ? ? ? ? ? v ? p ? r ? ? ? ? ? ? yv lv nv 0 0 yp lp np 1 0 yr lr nr 0 1 y? 0 0 0 0 y? 0 0 0 0 v p r ? ? ? ? y? A l? A n ? A 0 0 y? R l? R n ? R 0 0 ? ? ? ? ? ? ? A ? R (3. 6) ? ? ? ? ?=? ? ? ? ? ? ? ? ? ?+? ? ? ? ? 31 32 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS When the derivatives are referred to airplane wind axes, ? e = 0 (3. 7) from Appendix 1, it can be seen that y? = 0. Thus all the parts of the ? fth column in the system matrix are zero. This implies that ? has no in? uence on all other variables. To simplify analysis, in most of the cases, the following fourth order model is used ? ? ? ? ? v ? v y? A y? R yv yp yr y? ? p ? ? lv lp lr 0 ? ? p ? ? l? A l? R ? ?A ? ? ? ? ? ? =? (3. 8) ? r ? ? n v n p n r 0 ? ? r ? + ? n ? A n ? R ? ? R ? ? ? 0 1 0 0 0 0 ? (It should be noticed that the number of the states is still ? ve and this is vertical for the purpose of simplifying analysis). Obviously the above equation can also be put in the general state space equation ? X = AX + BU with the state variables ? v ? p ? ? X=? ? r ? , ? ?A ? R yp lp np 1 yr lr nr 0 ? (3. 9) (3. 10) the input/control variables U= the system matrix yv ? lv A=? ? nv 0 and the input matrix ? ? , ? y? 0 ? ? 0 ? (3. 11) (3. 12) y ? A ? l? A B=? ? n ? A 0 ? y? R l? R ? ? n ? R ? 0 (3. 13) For the lateral dynamics, other widely used choice of the state variables (American system) is to replace the lateral velocity v by the cutting angle ? and keep all others. have in mind that v (3. 14) Ue The relationships between these two representations are easy to identify. In some textbooks, primed derivatives, for example, Lp , Nr , so on, are used for state space representation of the lateral dynamics. The primed d erivatives are the same as the concise small letter derivatives used in above and in Appendix 1.For stability augmentation systems, di? erent from the state space model of the longitudinal dynamics where only one input elevator is considered, there are two inputs in the lateral dynamic model, i. e. the aileron and rudder. 3. 2. TRANSIENT RESPONSE TO AILERON AND RUDDER Table 3. 1 Dimensional Derivatives B747 jet Y(lb) L(ft. lb) N(ft. lb) v(ft/s) ? 1. 103 ? 103 ? 6. 885 ? 104 4. 790 ? 104 p(rad/s) 0 ? 7. 934 ? 106 ? 9. 809 ? 105 r(rad/sec) 0 7. 302 ? 106 ? 6. 590 ? 106 ? A (rad) 0 ? 2. 829 ? 103 7. 396 ? 101 ? R (rad) 1. 115 ? 105 2. 262 ? 103 ? 9. 607 ? 103 33 3. 2 3. 2. 1 Transient response to aileron and rudderNumerical example Consider the lateral dynamics of Boeing 747 under the same ? ight condition as in Section 2. 3. 1. The lateral aerodynamic derivatives are listed in Table 3. 1. Using the expression in Appendix 1, all the parameters in the state space model can be calculated , given by ? ? ? 0. 0558 0. 0 ? 774 32. 2 ? ?0. 003865 ? 0. 4342 0. 4136 0 ? ? A=? (3. 15) ? 0. 001086 ? 0. 006112 ? 0. 1458 0 ? 0 1 0 0 and 0. 0 ? ?0. 1431 B=? ? 0. 003741 0. 0 ? ? 5. 642 0. 1144 ? ? ? 0. 4859 ? 0. 0 (3. 16) Stability Issue ? 0. 0330 + 0. 9465i ? 0. 0330 ? 0. 9465i eig(A) = ? 0. 5625 ? 0. 0073 (3. 17)All the eigenvalues have negative real part hence the lateral dynamics of the Boeing 747 jet transport is stable. 3. 2. 2 Lateral response and transfer functions ? v p ? ?+B r ? ? State space model of lateral dynamics ? ? ? v ? ? p ? ? ? ? ? = A? ? r ? ? ? ? ? ?A ? R (3. 18) This is a typical Multi-Input Multi-Output (MIMO) system. For an MIMO system like the lateral dynamics, similar to the longitudinal dynamics, its corresponding transfer function can be derived using the same technique introduced in Chapter 2. However, in this case the corresponding Laplace transmute of the state space model, 34 CHAPTER 3.LATERAL RESPONSE TO THE CONTROLS G(s) ? Rr? m is a daedal f unction matrix which is referred as a transfer function matrix where m is the number of the input variables and r is the number of the output variables. The ijth element in the transfer function matrix de? nes the transfer function between the ith output and jth input, that is, Gyij (s) = u yi (s) . uj (s) (3. 19) For example, GpA (s) denotes the transfer function from the aileron, ? A , to the roll ? rate, p. Its corresponding transfer function matrix is given by ? ? ? ? v G? A (s) GvR (s) v(s) ? ? p(s) ? ? Gp (s) Gp (s) ? ?A (s) ? R ? ? ? ? ?A (3. 20) ? r(s) ? ? Gr (s) Gr (s) ? ?R (s) ? A ? R ? p ? (s) G? A (s) G? R hi(s) With the data of Boeing 747 lateral dynamics, these transfer functions can be found as ? 2. 896s2 ? 6. 542s ? 0. 6209 GvA (s) = 4 fps/rad (3. 21) ? s + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 ? 0. 1431s3 ? 0. 02727s2 ? 0. 1101s rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 22) 0. 003741s3 + 0. 002708s2 + 0. 0001394s ? 0. 00453 4 GrA (s) = rad/s/rad, deg/s/deg ? s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 23) ? 0. 1431s2 ? 0. 02727s ? 0. 1101 ? rad/rad, or deg/deg (3. 24) G? A (s) = 4 s + 0. 6344s3 + 0. 9375s2 + 0. 097s + 0. 003658 and GpA (s) = ? GvR (s) = ? 5. 642s3 + 379. 4s2 + 167. 5s ? 5. 917 fps/rad s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 25) GpR (s) = ? 0. 1144s3 ? 0. 1991s2 ? 1. 365s rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 26) ? 0. 4859s3 ? 0. 2321s2 ? 0. 008994s ? 0. 05632 rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 27) 0. 1144s2 ? 0. 1991s ? 1. 365 rad/rad, or deg/deg (3. 28) s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 GrR (s) = ? G? R (s) = ? The denominator polynomial of the transfer functions can be factorised as (s + 0. 613)(s + 0. 007274)(s2 + 0. 06578s + 0. 896) (3. 29) 3. 3. REDUCED ordination MODELS 35 It has one large real root, -0. 5613, one small real root, -0. 0073 (very close to ori gin) and a pair of analyzable roots (-0. 0330 + 0. 9465i, -0. 0330 0. 9465i). For most of the aircraft, the denominator polynomial of the lateral dynamics can be factorized as above, ie. , with two real roots and a pair of complex roots. That is, 2 (s + 1/Ts )(s + 1/Tr )(s2 + 2? d ? d s + ? d ) = 0 (3. 30) where Ts Tr is the spiral time constant (for spiral mode), Tr is the roll subsiding time constant (for roll subsidence), and ? d , ? are damping ratio and natural frequency of Dutch roll mode. For Boeing 747, from the eigenvalues or the roots, these parameters are calculated as Spiral time constant Ts = 1/0. 007274 = 137(sec) (3. 31) Roll subsidence time constant Tr = 1/0. 5613 = 1. 78(sec) and Dutch roll natural frequency and damping ratio ? d = 0. 95(rad/sec), ? d = 0. 06578 = 0. 0347 2? d (3. 33) (3. 32) The basic ? ight condition is steady symmetric ? ight, in which all the lateral variables ? , p, r, ? are identically zero. Unlike the elevator, the lateral controls are not used individually to arouse changes in steady state.That is because the steady state values of ? , p, r, ? that result from a constant ? A and ? R are not of interest as a profitable ? ight condition. Successful movement in the lateral channel, in general, should be the combination of aileron and rudder. In view of this, the impulse response, rather than step response used in the lateral study, is employed in study the lateral response to the controls. This can be considered as an idealised state of affairs that the control surface has a sudden move and then back to its normal position, or the recovering period of an airplane deviated from its steady ? ght state out-of-pocket to disturbances. The impulse lateral responses of Boeing 747 under unit aileron and rudder impulse action are shown in Figure 3. 1 and 3. 2 respectively. As seen in the response, the roll subsidence dies away very quickly and mainly has the in? uence at the beginning of the response. The spiral mode has a large time constant and takes quite long time to respond. The Dutch roll mode is quite poorly damped and the oscillation caused by the Dutch roll dominates the on the whole lateral response to the control surfaces. 3. 3 Reduced order models Although as shown in the above ? gures, there are di? rent modes in the lateral dynamics, these modes interact each other and have a strong uniting between them. In general, the approximation of these models is not as accuracy as that in the longitudinal dynamics. However to simplify analysis and design in Flight Control Systems, minify order models are still useful in an initial stage. It is suggested that the full lateral dynamic model should be used to verify the design based on bring down order models. 36 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS Lateral response to impluse aileron deflection 0. 1 Lateral velocity (f/s) 0. 05 0 ? 0. 05 ? 0. 1 ? 0. 5 0 10 20 30 Time(s) 40 50 60 0. 05 Roll rate (deg/sec) 0 ? 0. 05 ? 0. 1 ? 0. 15 0 x 10 ? 3 10 20 30 Time (s) 40 50 60 5 Yaw rate(deg/sec) 0 ? 5 ? 10 ? 15 0 10 20 30 Time (s) 40 50 60 0 Roll angle (deg) ? 0. 05 ? 0. 1 ? 0. 15 ? 0. 2 ? 0. 25 0 10 20 30 Time (s) 40 50 60 Figure 3. 1 Boeing 747-100 lateral response to aileron 3. 3. REDUCED ORDER MODELS 37 Lateral response to unit impluse rudder deflection 10 Lateral velocity (f/s) 5 0 ? 5 ? 10 0 10 20 30 Time (s) 40 50 60 2 Roll rate (deg) 1 0 ? 1 ? 2 0 10 20 30 Time (s) 40 50 60 0. 4 Yaw rate (deg) 0. 2 0 ? 0. 2 ? 0. 4 ? 0. 6 0 10 20 30 Time (s) 40 50 60 Roll angle (deg) 0 ? 1 ? 2 ? 3 ? 4 0 10 20 30 Time (s) 40 50 60 Figure 3. 2 Boeing 747-100 lateral response to Rudder 38 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS 3. 3. 1 Roll subsidence Provided that the perturbation is small, the roll subsidence mode is observed to involve almost pure rolling motion with little coupling into sideslip and yaw. A reduced order model of the lateral-directional dynamics retaining only roll subsidence mode follows by removing the side force and yaw moment equations to give p = lp p + l? A ? A + l? R ? R ? (3. 34) If only the in? uence from aileron de? ction is concerned and sorb that ? R = 0, taking Laplace transform on Eq. (3. 34) obtains the transfer function p(s) l ? A kp = = ? A s ? lp s + 1/Tr where the gain kp = l? A and the time constant Tr = 1 Ix Iz ? Ixz =? lp Iz Lp + Ixz Np (3. 36) (3. 37) (3. 35) Since Ix Ixz and Iz Ixz , then equation (3. 37) can be further simpli? ed to give the classical approximation expression for the roll mode time constant Tr = ? Ix Lp (3. 38) For the Boeing 747, the roll subsidence estimated by the ? rst order roll subsidence approximation is 0. 183e + 8 Tr = ? = 2. 3sec. (3. 39) ? 7. 934e + 6 It is close to the real value, 1. sec, given by the full lateral model. 3. 3. 2 Spiral mode approximation As shown in the Boeing 747 lateral response to the control surface, the spiral mode is very slow to develop. It is usual to assume that the motion variables v, p, r are quasi-steady relat ive to the time scale of the mode. Hence p = v = r = 0 and the ? ? ? lateral dynamics can be written as ? ? ? 0 yv ? 0 ? ? lv ? ? ? ? 0 ? = ? nv ? 0 ? yp lp np 1 yr lr nr 0 y? v 0 p 0 r 0 ? ? y? A ? ? l ? A ? +? ? ? n ? A 0 ? ? y ? R l? R ? ? n ? R ? 0 ?A ? R (3. 40) If only the spiral mode time constant is concerned, the unforced equation can be used.After solving the ? rst and 3rd algebraic equations to yield v and r, Eq. (3. 40) reduces to lp nr ? l n l np ? lp n 0 p yv lr nv ? lr np + yp + yr lv nv ? lv nv y? v r r r (3. 41) ? = ? ? 1 0 3. 3. REDUCED ORDER MODELS 39 Since the terms involving in yv and yp are assumed to be insigni? cantly small compared to the term involving yr , the above expression for the spiral mode can be further simpli? ed as ? y? (lr nv ? lv nr ) ? = 0 ? + (3. 42) yr (lv np ? lp nv ) Therefore the time constant of the spiral mode can be estimated by Ts = yr (lv np ? lp nv ) y? (lr nv ? lv nr ) (3. 43)Using the aerodynamic derivatives of Boeing 747, th e estimated spiral mode time constant is obtained as Ts = 105. 7(sec) (3. 44) 3. 3. 3 Dutch roll ? p=p=? =? =0 ? v ? r ? = yv nv yr nr v r + 0 n ? A y? R n ? R ? A ? R (3. 45) (3. 46) Assumptions From the state space model (3. 46), the transfer functions from the aileron or rudder to the lateral velocity or roll rate can be derived. For Boeing 747, the relevant transfer functions are given by GvA (s) = ? GrA (s) = ? GvR (s) = ? GrR (s) = ? ?2. 8955 s2 + 0. 2013s + 0. 8477 0. 003741(s + 0. 05579) s2 + 0. 2013s + 0. 8477 s2 5. 642(s + 66. 8) + 0. 013s + 0. 8477 (3. 47) (3. 48) (3. 49) (3. 50) ?0. 4859(s + 0. 04319) s2 + 0. 2013s + 0. 8477 From this 2nd order reduced model, the damping ratio and natural frequency are estimated as 0. 1093 and 0. 92 rad/sec. 3. 3. 4 Three degrees of freedom approximation Assume that the following items are small and negligible 1). The term due to gravity, g? 2). Rolling acceleration due to yaw rate, lr r 3). Yawing acceleration as a result of roll rate, np p leash order Dutch roll approximation is given by ? ? ? ? ? ? v ? yv yp yr v 0 y ? R ? p ? = ? lv lp 0 ? ? p ? + ? l? A l? R ? ? r ? nv 0 nr r n? A n?R ?A ? R (3. 51) 40 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS For Boeing 747, the corresponding transfer functions are obtained as GvA (s) = ? GpA (s) = ? GrA (s) = ? ?2. 8955(s + 0. 6681) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) ? 0. 1431(s2 + 0. 1905s + 0. 7691) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 0. 003741(s + 0. 6681)(s + 0. 05579) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 5. 642(s + 0. 4345)(s + 66. 8) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 0. 1144(s ? 4. 432)(s + 2. 691) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) ? 0. 4859(s + 0. 4351)(s + 0. 04254) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) (3. 52) 3. 53) (3. 54) and GvR (s) = ? GpR (s) = ? GrR (s) = ? (3. 55) (3. 56) (3. 57) The poles corresponding to the Dutch roll mode are given by the roots of s2 + 0. 1833s + 0. 8548 = 0. Its damping ratio and natural frequency are 0. 0995 and 0. 921 rad/sec. Compared with the values given by the second order Dutch roll approximation, i. e. , 0. 1093 and 0. 92 rad/sec, they are a little spell closer to the true damping ratio ? d = 0. 0347 and the natural frequency ? d = 0. 95 (rad/sec) but the estimation of the damping ratio still has quite poor accuracy. 3. 3. 5 Re-formulation of the lateral dynamicsThe lateral dynamic model can be re-formulated to emphasise the structure of the reduced order model. ? ? v ? yv ? r ? ? nv ? ? ? ? ? p ? = ? lv ? ? 0 ? ? yr nr lr 0 yp np lp 1 g v 0 r 0 p 0 ? ? 0 ? ? n ? A ? +? ? ? l? A 0 ? ? y? R n ? R ? ? l? R ? 0 ? A ? R (3. 58) The system matrix A can be partitioned as A= Directional e? ects Directional/roll coupling e? ects Roll/directional coupling e? ects Lateral or roll e? ects (3. 59) Tutorial 2 1. Using the data of Boeing 747-100 at Case II, form the state space model of the lateral dynamics of the aircraft at this ? ight condition.When the sideslip angle and roll angle are o f interest, ? nd the output equation. 2. Find the second order Dutch roll reduced model of this airplane. come down the transfer function from the rudder to the yaw rate based on this reduced order model. 3. 3. REDUCED ORDER MODELS 41 3. Using MATLAB, assess the approximation of this reduced order model based on time response, and the damping ratio and natural frequency of the Dutch roll mode. 4. Based on the third order reduced model in (3. 51), ? nd the transfer function from the aileron to the roll rate under the assumption y? A = yp = 0.

Jefferson vs. Hamilton: Confrontations That Shaped a Nation

April 13, 1743 Albemarle County in the English colony of Virginia was the start of an the Statesn historical giant. Thomas Jefferson was born in affluence to his father, Peter Jefferson, a rising young planter in the Virginia colony, and his mother, Jane Randolph, who held a high berth within the colony as well.Due to his fathers prosperity Jefferson was afforded the absolute best in the ways of education, starting with private tutors at the age of five, then moving on to learn how to read classical and Roman in there original text and finally fetching his studies to the College of William and Mary in Williamsburg which he would phrase is what probably fixed the destinies of my e inquiryal state pg 5. On the other side of the spectrum, a fewer years later(prenominal) a nonher huge American historical figure is born.Presumably on January 11, 1755, Alexander Hamilton the bastard son of his father, pack Hamilton, a Scotsman of a well-k analogous a shotn(a) family only when n ever flourished on his own, and his mother, Rachel Fawcett Lavien, who had left her husband, ass Lavien, to live with James Hamilton. There is very little said astir(predicate) Hamiltons early life just that his father drifted away and his mother passed in 1768. missing wealth, Hamiltons educational opportunities in his young life were nonexistent, this is not to advance though that his youth was wasted it was here that he gained a extensive knowledge of line of merchandise and finance that he would later use in his service to President George Washington. Even in their early lives it was easy to take on the great dissimilarities between these two patriarchs, now I will dissertate further more(prenominal) issues that Jefferson and Hamilton shared some differences of opinions.During the Philadelphia convention of 1787, which we now refer to as the Constitutional Convention, James capital of Wisconsin, representative from Virginia, in his notes of Hamiltons lengthy speech on June 18, 1787 he writes, Mr. Hamilton, had been hitherto silent on the business before the Convention pg 17 Madison had added it to Hamiltons char motioner to be so, suggesting that it was out of respect of the men who were superior in age, abilities, and experience to him, but also making it clear that the matters before the convention were further excessively outstanding for Hamilton to remain that way.Hamilton expressed dissatisfaction with two of the plans brought forward to the convention. The Virginia plan, which was a plan to completely abandon the Articles of Confederation, and replace it with a bicameral national legislature, an executive director branch selected by the legislature, a judiciary, and a council of revision with the top executive veto, and the bare-ass Jersey plan, which suggested to keep the Articles of Confederation, but revise it to give sex act the power to tax, regulate commerce, and choose plural executive and members of a supreme court. Hamilt on mat both these plans lacked a strong central g all overnment.He was specifi titley displease with the New Jersey plan being fully convinced Madison writes that no amendment of the Confederation, leaving the States in possession of their Sovereignty could possibly resolvent the purpose. pg 18-19. He feared that each of these plans would leave the newly founded country weakened and would in the long eliminate destroy the confederation pg 20. Finally, though Hamilton would given the federal government more power, he supported what was to be and is now the Constitution of the United States of America and he became one of its leading supporters during the ratification process.Though Jefferson was unable to reefer the convention due to his duties in France as the United States Minister, he garnered of its resolution in November 1787, from a copy sent to him by John Adams. After receiving a letter from Madison explaining the proceedings of the convention he articulated his like s and dislikes of the Constitution. Jefferson liked the organization of the government into legislative, Judiciary and Executive pg 23 and the powers given to for each one branch.Among his dislikes, the greatest seeming to be the omission of a bill of rights providing clearly and without the aid of sophisms pg 23 the freedoms he matt-up intrinsic to the human race. During the cut Revolutionary war in 1789, Jefferson and Hamilton found each other on opposing sides once again. Jefferson felt that while the wildness in the revolt was deplorable but he would shake off seen fractional the earth desolated. pg 109 than to see the cause of liberty fail, feeling that the liberty of the self-coloured earth was depending on the issues of the contest. pg 109. While Hamilton, disagreed almost wholly on Jeffersons justification for the violence in France. He as a comrade to mankind and to liberty pg 106-107 rejoiced in the efforts made by Marquis de Lafayette, habitual of the French Rev olutionary National Guard and after serving with Hamilton in 1781 a friend of his as well, but feared the steps in motion to gain the freedom that Lafayette and all of France desired. It was said that Hamilton never commented on the French revolution without it bringing the horror, abhorrence, and repulsion to his mind.In the presidential election in 1800, Hamilton not being a native born citizen could never run for the presidency, but this didnt stop him from doing a helping of work in the background to protect his concur-sized government federalist views. When he first lettered of the federalists loss in New York he wrote a letter to Theodore Sedgwick urging him and the other federalists in the legislature to support Adams and Pinckney, equally pg one hundred thirty feeling that it was their only shot at a federalist for president.Hamilton went as far as to contact John Jay, federalist and the current governor of New York, to call a special session of the federally run legisl ature lettered that there were m whatever objections to this proposal but feeling that the reasons for (the special session) outstrip the objections pg 131. When it came time to vote the results ended as Jefferson 73, Burr 73, Adams 65, Pinckney 64, and John Jay 1. Hamilton now had a much different fight to withstand, while Jefferson was less than coveted as a candidate, Aaron Burr was an unthinkable choice in Hamiltons mind.Knowing now that it either was going to be Burr or Jefferson, Hamilton started contacting people persuading them to vote Jefferson over Burr because Jefferson is to be preferred. He is by far not so dangerous a man and he has pretensions to his character. pg 133. In April 1802, Spain had retroceded Louisiana to France, giving the French a valuable power point on the continent with the Mississippi river connecting the south U. S. to the rest of the commercial U. S. The big turning point was in October of 1802 when Spain invalidates the United States use of the ports on the river.Jefferson seemed to be very torn by this predicament having been the U. S. minister to France a few years previously saying this is not a state of things we render or desire. pg 159. On April 30, 1803 James Monroe, a diplomat to France, left fain to offer France up to fifty million livres, upon arrival he and Robert R. Livingston initialed an commensurateness that ceded all of Louisiana to the United States for $15 Million, the settling of which is looked on as the peak of Jeffersons first term as president. Hamilton had a lot to say about this purchase feeling that Spains direct nfraction of an important article of (Spain and the U. S. ) treaty pg 164 was a call for war and should we have vox populi it advisable to terminate hostilities by a purchase, we might then have done it on almost our own terms pg 164. In 1798 Congress was well on the way to signing into law two acts that they felt at the time were for internal security of the U. S. The naturaliz ation act and a sedition act, the idea behind the naturalization act was the president reserved the right to deport any resident alien he deemed dangerous, or if the U. S. was at war with an aliens home country.The sedition act made it a crime to combine, conspire or to oppose any lawful measure of the government, it also made it punishable to write, say, or write anything to defame the president, Congress or the government of the U. S. Jefferson saw these acts as a slap in the face to the constitution feeling like it was an investigate on the American mind to see how far it will bear an avowed violation of the constitution. pg 115. Hamilton felt just a strongly about these acts in his letter to Theodore Sedgwick he asks What, My Dear Sir, are you going to do with Virginia? pg 118 after he learns of Virginias protest to the acts. He felt as congress did that the acts were for the best of the country feeling that the government will not be the dupes of an insidious plan to disuni te the people of America pg 119. To say that Jefferson and Hamilton were diametrically opposed on all subjects some would say is an over simplification, while the two disagreed on some(prenominal) topics there were also many things that they could come together in agreement on.The biggest example was the way both men looked at the Constitution, not to say that it was perfect in either mans eyes but it was definitely an area where they came to a very nice compromise with each others beliefs about how this country should be run. Both of these men fought hard and struggled against opposing views to make this country what it is today, there is no telling what our present would look like without these two gentlemen who did their best to make this a free nation.

Sunday, February 24, 2019

Marketing strategy of the organisation

In instal for us to understand every administration we have to either cognize about or call some research on it. So in this concession I have chosen an administration. The administration I am making my assignment on is Polyflor entropy Africa. The administration allow be described, including their products/services and their clients. In order to understand the administration we need to carry on a trick out analysis, besides looking at barriers. If we have all the require information from above we will understand the administration better and nominate get down with the selling program.Question 1The administration I am travel to depict is Polyflor Gross saless Africa otherwise have a go at itn as Polyflor South Africa. Polyflor S.A is an industrial shock company. The South African subdivision was established in 2004 by Denver Coleman and Sheila Coleman. The administration really originates from the unite Kingdom where the first branch of was established in Manchester.Poly flor is all oer the universe, they have subdivisions all over the universe. There atomic number 18 subdivisions all over Europe, Asia, North America and Australia. So they be fundamentally on incessantlyy continent. Polyflor Manchester is the chief ca raise office.Polyflor S.A is a house feed concern, where they all work together. The administration believes strongly in team Work . And is all about profiting the con sum totaler.Polyflor S.A has a figure of different wares. any last(predicate) their merchandises are heavy responsibility, industrial merchandises. They do chiefly shocking simply they got some other merchandises that is non flooring, it s really got to make with plumbing.Their chief merchandise is heavy responsibility, industrial vinyl flooring. Another merchandise of theirs is industrial lucre gum stretchable tiles. They besides do wooden block mosaics. The merchandise that is non shocking related that is comparatively new at Polyflor S.A is specialized in dustrial drains.As you can see all their merchandises are industrial merchandises so they meant to be used in topographic points where the merchandises will be introduced to a batch of wear and tear. You will happen their merchandises in mills, large edifices etcetera The vinyl flooring can really be seen chiefly in infirmaries and in many of the luxury coach line drives. The merchandises are non sincerely yours used in a clannish capacity.Polyflor does nt reach their merchandises from and unity topographic point, their merchandises come all over the universe. Their vinyl shocking comes from the United Kingdom, from Manchester. That s the majority of their merchandises they do. The specialised drains are seconded from Germany. They do hold a local manufactured merchandise and that is the meshing gum elastic tiles that they really get from down the route from them.They do nt truly total a service as they sell shocking and other merchandises, save they do hold support system s where they will the forethought at anytime. They ever in that location for their clients. They besides got a station sale service where they will help refering their merchandises. If a client is necessitating aid or general aid refering their merchandises, the employees of Polyflor S.A will ever be at that place to assist every bit much as they can.For these types of merchandises there is non a immense pelf market, as these merchandises are non used in a personal and private capacity. They are used in a industrial capacity, intending mills, infirmaries and so on. Polyflor S.A does non put in the shocking they sell their merchandises to the administrations who install it.So Polyflor s has non got an limitless sum of clients, but they do hold a big client base. Which the deal of them are regular, on traveling purchasers of their merchandises. Basically, Polyflor merely sells to shocking contractors. Polyflor is non allowed to put in their flooring. So other shocking companies wh o do besides sell flooring and put in it. Purchase from Polyflor. Some of the chief flooring contractors that are regular clients are Peter Bates, Turner Piercing, Kevin Bates, Albert Carpets and many others.Question 2Looking at motley administrations we need to understand them more(prenominal). We can make this by carry oning a SWOT analysis. Basically a SWOT analysis is structuring of information, which is collected from the native and external environment. It gives us a theoretical account in order to get down the selling planning procedure. Shows us the strengths, failings, chances and menaces of the administration.Cloete ( 2010,21 )So the SWOT analysis of Polyflor isStrengthsThey are a recognized administration, intimately known in the flooring trade.They are known for holding the best quality merchandises.They are an international administration, so they are recognised all over the universe.Polyflor has one of the best proficient backgrounds in the flooring trade.They are a folk concern so the squad of employees work truly expeditiously and good together.Keeping up in the new manner tendencies of today with their new designs on the flooring.They are a really environmentally well-disposed company.FailingsFinding reps that know about shocking who will be able to sell their merchandises.Finding employees who are able to work with the merchandises and know about flooring.Not holding plenty South African manufactured merchandises.Geting bulk merchandises from abroad states, tidy sum need to acquire the merchandises and start assorted occupations utilizing Polyflor s merchandises but they cant because Polyflor has non received the merchandises ascribable to detain being imported from abroad.OpportunitiesThe manner tendencies are altering all the clip, so Polyflor can hold new manner tendencies in the designs of their flooring.Most people and companies are going Green , so they want to utilize more environmentally friendly merchandises. So Polyflor ca n sell environmentally friendly merchandises and present new Green merchandises.Besides they could sell more locally produced goods, .Polyflor could non merely merely sell to shocking contractors for industrial usage, but besides open up up a new market for people to utilize it in a private capacity.MenacesOf class their chief menace is competition there are other shocking companies besides Polyflor. So they have to constantly maintain happening ways to remain on top and maintaining competitory advantage.Another menace would be that most merchandises come from other international states, which sometimes the merchandises are nt delivered in clip. So clients could travel looking other topographic points.Besides coming from abroad, the monetary value of the merchandises could alter easy, due to currency, revenue enhancements, import and export responsibilities.All of these facets can hold deductions on the selling scheme of the administration. therefore it is of import to make th is analysis.