7 Proven Ways to Manage Stress at Work

7 Proven Ways to Manage Stress at Work Stress. I have it, you have it, we all keep sending it around and around like that nasty cold everyone in your office got last month. Stress is one of the biggest culprits in workplace dissatisfaction- and more than that, it takes its toll on your health and well-being. Who needs that? Here are seven ways to counteract some of the stress that pops up in your everyday life. Treat YourselfSure, a Rolex would be nice, but not terribly feasible for most of us. You know what you can do instead? Take half an hour to treat yourself to a latte. Block out an evening to go see that movie you’ve been eyeing for weeks. Anything that breaks the routine and lets you do something you enjoy works here.DIY AromatherapyStudies have shown that fruit scents (especially green apple) can lessen pain in migraine sufferers. Fruits like apples and citrus are great for calming nerves. Also, herbal scents like peppermint, eucalyptus, rosemary, and lavender often have calming, anti-depressant effects. If your local farmer’s market isn’t in season (or if you’d feel weird skulking around the produce section of your local grocery store, sniffing everything in sight), a nice candle and a few quick moments of deep breathing can help get you back to your happy place when things are hairy.Beach StaycationYou’re on a calm, sandy beach, watching the waves. Is that a dolphin frolicking in the distance? Some gentle gull calls in the distance? Watching something repetitive like waves can help your brain zone out and decompress. The tropical vacation may not be an option right now, but you can find some good temporary (and indoor solutions).Find a Leafy FriendDid you ever do that science experiment when you were a kid, where you see whether plants grow better when they’re talked to? Well, regardless of whether the plant does better, having a plant on your desk can help you de-stress at work. Studies have shown that adding some greenery can lower one’s b lood pressure. Plus, the occasional care (watering pruning, shifting to a sunbeam) gives you a welcome distraction from the daily grind. And it’s a great option if you’re allergic to cuddlier stress-busters like pets.Pick up a Coloring BookYou may have noticed this whole â€Å"adult coloring book† trend lately. Publishers are coming out with all sorts of coloring formats for grownups, busting the conventional wisdom that coloring is a kids’ game. Coloring is great for the adult brain too†¦repetitive motions and patterns let the brain decompress.Massage Your EarsOkay, I know this one sounds weird. But massaging your ears for a minute or two releases endorphins throughout your body and promotes relaxation throughout.Cute Animal VideosIf all else fails, I dare you to stay stressed and unhappy while watching videos of adorable animals doing adorable things. YouTube is the cute animal video capital of the world, and is always great for a two minute pick-m e-up when you’re having a rough day.

40 Topic Suggestions for a Descriptive Paragraph

40 Topic Suggestions for a Descriptive Paragraph If you want to be a successful writer, you must be able to describe [your subject], and in a way that will cause your reader to prickle with recognition....Thin description leaves the reader feeling bewildered and nearsighted. Overdescription buries him or her in details and images. The trick is to find a happy medium.(Stephen King, On Writing, 2000) Descriptive writing calls for close attention to factual and  sensory details: show, dont tell. Whether your subject is as small as a strawberry or as large as a fruit farm,  you should begin by observing your subject closely. Examine it with all five senses, and write down any details and descriptions that come to mind. Next go a little further afield with your list and associate your chosen topic or object with memories, opinions, and impressions. This list may give you some ideas for metaphors and possibly even a direction for your  paragraph or essay. Then make a list of verbs that could be  associated with your topic or object. This will help you have more variety than just buzzing be verbs and keep the writing and imagery descriptive and active. After your brainstorming phase, go through your list and decide which details and descriptions you like the most and are significant. Dont cross off the others, though. At this point in the project, you want to be open to any direction your imagination and writing take you. 40 Topic Suggestions: Description To get you  started,  here are 40 topic suggestions for a descriptive paragraph, essay, or speech.  These suggestions should help you discover a subject that especially  interests  you. If you dont start out with a topic that youre willing to spend some time with, your writing will show your lack of enthusiasm. If 40 is not enough, try this list of  400 writing topics. If you need some advice for the drafting phase, see   Composing Descriptive Paragraphs and Essays and   How to Write a Descriptive Paragraph. a waiting rooma basketball, baseball glove, or tennis racketa smartphonea treasured belonginga laptop computera favorite restaurantyour dream houseyour ideal roommatea closetyour memory of a place that you visited as a childa lockeran accident scenea city bus or subway trainan unusual rooma childs secret hiding placea bowl of fruitan item left too long in your refrigeratorbackstage during a play or a concerta vase of flowersa restroom in a service stationa street that leads to your home or schoolyour favorite foodthe inside of a spaceshipthe scene at a concert or athletic eventan art exhibitan ideal apartmentyour old neighborhooda small town cemeterya pizzaa peta photographa hospital emergency rooma particular friend or family membera paintinga storefront windowan inspiring viewa work tablea character from a book, movie, or television programa refrigerator or washing machinea Halloween costume

Public relations thrives on public opinion Research Paper

Public relations thrives on public opinion - Research Paper Example Soon after, a large number of companies and organizations started employing publicity tactics to attract large audiences. The Excellence Project Vercic, L.A. Grunig (1996) states that nine generic principles govern the basis of setting up global public relations. Edward Louise Barneys is considered an America pioneer in the development of techniques relating to public relations and propaganda. Edward Barneys is considered as the ‘father of public relations’. Edward Barneys developed many influential PR techniques including press release and third party advocacy. Barneys helped remove the taboos surrounding the idea of women smoking in public, thorough his famous campaign of 1920 known as the Women’s Smoking Campaign. Barneys educated the industries regarding the importance of news and stated that it was the most effective method of conveying message to the public. Edward Barneys summarized the importance of PR techniques in his famous quotation given below: This quotation fully explains the importance of public relations and public opinions in setting up a democratic system. In modern world, setting up a democratic system encompasses various aspects. One of the major aspects in establishing democracy is public relations. The research following the development of Excellence theory shows that public relations serve as a pivotal condition for the establishment of a liberal democracy. During the 20th century, public relations gained the status of becoming a powerful and influential industry, not only in the United States of America but across the whole world. In modern society, public relation operations are not just limited to the in-house activity of public corporations and public institutions, but are also being increasingly established as independent consultation firms. Thus, public relation operations have become a prominent constituent in

21 days Essay Example | Topics and Well Written Essays - 1000 words

21 days - Essay Example What to do? Today started well after two of my assignments received maximum points. However, my mentor put a dampener on my spirits by telling me I was falling behind my schoolwork. At times, I think I do not need a mentor, but she helped me a lot last semester. I also reconnected with an old friend Chris on Facebook, which was fun. Today I feel in need of inspiration, and I decide not to visit my mentor for two weeks. I attend a poetry club meeting, which was impressive, and it went some way in inspiring me. I also registered as a member of the drama club as I feel I need something to inspire me at school. Finally, Chris wanted financial help, and I feel I can trust him. Today my friends and I went bowling, and I did well. I bowled a 150, 166, and 160. However, I was still feeling downcast, especially during class. Another of my assignments came back with a C. I revised it and resubmitted it in the hope of getting a better score. I attended my first drama club meeting and felt lost, as there was such camaraderie among the other members. I forgot to write another assignment last night and had to do it in class today. After handing it in, I tried to write some poetry in class to lighten my spirits. On my way from class, I came across some lottery tickets, which had a jackpot of 140 million. Although, I purchased three tickets, I did not win, and neither did anyone else at that time so, I did not feel too downhearted. My mentor called me today and asked me to see her, which I did. I was not aware I was supposed to see her once every two days. I also lent my friend some $300, which I pray is not a mistake as he sounded desperate. The poetry is not improving my spirits, and I decided to try the drama club instead. Today, I managed to make two friends there and got to know how they interact. It was fun. Today was a very busy day with a surprise CAT test. One of my classmates was caught copying my work and

Report on the Analysis of Ineffective Communication in the Workplace Essay Example for Free

Report on the Analysis of Ineffective Communication in the Workplace Essay This report will analyse and examine issues of interpersonal behaviour in the workplace. It will describe a scenario observed concerning communication and will include an analysis of the problems that occurred. A conclusion will be made which will lead to recommendations to prevent this situation from recurring. 2. 0 The scenario The main conflict in this scenario transpired between persons B and C (see appendix 1) on the shop floor of B Q. Person B had previously spoken rudely about person C to person D. Persons D and C are good friends, therefore person D informed C about the incident. Person C then discussed the issue with Person A who had a one-to-one meeting with person B. The outcome of the meeting was that Person B should have an informal meeting with person C to resolve the issue. However, person B avoided holding this meeting and instead chose to speak to person C on the shop floor in the presence of customers. (See appendix 2 for the transcript of the scenario). 3. 0 Transactional Analysis and Effective Communication Transactional Analysis assists when evaluating this situation as the model is a popular way of explaining the dynamics of interpersonal communication. It was developed by Eric Berne in 1949 and has two fundamental assumptions; all the events and feelings people experience are stored within them and can be replayed, and that personality is made up of three ego states that manifest themselves in gesture, tone of voice and actions. The child ego state is described as the ‘feelings state’ and involves people behaving as they did when they were a child. This includes three sub-states which are the ‘free or natural child’, the ‘little professor’ and the ‘rebellious child’. The free or natural child state focuses on genuine feelings, acting on impulse and letting others know how we feel. The little professor state is creative, questioning and experimental. As the name suggests, the rebellious child state invokes rebellion, frustration and withdrawal. The adult ego state involves behaviour that concerns thought processes and can be defined as ‘the thoughtful’ state. This state focuses on data collection, reality testing and objectiveness. The parent state is described as the ‘taught’ state and consists of two sub-states; the nurturing and the critical parent. In this state, people take responsibility and tend to behave in ways learnt from parental figures. The nurturing parent state involves caring for other people, whereas in the critical or controlling parent state people have a tendency to lay down rules and boundaries and insist on their own method of getting the job done. Exclusions of ego states occur when someone is permanently using one ego state and cuts off the others (see appendix 7). There are three types of transactions in communication; complementary, crossed and ulterior (see appendix 3). When both parties’ ego states match, this is a complementary transaction and communication can continue. Crossed transactions occur when one party addresses a different ego state to the one the other party is currently in. The communication in crossed transactions disintegrates and can result in bad feelings. Ulterior transactions involve a crossed transaction on a psychological level, however on the surface the ego states seem to match leading to people playing games with one another. Strokes are units of recognition and are given and received via the five senses. Positive strokes are life and growth encouraging, whereas negative strokes are the opposite and cause the recipient to feel dejected. Transactional analysis assumes that our characteristic ways of feeling and behaving derive from the way we feel about ourselves in relation to other people. These are referred to as the four life positions and consist of â€Å"I’m not OK, You’re OK†, â€Å"I’m not OK, You’re not OK†, â€Å"I’m OK, You’re not OK† and â€Å"I’m OK, You’re OK† (see appendix 4). Body language is another method used to communicate and can assist when deciphering an underlying message that someone is trying to purvey. According to Pivcevic, â€Å"it is commonly agreed that 80 per cent of communication is non-verbal† (Mullins, L. J, 2010, pp 235). Effective communication is achieved by attending, reflecting and following (see appendix 5). This benefits both the listener and the speaker as it aids the listener in thoroughly understanding what the speaker is saying. Attending is non-verbal communication that signifies someone is paying careful attention to the person talking. Attending includes body posture, gestures, eye contact and an environment free of distractions. Following skills require the listener to offer openers and encouragements. Openers are non-coercive invitations for the speaker to talk and include judgemental, reassuring and advice statements. Opening questions and silence can be used as they encourage and concentrate on the concerns of the speaker rather than the listener. Reflecting skills avoid both speaker and listener problems. Words are perceived differently to people and listeners can often become distracted. Reflective responses are non-judgmental and help the listener to grasp the feelings of the speaker. Guirdham’s cycle of perception and behaviour can also aid in analysing communication as perceptions can alter the way in which we behave, thus having an effect on communication (see appendix 8). 4. 0 Analysis of the scenario By applying the Transactional analysis model, it is evident that when person B approached C, she was speaking from her critical parent ego state. This state is condescending and admonishing and can cause the addressee to feel discouraged. When replying, person C speaks from her adult ego state which is objective and rational, presenting a crossed transaction as B was addressing a different ego state to that of which C is currently in (see appendix 3). Person B should have shifted to an adult ego state to ensure that the states matched, amending it to a complementary transaction. However, B replies she has no time denoting that she is speaking from her critical parent ego state and sending out negative strokes. Her abrupt and loud tone insinuates she is defensive and angry. Her body language also gives an implication of her underlying message as she is walking away from the situation with her arms crossed, suggesting she is uninterested. Person C is rational and relaxed with her body language, making constant eye contact and positioning herself closely to person B, signifying she is listening intently. C’s ego state shifts to a rebellious child state when B’s body language and attitude is perceived as rude, abrupt and unconcerned. This subliminal communication causes an argument to break out and C begins to speak vociferously. The clenching of her fists and words spoken infer this shift in ego state. A change in behaviour occurs due to C’s perceptions of B’s behaviour (see appendix 8). Person A then interrupts the conversation and speaks from a nurturing parent ego state; this is presumed as he interjects with a question, â€Å"are you okay guys? † He places a hand on person C’s shoulder, signalling a display of power over her. At this point, person B begins to fiddle with her pen, suggesting a transition out of her comfort zone and showing she is uncomfortable in the situation. By this point, person C is very distressed and is deep in a rebellious child ego state. Her body language conveys feelings of anger and frustration as she is frantically waving her arms. Person B is reluctant to apologise or be sympathetic throughout the incident, indicating her ego state has not changed. This implies that she is currently in an arrogant life position as she feels she is not in the wrong (see appendix 4). She walks away, with her arms crossed expressing hostility and disregard to the situation. Person C reverts back to an adult ego state towards the end of the conversation and realises that she needs to calm down and clear her head. She also displays anxiety as she begins to bite her lip. Person A has maintained a nurturing parent ego state throughout as he is caring and tries to control and pacify the situation. 5. 0 Conclusion In conclusion, person B has inadequate communication skills. The crossed transaction, exclusion of other ego states and current life position (see appendix 4) of person B combine together to make her appear arrogant and uninterested, leading to conflict between the two parties. Attending, following and reflecting skills (see appendix 5) should have been applied to the conversation on B’s part to ensure effective communication took place. Person B’s disregard to instructions given to her by A could be due to the age gap between the two. According to Hart (Mullins, L. J, 2010, pp 101), age gaps can lead to conflict in the workplace as there is a dispute between age and experience. 6. 0 Recommendations To avoid this situation recurring, person B should receive training on interpersonal skills (see appendix 6), attending, following and listening (see appendix 5), enabling her to understand her own behaviour, other points of view and improve communication skills. Person A should hold an informal, one-to-one meeting with B and discuss possible outcomes of the meeting, such as training. Person A should identify whether B is in a constant ‘arrogant or cosmetic’ life position as she could have been having a bad day when the argument broke out. If it is found that her constant life position is ‘I’m OK, you’re not OK’ then an attempt should be made to modify this as it has a negative effect on communication. Person A should ensure this is carried out in a conscientious manner to prevent another conflicting situation from occurring. Person A should avoid singling out B as this could demotivate her from joining work shop training, so should offer the opportunity to every employee. This informal, fun atmosphere may help to improve person B’s opinions of others and alter her current life position. Another method of altering person B’s life position is to offer counselling but should be suggested at a later date if workshops fail.

Frankenstein as a gothic novel Essay -- English Literature

Frankenstein as a gothic novel The gothic tradition highlights the grotesque, relies on mysterious and remote settings, and is intended to evoke fear. All of these are evident in Mary Shelley's Frankenstein, especially in chapter five. The settings in the novel are striking and distinctively gothic. Appropriately, the creature first breathes on a "dreary night of November," in a remote laboratory at Ingolstadt. The eerie atmosphere is typical of the gothic tradition. Victor, unafraid of the dark, spends his time in "vaults and charnel-houses,† he boldly visits the cemetery at the dead of night. details such as the creaking doors, the soft blowing of the wind in the still of the night, and the quiet footsteps in the house all lead to a feeling of fear and suspense. On a certain level, Victor's interest in creating life is an extension of his desire to escape death. By assembling the body parts of the dead, Victor makes a "monster", a massive, grotesque being, with the mind of a new born baby; and like a tormented spirit, the creation haunts Victor’s mind. Analysis: Chapters 3–5 The first three chapters give the reader a sense of impending doom, and chapter four depicts Victor on the way to tragedy. The creation of the monster is a grotesque act, far removed from the triumph of scientific knowledge for which Victor had hoped. His nightmares reflect his horror at what he has done and also serve to foreshadow future events in the novel. The images of Elizabeth â€Å"livid with the hue of death† prepare the reader for Elizabeth’s eventual death and connect it, however indirectly, to the creation of the monster. Victor’s pursuit of scientific knowledge reveals a great deal about his perceptions of sc... ... comments such as â€Å"I fear, my friend, that I shall render myself tedious by dwelling on these preliminary circumstances† both remind the reader of the target audience (Walton) and help indicate the relative importance of each passage. Shelley employs other literary devices from time to time, including apostrophe, in which the speaker addresses an inanimate object, absent person, or abstract idea. Victor occasionally addresses some of the figures from his past as if they were with him on board Walton’s ship. â€Å"Excellent friend!† he exclaims, referring to Henry. â€Å"How sincerely did you love me, and endeavor to elevate my mind, until it was on a level with your own.† Apostrophe was a favorite of Mary Shelley’s husband, Percy Bysshe Shelley, who used it often in his poetry; its occurrence here might reflect some degree of Percy’s influence on Mary’s writing.

Head & Shoulder Marketing Essay

Benefits that consumers get when using Head & Shoulders Anti Dandruff (H&S) shampoo include eliminating dandruff, leaving consumer with 100% flake free hair, nourished and healthy scalp and also oiliness control over their hair. Apart from that, consumers using H&S shampoo can solve any scalp itchiness associated with dandruff. Besides that, the menthol content in the shampoo gives consumer a cooling effect and refreshing after washing their hair (The Benefits of H&S 2011). The H&S Anti dandruff shampoo bottle is a white rectangular shaped bottle with a royal blue upper part where the cover is, having a few different colored strips on it depending on the variant of the product. Using attractive white and royal blue as its base packaging and also freshness essence helps create a positive impression for consumers. The labeling of this product can be found at the back of the bottle. It includes some information for the consumer such as a direction on how to use the shampoo to achieve the best result and also a caution part where consumers are reminded of their safety when using the shampoo. Furthermore, the labeling includes additional information and reminder to consumers that this product helps fight five signs of dandruff which are flakes, itchiness, dryness, oiliness and irritation. Lastly, a list of ingredients used for the shampoo can be found in the labeling and also information of H&S’s manufacture- Procter and Gamble. H&S shampoo has a unique feature that is it is rich in Pyrithione Zinc (PTZ), the only active ingredient, which is highly effective at fighting and also preventing dandruff symptoms because of its anti-fungal properties. With PTZ, H&S created HydraZinc complex which is found in all H&S products. This formula allows consumers to control and also act on dandruff quickly (Our formula, 2011). Other than that, the ActiZinc formula which adds natural cool menthol to H&S’s menthol shampoo gives consumer a cooling and refreshed effect after washing their hair as mentioned earlier (Refreshing 2011). Due to H&S shampoo is a convenience product, there are no extra services consumer get from this product but online members of Procter & Gamble are able to enjoy free samples of H&S Anti Dandruff shampoo and also free expert tips and advice of everyday life (Coupons, Samples and Savings, 2011). As for product classification, H&S Anti Dandruff shampoo is a consumer product and normally bought by consumers  for personal use. Consumers, especially those with dandruff symptoms, are willing to spend some time in looking for the availability of this product because of its unique function in treating dandruff. In the anti dandruff shampoo market, H&S has a few competitors such as Clear anti dandruff shampoo and Sunsilk anti dandruff shampoo, which are considered as acceptable substitute products for consumers. Normally brand conscious and loyal customers would spend more time in seeking this product before they turn to other brands or even do not buy at all compared to normal consumers. H&S anti dandruff shampoo comes with 11 varieties in the market based on consumer needs. The product is also divided into regular or menthol content. Each product has 3 SKUs (Stock-Keeping Unit) consisting 80ml, 200ml and 380ml. The introduction of PTZ formula into the product has differentiated H&S Anti Dandruff from other anti dandruff shampoos. 1.2 Current Pricing In Malaysia, anti-dandruff shampoo’s market is slightly saturated with a number of sellers of different brands; therefore H&S shampoo is in a monopolistic competition market. This is the case because there are different brands offering products with the same function and almost similar price as H&S. Examples of the competitors are Sunsilk Anti Dandruff shampoo, Pantene Anti Dandruff 2 in 1 shampoo, Clear Anti Dandruff Scalp Oil Control shampoo and also Dove Anti Dandruff shampoo (Shampoo Catalogue 2011). A change in price of this product would definitely change consumers’ demand but preference is also a factor in this case. An increase in price will lead to a slight decrease in the quantity demanded although there are available substitute products with the same function but some consumers will still prefer H&S. An example on how the product’s price can affect its volume is shown when Procter & Gamble with the intention of increasing sales volume of H&S in India cuts down the products prices (Sharma Shailaja 2011). The current price of the product is slightly higher than all its competitors in the market, close to around Ringgit Malaysia (RM) 1.00. Consumers of H&S Anti Dandruff shampoo normally based their purchase decision on the quality and feature of this product. It is clear that this product costs higher than its competitors, the reason why consumers choose this product because of its effectiveness in treating dandruff issues and most victims of dandruff symptoms find this product a solution to their problems. Besides that, the  company has successfully linked H&S shampoo’s to anti dandruff through its advertisement. This creates a brand image among consumers that when it comes to dandruff they would refer it to H&S shampoo. This established brand image also plays a part in influencing consumers’ purchasing decision. Business’s costs are not very important when pricing this product because Procter and Gamble (P&G), a multinational company has 12 different hair care brands under it and also other products (Hair Care Brands 2011), because of P&G’s products diversification, the pricing of H&S is not based on the business’s costs. H&S is using price-quality inference strategy therefore it has a higher price compared to other brands. 1.3 Current Distribution This product is available for consumers in many locations around Malaysia. In Miri, there are a few locations and shops where this product can be found. At Bintang Plaza, convenience stores such as Guardian and Watsons are selling the product and also hypermarkets such as Giant Hypermarket and Survey Hypermarket. Supermarkets selling H&S Anti Dandruff shampoo are the GK supermarket at Curtin University and also the Pick & Save Mart found at the shop lots in Senadin area, Miri (Supermarkets Directory 2011). In Malaysia, Procter & Gamble (M) Sdn. Bhd. is the wholesaler of H&S shampoo and the products are imported by the company before distributing to retailers around the country. All the products are imported from Procter & Gamble (Thailand) Ltd. which is the manufacturer for H&S shampoos of Malaysia and the Philippines (Manufacturers 2011). However, consumers who wish to buy directly from the wholesaler could do so but normally very few would go by this mean because of the availability of the products in most stores and online purchasing services. The factor that influences the wide distribution of this product includes the price of the product which is low and affordable for most consumers and also a convenience product. Apart from that, another characteristic is that the product comes in three bottle sizes ranging from 80ml, 200ml and 380ml. The small product size requires only a little space for display in most stores, which contributes to why it is widely distributed. 1.4 Current Promotion One of the promotional message for H&S shampoo is â€Å"All-day itch relief. Guaranteed.†. This message is promoted at the product’s official website (Head & Shoulders 2011). The message is to assure and guarantee consumers that the product will solve any itchiness or similar kind of issues on their hair. The other promotional message of the brand is â€Å"100% flake-free† that is found in a commercial advertisement on TV. This message is showed at the last part of the advertisement together with H&S anti dandruff shampoo bottles. The message H&S is trying to bring out is the effectiveness and confidence it has in removing dandruff or flakes on people’s hair. The promotion of this product through advertisement and also internet has the intention of establishing the brand in the market so that consumers realize the existence of such product that comes with special functions. It is obvious that H&S are using some promotional tools in promoting its product such as advertising through TV, advertising online and also discounts. The purpose of promotion is that the company wants to send a message to consumers that H&S anti dandruff shampoo is a solution for dandruff symptoms patients and people who are fed up with hair itchiness. By doing this, not only the company can establish the brand in the market but also increase sales volume as promotion can be a big tool in ensuring success. The theme of the current promotions is emphasizing on the effectiveness of H&S anti dandruff in removing dandruffs and solving itchiness on the hair. In all the promotional tools that the company uses, the company has been consistent in shaping messages that reflect the theme of the promotions. Another example of the company’s promotional tool is a slogan â€Å"I Never Knew You Had Dandruff† (List of Advertising Slogans 2011). 2.0 Segmentation Theory 2.1 Define and Explain Segmentation Market segmentation is a strategy that involves dividing a larger market into subsets of consumers with common needs. This can be done through the process of splitting potential customers into different groups, within which customers share similar interests for the goods or services offered in the market (Market Segmentation 2011). Market segmentation strategies can be developed over a wide range of characteristics found among consumers, for example one group may be identified by gender while the other group composes of consumers within a given age group (Malcolm Tatum 2011). Market  segmentation has allowed H&S to benefit from the anti dandruff market in a few ways. The first benefit gained is H&S was able to better understand consumers’ needs in the market (Rupal Jain 2011). This is shown when H&S differentiate its product and came out with H&S anti dandruff shampoo, obviously wanting to help consumers that have a need in solving dandruff issues after the company found out the problem through market segmentation. The next benefit to the brand from using segmentation strategy is H&S through understanding and focusing on the needs of customers, was able to get ahead of its competitors, in this case anti dandruff. H&S anti dandruff shampoo succeeded in being the market leader by becoming the world’s No.1 brand in anti dandruff shampoo market (About Head and shoulders 2011). The third benefit from segmentation that the brand enjoyed is consumers now buy their product because they are able to focus their marketing on consumers who are most likely to buy H&S anti dandruff shampoo. On emphasizing on the ability of the product to remove dandruff, refreshing and also prevent itchiness, H&S was able to focus particularly on a group of consumers in the market which leads to a possible higher profit from the sales to these consumers who share the same interest. In 2007, the introduction of PTZ as an active ingredient to the shampoo, which is highly effective in fighting dandruff because of its anti-fungal properties is an effort of H&S to further focus on marketing this product to its segmented market. The three benefits mentioned reflects that market segmentation is used as a strategy by H&S which has successfully benefited the brand so far. 3.0 Target Market Identification 3.1 Geographic Segmentation H&S can practice geographic segmentation for its product in Malaysia. The company should extend its product marketing into the state of Johor, the southern state of West Malaysia. The capital city of Johor is Johor Bahru and this would be the focused city for H&S, with a city size of 1.37million. A density of 7,409 per km2 makes Johor Bahru the second largest urban area in Malaysia (Introduction to Johor Bahru, 2011). The main reason H&S should target the above city mentioned is because in urban areas, consumers have higher purchasing power and are more health conscious compared to consumers from rural areas. The state of Johor has an average temperate ranging from  73 °F to 89 °F each year and has high humidity (Monthly averages for Johor Bahru, 2011). The hot and humid weather are causes for dandruff or itchiness for most people. With a high density city, hot and humid weather, Johor Bahru is a potential market to be targeted by H&S that provides refreshing and healthy hair for its users. 3.2 Demographic Segmentation Demographic factors are popular bases for segmentation analysis. Based on other markets of H&S, the target market should be focused on Generation-Y, the age group of 18-30 years old male Malaysian consumers, with middle or high income level. This target group is a group that is brand conscious and care about the health of their hair. They are willing to spend more on a product as long as the product benefits them or provides a solution to their problem. The H&S anti dandruff shampoo will be a good product to this targeted group. Not only the product has many benefits to the user, considered expensive among its competitors and fights dandruff or itchiness effectively, H&S anti dandruff shampoo gives users a healthy hair, which is suitable for those with active social life. Besides, the fact that H&S is the No.1 anti dandruff brand in the world makes it attractive to the targeted group which is brand conscious. In terms of occupation and education level in an urban area, those which will be targeted are most likely professionals, managers, officers, high school graduates, businessmen and also college students. In short, the young adults are going to be the primary target of H&S. Reference List About Head and Shoulders. 2011. Head and Shoulders. Accessed 5 April,http://www.headandshoulders.com/en-US/about-us.jspx. Benefits of Head and Shoulders. 2011. Head and Shoulder. Accessed April 4,http://www.headandshoulders.com/en-US/sevenBenefits.jspx. Coupons, Samples and Savings. 2011. Procter and Gamble Brand Sampler. Accessed 4April,https://www.pgeverydaysolutions.com/pgeds/headshoulderbrandsampler.jsp. Hair Care Brands. 2011. P&G. Accessed 5 April,http://www.pg.com/en_US/brands/beauty_grooming/index.shtml. Head and Shoulders. 2011. Accessed 5 April, http://www.headandshoulders.com/en-US/index.jspx. Introduction to Johor Bahru. 2011. Popular Places. Accessed 5 April,http://www.marimari.com/content/malaysia/popular_places/cities/johor_bahru/johor_bahru.html. Jain, Rupal. 2011. Market segmentation. 123oye. Accessed 5 April,http://www.123oye.com/. List of Advertising Slogans. 2011. Sharing The Secrets of Internet Marketing Success. 5April,http://www.nowsell.com/marketing-guide/list -of-advertising-slogans.html#Personal-Care. Manufacturers. 2011. Procter & Gamble (Malaysia) Sdn. Bhd.. Accessed 5 April,http://www.tradenex.com/sites/proctergamble/. Market Segmentation. 2011. The Market Segmentation Company. Accessed 5 Aprilhttp://www.marketsegmentation.co.uk/segmentation_tmsc.htm. Monthly Averages for Johor Bahru. 2011. The weather channel. Accessed 5 April,http://www.weather.com/weather/wxclimatology/monthly/graph/MYXX0004. Murugiah, Surin. 2008. â€Å"Average household income in Malaysia.† The Edge

Flight Control Systems

Flight Control Systems W. -H. Chen Department of Aeronautical and Automotive Engineering Loughborough University 2 Flight Control Systems by W. -H. Chen, AAE, Loughborough Contents 1 Introduction 1. 1 Overview of the Flight Envelope 1. 2 Flight control systems . . . . . . 1. 3 Modern Control . . . . . . . . . . 1. 4 Introduction to the course . . . . 1. 4. 1 Content . . . . . . . . . . 1. 4. 2 Tutorials and coursework 1. 4. 3 Assessment . . . . . . . . 1. 4. 4 Lecture plan . . . . . . . 1. 4. 5 References . . . . . . . . . 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 Longitudinal response to the control 2. 1 Longitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2 State space description . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 State variables . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 2 General state space model . . . . . . . . . . . . . . . . . . . 2. 3 Longitudinal state space model . . . . . . . . . . . . . . . . . . . . 2. 3. 1 Numerical example . . . . . . . . . . . . . . . . . . . . . . . 2. 3. 2 The choice of state variables . . . . . . . . . . . . . . . . . . 2. 4 Aircraft dynamic behaviour simulation using state space models . 2. 4. 1 Aircraft response without control . . . . . . . . . . . . . . . 2. 4. 2 Aircraft response to controls . . . . . . . . . . . . . . . . . 2. 4. 3 Aircraft response under both initial conditions and controls 2. 5 Longitudinal response to the elevator . . . . . . . . . . . . . . . . 2. 6 Transfer of state space models into transfer functions . . . . . . . . 2. 6. 1 From a transfer function to a state space model . . . . . . . 2. 7 Block diagram representation of state space models . . . . . . . . . 2. 8 Static stability and dynamic modes . . . . . . . . . . . . . . . . . . 2. 8. 1 Aircraft stability . . . . . . . . . . . . . . . . . . . . . . . . 2. 8. 2 Stability with FCS augmentation . . . . . . . . . . . . . . . 2. 8. 3 Dynamic modes . . . . . . . . . . . . . . . . . . . . . . . . . 2. 9 Reduced models of longitudinal dynamics . . . . . . . . . . . . . . 2. 9. Phugoid approximation . . . . . . . . . . . . . . . . . . . . 2. 9. 2 Short period approximation . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Lateral response to the controls 3. 1 Lateral state space models . . . . . . . . . . . . 3. 2 Transient response to aileron and rudder . . . . 3. 2. 1 Numerical example . . . . . . . . . . . . 3 . 2. 2 Lateral response and transfer functions 3. 3 Reduced order models . . . . . . . . . . . . . . 3. 3. 1 Roll subsidence . . . . . . . . . . . . . . 3. 3. Spiral mode approximation . . . . . . . 3. 3. 3 Dutch roll . . . . . . . . . . . . . . . . . 3. 3. 4 Three degrees of freedom approximation 3. 3. 5 Re-formulation of the lateral 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 systems . . . 4. 2. 1 The choice of feedback variables . . . . 4. 2. 2 SAS for short period dynamics . . . . . . 4. 3 Lateral stability augmentation systems . . . . . . 4. 3. 1 Yaw rate feedback for rudder control . . . 4. 3. 2 Roll feedback for aileron control . . . . . 4. 3. 3 Integration of lateral directional feedback 5 Autopilots 5. 1 Pitch holding autopilot . . . . . . . . . . . . . . . . . . . . . . . 5. 1. 1 phugoid suppress . . . . . . . . . . . . . . . . . . . . . . 5. 1. 2 Eliminate the steady error with integration . . . . . . . 5. 1. 3 Improve transient performance with pitch rate feedback 5. 2 Height holding autopilot . . . . . . . . . . . . . . . . . . . . . . 5. . 1 An intuitive height holding autopilot . . . . . . . . . . . 5. 2. 2 Improved height holding systems . . . . . . . . . . . . . 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 system . . 6. 2. 3 Pilot-in-loop . . . . . . . . . . . . . . . . . 6. 3 Flying qualities requirements . . . . . . . . . . . . 6. 4 Aircraft role . . . . . . . . . . . . . . . . . . . . . . 6. . 1 Aircraft classi? cation . . . . . . . . . . . . . 6. 4. 2 Flight phase . . . . . . . . . . . . . . . . . . 6. 4. 3 Levels of ? ying qualities . . . . . . . . . . . 6. 5 Pilot opinion rating . . . . . . . . . . . . . . . . . . 6. 6 Longitudinal ? ying qualities requirements . . . . . 6. 6. 1 Short perio d 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 subsidence 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 data . . . . . . . . . . . 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 Flight Envelope †¢ Flight planing †¢ Aircraft checking †¢ Taxi †¢ Take-o? – Rotate, â€Å"select† an attitude – Clean up (gear, ? aps, etc) – Emergencies (engine failure, ? re, etc) †¢ Climb – Speed control – Procedure (manual, autopilot) †¢ Mission Tasks – Cruise – Combat (air to air) – Strike (air to earth) – General handling (stalling, spinning, aerobatics) – Formation ? ing (Navigation, procedure etc) – Emergencies – Con? guration (weapons, tanks, fuel load) †¢ Recovery – Descent – Instrument approach – Landing – Overshoot 7 8 CHAPTER 1. INTRODUCTION Stick – Linkage 6 Trim ? -? Servo Actuator – Aircraft dynam ics Figure 1. 1: Manual pilot control aircraft – Formation – Procedures – Emergencies †¢ Taxi Longitudinal and lateral dynamics thus Flight control systems are involved in Take o? , Climb, Mission tasks and Recovery. †¢ Di? erent aircraft (aircraft class) †¢ Di? erent ? ight phase Manual– handling 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 fully †¢ To increase the performance of aircraft or missiles Types of Flight Control Systems (FCS) 1. Open-loop control 2. Stability augmentation systems 3. Autopilot 4. Integrated Navigation systems and Autopilots (? ight management systems) 1. 3 Modern Control †¢ Classic control– transfer function – frequency domain †¢ Limitation of classic design method: single input, single output (SISO), only conc ern the output behaviour, linear systems (saturation) †¢ System description in state space form. 1. 4.INTRODUCTION TO THE COURSE 9 Stick Trim – Aircraft dynamics – + ? + -Linkage – ? – ? – Servo Actuator 6 6  Stability Aug. Systems  Sensor  ? Figure 1. 2: Stability Augmentation Systems Reference Command + -? Autopilot – 6 6 + -? 6 – SAS – Actuators – Aircraft dynamics – Sensor  6  Navigation Systems ? ? Figure 1. 3: Autopilot con? guration †¢ Describe aircraft or other dynamics systems in a set of ? rst order di? erential equations. Expressed in a matrix form †¢ State space analysis and design techniques– very powerful technique for control systems †¢ Matrix manipulation knowledge required 1. 4 1. 4. 1 Introduction to the courseContent This course will cover †¢ state space analysis and design techniques for aircraft †¢ simple ? ight control systems including stability aug mentation 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 Week 3 †¢ One tutorial section in each week †¢ One coursework based on MATLAB/Simulink simulation, must be handed in before 4:00 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, references 1. Introduction 2. Response to the controls (a) State space analysis (b) Longitudinal response to elevator and throttle (c) Transient response to aileron and rudder 3. Aircraft stability augmentation systems 1. 4. INTRODUCTION TO THE COURSE (a) Performance evaluation †¢ †¢ †¢ †¢ stability Time domain requirements Frequency domain speci? ations Robustness 11 (b) Longitudinal Stability Augmentation Systems †¢ Choice of the feedback variables †¢ Root locus and gain determination †¢ Phugoid suppress (c) Lateral stability augmentation systems †¢ Roll feedback for aileron control †¢ Yaw rate feedback for rudder control 4. Simple autopilot design †¢ Augmented longitudinal dynamics †¢ Height hold systems 5. Handling Qualities (a) Time delay systems (b) Pilot-in-loop dynamics (c) Handling qualities (d) Frequency domain analysis (e) Pilot induced 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. McL ean. 1990. Prentice Hall International Ltd. Chaps. 2, 3,6,9. 3. Introduction to Avionics Systems. Second edition. R. P. G. Collinson. 2003. 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 can 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 are de? ned as u: Perturbation about steady state velocity Ue w: Perturbation on steady state normal velocity We q: Pitch rate ? : Pitch angle Under the assumption that the aeroplane is in level straight ? ight and the reference axes are wind or stability axes, we have ? e = We = 0 (2. 4) The main controls in longitudina l dynamics are the elevator angle and the engine trust. The small perturbation terms in the right side of the above equations can 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 RESPONSE TO THE CONTROL ? e : the elevator de? ection (Note ? is used in Appendix 1) ? : engine thrust perturbation Substituting the above expression into the longitudinal symmetric motion 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) can be put in a more concise vector and matrix format. The longitudinal dynamics can 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 DYNAMICS Equation (2. 12) becomes 15 ? MX = A X + B U (2. 18) It is custom to convert the above set of equations into a set of ? rst order di? erential equations by multiplying both sides of the above equation by the inverse of the matrix 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 can be written in a concise 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 between the new stability and control derivatives in the matrix 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 are derived based on the Cramer’s rule and hold for general body axes. In the case when the derivatives are 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 matrix-vector format as in (2. 19) can be extended to represent all general dynamic systems. Consider a system with order n, i. e. , the system can be described by n order di? rential equation (as it will be explained later, this is the same as the highest order of the denominator polynomial in the transfer function is n). In the representation (2. 22), A ? Rn? n is the system matrix ; B ? Rn? m is the input matrix ; X ? Rn is the state vector or state variables and U ? Rm the input or input vector. The equation (2. 22) is called state equation. For the stability augmentation system, only the in? uence of the variation of the elevator angle, i. e. the primary aerodynamic control surface, is concerned. The above equations of motion can be simpli? ed. The state space representation remains the 6 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL same format as in eq. (2. 22) with the same matrix A and state variables but with a di? erent B and input U as given 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 are used. For the state space representation of longitudinal dynamics, sometime 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 expression of the small letter derivatives under certain assumptions, i. e. using stability axis. 2. 2 2. 2. 1 State space description State variables A minimum set of variables which, when known at time t0 , together with the input, are su? ient to describe the behaviours of the system at any time t > t0 . State variables may 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 one input or control variable, the elevator de? ection, U = ? e (2. 33) 2. 3. LONGITUDINAL STATE SPACE MODEL Thus 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 angle of attack ? is of concern, it can be written as ? = which can 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 path 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 interest including the ? ight path angle ? , the angle of attack ? and the normal acceleration az (nz ).Putting all variables together, the output vector can be written a s 18 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL ? ? ? ? ? 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 concise derivative format. Hence 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 747 jet transpor t at ? ight condition cruising in horizontal ? ight at approximately 40,000 ft at Mach number 0. 8. Relevant data are given in Table 2. 1 and 2. 2.Using tables in Appendix 1, the concise small derivatives can be calculated 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 determined. 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) 5500 ft2 (511. m2 ) 27. 31 ft (8. 324 m) 195. 7 ft (59. 64 m) 0. 183 ? 108 slug ft2 (0. 247 ? 108 kg m2 ) 0. 331 ? 108 slug 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 jet 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), another widely used choice (in American) is ? u ? ? ? ? X=? ? q ? ? ? (2. 49) Certainly, when the logitudinal dynamics of the aircraft are represented in terms of the above state variab les, di? rent A, B and C are resulted (see Tutorial 1). 2. 4 Aircraft dynamic behaviour simulation using state space models State space model developed above provides a very powerful tool in investigate dynamic behavious of an aircraft under various condition. The idea 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 dynamics are descri bed 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 velocity u, normal velocity w (angle of attack) and ? ight path angle ? under the step movement of the levator are displayed in Fig 2. 1–2. 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 seriously interfered with it. A good operation performance cannot be achieved by simply changing the angle of elevator. Clearly, longitudinal control, whether by a human pilot or automatic pilot, demands a more sophisticated control activity than open-loop strategy. 2. 6 Transfer of state space models into transfer functions Taking Laplace transform on both sides of Eq. (2. 2) under the zero initial assumption yields sX(s) = Y (s) = where X(s) = L{X(t)}. AX(s) + BU (s) CX(s) (2. 54) (2. 55) 2. 6. TRANSFER OF STATE SPACE MODELS INTO TRANSFER FUNCTIONS21 Step 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) 400 500 600 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 50 0 600 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) = C[sI ? A]? 1 BU (s) Hence the transfer function of the state space representation is given by G(s) = C[sI ? 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 determined 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 outputs, u, w, q, ? , ? , az . Using the techniq ue 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 derivation of the corresponding transfer function from its state space model can be completed manually. For complicated systems with high order, it can be done by computer software like MATLAB. It can be found that although the transfer functions from the elevator to di? erent outputs are di? erent but they have the same denominat or, i. e. s4 + 0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959 for Beoing 747-100. Only 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 original aircraft dynamics concerns if there is no any control e? ort,whether the uncontrolled motion is stable. It is also referred as openloop stability in general control engineeri ng. 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 roots of the characteristic 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 purely 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 aircraft under ? ight co ntrol systems is refereed as closed-loop stability. 26 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Then the closed-loop system under the control law 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 fed 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 complex roots, the denominator can be written in the typical second order system’s 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 poor 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 quickly and only has the in? uence at the beginning of the response. 2. 9 Reduced mode ls 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 similar response behaviour as Boeing 747. This makes it is possible to simplify 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 full 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 assume 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 are rel atively 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 transform 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 substit uting 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 slightly 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 while 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 t hose obtained 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 Normal 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 di splacement. 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. Find 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 Engineeri ng (When Routh stability criterion is applied, you can study the stability using the transfer function from ? to q or that from ? e to ? (why? )). Compare 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 relationships ? ?= 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 elements 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 just 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, another widely used choice of the state variables (American system) is to replace the lateral velocity v by the sideslip angle ? and keep all others. Remember 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 derivatives ar e 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, gi ven 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 transform of the state space model, 34 CHAPTER 3.LATERAL RESPONSE TO THE CONTROLS G(s) ? Rr? m is a complex functi on 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. 004534 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 ORDER MODELS 35 It has one large real root, -0. 5613, one small real root, -0. 0073 (very close to origin) and a pair of complex 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 subsidence 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 produce 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 useful ? 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 investigating the lateral response to the controls. This can be considered as an idealised situation 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 due 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 a nd takes quite long time to respond. The Dutch roll mode is quite poorly damped and the oscillation caused by the Dutch roll dominates 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 coupling 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, reduced 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 reduced 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 giv e p = lp p + l? A ? A + l? R ? R ? (3. 34) If only the in? uence from aileron de? ction is concerned and assume 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 relative to the time scale of the mo de. 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 third 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, the estimated spiral mode time c onstant 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 Third 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 wit h the values given by the second order Dutch roll approximation, i. e. , 0. 1093 and 0. 92 rad/sec, they are a little bit 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 of interest, ? nd the output equa tion. 2. Find the second order Dutch roll reduced model of this airplane. Derive 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.

Ferragamo Case Analysis Essays

Ferragamo Case Analysis Essays Ferragamo Case Analysis Essay Ferragamo Case Analysis Essay Essay Topic: Marketing Ferragamos strategy in Salvatores times was a narrow, differentiation strategy by Porters definition. They had upper class customers limited in region, and sold shoes differentiated in innovative design and craftsmanship. Now in the family times, Ferragamo has a broad, differentiated strategy. They still have upper class people as their customers and differentiate themselves in quality, but they serve customers from all around the world and produce clothes and fashion accessories as well as shoes. Strategy Recommendation I recommend that Ferragamo should keep focusing on the global upper class customers and offer quality and luxurious goods, but also broaden their strategy in terms of customer age and gender, and different product lines. External Factors : Porters Five Forces Model (Exhibit 1) shows the external environment that Ferragamo is in. The luxurious fashion industry is a highly competitive industry. There are many companies competing in the same industry. Also, in the system of external manufacturing and the quality of the product being very important, the suppliers have high bargaining power. Rivals playing in the field must create strategies to differentiate oneself from other competitiors since the customers are not very sensitive to price, but demands value-added products. Internal Factors Using Barneys VIRO Model (Exhibit 2), we can see that Ferragamos strengths are mainly created in production, design, and organizational structure. Their stringent manufacturing standards and close relationships with the manufacturing suppliers resulted in high quality products. The consistent and practical, yet innovative designs and broad range of sizes gave the customers satisfaction, making them repeat customers. Also, Ferragamos informal organizational structure made the employees to be open-minded and creative. The current problems that Ferragamo have is that the Ferragamo family is valuing family harmony so much that their decision making is very slow and it is not a good factor for a such a competitive industry. Also, they are focusing on product quality so much that they dont think about the customers very much. They dont know exactly who their customers are and what they want from Ferragamo. Just making high quality products with a good design is appealing to customers so far, but you never know when they want something more, so Ferragamo should keep an eye on the market. Specific Implementation Strategies Ferragamo is doing a good job in creating value for the customers, thus building a good brand identity, starting in shoes. Now what they need to do is to move on to other fashion product lines, like products for all ages. For example, the sound of Ferragamo producing baby clothes and shoes will tempt many current customers since they are well known for their quality. Making products for the whole family would make synergies for the whole company. But it all has to be consistent with the whole Ferragamo image. Moving into cosmetics and watches would be a little too early, considering that those products need much RD as well as the design and aesthetic point. Implementation Threats The potential threats that can emerge during implementation are that going into too many products at once can decrease the quality of Ferragamos products. Thats exactly what Wanda thought about some of Ferragamos rivals and it could happen to them also. The quality and the number of product lines and quantity of the products can have a trade-off relationship if not managed properly. Also, the customers might think Ferragamo is losing their specialty in shoes expanding into other products lines.

Second Battle of Manassas - Civil War Second Battle of Manassas

Second Battle of Manassas - Civil War Second Battle of Manassas Second Battle of Manassas - Conflict Dates: The Second Battle of Manassas was fought August 28-30, 1862, during the American Civil War. Armies Commanders Union Major General John Pope70,000 men Confederate General Robert E. Lee55,000 men Second Battle of Manassas - Background: With the collapse of Major General George B. McClellans Peninsula Campaign in the summer of 1862, President Abraham Lincoln brought Major General John Pope east to take command of the newly created Army of Virginia. Consisting of three corps led by Major Generals Franz Sigel, Nathaniel Banks, and Irvin McDowell, Popes force was soon augmented by additional units taken from McClellans Army of the Potomac. Tasked with protecting Washington and the Shenandoah Valley, Pope began moving southwest towards Gordonsville, VA. Seeing that Union forces were divided and believing that the timid McClellan posed little threat, Confederate General Robert E. Lee sensed an opportunity to destroy Pope before returning south to finish off the Army of the Potomac. Detaching the left wing of his army, Lee ordered Major General Thomas Stonewall Jackson to move north to Gordonsville to intercept Pope. On August 9, Jackson defeated Banks corps at Cedar Mountain and four days later Lee began moving the other wing of his army, led by Major General James Longstreet, north to join Jackson. Second Battle of Manassas - Jackson on the March: Between August 22 and 25, the two armies squared off across the rain-swollen Rappahannock River, with neither able to force a crossing. During this time, Pope began receiving reinforcements as McClellans men were withdrawn from the Peninsula. Seeking to defeat Pope before the Union commanders force grew much larger, Lee ordered Jackson to take his men and Major General J.E.B. Stuarts cavalry division on a bold flanking march around the Union right. Moving north, then east through Thoroughfare Gap, Jackson severed the Orange Alexandria Railroad at Bristoe Station before capturing the Union supply base at Manassas Junction on August 27. With Jackson in his rear, Pope was forced to fall back from the Rappahannock and reconcentrate near Centreville. Moving northwest from Manassas, Jackson moved through the old First Bull Run battlefield and assumed a defensive position behind an unfinished railroad grade below Stony Ridge on the night of August 27/28. From this position, Jackson had a clear view of the Warrenton Turnpike which ran east to Centreville. Second Battle of Manassas - Fighting Begins: The fighting began at 6:30 PM on August 28 when units belonging to Brigadier General Rufus Kings division were seen moving east on the turnpike. Jackson, who learned earlier in the day that Lee and Longstreet were marching to join him, moved to the attack. Engaging on the Brawner Farm, the fight was largely against the Union brigades of Brigadier Generals John Gibbon and Abner Doubleday. Firing for around two and half hours, both sides took heavy losses until darkness ended the fighting. Pope misinterpreted the battle as Jackson retreating from Centreville and ordered his men to trap the Confederates. Second Battle of Manassas - Assaulting Jackson: Early the next morning, Jackson dispatched some of Stuarts men to direct Longstreets approaching troops into pre-selected positions on his right. Pope, in an effort to destroy Jackson, moved his men to the fight and planned attacks on both Confederate flanks. Believing that Jacksons right flank was near Gainesville, he directed Major General Fitz John Porter to take his V Corps west to attack that position. At the other end of the line, Sigel was assault the Confederate left along the railroad grade. While Porters men marched, Sigels opened the fighting around 7:00 AM. Attacking Major General A.P. Hills men, the Brigadier General Carl Schurzs troops made little progress. While the Union did achieve some local successes, they were often undone by vigorous Confederate counterattacks. Around 1:00 PM, Pope arrived on the field with reinforcements just as Longstreets lead units were moving into position. To the southwest, Porters corps was moving up the Manassas-Gainesville Road and engaged a group of Confederate cavalry. Second Battle of Manassas - Union Confusion: Shortly thereafter, its advance was halted when Porter received a confusing Joint Order from Pope which muddied the situation and did not provide any clear direction. This confusion was worsened by news from McDowells cavalry commander, Brigadier General John Buford, that large numbers of Confederates (Longstreets men) had been spotted in Gainesville that morning. For an unknown reason, McDowell failed to forward this to Pope until that evening. Pope, waiting for Porters attack, continued to launch piecemeal assaults against Jackson and remained unaware that Longstreets men had arrived on the field. At 4:30, Pope sent an explicit order for Porter to attack, but it was not received until 6:30 and the corps commander was not in a position to comply. In anticipation of this attack, Pope threw Major General Philip Kearnys division against Hills lines. In severe fighting, Kearnys men were only repelled after determined Confederate counterattacks. Observing Union movements, Lee decided to attack the Union flank, but was dissuaded by Longstreet who advocated a reconnaissance in force to set up an assault in the morning. Brigadier General John B. Hoods division moved forward along the turnpike and collided with Brigadier General John Hatchs men. Both sides retreated after a sharp fight. Second Battle of Manassas - Longstreet Strikes As darkness fell, Pope finally received McDowells report regarding Longstreet. Falsely believing that Longstreet had arrived to support Jacksons retreat, Pope recalled Porter and began planning a massive assault by V Corps for the next day. Though advised to move cautiously at a council of war the next morning, Pope pushed Porters men, supported by two additional divisions, west down the turnpike. Around noon, they wheeled right and attacked the right end of Jacksons line. Taken under heavy artillery fire the assault breached the Confederate lines but was thrown back by counterattacks. With the failure of Porters attack, Lee and Longstreet moved forward with 25,000 men against the Union left flank. Driving scattered Union troops before them, they only encountered determined resistance at a few points. Realizing the danger, Pope began moving troops to block the attack. With the situation desperate, he succeeded in forming a defensive line along the Manassas-Sudley Road at the foot of Henry House Hill. The battle lost, Pope began a fighting withdraw back towards Centreville around 8:00 PM. Second Battle of Manassas - Aftermath: The Second Battle of Manassas cost Pope 1,716 killed, 8,215 wounded and 3,893 missing, while Lee suffered 1,305 killed and 7,048 wounded. Relieved on September 12, Popes army was incorporated into the Army of the Potomac. Seeking a scapegoat for the defeat, he had Porter court-martialed for his actions on August 29. Found guilty, Porter spent fifteen years working to clear his name. Having won a stunning victory, Lee embarked on his invasion of Maryland a few days later. Selected Sources National Park Service: Manassas National BattlefieldLibrary of Congress: Second Battle of ManassasHistoryNet: Second Battle of Manassas