Aircraft Maintenance Licence Category B1 Physics Module 2 PDF

Maintenance Training Organisation Part -147 MODULE 02 Physics for Aircraft Maintenance Licence...

Maintenance Training Organisation Part -147 MODULE 02 Physics for Aircraft Maintenance Licence Category B1 www.aviotraceswiss.com  Cat. B1 - Table of Contents Table of Contents 2.1 Matter..................................................................................................................................... 3 2.1.1 Nature of matter: the chemical elements, structure of atoms, molecules..................... 3 2.1.2 Chemical compounds....................................................................................................... 8 2.1.3 States: solid, liquid and gaseous...................................................................................... 9 2.1.4 Changes between states................................................................................................ 10 2.2 Mechanics............................................................................................................................. 13 2.2.1.1 Forces, moments and couples, representation as vectors......................................... 13 2.2.1.2 Center of gravity.......................................................................................................... 16 2.2.1.3 Elements of theory of stress, strain and elastic: tension, compression, shear and torsion..................................................................................................................................... 17 2.2.1.4 Nature and properties of solid, fluid and gas.............................................................. 19 2.2.1.5 Pressure and buoyancy in liquids................................................................................ 20 2.2.2.1 Linear movement: uniform motion in a straight line, motion under constant acceleration (motion under gravity)....................................................................................... 24 2.2.2.2 Rotational movement: uniform circular motion (centrifugal/centripetal forces)...... 28 2.2.2.3 Periodic motion: pendular movement........................................................................ 30 2.2.2.4 Simple theory of vibration, harmonics and resonance............................................... 31 2.2.2.5 Velocity ratio, mechanical advantage and efficiency.................................................. 34 2.2.3.A.1 Mass......................................................................................................................... 40 2.2.3.A.2 Force, inertia, work, power, energy (potential, kinetics and total energy), heat, efficiency................................................................................................................................. 40 2.2.3.B.1 Impulse..................................................................................................................... 45 2.2.3.B.2 Momentum, conservation of momentum............................................................... 46 2.2.3.B.3 Gyroscopic principles............................................................................................... 47 Edition 22/01/2020 - Rev. 01 1  Cat. B1 - Table of Contents 2.2.3.B.4 Friction: nature and effects, coefficient of friction (rolling resistance)................... 48 2.2.4.A.1 Specific gravity and density..................................................................................... 50 2.2.4.B.1 Viscosity, fluid resistance, effects of streamlining................................................... 52 2.2.4.B.2 Effects of compressibility on fluids.......................................................................... 53 2.2.4.B.3 Static, dynamic and total pressure: Bernoulli’s theorem, Venturi.......................... 53 2.3 Thermodynamics................................................................................................................... 55 2.3.A.1 Temperature: thermometers and temperature scales: Celsius, Fahrenheit and Kelvin................................................................................................................................................. 55 2.3.B.1 Heat capacity, specific heat........................................................................................ 56 2.3.B.2 Heat transfer: convection, radiation and conduction................................................. 57 2.3.B.3 Volumetric expansion................................................................................................. 58 2.3.B.4 First and second laws of thermodynamics.................................................................. 58 2.3.B.5 Gases: ideal gases law; specific heat at constant volume and constant pressure, work done by expanding gas............................................................................................................ 59 2.3.B.6 Isothermal, adiabatic expansion and compression, engine cycles, constant volume and constant pressure, refrigerators and heat pumps........................................................... 61 2.3.B.7 Latent heats of fusion and evaporation, thermal energy, heat of combustion.......... 64 2.4 Optics (light).......................................................................................................................... 65 2.4.1 Nature of lights, speed of light....................................................................................... 65 2.4.2 Laws of reflection and refraction: reflection at plane surfaces, reflection by spherical mirrors, refraction, lenses....................................................................................................... 66 2.4.3 Fiber optics..................................................................................................................... 69 2.5 Wave motion and sound....................................................................................................... 73 2.5.1 Wave motion: mechanical waves, sinusoidal wave motion, interference phenomena, standing waves........................................................................................................................ 73 2.5.2 Sound: speed of sound, production of sound, intensity, pitch and quality, Doppler effect....................................................................................................................................... 74 Edition 22/01/2020 - Rev. 01 2  Cat. B1 - 2.1 Matter 2.1 Matter 2.1.1 Nature of matter: the chemical elements, structure of atoms, molecules By definition, the matter is anything that takes up space and has a mass. The Law of Conservation of the matter says that the matter can’t be created, or destroyed. Only its characteristics can be changed. Whenever the matter changes its state there is an energy conversion, which is the matter ability to do a work. The Greek Philosopher Democritus of Abdera spoke about the atom in 430 before Christ. He proposed the idea that indivisible atoms were the basis of all material bodies, but his idea remained only a philosophical speculation until the17th century. Only at that moment the looking for the nature of compressed gasses, magnetism and static electricity, has gradually collected observations explained only by the presence of atoms, and of parts composing atoms. Among the first inquires in this field, we can remember Robert Boyle, an English man who published, in 1661 the book “The Sceptical Chemist”. In his book, he quoted results of some experiments on gasses and gave the notion of the chemical element. According to Boyle, the chemical element was, one of the simple substances, of which the world was made up, and it wasn’t possible to divide it into simpler chemical elements. A century and half after Boyle, Dalton gave the modern atomic theory. Based on an accurate chemical analysis of the past two centuries, Dalton could demonstrate that each chemical element must be composed of atoms of the same weight, and that two different elements like copper and silver for example, are different because composed of atoms of different weight. Approximately while chemistry was moving forward the atomic theory, the scientific study on electricity started. From first works of the German Physicist, Otto von Güricke, whose works on air pump inspired Boyle's searches, a slow progress brought to Benjamin Franklin, with his famous kite. Franklin thought that there was an electrical “fluid”, which can be added or subtracted to things only rubbing them. The addition or the subtraction of this electric fluid is manifested by an electric charge, depending on the object from which it was removed or added. Franklin defined those objects on which it considered that the fluid was added, as positively charged. On the contrary defined negative charged objects on which he considered the fluid was subtracted, like for example a rubber stick, rubbed with a wool duster. Edition 22/01/2020 - Rev. 01 3  Cat. B1 - 2.1 Matter He also noted that the objects, having the same kind of charge, rejected each other, while objects with opposite charge exerted an attractive force, which decreased rapidly increasing the distance between the objects. He deduced that the space around an object electrically charged, changed in a certain way its characteristics, and showed this fact, bringing a second electric charge near the object electrically charged. In this way, the concept of electrical field was born19. Franklin deduced that connecting objects of different charge, the fluid would move from the object positive charged to the negative one, passing through the thread connecting them. The experiment supposed the concept of current, or anyway of fluid flow, from the positive to the negative. Nowadays we know that Franklin theories contained errors, in fact, when two objects electrically charged are connected with a thread, the fluid of electrons starts from the object negative charged, moving to the positive one. Moreover, adding electrons to an object, we obtain a negative charge, whereas subtracting electrons to an object, we obtain a positive charge. Unfortunately this discovery was made after lots of important discoveries in electrical field, so Franklin convention of current flow from positive to negative still remains, although the majority of electrical circuits are based on the movement of electrons. Before the experiments of English Physicists Thompson and Rutherford at the beginning of the 20th Century, the atom was considered a solid indestructible corpuscle. The experiments of these two scientists have given a different knowledge of atom, now composed of a nucleus, in which nearly the whole atomic mass is concentrated, surrounded by a swarming of lighter particles, rotating all around. However the disagreement about some following theories, the concept of atom of Rutherford has made possible the explanation of lots of observations made in different fields, like Electrotechnics, Chemistry and Magnetism. Rutherford noticed that the atom was kept together thanks to electrical fields generated by the nucleus, characterized by a positive charge, and by electrons which surrounded it, negative charged. The electrical charge, phenomenon studied in different ways, is bonded to atom‘s characteristics. In 1913, the physicist Bohr suggested some additions to the atomic model of Rutherford. Even though almost a century has passed since Bohr published his atomic model, this one wasn't improved even more. Kept together, Rutherford and Bohr's theories speak of an atomic nucleus consisting of one or more corpuscles, positive charged, called proton. All around this nucleus rotate the electrons, at a precise distance, negative charged and equal in number to the protons of the nucleus. An electron is lighter than a proton, and its mass is approximately 1850 time less than the proton Edition 22/01/2020 - Rev. 01 4  Cat. B1 - 2.1 Matter one. The negative charge of the electrons in the atom, balances the equal and opposite charge of the protons. The atom, kept alone, doesn't present outside any electrical charge. As bodies of same charge repel each other, repulsive forces would tend to break the nucleus, separating its parts. But these forces are compensated by the presence of some corpuscles in the nucleus without any charge. These corpuscles, called neutrons, have a mass almost equal to the mass of the protons, and it seems they act as a glue to keep the nucleus compact. Every certain number of protons, a minimum number of neutrons in the nucleus is needed, in order to avoid that the nucleus becomes unstable, breaking and releasing a big energy. Only the nucleus of the hydrogen, constituted by a lonely proton, doesn't need any neutron. Bohr suggested that the path of every single electron around the nucleus is strictly determinate by the number of electrons in the nucleus and by the energy level of each electron of the atom. Paths are organized in shells. Each shell contains a certain number of paths, called energetic levels. Each energetic level can contain a maximum number of electrons, well defined. The shells are identified with capital letters K, L, M and N, starting from the shell nearest to the nucleus. K shell has one energetic level, called s, which at most can contain two electrons. The simplest atom, the atom of the Hydrogen, whose nucleus has only 1 proton, has only 1 electron at s energetic level, orbiting in the K shell. The next atom, Helium, is more complex. Its nucleus contains 2 protons and 2 neutrons, with 2 electrons at s energetic level, orbiting in the K shell. The maximum capacity of the energetic level is 2 electrons. The next element, Lithium, has 3 protons in the nucleus, so 3 electrons too. The first two electrons complete the energetic level s, inside K shell. As K shell has only an energetic level (level s) the third electrons of Lithium takes place in the energetic level s of the second shell, as to say Shell L. The shell L has two energetic levels: level s, which can contain at most two electrons, and level p, which can contain at most 6 electrons; in total shell L can contain 8 electrons, 2 at level s, and 6 at level p, then it starts to take next shell. The element whose number of electrons completely occupies the shell L is Neon, with 10 electrons in total, 2 at level s in shell K, 2 at level s in shell L, and 6 at level p still in shell L. To resume the main aspects of the nature of matter some definitions are suggested again (the chemical element, the atom and the molecule) together with a little explanation of the periodic table. A chemical element is a pure chemical substance consisting of one type of atom distinguished by its atomic number. We can define the atom as the unit of the matter. Three elementary particles existing into the atom are protons, neutrons and electrons; these define atom characteristics forming all atoms. Protons, with positive charge, and neutrons, with neutral charge, coexist together in the nucleus of the atom. Otherwise, the electrons, with their negative charge, orbit around the nucleus along their orderly orbital called shell. Nowadays we know 109 different elements or atoms. Each of them has a precise number of protons, neutrons and electrons. Each of them is identified by a proper atomic number and has Edition 22/01/2020 - Rev. 01 5  Cat. B1 - 2.1 Matter a proper atomic mass. The atomic number corresponds to the number of the protons in the nucleus. The atomic mass is the mass of an atom of a single element. The matter is composed of lots of particles called molecules. Usually, when atoms bind themselves together they form the molecules. Molecules are defined as the smaller particle in which a substance can be reduced, preserving its chemical and physical characteristics. Moreover all molecules of a certain substance are all identical, and are unique for that substance. Molecules are bound together, and the matter can exist in one of the following state, solid, liquid, and gaseous. Among molecules some attractive and repulsing electromagnetic forces exist, which vary, according to the state of the matter. All atoms and molecules of the matter are constantly moving; the quantity of the movement or vibration depends on the temperature of the material, in that moment. This movement is caused by the energy, like heat, existing in the material. The quantity of movement defines the physical state of the matter. All matter is always in one of the three previous states. The physical state refers to compound conditions and has no influence on the chemical structure of the compound. Atom’s characteristics are summed up into the periodic table of chemical elements, made according to progressive numbers of protons contained in element nucleus. Vertical columns are arranged according to the valence number of electrons, which can reach at most 8 electrons (the external electrons, called valence electrons, can be assigned to another atom, necessary to complete its shell). For that reason we can expect that elements with similar chemical properties fall in the same columns. Edition 22/01/2020 - Rev. 01 6  Cat. B1 - 2.1 Matter In the periodic table of chemical elements, each box represents each single element and gives the name of the element, the chemical symbol, its atomic number and the atomic mass. In some periodic tables, we can find also electrons distribution into theirs shell and energetic level. Observing the table we can notice that electrons distribution into elements following Argon, with an atomic number 18, starts to be difficult. The 19th electron of Potassium, with atomic number 19, instead to be inserted in the energetic level of shell M, takes place into the energetic level of shell N. It is only with element 21, called Scandium, that electrons start to fill up energetic the level of shell M, shaping a series of transition elements, having similar physical and chemical characteristics. It must be noticed that all these elements of transition have the same electrons valence number. The first group of these transition elements includes the most of elements with precious characteristics for the realization of electrical and magnetic circuits. Almost all elements usually used to transport electricity or for their magnetic characteristics, find themselves in one of the transition elements family. We can also notice that two elements used in transistor constructions, like Silicon, with atomic number 14, and Germanium, with atomic number 32, are both in the column of elements with valence of 4 electrons. We can wonder why electrons orbits in different shells are called energetic levels. The reason stays in the concept of atom proposed by Bohr. Bohr thought that the particular path of one electron depends not only on characteristics of the shell including the orbital, but also on the quantity of the owned energy of the electron. This energy at atoms level doesn’t vary in a continuous way, but in defined quantities, as to say in steps of a certain value. Steps of energy, as to say the defined energy quantity is called “quantum”. If an electron is struck by an energy quantum, the electron absorbs the energy and jump to a path farther from the atom nucleus. Eventually this electron gives back this energy pack and Edition 22/01/2020 - Rev. 01 7  Cat. B1 - 2.1 Matter jump back to a lower energetic level, closer to the nucleus. The energy quantum released can be absorbed by another electron, which repeats the same cycle. In some cases, especially in the case of very close atoms, as ones of metallic substances, an electron can receive enough energy to jump into a position so distant from its nucleus that the attraction of the opposite charge from a nucleus can't be able to hold it. In this case is said that the electron is gone into the energetic level of conduction, said conduction band too, becoming a free electron, or a conduction electron. 2.1.2 Chemical compounds A chemical compounds is a pure chemical substance consisting of two or more different chemical elements that can be separated into simpler substances by chemical reactions. Chemical compounds have a unique and defined chemical structure; they consist of a fixed ratio of atoms that are held together in a defined spatial arrangement by chemical bonds. Chemical compounds can be compound molecules held together by covalent bonds, salts held together by ionic bonds, metallic compounds held together by metallic bonds, or complex held together by coordinate covalent bonds. Elements form compounds to become more stable. A complete shell presents a state of preferential stability. On the other side, the Lithium which has the shell K completed and only 1 electron in the shell L, presents a considerable chemical affinity for the other elements. The external electrons, called valence electrons, can be assigned to another atom, necessary to complete its shell. For example, Lithium forms a compound with hydrogen, called Lithium hydride. In this compound the external electron of Lithium goes in K shell of Hydrogen's atom, creating a comprehensive charge of Lithium's atom +1, and the comprehensive charge of Hydrogen -1. These charges of opposite charge keep the atoms together to constitute a single molecule of the compound Lithium hydride. Compounds bounded together through this mechanism, are called electrovalent or bounded by an ionic bond. In nature, these substances, comprehending also kitchen salt (sodium chloride), usually find in crystalline structures consisting of atoms with a charge, called ions, arranged alternately to form the crystal. Another kind of bond can be verified when the outer shell is filled partially, as in the case of the coal, silicon and germanium, elements of basic importance in the electronics industry. An atom of carbon is able to fill its external shell sharing its electrons with other atoms. In the case of a diamond, each atom of carbon shares two of its electrons in the external shell, with each of the three atoms of carbon near it, in order to transform the so called covalent bond. Edition 22/01/2020 - Rev. 01 8  Cat. B1 - 2.1 Matter Considering that in the covalent bond the external shells of the atoms, come in contact and the bond is usually closer and stronger than ionic bond. As already seen in the case of Lithium, the number of the valence electrons, as to say in the outer energetic level or levels of the atom, determines most of the physical and chemical features of the element. 2.1.3 States: solid, liquid and gaseous A solid is defines as a portion of the matter finding in a condensed state. In these conditions, the particles composing it, as to say atoms, molecules and ions, are strongly packed up among them, oscillating around fixed positions in the space and reacting to changes in shape and volume, with stresses depending on the function of the entity of the deformation suffered. Giving energy as heat to a solid material, the movement of the molecules increases. This energy forces molecules to go away from their fixed positions, which characterize the precise shape of the solid. When a material changes from solid to liquid, the volume of the material doesn’t change so much. But the material adapts itself to the form of the container. Liquids are considered incompressible. However molecules of one liquid are more distant among them, compared to a solid ones, they are not enough distant to be able to compress a liquid. In a liquid molecules are still partially bound. This bond force is note as surface tension and avoid that the liquid expands itself all around. Going on giving energy to one material, as heat, the molecular movement increases even more, until the liquid reaches the condition in which the surface tension is not more able to hold the molecules of the material. At this point molecules go away as gas or steam. The heat quantity necessary to transform a liquid into a gas depends on the type of the liquid, and on pressure to which a liquid is subjected. The boiling point of a liquid is directly proportional to the pressure to which it is subjected. Gasses are very different from solids and liquids, because they have no definitive shape and volume. Chemically, gas molecules are identical to liquid and solid ones. But a gas, unlike liquid and solid, is compressible. Edition 22/01/2020 - Rev. 01 9  Cat. B1 - 2.1 Matter 2.1.4 Changes between states Among molecules, in solid, liquid or gaseous state exists a lot of empty space. This empty space allows to compress the material, forcing molecules moving closer, taking up less volume. As reducing the volume, gradually the compression force must be increased, and this fact shows that a repulsive force between molecules exists. On the contrary, in order to produce the extension of a solid body, bar shaped, must be applied an attractive force or a voltage force gradually growing. This fact points out that an attractive force among molecules exists. The force keeping together molecules of same substance is called cohesion force. As said, the matter exists at solid, liquid and gaseous state. In a solid, molecules are kept into fixed positions during a vibration. Increasing the solid temperature, increases also the actual vibration of molecules, and consequently increases the speed of vibration, as to say the kinetic or moving energy of molecules of the body. If the temperature goes on increasing, bonds keeping together molecules weaken, and so molecules can move from their fixed positions, which identified the solid state of the body. In these conditions the substance becomes a liquid. Increasing more the temperature, a bigger energy to molecules is provided, until a second point or energy level is reached, in which the attraction bond among molecules is not more able to keep them together and at this point, the liquid becomes a gas. State changes, whether from solid to liquid or from liquid to gas, happen at a definite and constant temperature. This phenomenon is due to the fact that the energy absorbed during the transformation phase, is used to break the attraction bonds. The necessary heat to reach the change from the substance solid state to the liquid one, is called “fusion latent heat”, instead the one necessary to transform a liquid into a gas is called “evaporation latent heat”. On the contrary, each time that absorbed energy causes a temperature variation, the given heat is called “sensible heat”. Usually, sensible heat and latent heat are called “enthalpy”, and “specific enthalpy”, during state changes. The changes between states can be classified as: 1. Fusion: that is the transformation process between the solid state and the liquid state 2. Solidification: that is the transformation process between the liquid state and the solid state 3. Evaporation: that is the transformation process between the liquid state and the gaseous state Edition 22/01/2020 - Rev. 01 10  Cat. B1 - 2.1 Matter 4. Condensation: that is the transformation process between the gaseous state and the liquid state 5. Sublimation: that is the transformation process between the solid state and the gaseous state 6. Sublimation: that is the transformation process between the gaseous state and the solid state. Edition 22/01/2020 - Rev. 01 11  Cat. B1 - 2.1 Matter PAGE INTENTIONALLY LEFT BLANK Edition 22/01/2020 - Rev. 01 12  Cat. B1 - 2.2 Mechanics 2.2 Mechanics 2.2.1.1 Forces, moments and couples, representation as vectors When it is necessary to extend, bend, compress, twist or break a body, for example a metal one, we must apply to the body a force. Applying a force is necessary to generate the movement of an initially still body or to produce an acceleration, a slowdown or to change object’s direction. Usually, a force is taken into consideration, in relation to the effects it will have on the body. From this point of view the force is defined as "what changes or tries to change the state of rest of a body or its state of uniform motion on a straight line". The unit of measurement of the force, in all its aspects, is the Newton [N]. The Newton is the unit of the force in the International System, adopted in honor of Sir Isaac Newton. This unit is defined as: 𝑚 [𝑁] = [𝑘𝑔 ∙ ( 2 )] 𝑠 A force can be used to produce a rotation, for example as happens when you open a door. This movement requires that the door rotates, hinged in a certain position and that the force is applied at a certain distance from the point of rotation. The effect of rotation of the force is knownas moment of the force and its intensity is determined by the multiplication of the intensity of the force (F), to the distance (d) between the force and the point of rotation or hinge. Anyway note that the distance is taken on the perpendicular to the line of action of the force, perpendicular passing through the rotation center or hinge. This distance forms always a right angle with the action line of the force. Edition 22/01/2020 - Rev. 01 13  Cat. B1 - 2.2 Mechanics Thus it is called moment of the force: 𝑀 = 𝐹 ∙ 𝑑 = [𝑁 ∙ 𝑚] In some cases, for example tightening or unloosing a throttle screw or a wing nut, the two tightening forces are applied with the same intensity, but in different directions. In this case the resulting force on the rotation center is worthless, because only the rotation force remains, without any tendency to move. The resulting value of the moment produces only the rotation. This disposition of the forces, and the resulting moment of these, is called "torque". Established a system of reference, we will consider the position of a body in two different moments, t0 and t1. At the moment t0 the body is in point A and at the moment t1 in point B. The distance between point A and point B is called displacement. Obviously the fact that the body has made a displacement from A to B in the time interval T, doesn’t say nothing about the path or about the real trajectory, the body has followed to do that displacement. In fact, the displacement connecting two points in the space is always represented by a segment; instead the trajectory can be represented by any curve. If a displacement happens in a generic place, isn’t enough to tell its length but are also necessary direction and orientation. The magnitudes as displacements, expressed by numbers, by directions and by orientations are called vectorial magnitudes. For example besides the displacement, the velocity, the acceleration and the force are vectorial magnitudes. Vectorial magnitudes are graphically represented by an arrow, whose length is the vector module. In order to distinguish a vectorial magnitude, an arrow or a little line is put up its identifying letter. For example, a force can be represented by a straight line of the action, a length indicating its magnitude, and by a symbol of the arrow, showing the direction. Edition 22/01/2020 - Rev. 01 14  Cat. B1 - 2.2 Mechanics Vectorial magnitudes have a quality compared to scalar ones in fact, vectors can be added graphically, and this characteristic simplifies problems. Magnitudes expressed only by a number are called scalars. Some scalar magnitudes are time, temperature and mass. A scalar magnitude can be represented by a marked length on an opportune scale. A scalar quantity is characterized by its magnitude, and to define it nothing else is necessary. It can be represented by a length drawn on a straight line quoting a scale. The difference between vectorial and scalar magnitudes is evident, when we must add two scalar magnitudes, rather than two vectorial ones. While for scalar magnitudes is enough to make a simple arithmetical sum, for vectors is necessary to make a vectorial sum considering directions and orientations. Considering two vectors A and B with different directions. The easier method to make the vectorial sum is to draw vectors in a consecutive way and mark the resulting vector, connecting with an oriented segment, the end of the first displacement, with the end of the second one. Consider for example the case of two forces applied to a body. In this case, it's not important which force is the first to be drawn as a vector, whatever is selected to this one must be added the other one, as to say that the second vector shall start where the first one ends. The line, which connects the starting point to the final one of the diagram, is the vector representing the resultant of forces applied simultaneously to the body. Using the scale of the diagram, we can find the width corresponding to the resultant length, the direction using a goniometer to measure the angle created with horizontal plane taken as reference and finally the direction looking at the direction of the resultant arrow. Resultant is called the vector, showing a single force replacing the initial system of forces, able to produce on the body the same effects. Edition 22/01/2020 - Rev. 01 15  Cat. B1 - 2.2 Mechanics Normally a single force is necessary to keep in balance the system of forces applied originally. This force is known as balancing force. The balancing force has the same width and direction of the resultant force, but in opposite orientation or direction. So, to maintain the system of forces applied initially is necessary to apply the balancing force. In order that a system with three forces acting on the same plane is in balance, it is necessary that the lines of actions of forces pass through the same point as to say that forces must compete and once represented as vectors and drawn one following the other, these will form a close triangle. This disposition is known as "triangle of forces". 2.2.1.2 Center of gravity The center of gravity CG is defined as the point where the mass of a body is concentrated, or seems to be concentrated. When a body is involved by a gravitation field, the whole body weight "seems" to act in this point, regardless of the position or "asset" of the body. It's really important to say seems, because in the case of a cave body, the center of gravity falls externally to the body just think about a horseshoe, where the center of gravity is situated in the space between iron arms. Pay attention not to confuse the centre of gravity with the barycenter of a plane picture: in fact a body, to have a mass, must necessarily have three dimensions. Edition 22/01/2020 - Rev. 01 16  Cat. B1 - 2.2 Mechanics 2.2.1.3 Elements of theory of stress, strain and elastic: tension, compression, shear and torsion Each time an external force acts on a body, it is contrasted by an internal force of the body, called stress: 𝐸𝑥𝑡𝑒𝑟𝑛𝑎𝑙 𝑓𝑜𝑟𝑐𝑒 𝑆𝑡𝑟𝑒𝑠𝑠 = 𝐹𝑜𝑟𝑐𝑒 𝑎𝑝𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑎𝑟𝑒𝑎 The English measurement unit is pound per square foot [lb/ft2] or pound per square inches [lb/in2], while in the I.S. it’s measured by Pascal [Pa]. Mechanics distinguishes between five different types of stress applied to bodies: tension, compression, torsion, bending and shear. The tension indicates the force tending to separate an object. A steel flexible hose, used to control an airplane system, is an example of an element designed to bear tension loads. The compression is the characteristic to resist to an external force that tends to push the object inside. Aeronautical rivets are inserted using a force of compression. The torsion stress is applied to a material when it is twisted. The crankshaft of an engine is a piece, whose principal stress is the torsion. The bending stress is applied to a material, when it is subjected to two simultaneous stresses, tension and compression. During the flight, lift forces push the wings of the airplane to bend upwards. When this stress happens, the upper side of the wing is subjected to a compression, and the under one is subjected to a tension. The stress of shear tends to divide the body into slices. A bolt subjected to shear inserted in an airplane control system, is designed to bear stresses to cut. Each time that a force is applied to a solid, there is a deformation. Deformation is the change of dimensions of an object after a stress. Sometimes this deformation results definitive, as bending a safety wire. Other times the deformation is temporary and when the action of the force ends the solid returns to its previous form, as happens with springs. When the deformation is temporary, it is said that the material finds in the elasticity field, and that the material response is an elastic way. The response of a material to the application of a force depends on different factors as the intensity, the direction of the force, the interval during which it acts, the section on which the force works and the type of the material subjected to the force. Edition 22/01/2020 - Rev. 01 17  Cat. B1 - 2.2 Mechanics The material tends to neutralize the force effect, exerting an opposite reaction force. If the force applied exceeds the reaction one, the material can break. Applying a moderate force to the most of the materials, they behave elastically at force removing. Instead, if a force exceeds a certain value, then the material changes form permanently, as to say it warps. When a material changes its form, both elastically or deforming permanently, it is said that the material stretches or that it is subjected to a stretch. Subjected to a stress load, lots of materials behave initially as they were elastic, but increasing the load they acquire at the end a permanent deformation. The first connection between load and extension was made by Robert Hooke in 1676. Hooke's law states that "the extension produced by an elastic material is directly proportional to the load which has produced it." Hooke’s law establishes that: if the deformation (ε) doesn't overcome the elasticity limit of the body, there is directly proportional to the stress applied (σ). The formula of the Hooke’s law, that is true in the elastic field of the materials, is: 𝜎 =𝐸∙𝜀 The constant E is known as "Young’s elasticity module", with same dimensions of stress [Pa]. This module is a constant characteristic of each material. Representing with a graph this statement we obtain a straight line extending to a certain point, called "proportionality limit". Going beyond this point, there is a deformation no more proportional to the applied force, and the graph starts to bend. The point where the material loses its elasticity is called "elastic limit" of the material. In some material this point coincides with "limit of proportionality", but in other ones it occurs in curve part of the graph no more proportional. Wherever the "limit of elasticity" occurs, when it is passed, the material doesn't return to its original length, because it acquires permanently a new form or dimension. Drawing the diagram of stresses and deformations, the form of diagram follows initially the one of the diagram load-extension. This happens until the elasticity limit. Edition 22/01/2020 - Rev. 01 18  Cat. B1 - 2.2 Mechanics It is important to remember that each time you make a repair on the structure of an airplane, is important not to make sudden variations in the section of structural elements, because where section suddenly changes, here stress are concentrated, with consequently probably breaking in the structure. 2.2.1.4 Nature and properties of solid, fluid and gas A solid is a portion of matter that has a condensed state; it is characterized by a structural rigidity and resistance to deformation (that is changes of shape and/or volume). Solids can be classified as crystalline and amorphous, according to the presence of a regular structure of particles’ positions. If it analyzes a solid in the microscopic scale, it observes that: 1. The atoms or molecules composing the solid are packed closely together 2. The atoms or molecules vibrate around some fixed positions in the space (motion of thermal perturbation) 3. The atoms or molecules respond to shape/volume changes with stress that are related to the suffered deformation. A gas is an aeriform characterized by a critical temperature less than the room temperature; the aeriform substances, for which this doesn’t happen, are at vapor state. A gas can be also defined as an aeriform that isn’t condensable at room temperature. In the aeriform state the interaction of particles are very weak and so atoms and molecules can freely move; gasses haven’t a definite shape or volume and they are prone to expand themselves until they occupy all available volume. They are very compressible. Edition 22/01/2020 - Rev. 01 19  Cat. B1 - 2.2 Mechanics A liquid is a fluid that has free particles and it can form a distinct surface at the boundaries of its bulk material. A liquid has some important properties: 1. Fluidity: a liquid is a fluid. Liquids are fluent because there are weak cohesion forces between atoms and molecules. But not all liquids are equally fluid, in fact the ether is more fluent than the water or the water is more viscous than the ether 2. Elasticity: liquids are very elastic. They easily deform themselves under a force and then they immediately take again their original shape when the force stops 3. Incompressibility: a liquid is a fluid whose volume is constant when the temperature and the pressure are constant. The compressibility of liquids is generally lower than the compressibility of gasses and so liquids can be considered as incompressible. Normally a liquid is less dense than a solid, but the water is an important exception. A fluid is a substance that deforms itself, if it is submitted to shear (it isn’t important the entity). This is a particular state of matter that includes liquids, gasses and plastic solids. Fluids haven’t a typical shape as solids. 2.2.1.5 Pressure and buoyancy in liquids In ordinary life, often the concept of pressure is identified as a force constantly applied, but from a scientific point of view, the pressure is defined as a force per unit of surface, and is expressed in Newton per square meter [N/m2] or in Pascal [Pa]. In the Imperial System, the pressure is defined as pound per square inches [lb/in2] or in [psi]; this unit of measurement is very used in aeronautics. Take note that one psi corresponds to 6894 Pascal. In order to have an idea of the magnitude, the normal pressure of a tire is about 2.05 atmospheres, corresponding to about 30 psi, equivalent to 206.820 Pascal. Pressure has the same dimensions or unit of measurement of the material stress. Usually we can be found also another unit of measurement of pressure the [bar]. Particularly interesting is the pressure in fluids. As "fluid" we intend liquids and gas. A fluid has the property to flow, occupying a space of any form. A fluid, at liquid or gaseous state, has the following properties: 1. The pressure applied on a point, inside a fluid, is the same in all directions. 2. The pressure applied by a fluid is always directed perpendicularly to the container sides, containing the fluid 3. A fluid in pressure exerts the same pressure in all fluid points, without important losses. Edition 22/01/2020 - Rev. 01 20  Cat. B1 - 2.2 Mechanics The indicated pressure, psig, is the pressure shown directly on instrument dial, and represents pressure which overpasses the atmospheric one. This happens, for example, every time that a valve of an oxygen cylinder valve is opened. After this action, oxygen of the cylinder escapes quickly until the pressure inside matches the atmospheric pressure, as to say 14.7 psi. When the two pressures the internal of the cylinder and the atmospheric one match, the gauge indicates 0. When the pressure is referred to the vacuum, as to say the absence of pressure, instead to be referred to the atmospheric pressure, we speak about absolute pressure or psia. By definition, the absolute pressure is equivalent to the sum of pressure indicated and the atmospheric one. The differential pressure, indicated by the initials psid, isn’t nothing else that the pressure equivalent to the difference between two pressures. One of the most popular manometers, on board an airplane, is the anemometer. This instrument measures the difference between the pressure of the impact of the air on the input of Pitot tube and the static air pressure. In all fluids, the pressure depends on the depth of the fluid, because the fluid is subjected to a gravitation field. This fact is evident when we consider the atmospheric depth, at which we have a pressure of about 101.32 [kN/m2], as to say at sea level. In liquids, the pressure created with depth may be is more evident. Lots of measuring instruments use this principle to work. The mercury column of a barometer is a good example. This instrument essentially consists of a glass pipe, sealed at one extremity. We make the air escape from the pipe, plunging it in a mercury bath. Once overturn the pipe, keeping the open side plunged in the mercury bath, we note that the mercury inside of the pipe goes down slightly under the sealed extremity, generating in this zone a vacuum. The mercury column in the pipe is supported by the pressure of the air, applied on the surface of the mercury bath. The rising and downing of the mercury column, as consequence of air pressure variations, allows the direct reading of the pressure of the air. Edition 22/01/2020 - Rev. 01 21  Cat. B1 - 2.2 Mechanics The formula which allows calculating the pressure depending on the depth is the product of the fluid density, the gravitation acceleration, to the height or depth: 𝑃 =𝜌∙𝑔∙ℎ The reading of the pressure on the instrument, psig, can be transformed into absolute pressure, psia, adding the atmospheric pressure. This is defined as "standard atmospheric pressure". The standard atmospheric pressure at mean sea level, and at 59° F, is equivalent to a mercury column of 29.92 inches or 14.7 lb/inch2 or psi. In the metric system, the pressure is expressed in [millibar]. A millibar corresponds to about 0.02953 inches of mercury. In the metric system, standard atmospheric pressure at sea level at 15°C is equivalent to 1013.2 millibar. Putting a body into the water, it can sink or float, but in any case, whatever it does, it receives a lift upward. This lift is equal to the magnitude of the weight of the water displaced by the body. This statement is known as Archimedes' principle. The Archimedes’ principle is a theorem concerning the interaction between a fluid and a body immersed in it: any object, wholly or partly immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object. That force is called Archimedes' force or Archimedes's buoyancy or buoyancy. A simplest formulation of this principle is: any object is buoyed up by a force equal to the weight of the fluid displaced. The buoyancy is applied to the center of gravity of the body immersed and is directed upwards, to the plane of hydrostatics loads, according to the fundamental equation of hydrostatics. The formula of Archimedes' principle is: 𝐹𝑢𝑝 = 𝜌 ∙ 𝑉 ∙ 𝑔 where ρ is the density of the solid immersed, V is the volume and g is the force of gravity. The buoyancy is independent of the body depth. The weight of a body wholly or partially immersed, is not the total one measurable out of the liquid, but is the fluid volume weight displaced by the part immersed. This quantity reduces the body weight, when it is hanging by a thread in the empty space. If the body sunk, it means that the weight of the body overcomes the upward lift. While, if the body floats, it means that the upward lift, caused by the liquid displaced by the body immersed, balances the body weight. In this last case, the resultant of the weight force and upward lift is equal to zero. Edition 22/01/2020 - Rev. 01 22  Cat. B1 - 2.2 Mechanics Naturally this principle is true not only for bodies floating in liquids, but also for bodies "suspended in the air". For example, an aerostat or balloon, during a flight displaces its volume of air, and consequently receives a lift upward equal to the weight of the air displaced. In the case we should make a comparison, between the density of a substance and the water one, are used quantities called "relative density" and "specific weight". The densimeter is an instrument, which uses Archimedes' principle, to give a direct reading of the relative density of one liquid. The instrument is composed by weighted pipe ending with an extremity calibrated, such to give a direct reading. The densimeter floats in a vertical position and the part immersed displaces a liquid mass, whose weight is equal to the densimeter weight. Bigger is the density of the liquid, smaller will be the mass of the liquid necessary to balance the weight, so smaller will result the immersed part. The Pascal's law states that when a liquid enclosed in a container is in pressure, the liquid exerts the same pressure on the sides of the container, directed to the perpendicular of the container surfaces. An example can help us to understand the concept. We have a cylinder filled with a liquid and fitted with a square inch piston. Applying a force of 1 lb to the piston, the resultant pressure of the liquid enclosed is equal to 1lb/inch2 per square inch, and this pressure exerts on the sides of cylinder according to the Pascal's law. The whole force of a hydraulic piston can be calculated multiplying the piston surface to the pressure exerted on the fluid: 𝐹 =𝐴∙𝑃 where A is the area of the piston and P is the pressure. Edition 22/01/2020 - Rev. 01 23  Cat. B1 - 2.2 Mechanics 2.2.2.1 Linear movement: uniform motion in a straight line, motion under constant acceleration (motion under gravity) The motion of an object is never absolute, but relative. To study the motion of an object is necessary to establish a system of reference for example, the system of Cartesian coordinates, in order to specify positions comparing to it. Moreover is necessary to have a clock in order to register different positions of the body in different moments. The line drawn during its motion is called trajectory. Obviously also the trajectory of an object depends on the selected reference system and on the point from which the motion is seen. If S is the measure of the displacement of a body and t is the time spent to do it, average velocity can be defined as the ratio of displacement and time: 𝑆 𝑉= 𝑡 In the IS the unit of measurement is the meter per second [m/s]. Considering that usually velocity is expressed in kilometers per hour [km/h], learning now this conversion is necessary. So, to transform velocity from [km/h] into [m/s], is enough to divide it for 3.6. Velocity described before, is only an average velocity. The instant velocity can be obtained considering short time intervals and consequently very little displacements. If instant velocity remains constant during considered time, this one will coincide with average velocity and we can talk about uniform motion. When instant velocity isn't constant but varies in time, there is an acceleration. Average acceleration can be defined as the variation of the average velocity in a time interval: 𝑉 𝑎= 𝑡 In the International System, the unit of measurement of acceleration is the meter per square second [m/s2]. As for instant velocity, there is also an instant acceleration. Obviously, if a body is slowing down, as to say decelerates, we obtain a negative value of acceleration. Edition 22/01/2020 - Rev. 01 24  Cat. B1 - 2.2 Mechanics If the velocity vector keeps constant module, direction and orientation, there is a uniform linear motion. From the definition of velocity we can formulate the equation of motion, which gives covered spaces at time variation: 𝑆 =𝑉∙𝑡 If we represent the equation of motion with the Cartesian coordinates system, we put on the horizontal axis the time and on the vertical one the space covered. A graph like this is called time-space graph. We note that the curve representing a uniform linear motion in a time-space graph is a straight line. The gradient of this line shows the motion velocity. This means that faster an object is, higher will be the gradient of its representing line in a space-time graph. When velocity isn't constant in time, there is an acceleration that can be variable or constant. When the acceleration in a motion is constant, there is a uniformly accelerated motion. If acceleration is constant, the average acceleration will coincide with instant acceleration. Using the definition of acceleration, it is possible to find out the velocity law: 𝑉 =𝑎∙𝑡 In case of a uniformly accelerated motion, this law states that velocity variations are proportional to time intervals, so in equal time intervals we will have same velocity variations. The graphic representation of velocity law in a velocity-time graph, brings us another time to a straight line, whose gradient shows a constant acceleration; bigger is the motion acceleration, bigger will be the line gradient. Edition 22/01/2020 - Rev. 01 25  Cat. B1 - 2.2 Mechanics In case the body doesn't start moving from a standstill condition, but starts to accelerate when has already a certain velocity; in this case the velocity law must be appropriately modified: 𝑉 = 𝑉0 + 𝑎 ∙ 𝑡 In the graph, this condition brings to a displacement of the line, throughout the velocity axis. Supposing that the body in motion at one instant slows down, in this case there is a decelerated motion. If this decreasing in velocity is constant, we will have a uniformly decelerated motion. In this case, in a velocity-time graph, the gradient of the line representing the velocity law will be negative, as to say the line will be directed downwards. Velocity-time graphs let us determine covered space during a uniformly accelerated motion. Let's represent in a velocity-time graph the velocity of a uniform linear motion. This will be Edition 22/01/2020 - Rev. 01 26  Cat. B1 - 2.2 Mechanics shown by a horizontal line corresponding to the constant velocity. Observing the equation of motion for uniform linear motion, we can easily verify that the covered space results from the area between the ray representing the velocity time axis and a vertical segment drawn reflecting the final instant of the motion. The same is valid for uniformly decelerated motion. The equation for a uniformly accelerated motion is: 1 𝑆 = 𝑆0 + 𝑉0 ∙ 𝑡 + (2 ∙ 𝑎 ∙ 𝑡) A particular case of uniformly accelerated motion is a free falling object. The falling of a body is determined by the body weight, as to say by the earth attracting force. In this case the acceleration of the motion is given by the gravity acceleration (g), which corresponds approximately to 9.81 m/s2. Edition 22/01/2020 - Rev. 01 27  Cat. B1 - 2.2 Mechanics For a free falling object the motion is naturally accelerated in the weight force direction, whose laws are: 𝑉 =𝑔∙𝑡 1 𝑆= 2 ∙ 𝑔 ∙ 𝑡2 From the second expression we can derive the time spent to fall from a certain height (h): 2ℎ 𝑡= √ 𝑔 Moreover, relating both laws we can derive the arriving velocity at earth, falling from a certain height. It will be enough to substitute t in the first expression: 2ℎ 𝑉 =𝑔∙𝑡 =𝑔∙√ = √2𝑔ℎ 𝑔 2.2.2.2 Rotational movement: uniform circular motion (centrifugal/centripetal forces) An object moving throughout a circumference makes a circular motion. If that movement has a constant module velocity, it is a uniform circular motion. In uniform circular motion we define some magnitudes. The time spent by the object to make a whole revolution on the circumference, is called period T. Edition 22/01/2020 - Rev. 01 28  Cat. B1 - 2.2 Mechanics The number of revolutions that the object makes in a second is called frequency (f). The frequency is equal to the multiplicative inorientation of the period. In the International System the unit of the frequency is Hertz [Hz]. If the velocity remains constant, the object will make a whole revolution, so the path of 2πr, in the time corresponding to a period. That velocity is called tangential, because the direction of the vector is tangential to the circumference: 2𝜋𝑟 𝑉= 𝑇 As tangential vector changes continuously direction, that variation determines an acceleration, called centripetal acceleration, always directed inward to the center of the circumference: 𝑉2 𝑎𝑐 = 𝑟 If we analyze the circular motion according to an inertial system of reference (that is supportive with the motion), we observe another force acting on the mass: the centrifugal force. This is an apparent force and it pushes the mass towards the curve outside. Another type of velocity is used for circular motions. While a point moves on a circumference with a constant velocity, the radius connecting that point to the center describes a certain angle 2π obviously that angle will be proportional to the arc defined by the point. That's because we can refer the velocity to the angle described by the radius, instead of the arc covered by the point. The angular velocity is the ratio between the angle described and the time spent to do it: 2𝜋 𝜔= 𝑇 In the International System, the unit of the angular velocity is radiant per second [rad/s]. The radiant is the measure of one angle between two radii, subtended by an arc equal to the radius. As the circumference of radius r, corresponds to 2πr, the round angle corresponds to 2π radian. Comparing angular velocity value to tangential velocity one, we can deduce a relation connecting the two magnitudes: 2𝜋𝑟 𝑉= = 𝜔𝑟 𝑇 Edition 22/01/2020 - Rev. 01 29  Cat. B1 - 2.2 Mechanics 2.2.2.3 Periodic motion: pendular movement A periodic motion is a motion that after a certain period T gets back to its starting point. Let's consider a point P that moves with a uniform circular motion on a circumference of radius r. It is easy to note that its projection is at a maximum level when the point is in A, cancels it when passes through B, than becomes minus r and cancels itself another time in D. Let's represent this trend in a Cartesian coordinates system. On the horizontal axis we put time course, as fractions or multiples of period T and on the vertical axis the projection length. The result is a harmonic motion. The harmonic motion is a periodic motion and all periodic motions can be represented as harmonic motions. If we consider little displacements, the pendulum oscillates of harmonic movement. It is composed by a marble of mass m, which is hooked to a thread with length l. The thread is fixed in a point O. Edition 22/01/2020 - Rev. 01 30  Cat. B1 - 2.2 Mechanics The force that moves the pendulum is the component F of the weight force. The force F is tangent of the marble trajectory. The component parallel to the thread hasn’t any effect, because it is annulled by the thread bond. Through mathematical processes we can find the formula of pendulum’s period. It doesn’t depend on the wideness of oscillation and on the mass of the marble. It is function of the thread length and of the gravity acceleration (g): 𝑙 𝑇 = 2𝜋 ∙ √ 𝑔 2.2.2.4 Simple theory of vibration, harmonics and resonance The term “vibration” indicates a particular mechanical oscillation around an equilibrium point. The oscillation can be periodic, as the motion of the pendulum, or random as the movement of a tire on a paved road. In some case the vibrations are a “wanted phenomenon”: they are the cases of a tuning fork or others musical instruments. However the vibrations are more often undesired: in fact they can disperse energy and create tiresome sounds and noises. An example is the engine of a vehicle when it is working. It is important to remember that the study of sounds and the analysis of vibrations are closely related. The sound is pressure wave generated by a pulsing structure (for example vocal cords). If it wants to reduce a noise, it has to study the way to eliminate vibrations. The vibrations are classified as: 1. Free vibrations: they are produced by a mechanical pulsing system that isn’t subjected to any forces. Ideally if there is no friction and any energy dispersions, the system endlessly will continue to vibrate. A system without a force oscillates because its initial conditions weren’t null. A simple example is the case of a mass connected with a loom through a spring. At the initial moment the spring is compressed. 2. Forced vibrations: they happen when there is a force applied to the system. The mass-spring-damper is a simple model to study the phenomena of vibrations. This model is an example of simple harmonics oscillator. Edition 22/01/2020 - Rev. 01 31  Cat. B1 - 2.2 Mechanics  Free vibration without damping. To start the investigation of the mass–spring–damper for this case it will assume the damping is negligible and that there is no external force applied to the system. The only force applied to the system is the spring force. The mass of the system is called m and the stiffness of the spring is indicated by k; the number fn is one of the most important quantities in vibration analysis and is called the “undamped natural frequency”. For the simple mass–spring system, fn is defined as: 1 𝑘 𝑓𝑛 = (2 𝜋) ∙ √ 𝑚 With this formula it can determine the frequency at which the system will vibrate once it is set in motion by an initial disturbance. Every vibrating system has one or more natural frequencies. This simple relation can be used to understand what will happen to a more complex system.  Free vibration with damping. To study this case it is necessary to add a "viscous" damper to the model. The damping models the effects of an object within a fluid. The proportionality constant c is called the damping coefficient. Edition 22/01/2020 - Rev. 01 32  Cat. B1 - 2.2 Mechanics The frequency in this case is called the "damped natural frequency", fd. The damped natural frequency is related to the undamped natural frequency of the previous case thought the following formula: 𝑓𝑑 = √(1 − ζ2 ) ∙ 𝑓𝑛 where ζ is the coefficient of the damping ratio of the mass–spring–damper model. The damped natural frequency is generally less than the undamped natural frequency.  Forced vibration with damping. In this case it studies the behavior of the spring-mass-damper model when a harmonic force is applied. A force of this type can be expressed by the following formula: 𝐹 = 𝐹0 ∙ cos(2𝜋𝑓) The mass will oscillate at the same frequency, f, of the applied force, but with a phase shift (φ). The amplitude of the vibration is made of two contributes: the steady element, that is the displacement done by the system if there is a constant force (F0) and the factor of dynamic amplification, that is the increase of displacement caused by the variation of the force in the time. In a lightly damped system when the forcing frequency (f) is close to the natural frequency (which is the frequency of free vibrations) the amplitude of the vibration can reach high value. This phenomenon is called resonance. In a mechanical system, the resonance can produce important damages, which can lead to eventual failure of the system. Consequently, one of the main reasons for vibration analysis is to predict the resonance may occur on the system, and to determine what steps it have to take in order to prevent it. In other words, the resonance is the tendency of a system to oscillate with larger amplitude. This phenomenon more occurs at some frequencies than at others. The frequencies at which the resonance happens are known as the system's resonant frequencies (or resonance frequencies). At these frequencies, the system stores vibrational energy and large amplitude vibrations are produced. Resonance phenomena occur in different field, in which there are vibrations and waves. Edition 22/01/2020 - Rev. 01 33  Cat. B1 - 2.2 Mechanics 2.2.2.5 Velocity ratio, mechanical advantage and efficiency Velocity ratio The ratio of two velocities hasn’t a unit of measurement. An example is the Mach number: 𝑉 𝑀𝑎𝑐ℎ = 𝑐 where V is the speed of the aircraft and c is the velocity of sound in the air. Mechanical advantage and efficiency Generally the efficiency can be defined as the level of use of a substance during a process. If we study the energetic case, it is the ratio of the output to the mediums utilized. To increase the efficiency means to produce more goods/services with smaller use of resources. Lots of machines, realizable in practice, use the so-called "mechanical advantage", to modify the force necessary to move an object. Simple devices usually used to exploit this "mechanic advantage" are the lever, the inclined plane, pulley, and gears. The mechanical advantage is calculated dividing the weight, or resistance of an object, by the stress necessary to move the object: 𝑊𝑒𝑖𝑔ℎ𝑡 𝑀𝐴 = 𝑆𝑡𝑟𝑒𝑠𝑠 A mechanical advantage of 4 indicates that for each pound of applied force, 4 pounds of resistance can be contrasted. The lever is a device used to realize a mechanical advantage. In its simpler form, it is composed by a beam put in balance with two loads at the extremes. One of them tends to rotate the beam in clockwise direction, instead of the other one which tends to rotate in counterclockwise direction. Each load creates a "moment" or a torsion force. Three classes of lever exist: 1. Lever of first class 2. Lever of second class 3. Lever of third class. Edition 22/01/2020 - Rev. 01 34  Cat. B1 - 2.2 Mechanics  Lever of first class. In the following example, the final part of a 4 ft pole is put under a load of 100 lb. Moreover, the fulcrum finds at 0.50 ft from the CG of the load. In this way, there are 3.50 ft between the fulcrum and the point where the force or stress of lifting is applied. When a stress S is applied, it takes an effect in the opposite direction to the one of the load. In order to deduce the required stress to lift the load is necessary to calculate, the moments at the two opposite sides of the fulcrum: 𝑅∙𝐿 =𝑆∙𝐼 where I is the length of the arm where the stress is applied, S is the stress applied, R is the resistant force and L is the length of the lever arm between fulcrum and resistant force. In this case our formula becomes: 𝑅∙𝐿 𝑆= = 14.28 𝑙𝑏 𝑙  Lever of second class. Unlike lever of first class, a second class lever has the fulcrum at one extreme of the lever, and the force is applied on the other extreme. The resistance, as to say the load to lift, is applied on the lever near the fulcrum. The most common example of a lever of second class is the wheelbarrow. Using a wheelbarrow, the lever, as to say the handles of the wheelbarrow, are used to gain a mechanical advantage and to reduce the effort necessary to carry the load. For example, if in the wheelbarrow there is a load of 200 lb, concentrated at 12 in from the axis of the Edition 22/01/2020 - Rev. 01 35  Cat. B1 - 2.2 Mechanics wheel, and the force applied is at 48 in from the axis of the wheel, only 50 lb of force are necessary to lift the load. The necessary force can be calculated using the same formula of first class lever: 𝑅∙𝑙 =𝑆∙𝐿 𝑅∙𝑙 𝑆= = 50 𝑙𝑏 𝐿 The resulting mechanical advantage using a second class lever is the same obtained using a first class lever. The only one difference is that the force and the resistant move in the same direction.  Lever of third class. In aeronautical field, a third class lever is mainly used to move a resistance to an exceeding distance, compared to the one of the force applied. This is obtained applying a force between the fulcrum and the resistance. Each time this is done, a superior force must be applied to obtain a movement. An example of third class lever is given by the mechanism that extends and retracts the landing gear. The force necessary to retract the landing gear is applied near the fulcrum, while the resistance is applied in the opposite side of the lever. Another way to obtain a mechanical advantage is using an inclined plane. The inclined plane gives the mechanical advantage of moving a big resistance with a little force in a long distance. The necessary force is given by: 𝑆∙𝐿 =𝑅∙𝑙 where L is the length of the ramp, I is the high of the ramp, R is the load or resistance of the object to move and S is the force necessary to lift or to lower the object. Edition 22/01/2020 - Rev. 01 36  Cat. B1 - 2.2 Mechanics For example, the force required to make a 500 lb barrel roll, on an inclined plane of 12 in to reach a platform 4 ft hight, is: 𝑅∙𝑙 𝑆= = 166.7 𝑙𝑏 𝐿 Using an inclined plane, 500 lb of resistance are moved with a force of 166.7 lb. The wedge is a special application of the inclined plane, consisting of two inclined planes, placed one against the other. Making a wedge penetrate in a material, all along its length, the material is forced to split off for a distance equal to the wedge width. The mechanical advantage is more notable in thin and long wedge. Pulleys or sheaves are another type of simple machine, which allow obtaining a mechanical advantage. A single pulley joint to a fix point, is equivalent to a lever of first class. The fulcrum is the pulley center, and the arms, extending to the two different sides of fulcrum, are of the same length, radius of the pulley. The mechanical advantage of a single pulley franked to a fixed point is equal to one. When a pulley connected in this system is used, the force necessary to lift an object is equal to the object weight. If the pulley is not franked, works as a second class lever, as to say that both force and weight are in the same direction. Edition 22/01/2020 - Rev. 01 37  Cat. B1 - 2.2 Mechanics Another thing to remember using a pulley is that, the more one gains in a mechanical advantage, the most the distance to which apply the force increases. In other words if a mechanical advantage of 2 is obtained, for each foot of resistance shift, a force of 2 feet of rope must be applied. One of simple machines application is represented by the gears. Gears are used in wall clocks and wristwatch, in cars and airplanes and in almost all mechanical devices. Gears can be used to provide a mechanical advantage and to change direction. To gain the mechanical advantage using gears, the number of teeth of the conductor gear is different from the one of the gear conducted. In practice lots of different gears are used. For example spur gears have right angle cut teeth along the circumference, and are used to join two parallel shafts. When all gears have teeth outside, shafts rotate in opposite orientation. If the two shafts, the conductor and conducted one, are not parallel, in that case bevel gears are used. As teeth of bevel gears are outside, the direction of the motion of a shaft is always opposite to the other one. A common example of the use of bevel gears is tail rotor gearbox in helicopters. Edition 22/01/2020 - Rev. 01 38  Cat. B1 - 2.2 Mechanics When a big mechanical advantage is needed a worm screw can be used. The worm screw consists of a shaft surrounding by a spiral, as a screw, and by the gear, which meshes with the worm. Commonly, the two shafts of worm screw and of the gear are positioned at right angle. If the worm screw is a single-threaded, the worm screw makes the gear move forward of one tooth per each tour. In airplane propeller, with powerful engine, an epicyclical reduction is used to reduce the speed of the propeller shaft. The epicyclical reduction lets the engine develop all its power at optimal revolutions, adapting it to optimal number of revolutions of the propeller. In the system of epicyclical gears, the propeller is moved by a cage, spider shape, called planetary gear. In different epicyclical reduction systems, the solar gear is the conducting gear, and the crown gear constituting the external ring, is integral with the frontal part of the engine, as in the case of airplanes propeller, or makes part of the main transmission box, as in the case of helicopters. In these situations, gears mounted on a planetary one, are intermediate gears, without shaft, but with the task to drag the engine on which they are mounted. In this case the gear ratio between solar gear and planetary one is the ratio between tooth of the crown gear, external and fixed, and the central or solar one. Edition 22/01/2020 - Rev. 01 39  Cat. B1 - 2.2 Mechanics 2.2.3.A.1 Mass The mass is the magnitude that permits to calculate how much matter is contained in a body or in an object. Moreover, in physical sector, the mass is also considered as the measurement of the resistance, which is made by a body when we try to accelerate it. In this way we can define the inertial mass of the body. The mass is a scalar because it is detected only by a number and because masses of two objects are added as two numbers. The weight of objects is proportional to their masses: 𝑊 =𝑚×𝑔 It can be expressed in Newton, because it is a force. The objects of same weights have also same masses because on the Earth, the g value is constant: 9.81 m/s2. 2.2.3.A.2 Force, inertia, work, power, energy (potential, kinetics and total energy), heat, efficiency The force is the magnitude that measures the interaction among physical systems. The force can be defined as the cause that produces or modifies the movement of a body: that is the cause that generates changes of velocity. If an object is steady and it remains steady, the total force that the body is suffering is null. Instead, if an object is steady and then it gets a move on and gains velocity, there is a force that produces the movement. The force is a vector. The heat is a transfer of energy (at macroscopic level) between two bodies, initially at different temperatures. The heat is energy in transit. It is measured in Joule, but it can also expressed by the calorie: 1 𝑐𝑎𝑙 = 4.186 𝐽 Edition 22/01/2020 - Rev. 01 40  Cat. B1 - 2.2 Mechanics The energetic efficiency is a dimensionless number with a value included from 0 to 1. This value can be multiplied by 100 and so we can speak about percentages. The efficiency of a process is defined as: in power η= out power The work done by a force F is the product of the intensity of the component of the parallel force, at displacing, to the displacement. In the International System, the work is measured in Newton meter [N·m]. This unit is the Joule [J], which is defined as the work done by a force of one Newton, acting on a distance of one meter. From this definition derives that the work done by a force depends on the angle included between the displacement vector and the vector of the force. Obviously, follows that a work can be positive, negative or zero. A positive work is called motive work, instead the negative one is called resisting work. In the particular case of a 90° angle between direction of force and direction of displacement, the parallel component to the displacement is zero, and consequently also the work is zero. The power is the ratio of the work done, to the time interval spent to do it. The unit of the power in the International System is Joule per second [J/s], called Watt [W]. One Watt is hence the power of a device which does a work of one joule in one second. If the engine of an elevator has to lift a total load of 5000 N for a path of 20 m in 40 s, it has to develop a power of 2500 W. Considering that the watt is a little unit, usually its multiples are used. The kilowatt is, probably, the most common multiple of the watt; the power of electrical devices, in fact, is expressed in kilowatt. Knowing that the work is the product of the force to the displacement, it is possible to give another definition to the power often very useful. The power necessary to a force to make a body move in a constant velocity is equal to the product of force to velocity: 𝑃𝑜𝑤𝑒𝑟 = 𝐹𝑜𝑟𝑐𝑒 ∙ 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 Edition 22/01/2020 - Rev. 01 41  Cat. B1 - 2.2 Mechanics With the word machine, we identify all devices able to do a work. A car, a lift, a crane, a drill, a blender, a hydraulic hoist and a pump are all examples of machines. According to the given definition, also the human body has to be considered as a machine. Every machine is characterized by a power, as to say by the rate of work it can do in the time unit. A machine is more powerful than another of the same type if it does a work in less time. Let's consider two cases a trolley bumps into a ball while moving, and makes it move, and the same trolley bumps into a spring compressing it. In both cases, the trolley moving does a work on the hurt object and we say it has energy. The energy is the capacity to perform a work. The energy is clearly bound to the trolley movement, if this one was standstill, it wouldn't have the capacity to move objects and to deform them. The capacity to perform a work of a moving object is called, kinetic energy: 1 𝐸𝑘 = 2 𝑚𝑉 2 From kinetic energy relation follows that higher the velocity of an object is, bigger its kinetic energy will be, and also its capacity to do a work. That shows also that the kinetic energy increases at the mass object increasing. In the International System kinetic energy is expressed in Joule, like the work. In fact, the unit of the energy is the product of the unit of the mass to the square of the unit of the velocity. As already seen, an object falling from a certain height performs a work, because weight force and displacement have the same direction. To do a work every object must have energy. In this case, it’s not kinetic but potential energy. Edition 22/01/2020 - Rev. 01 42  Cat. B1 - 2.2 Mechanics Generally, an object put at a certain height, has a potential energy caused by its position. The potential gravitational energy can be written as follows: 𝐸𝑝 = 𝑚𝑔ℎ The unit of measurement of potential energy is the Joule, the same as for work and energy. An object standstill at a certain height from the ground, has potential energy, but not kinetic. During the fall, the height decreases and the velocity increases. So, the potential energy decreases and the kinetic one increases. Near the ground the height is zero so the object has no potential energy but only kinetic. The sum of kinetic energy and potential energy of a body, calls mechanical energy, and is measured in joule. During the movement of an object, the sum of kinetic and potential energy remains constant, as to say that the mechanical energy preserves itself. That law obviously is valid until friction can be neglected. Aristotle thought that the natural state of bodies was the rest, and that to keep the velocity of a body constant, a force was necessary. Daily experience lets us wrongly believe that Aristotle’s idea was right. A cyclist must cycle to keep the velocity constant, so he must apply a force. A car driver must keep the foot on the accelerator, so that the engine applied a force in order to avoid that the car will turn off. At the beginning of 17th century Galileo Galilei contested Aristotle’s intuition: a body keeps its velocity, only if no forces act on it. In the opinion of Galileo, a force is necessary to fight the friction of the body while is moving, but if the friction could be removed the body will continue to move with a uniform motion. Galileo discovered the inertia law with an ideal experiment, imagining the limit case of a body moving on a horizontal plane without friction. On the basis of Galileo's ideas, Isaac Newton stated the first law of dynamics: a body persists in its state of rest or of uniform motion, unless acted upon by an external unbalanced force. Up to this statement, if a body is in rest and all forces acting on it have a resultant equal to zero, the body will be standstill and if it has a uniform motion, will keep its motion. Edition 22/01/2020 - Rev. 01 43  Cat. B1 - 2.2 Mechanics The first law of dynamics is called also law of inertia. Inertia is the property of the substance which determines its resistance to accelerate if subjected to a force. The first law states also that a force on a body generates acceleration. Experimentally it is possible to verify that the force applied and the acceleration generated, are directly proportional, reducing at minimum the friction. Instead, keeping the force constant and varying the mass of the body, the acceleration generated will result inorientationly proportional to the mass of the object. These two statements can be resumed in one law, the second law of dynamics: 𝐹 =𝑚∙𝑎 The resultant of the forces applied on a body is equal to the product of the mass (m) of the body to the acceleration (a) it acquires. We make immediately notice that the second law of dynamics contains the first law as particular case: if the resultant force is equal to zero, the acceleration will be zero too and so the body will keep its motion state. Let's apply this second law of dynamics to a very simple case, the one of a body free falling. In the free fall, the single force acting on the body is the gravity, as to say the weight W. Acceleration of the body is the acceleration of gravity. Up to second law of dynamics, as the resultant force is equal to the product of the mass to acceleration, we can write: 𝑊 =𝑚∙𝑔 The third law of dynamics says that to every action corresponds always an equal and opposite reaction. So mutual actions between bodies are always equal and directed to the contrary. More precisely when a body A exerts a force on a body B, B exerts a force on A, too. The two forces have the same module, same directions, but opposite orientations. Edition 22/01/2020 - Rev. 01 44  Cat. B1 - 2.2 Mechanics 2.2.3.B.1 Impulse The second principle of dynamics, replacing acceleration with average acceleration definition, can be written: ∆𝑉 𝐹 =𝑚∙𝑎 =𝑚∙ ∆𝑡 Multiplying both equation members to the time interval Δt, we obtain: 𝐹 ∙ ∆𝑡 = 𝑚 ∙ ∆𝑉 Observing this relation can be stated that velocity variation depends not only on force, but also on time interval. For example, to stop a moving object can be applied a great force during a short time interval or a little force during a long time interval. Taking into account both force and time interval, another magnitude can be considered: the impulse of the force. The impulse of the force is the product of force to time interval: 𝐼 = 𝐹 ∙ ∆𝑡 In the International system, the impulse unit is the Newton second [N·s]. As product of vector to scalar, the impulse is a vector with same direction and orientation of force. A bullet of little mass and high speed and a stone of big mass and low speed can produce the same damage, hurting an object. This means that the effects produced by an object in movement depend on mass and velocity. Edition 22/01/2020 - Rev. 01 45  Cat. B1 - 2.2 Mechanics 2.2.3.B.2 Momentum, conservation of momentum The momentum of an object is the product of the mass to the velocity, indicating with p the motion quantity: 𝑝 =𝑚∙𝑉 The unit of measurement is the [Kg·m/s]. As the impulse (I), the momentum is a vector. Considering that the velocity variation results from final and initial velocity difference, operating some appropriate replacing, we obtain: 𝐼 = 𝐹 ∙ ∆𝑡 = 𝑚 ∙ ∆𝑉 = 𝑚 (𝑉𝑓 − 𝑉𝑖 ) = 𝑚 ∙ 𝑉𝑓 − 𝑚 ∙ 𝑉𝑖 where F is the force and Δt is the time interval. Follows the law of impulse: 𝐼 = ∆𝑝 = 𝑝𝑓 − 𝑝𝑖 The impulse of an applied force during a time interval is equal to the momentum variation of the body during the same time interval. The theory of impulse is a different way to state the second law of dynamics instead to talk about force and acceleration there are the impulse and the momentum. Thanks to this theory, we can analyze situations, in which the acting force is difficult to know, for example some phenomena as explosions, collisions, or implosions. The theory of impulse allows us also to state the law of momentum conservation. Let's see first some definitions concerning bodies system. A system of bodies is a set of bodies we can consider as a single one. For example, two marbles that are about to collide, form a set of two bodies that we can consider as a single body. This system moves with a momentum equal to the vector sum of the momentum, of the two single bodies of the system. All the forces out of the system are external forces. A system is called isolated, when no external forces act on it, or when the resultant of the external forces is zero. In the case of two marbles moving on a horizontal table, the system is isolated if the friction is negligible and if at the collision act on it only forces inside the system. Consider an isolated system. As the resultant of external forces is zero, the impulse is zero too. Up to the theory of impulse, the final momentum is equal to the initial one. If no external forces act on a system, this can evolve from an initial to a final state, but its momentum keeps constant. The law of conservation of the momentum allows studying phenomena acting in short time with only internal forces, as per explosions and collisions. Edition 22/01/2020 - Rev. 01 46  Cat. B1 - 2.2 Mechanics 2.2.3.B.3 Gyroscopic principles The gyroscope consists of a spinning wheel, which, up to conservation laws, tends to keep its rotation axis fixed. Any mass rotating about an axis acquires certain properties, called gyroscopic. So, any rotating mass designed and set to exploit this propriety, is a gyroscope. To exploit the properties of a gyroscope is necessary to lock one or more axes of it, deleting one or two freedom degrees. These properties are exploited in aeronautic applications, particularly in gyroscopic instruments.  The inertia, or rigidity, of a gyroscope. In standstill conditions, and gyroscope stationary, is possible to rotate easily the gyroscope axis in fact is enough to apply a sufficient force to defeat frictions on the frame bearings. Anyway, if the body has a high rotation velocity a greater force is needed to do the same operation and the intensity of this force is proportional to the rotation velocity. That effect is called gyroscopic inertia; it explains the capacity of a gyroscope to keep unchanged the direction of rotation. In other words, once the wheel is rotating, is possible to take the system and flip it, make it rotate in any direction, observing that meanwhile the rotation axis remains constantly directed to the initial direction.  Thanks to gyroscopic precession if a torque is applied perpendicular to the gyro spin axis, it will move the gyroscope as though the torque had been at 90° removed from the actual point of application, in the direction of rotation. As it can be seen, considering the axis x as the gyroscopic rotation axis, if an applied torque causes a rotation about the axis y, the wheel will react with a rotation about the axis z, called rotation of precession. Looking at a top we can see a practical example of this phenomenon. In fact the top is a real gyroscope, because it is a mass rotating about an axis. Until it rotates at high velocity, the rotation axis remains fixed in the space, always perpendicular to the ground. At velocity decreasing, caused by the friction, the top follows a precession motion evenwider, and the rotation axis tilts even more, until it becomes parallel to th