1.
Dipole graph
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In graph theory, a dipole graph is a multigraph consisting of two vertices connected with a number of parallel edges. A dipole graph containing n edges is called the order-n dipole graph, the order-n dipole graph is dual to the cycle graph Cn. The honeycomb as a graph is the maximal abelian covering graph of the dipole graph D3. Jonathan L. Gross and Jay Yellen,2006, graph Theory and Its Applications, 2nd Ed. p.17. ISBN 1-58488-505-X Sunada T. Topological Crystallography, With a View Towards Discrete Geometric Analysis, Springer,2013, ISBN 978-4-431-54176-9 978-4-431-54177-6
2.
Graph (discrete mathematics)
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In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. The objects correspond to mathematical abstractions called vertices and each of the pairs of vertices is called an edge. Typically, a graph is depicted in form as a set of dots for the vertices. Graphs are one of the objects of study in discrete mathematics, the edges may be directed or undirected. In contrast, if any edge from a person A to a person B corresponds to As admiring B, then this graph is directed, because admiration is not necessarily reciprocated. The former type of graph is called a graph and the edges are called undirected edges while the latter type of graph is called a directed graph. Graphs are the subject studied by graph theory. The word graph was first used in this sense by J. J. Sylvester in 1878, the following are some of the more basic ways of defining graphs and related mathematical structures. In one very common sense of the term, a graph is an ordered pair G = comprising a set V of vertices, nodes or points together with a set E of edges, arcs or lines, which are 2-element subsets of V. To avoid ambiguity, this type of graph may be described precisely as undirected, other senses of graph stem from different conceptions of the edge set. In one more general conception, E is a set together with a relation of incidence that associates with each two vertices. In another generalized notion, E is a multiset of unordered pairs of vertices, many authors call these types of object multigraphs or pseudographs. All of these variants and others are described more fully below, the vertices belonging to an edge are called the ends or end vertices of the edge. A vertex may exist in a graph and not belong to an edge, V and E are usually taken to be finite, and many of the well-known results are not true for infinite graphs because many of the arguments fail in the infinite case. Moreover, V is often assumed to be non-empty, but E is allowed to be the empty set, the order of a graph is |V|, its number of vertices. The size of a graph is |E|, its number of edges, the degree or valency of a vertex is the number of edges that connect to it, where an edge that connects to the vertex at both ends is counted twice. For an edge, graph theorists usually use the shorter notation xy. As stated above, in different contexts it may be useful to refine the term graph with different degrees of generality, whenever it is necessary to draw a strict distinction, the following terms are used
3.
Dynamical system
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In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the models that describe the swinging of a clock pendulum, the flow of water in a pipe. At any given time, a system has a state given by a tuple of real numbers that can be represented by a point in an appropriate state space. The evolution rule of the system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a time interval only one future state follows from the current state. However, some systems are stochastic, in random events also affect the evolution of the state variables. In physics, a system is described as a particle or ensemble of particles whose state varies over time. In order to make a prediction about the future behavior. Dynamical systems are a part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly process. The concept of a system has its origins in Newtonian mechanics. To determine the state for all future times requires iterating the relation many times—each advancing time a small step, the iteration procedure is referred to as solving the system or integrating the system. If the system can be solved, given a point it is possible to determine all its future positions. Before the advent of computers, finding an orbit required sophisticated mathematical techniques, numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, the difficulties arise because, The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. The approximations used bring into question the validity or relevance of numerical solutions, to address these questions several notions of stability have been introduced in the study of dynamical systems, such as Lyapunov stability or structural stability. The stability of the dynamical system implies that there is a class of models or initial conditions for which the trajectories would be equivalent, the operation for comparing orbits to establish their equivalence changes with the different notions of stability. The type of trajectory may be more important than one particular trajectory, some trajectories may be periodic, whereas others may wander through many different states of the system. Applications often require enumerating these classes or maintaining the system within one class, classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes
4.
State-space representation
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In control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. State space refers to the Euclidean space in which the variables on the axes are the state variables, the state of the system can be represented as a vector within that space. To abstract from the number of inputs, outputs and states, additionally, if the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form. The state-space method is characterized by significant algebraization of general system theory, the capacity of these structures can be efficiently applied to research systems with modulation or without it. The state-space representation provides a convenient and compact way to model, with p inputs and q outputs, we would otherwise have to write down q × p Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the representation is not limited to systems with linear components. The internal state variables are the smallest possible subset of variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system, in electric circuits, the number of state variables is often, though not always, the same as the number of energy storage elements in the circuit such as capacitors and inductors. The state variables defined must be independent, i. e. no state variable can be written as a linear combination of the other state variables or the system will not be able to be solved. In this general formulation, all matrices are allowed to be time-variant, however, in the common LTI case, the time variable t can be continuous or discrete. In the latter case, the variable k is usually used instead of t. Hybrid systems allow for time domains that have continuous and discrete parts. The stability of a time-invariant state-space model can be determined by looking at the transfer function in factored form. It will then look something like this, G = k, the denominator of the transfer function is equal to the characteristic polynomial found by taking the determinant of s I − A, λ = | s I − A |. The roots of this polynomial are the transfer functions poles. These poles can be used to analyze whether the system is stable or marginally stable. An alternative approach to determining stability, which not involve calculating eigenvalues, is to analyze the systems Lyapunov stability. The zeros found in the numerator of G can similarly be used to determine whether the system is minimum phase, the system may still be input–output stable even though it is not internally stable
5.
Block diagram
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A block diagram is a diagram of a system in which the principal parts or functions are represented by blocks connected by lines that show the relationships of the blocks. They are heavily used in engineering in hardware design, electronic design, software design, block diagrams are typically used for higher level, less detailed descriptions that are intended to clarify overall concepts without concern for the details of implementation. Contrast this with the diagrams and layout diagrams used in electrical engineering. As an example, a diagram of a radio is not expected to show each and every connection and dial and switch. The schematic diagram of a radio does not show the width of each connection in the circuit board. To make an analogy to the map making world, a diagram is similar to a highway map of an entire nation. The major cities are listed but the county roads and city streets are not. When troubleshooting, this high level map is useful in narrowing down, block diagrams rely on the principle of the black box where the contents are hidden from view either to avoid being distracted by the details or because the details are not known. We know what goes in, we know what goes out and this is known as top down design. Geometric shapes are used in the diagram to aid interpretation. The geometric shapes are connected by lines to indicate association and direction/order of traversal, each engineering discipline has their own meaning for each shape. Block diagrams are used in every discipline of engineering and they are also a valuable source of concept building and educationally beneficial in non-engineering disciplines. It is possible to create block diagrams and implement their functionality with specialized programmable logic controller programming languages. In biology there is an use of engineering principles, techniques of analysis. There is some similarity between the block diagram and what is called Systems Biology Graphical Notation, as it is there is use made in systems biology of the block diagram technique harnessed by control engineering where the latter itself is an application of control theory. Directed lines are used to input variables to block inputs. Black box Bond graph Data flow diagram Functional flow block diagram One-line diagram Reliability block diagram Schematic diagram Signal-flow graph
6.
Signal-flow graph
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Thus, signal-flow graph theory builds on that of directed graphs, which includes as well that of oriented graphs. This mathematical theory of digraphs exists, of course, quite apart from its applications, SFGs are most commonly used to represent signal flow in a physical system and its controller, forming a cyber-physical system. Among their other uses are the representation of flow in various electronic networks and amplifiers, digital filters, state variable filters. In nearly all literature, a graph is associated with a set of linear equations. Wai-Kai Chen wrote, The concept of a graph was originally worked out by Shannon in dealing with analog computers. The greatest credit for the formulation of signal-flow graphs is normally extended to Mason and he showed how to use the signal-flow graph technique to solve some difficult electronic problems in a relatively simple manner. The term signal flow graph was used because of its application to electronic problems. Lorens wrote, Previous to Masons work, C. E. Shannon worked out a number of the properties of what are now known as flow graphs, unfortunately, the paper originally had a restricted classification and very few people had access to the material. The rules for the evaluation of the determinant of a Mason Graph were first given and proven by Shannon using mathematical induction. His work remained unknown even after Mason published his classical work in 1953. Three years later, Mason rediscovered the rules and proved them by considering the value of a determinant, Robichaud et al. identify the domain of application of SFGs as follows, All the physical systems analogous to these networks constitute the domain of application of the techniques developed. Trent has shown all the physical systems which satisfy the following conditions fall into this category. Variables of the first type represent quantities which can be measured, at least conceptually, variables of the second type characterize quantities which can be measured by connecting a meter in series with the element. Firestone has been the first to distinguish two types of variables with the names across variables and through variables. Variables of the first type must obey a law, analogous to Kirchhoffs voltage law. Physical dimensions of appropriate products of the variables of the two types must be consistent, for the systems in which these conditions are satisfied, it is possible to draw a linear graph isomorphic with the dynamical properties of the system as described by the chosen variables. The techniques can be applied directly to these graphs as well as to electrical networks. These relationships define for every node a function processes the input signals it receives
7.
Energy
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In physics, energy is the property that must be transferred to an object in order to perform work on – or to heat – the object, and can be converted in form, but not created or destroyed. The SI unit of energy is the joule, which is the transferred to an object by the mechanical work of moving it a distance of 1 metre against a force of 1 newton. Mass and energy are closely related, for example, with a sensitive enough scale, one could measure an increase in mass after heating an object. Living organisms require available energy to stay alive, such as the humans get from food. Civilisation gets the energy it needs from energy resources such as fuels, nuclear fuel. The processes of Earths climate and ecosystem are driven by the radiant energy Earth receives from the sun, the total energy of a system can be subdivided and classified in various ways. It may also be convenient to distinguish gravitational energy, thermal energy, several types of energy, electric energy. Many of these overlap, for instance, thermal energy usually consists partly of kinetic. Some types of energy are a mix of both potential and kinetic energy. An example is energy which is the sum of kinetic. Whenever physical scientists discover that a phenomenon appears to violate the law of energy conservation. Heat and work are special cases in that they are not properties of systems, in general we cannot measure how much heat or work are present in an object, but rather only how much energy is transferred among objects in certain ways during the occurrence of a given process. Heat and work are measured as positive or negative depending on which side of the transfer we view them from, the distinctions between different kinds of energy is not always clear-cut. In contrast to the definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness. The modern analog of this property, kinetic energy, differs from vis viva only by a factor of two, in 1807, Thomas Young was possibly the first to use the term energy instead of vis viva, in its modern sense. Gustave-Gaspard Coriolis described kinetic energy in 1829 in its modern sense, the law of conservation of energy was also first postulated in the early 19th century, and applies to any isolated system. It was argued for years whether heat was a physical substance, dubbed the caloric, or merely a physical quantity. In 1845 James Prescott Joule discovered the link between mechanical work and the generation of heat and these developments led to the theory of conservation of energy, formalized largely by William Thomson as the field of thermodynamics
8.
Directed graph
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In mathematics, and more specifically in graph theory, a directed graph is a graph that is a set of vertices connected by edges, where the edges have a direction associated with them. It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, more specifically, these entities are addressed as directed multigraphs. On the other hand, the definition allows a directed graph to have loops. More specifically, directed graphs without loops are addressed as directed graphs. Symmetric directed graphs are directed graphs where all edges are bidirected, simple directed graphs are directed graphs that have no loops and no multiple arrows with same source and target nodes. As already introduced, in case of arrows the entity is usually addressed as directed multigraph. Some authors describe digraphs with loops as loop-digraphs. Complete directed graphs are directed graphs where each pair of vertices is joined by a symmetric pair of directed arrows. It follows that a complete digraph is symmetric, oriented graphs are directed graphs having no bidirected edges. It follows that a graph is an oriented graph iff it hasnt any 2-cycle. Tournaments are oriented graphs obtained by choosing a direction for each edge in undirected complete graphs. Directed acyclic graphs are directed graphs with no directed cycles, multitrees are DAGs in which no two directed paths from a single starting vertex meet back at the same ending vertex. Oriented trees or polytrees are DAGs formed by orienting the edges of undirected acyclic graphs, rooted trees are oriented trees in which all edges of the underlying undirected tree are directed away from the roots. Rooted directed graphs are digraphs in which a vertex has been distinguished as the root, control flow graphs are rooted digraphs used in computer science as a representation of the paths that might be traversed through a program during its execution. Signal-flow graphs are directed graphs in which nodes represent system variables and branches represent functional connections between pairs of nodes, flow graphs are digraphs associated with a set of linear algebraic or differential equations. State diagrams are directed multigraphs that represent finite state machines, representations of a quiver label its vertices with vector spaces and its edges compatibly with linear transformations between them, and transform via natural transformations. If a path leads from x to y, then y is said to be a successor of x and reachable from x, the arrow is called the inverted arrow of. The adjacency matrix of a graph is unique up to identical permutation of rows. Another matrix representation for a graph is its incidence matrix. For a vertex, the number of head ends adjacent to a vertex is called the indegree of the vertex, the indegree of v is denoted deg− and its outdegree is denoted deg+
9.
Power (physics)
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In physics, power is the rate of doing work. It is the amount of energy consumed per unit time, having no direction, it is a scalar quantity. In the SI system, the unit of power is the joule per second, known as the watt in honour of James Watt, another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written, because this integral depends on the trajectory of the point of application of the force and torque, this calculation of work is said to be path dependent. As a physical concept, power requires both a change in the universe and a specified time in which the change occurs. This is distinct from the concept of work, which is measured in terms of a net change in the state of the physical universe. The output power of a motor is the product of the torque that the motor generates. The power involved in moving a vehicle is the product of the force of the wheels. The dimension of power is divided by time. The SI unit of power is the watt, which is equal to one joule per second, other units of power include ergs per second, horsepower, metric horsepower, and foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the required to lift 550 pounds by one foot in one second. Other units include dBm, a logarithmic measure with 1 milliwatt as reference, food calories per hour, Btu per hour. This shows how power is an amount of energy consumed per unit time. If ΔW is the amount of work performed during a period of time of duration Δt and it is the average amount of work done or energy converted per unit of time. The average power is simply called power when the context makes it clear. The instantaneous power is then the value of the average power as the time interval Δt approaches zero. P = lim Δ t →0 P a v g = lim Δ t →0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration T is given by, W = P t
10.
Velocity
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The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of its speed and direction of motion, Velocity is an important concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a vector quantity, both magnitude and direction are needed to define it. The scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI system as metres per second or as the SI base unit of. For example,5 metres per second is a scalar, whereas 5 metres per second east is a vector, if there is a change in speed, direction or both, then the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction, constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a path has a constant speed. Hence, the car is considered to be undergoing an acceleration, Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified, however, if the car is said to move at 60 km/h to the north, its velocity has now been specified. The big difference can be noticed when we consider movement around a circle and this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, average velocity can be calculated as, v ¯ = Δ x Δ t. The average velocity is less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, from this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity v is the displacement function x. In the figure, this corresponds to the area under the curve labeled s. Since the derivative of the position with respect to time gives the change in position divided by the change in time, although velocity is defined as the rate of change of position, it is often common to start with an expression for an objects acceleration. As seen by the three green tangent lines in the figure, an objects instantaneous acceleration at a point in time is the slope of the tangent to the curve of a v graph at that point. In other words, acceleration is defined as the derivative of velocity with respect to time, from there, we can obtain an expression for velocity as the area under an a acceleration vs. time graph
11.
Force
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In physics, a force is any interaction that, when unopposed, will change the motion of an object. In other words, a force can cause an object with mass to change its velocity, force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity and it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newtons second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. In an extended body, each part usually applies forces on the adjacent parts, such internal mechanical stresses cause no accelation of that body as the forces balance one another. Pressure, the distribution of small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of materials, or flow in fluids. In part this was due to an understanding of the sometimes non-obvious force of friction. A fundamental error was the belief that a force is required to maintain motion, most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved-on for nearly three hundred years, the Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known, in order of decreasing strength, they are, strong, electromagnetic, weak, high-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotles view, the sphere contained four elements that come to rest at different natural places therein. Aristotle believed that objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground. He distinguished between the tendency of objects to find their natural place, which led to natural motion, and unnatural or forced motion
12.
Momentum
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In classical mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object, quantified in kilogram-meters per second. It is dimensionally equivalent to impulse, the product of force and time, Newtons second law of motion states that the change in linear momentum of a body is equal to the net impulse acting on it. If the truck were lighter, or moving slowly, then it would have less momentum. Linear momentum is also a quantity, meaning that if a closed system is not affected by external forces. In classical mechanics, conservation of momentum is implied by Newtons laws. It also holds in special relativity and, with definitions, a linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory. It is ultimately an expression of one of the symmetries of space and time. Linear momentum depends on frame of reference, observers in different frames would find different values of linear momentum of a system. But each would observe that the value of linear momentum does not change with time, momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities, because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Below, the properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations, the momentum of a particle is traditionally represented by the letter p. It is the product of two quantities, the mass and velocity, p = m v, the units of momentum are the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity in meters per second then the momentum is in kilogram meters/second, in cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters/second. Being a vector, momentum has magnitude and direction, for example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg m/s due north measured from the ground. The momentum of a system of particles is the sum of their momenta, if two particles have masses m1 and m2, and velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2. If all the particles are moving, the center of mass will generally be moving as well, if the center of mass is moving at velocity vcm, the momentum is, p = m v cm. This is known as Eulers first law, if a force F is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t
13.
Mass
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In physics, mass is a property of a physical body. It is the measure of a resistance to acceleration when a net force is applied. It also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram, Mass is not the same as weight, even though mass is often determined by measuring the objects weight using a spring scale, rather than comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity and this is because weight is a force, while mass is the property that determines the strength of this force. In Newtonian physics, mass can be generalized as the amount of matter in an object, however, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy. In addition, matter is a defined term in science. There are several distinct phenomena which can be used to measure mass, active gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force, according to Newtons second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A bodys mass also determines the degree to which it generates or is affected by a gravitational field and this is sometimes referred to as gravitational mass. The standard International System of Units unit of mass is the kilogram, the kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the prototype kilogram. As of January 2013, there are proposals for redefining the kilogram yet again. In this context, the mass has units of eV/c2, the electronvolt and its multiples, such as the MeV, are commonly used in particle physics. The atomic mass unit is 1/12 of the mass of a carbon-12 atom, the atomic mass unit is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include, the slug is an Imperial unit of mass, the pound is a unit of both mass and force, used mainly in the United States
14.
Angular displacement
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Angular displacement of a body is the angle in radians through which a point or line has been rotated in a specified sense about a specified axis. When an object rotates about its axis, the motion cannot simply be analyzed as a particle, since in circular motion it undergoes a changing velocity, when dealing with the rotation of an object, it becomes simpler to consider the body itself rigid. A body is generally considered rigid when the separations between all the particles remains constant throughout the motion, so for example parts of its mass are not flying off. In a realistic sense, all things can be deformable, however this impact is minimal, Thus the rotation of a rigid body over a fixed axis is referred to as rotational motion. In the example illustrated to the right, a particle on object P is at a distance r from the origin, O. It becomes important to represent the position of particle P in terms of its polar coordinates. In this particular example, the value of θ is changing, if using radians, it provides a very simple relationship between distance traveled around the circle and the distance r from the centre. Therefore,1 revolution is 2 π radians, when object travels from point P to point Q, as it does in the illustration to the left, over δ t the radius of the circle goes around a change in angle. Δ θ = θ2 − θ1 which equals the Angular Displacement, in three dimensions, angular displacement is an entity with a direction and a magnitude. The direction specifies the axis of rotation, which exists by virtue of the Eulers rotation theorem. This entity is called an axis-angle, despite having direction and magnitude, angular displacement is not a vector because it does not obey the commutative law for addition. Nevertheless, when dealing with infinitesimal rotations, second order infinitesimals can be discarded, several ways to describe angular displacement exist, like rotation matrices or Euler angles. See charts on SO for others, given that any frame in the space can be described by a rotation matrix, the displacement among them can also be described by a rotation matrix. Being A0 and A f two matrices, the angular displacement matrix between them can be obtained as Δ A = A f, when this product is performed having a very small difference between both frames we will obtain a matrix close to the identity. In the limit, we will have a rotation matrix. An infinitesimal angular displacement is a rotation matrix, As any rotation matrix has a single real eigenvalue, which is +1. Its module can be deduced from the value of the infinitesimal rotation, when it is divided by the time, this will yield the angular velocity vector. Suppose we specify an axis of rotation by a unit vector, expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix ΔR is represented as, Δ R = + Δ θ = I + A Δ θ
15.
Angular momentum
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In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers
16.
Moment of inertia
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It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change the bodys rotation. It is a property, the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. One of its definitions is the moment of mass with respect to distance from an axis r, I = ∫ Q r 2 d m. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about a perpendicular to the plane. When a body is rotating, or free to rotate, around an axis, the amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared in SI units, moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, for a point-like mass, the moment of inertia about some axis is given by mr2, where r is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a shape and uniform density. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, the term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Eulers second law. Comparison of this frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, and in Newtons laws of motion for a rigid body as a physical parameter that combines its shape. There is a difference in the way moment of inertia appears in planar. The moment of inertia of a flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, if the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their arms or divers curl their bodies into a tuck position during a dive. For a simple pendulum, this yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as. Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation
17.
Electric current
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An electric current is a flow of electric charge. In electric circuits this charge is carried by moving electrons in a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionised gas. The SI unit for measuring a current is the ampere. Electric current is measured using a device called an ammeter, electric currents cause Joule heating, which creates light in incandescent light bulbs. They also create magnetic fields, which are used in motors, inductors and generators, the particles that carry the charge in an electric current are called charge carriers. In metals, one or more electrons from each atom are loosely bound to the atom and these conduction electrons are the charge carriers in metal conductors. The conventional symbol for current is I, which originates from the French phrase intensité de courant, current intensity is often referred to simply as current. The I symbol was used by André-Marie Ampère, after whom the unit of current is named, in formulating the eponymous Ampères force law. The notation travelled from France to Great Britain, where it became standard, in a conductive material, the moving charged particles which constitute the electric current are called charge carriers. In other materials, notably the semiconductors, the carriers can be positive or negative. Positive and negative charge carriers may even be present at the same time, a flow of positive charges gives the same electric current, and has the same effect in a circuit, as an equal flow of negative charges in the opposite direction. Since current can be the flow of positive or negative charges. The direction of current is arbitrarily defined as the same direction as positive charges flow. This is called the direction of current I. If the current flows in the direction, the variable I has a negative value. When analyzing electrical circuits, the direction of current through a specific circuit element is usually unknown. Consequently, the directions of currents are often assigned arbitrarily
18.
Voltage
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Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential energy between two points per unit electric charge. The voltage between two points is equal to the work done per unit of charge against an electric field to move the test charge between two points. This is measured in units of volts, voltage can be caused by static electric fields, by electric current through a magnetic field, by time-varying magnetic fields, or some combination of these three. A voltmeter can be used to measure the voltage between two points in a system, often a reference potential such as the ground of the system is used as one of the points. A voltage may represent either a source of energy or lost, used, given two points in space, x A and x B, voltage is the difference in electric potential between those two points. Electric potential must be distinguished from electric energy by noting that the potential is a per-unit-charge quantity. Like mechanical potential energy, the zero of electric potential can be chosen at any point, so the difference in potential, i. e. the voltage, is the quantity which is physically meaningful. The voltage between point A to point B is equal to the work which would have to be done, per unit charge, against or by the electric field to move the charge from A to B. The voltage between the two ends of a path is the energy required to move a small electric charge along that path. Mathematically this is expressed as the integral of the electric field. In the general case, both an electric field and a dynamic electromagnetic field must be included in determining the voltage between two points. Historically this quantity has also called tension and pressure. Pressure is now obsolete but tension is used, for example within the phrase high tension which is commonly used in thermionic valve based electronics. Voltage is defined so that negatively charged objects are pulled towards higher voltages, therefore, the conventional current in a wire or resistor always flows from higher voltage to lower voltage. Current can flow from lower voltage to higher voltage, but only when a source of energy is present to push it against the electric field. This is the case within any electric power source, for example, inside a battery, chemical reactions provide the energy needed for ion current to flow from the negative to the positive terminal. The electric field is not the only factor determining charge flow in a material, the electric potential of a material is not even a well defined quantity, since it varies on the subatomic scale. A more convenient definition of voltage can be found instead in the concept of Fermi level, in this case the voltage between two bodies is the thermodynamic work required to move a unit of charge between them
19.
Electric charge
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Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of charges, positive and negative. Like charges repel and unlike attract, an absence of net charge is referred to as neutral. An object is charged if it has an excess of electrons. The SI derived unit of charge is the coulomb. In electrical engineering, it is common to use the ampere-hour. The symbol Q often denotes charge, early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that dont require consideration of quantum effects. The electric charge is a conserved property of some subatomic particles. Electrically charged matter is influenced by, and produces, electromagnetic fields, the interaction between a moving charge and an electromagnetic field is the source of the electromagnetic force, which is one of the four fundamental forces. 602×10−19 coulombs. The proton has a charge of +e, and the electron has a charge of −e, the study of charged particles, and how their interactions are mediated by photons, is called quantum electrodynamics. Charge is the property of forms of matter that exhibit electrostatic attraction or repulsion in the presence of other matter. Electric charge is a property of many subatomic particles. The charges of free-standing particles are integer multiples of the charge e. Michael Faraday, in his electrolysis experiments, was the first to note the discrete nature of electric charge, robert Millikans oil drop experiment demonstrated this fact directly, and measured the elementary charge. By convention, the charge of an electron is −1, while that of a proton is +1, charged particles whose charges have the same sign repel one another, and particles whose charges have different signs attract. The charge of an antiparticle equals that of the corresponding particle, quarks have fractional charges of either −1/3 or +2/3, but free-standing quarks have never been observed. The electric charge of an object is the sum of the electric charges of the particles that make it up. An ion is an atom that has lost one or more electrons, giving it a net charge, or that has gained one or more electrons
20.
Ohm
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The ohm is the SI derived unit of electrical resistance, named after German physicist Georg Simon Ohm. The definition of the ohm was revised several times, today the definition of the ohm is expressed from the quantum Hall effect. In many cases the resistance of a conductor in ohms is approximately constant within a range of voltages, temperatures. In alternating current circuits, electrical impedance is also measured in ohms, the siemens is the SI derived unit of electric conductance and admittance, also known as the mho, it is the reciprocal of resistance in ohms. The power dissipated by a resistor may be calculated from its resistance, non-linear resistors have a value that may vary depending on the applied voltage. The rapid rise of electrotechnology in the last half of the 19th century created a demand for a rational, coherent, consistent, telegraphers and other early users of electricity in the 19th century needed a practical standard unit of measurement for resistance. Two different methods of establishing a system of units can be chosen. Various artifacts, such as a length of wire or a standard cell, could be specified as producing defined quantities for resistance, voltage. This latter method ensures coherence with the units of energy, defining a unit for resistance that is coherent with units of energy and time in effect also requires defining units for potential and current. Some early definitions of a unit of resistance, for example, the absolute-units system related magnetic and electrostatic quantities to metric base units of mass, time, and length. These units had the advantage of simplifying the equations used in the solution of electromagnetic problems. However, the CGS units turned out to have impractical sizes for practical measurements, various artifact standards were proposed as the definition of the unit of resistance. In 1860 Werner Siemens published a suggestion for a reproducible resistance standard in Poggendorffs Annalen der Physik und Chemie and he proposed a column of pure mercury, of one square millimetre cross section, one metre long, Siemens mercury unit. However, this unit was not coherent with other units, one proposal was to devise a unit based on a mercury column that would be coherent – in effect, adjusting the length to make the resistance one ohm. Not all users of units had the resources to carry out experiments to the required precision. The BAAS in 1861 appointed a committee including Maxwell and Thomson to report upon Standards of Electrical Resistance, in the third report of the committee,1864, the resistance unit is referred to as B. A. unit, or Ohmad. By 1867 the unit is referred to as simply Ohm, the B. A. ohm was intended to be 109 CGS units but owing to an error in calculations the definition was 1. 3% too small. The error was significant for preparation of working standards, on September 21,1881 the Congrès internationale délectriciens defined a practical unit of Ohm for the resistance, based on CGS units, using a mercury column at zero deg
21.
Henry (unit)
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The henry is the SI derived unit of electrical inductance. The unit is named after Joseph Henry, the American scientist who discovered electromagnetic induction independently of, the magnetic permeability of vacuum is 4π × 10−7 H⋅m−1. The henry is a unit based on four of the seven base units of the International System of Units, kilogram, meter, second. The United States National Institute of Standards and Technology recommends English-speaking users of SI to write the plural as henries
22.
Farad
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The farad is the SI derived unit of electrical capacitance, the ability of a body to store an electrical charge. It is named after the English physicist Michael Faraday, one farad is defined as the capacitance across which, when charged with one coulomb, there is a potential difference of one volt. Equally, one farad can be described as the capacitance which stores a one-coulomb charge across a potential difference of one volt, the relationship between capacitance, charge and potential difference is linear. For example, if the difference across a capacitor is halved. For most applications, the farad is a large unit of capacitance. Most electrical and electronic applications are covered by the following SI prefixes,1 mF =1000 μF =1000000 nF1 μF =0.000001 F =1000 nF =1000000 pF1 nF =0. In 1881 at the International Congress of Electricians in Paris, the name farad was officially used for the unit of electrical capacitance, a capacitor consists of two conducting surfaces, frequently referred to as plates, separated by an insulating layer usually referred to as a dielectric. The original capacitor was the Leyden jar developed in the 18th century and it is the accumulation of electric charge on the plates that results in capacitance. Values of capacitors are specified in farads, microfarads, nanofarads and picofarads. The millifarad is rarely used in practice, while the nanofarad is uncommon in North America, the size of commercially available capacitors ranges from around 0.1 pF to 5000F supercapacitors. Capacitance values of 1 pF or lower can be achieved by twisting two short lengths of insulated wire together, the capacitance of the Earths ionosphere with respect to the ground is calculated to be about 1 F. The picofarad is sometimes pronounced as puff or pic, as in a ten-puff capacitor. Similarly, mic is sometimes used informally to signify microfarads, if the Greek letter μ is not available, the notation uF is often used as a substitute for μF in electronics literature. A micro-microfarad, an obsolete unit sometimes found in texts, is the equivalent of a picofarad. In texts prior to 1960, and on capacitor packages even more recently. Similarly, mmf or MMFD represented picofarads, the reciprocal of capacitance is called electrical elastance, the unit of which is the daraf. The abfarad is an obsolete CGS unit of equal to 109 farads. The statfarad is a rarely used CGS unit equivalent to the capacitance of a capacitor with a charge of 1 statcoulomb across a potential difference of 1 statvolt and it is 1/ farad, approximately 1.1126 picofarads
23.
Pressure
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Pressure is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure is the relative to the ambient pressure. Various units are used to express pressure, Pressure may also be expressed in terms of standard atmospheric pressure, the atmosphere is equal to this pressure and the torr is defined as 1⁄760 of this. Manometric units such as the centimetre of water, millimetre of mercury, Pressure is the amount of force acting per unit area. The symbol for it is p or P, the IUPAC recommendation for pressure is a lower-case p. However, upper-case P is widely used. The usage of P vs p depends upon the field in one is working, on the nearby presence of other symbols for quantities such as power and momentum. Mathematically, p = F A where, p is the pressure, F is the normal force and it relates the vector surface element with the normal force acting on it. It is incorrect to say the pressure is directed in such or such direction, the pressure, as a scalar, has no direction. The force given by the relationship to the quantity has a direction. If we change the orientation of the element, the direction of the normal force changes accordingly. Pressure is distributed to solid boundaries or across arbitrary sections of normal to these boundaries or sections at every point. It is a parameter in thermodynamics, and it is conjugate to volume. The SI unit for pressure is the pascal, equal to one newton per square metre and this name for the unit was added in 1971, before that, pressure in SI was expressed simply in newtons per square metre. Other units of pressure, such as pounds per square inch, the CGS unit of pressure is the barye, equal to 1 dyn·cm−2 or 0.1 Pa. Pressure is sometimes expressed in grams-force or kilograms-force per square centimetre, but using the names kilogram, gram, kilogram-force, or gram-force as units of force is expressly forbidden in SI. The technical atmosphere is 1 kgf/cm2, since a system under pressure has potential to perform work on its surroundings, pressure is a measure of potential energy stored per unit volume. It is therefore related to density and may be expressed in units such as joules per cubic metre. Similar pressures are given in kilopascals in most other fields, where the prefix is rarely used
24.
Volume
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Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, Volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
25.
Ohm's law
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Ohms law states that the current through a conductor between two points is directly proportional to the voltage across the two points. More specifically, Ohms law states that the R in this relation is constant, independent of the current and he presented a slightly more complex equation than the one above to explain his experimental results. The above equation is the form of Ohms law. In physics, the term Ohms law is used to refer to various generalizations of the law originally formulated by Ohm. This reformulation of Ohms law is due to Gustav Kirchhoff, in January 1781, before Georg Ohms work, Henry Cavendish experimented with Leyden jars and glass tubes of varying diameter and length filled with salt solution. He measured the current by noting how strong a shock he felt as he completed the circuit with his body, Cavendish wrote that the velocity varied directly as the degree of electrification. He did not communicate his results to other scientists at the time, francis Ronalds delineated “intensity” and “quantity” for the dry pile – a high voltage source – in 1814 using a gold-leaf electrometer. He found for a dry pile that the relationship between the two parameters was not proportional under certain meteorological conditions, Ohm did his work on resistance in the years 1825 and 1826, and published his results in 1827 as the book Die galvanische Kette, mathematisch bearbeitet. He drew considerable inspiration from Fouriers work on heat conduction in the explanation of his work. For experiments, he initially used voltaic piles, but later used a thermocouple as this provided a stable voltage source in terms of internal resistance. He used a galvanometer to measure current, and knew that the voltage between the terminals was proportional to the junction temperature. He then added test wires of varying length, diameter, from this, Ohm determined his law of proportionality and published his results. Ohms law was probably the most important of the early descriptions of the physics of electricity. We consider it almost obvious today, when Ohm first published his work, this was not the case, critics reacted to his treatment of the subject with hostility. They called his work a web of naked fancies and the German Minister of Education proclaimed that a professor who preached such heresies was unworthy to teach science, also, Ohms brother Martin, a mathematician, was battling the German educational system. These factors hindered the acceptance of Ohms work, and his work did not become widely accepted until the 1840s, fortunately, Ohm received recognition for his contributions to science well before he died. While the old term for electrical conductance, the mho, is used, a new name. The siemens is preferred in formal papers, Ohms work long preceded Maxwells equations and any understanding of frequency-dependent effects in AC circuits
26.
Hooke's law
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Hookes law is a principle of physics that states that the force needed to extend or compress a spring by some distance X is proportional to that distance. That is, F = kX, where k is a constant factor characteristic of the spring, its stiffness, the law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram and he published the solution of his anagram in 1678 as, ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law already in 1660, an elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean. Hookes law is only a linear approximation to the real response of springs. Many materials will deviate from Hookes law well before those elastic limits are reached. On the other hand, Hookes law is an approximation for most solid bodies, as long as the forces. For this reason, Hookes law is used in all branches of science and engineering. It is also the principle behind the spring scale, the manometer. The modern theory of elasticity generalizes Hookes law to say that the strain of an object or material is proportional to the stress applied to it. In this general form, Hookes law makes it possible to deduce the relation between strain and stress for complex objects in terms of properties of the materials it is made of. Consider a simple helical spring that has one end attached to some fixed object, suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Let X be the amount by which the end of the spring was displaced from its relaxed position. Hookes law states that F = k X or, equivalently, X = F k where k is a real number. Moreover, the formula holds when the spring is compressed. According to this formula, the graph of the applied force F as a function of the displacement X will be a line passing through the origin. Hookes law for a spring is often stated under the convention that F is the force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F = − k X since the direction of the force is opposite to that of the displacement
27.
Transformer
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A transformer is an electrical device that transfers electrical energy between two or more circuits through electromagnetic induction. A varying current in one coil of the transformer produces a magnetic field. Power can be transferred between the two coils through the field, without a metallic connection between the two circuits. Faradays law of induction discovered in 1831 described this effect, Transformers are used to increase or decrease the alternating voltages in electric power applications. A wide range of designs is encountered in electronic and electric power applications. Transformers range in size from RF transformers less than a centimeter in volume to units interconnecting the power grid weighing hundreds of tons. For simplification or approximation purposes, it is common to analyze the transformer as an ideal transformer model as presented in the two images. An ideal transformer is a theoretical, linear transformer that is lossless and perfectly coupled, perfect coupling implies infinitely high core magnetic permeability and winding inductances and zero net magnetomotive force. A varying current in the primary winding creates a varying magnetic flux in the transformer core. This varying magnetic field at the secondary winding induces a varying EMF or voltage in the secondary winding due to electromagnetic induction. The primary and secondary windings are wrapped around a core of high magnetic permeability so that all of the magnetic flux passes through both the primary and secondary windings. With a voltage source connected to the winding and load impedance connected to the secondary winding. The primary EMF is sometimes termed counter EMF and this is in accordance with Lenzs law, which states that induction of EMF always opposes development of any such change in magnetic field. The transformer winding voltage ratio is shown to be directly proportional to the winding turns ratio according to eq. common usage having evolved over time from turn ratio to turns ratio. However, some use the inverse definition. The ideal transformer model assumes that all flux generated by the primary winding links all the turns of winding, including itself. In practice, some flux traverses paths that take it outside the windings, Such flux is termed leakage flux, and results in leakage inductance in series with the mutually coupled transformer windings. Leakage flux results in energy being alternately stored in and discharged from the fields with each cycle of the power supply
28.
Lever
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A lever is a machine consisting of a beam or rigid rod pivoted at a fixed hinge, or fulcrum. A lever is a body capable of rotating on a point on itself. On the basis of the location of fulcrum, load and effort and it is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide an output force. The ratio of the force to the input force is the mechanical advantage of the lever. The word lever entered English about 1300 from Old French, in which the word was levier and this sprang from the stem of the verb lever, meaning to raise. The verb, in turn, goes back to the Latin levare, itself from the adjective levis, the words primary origin is the Proto-Indo-European stem legwh-, meaning light, easy or nimble, among other things. The PIE stem also gave rise to the English word light, the earliest remaining writings regarding levers date from the 3rd century BC and were provided by Archimedes. Give me a place to stand, and I shall move the Earth with it is a remark of Archimedes who formally stated the correct mathematical principle of levers. The distance required to do this might be exemplified in astronomical terms as the distance to the Circinus galaxy - about 9 million light years. It is assumed that in ancient Egypt, constructors used the lever to move, a lever is a beam connected to ground by a hinge, or pivot, called a fulcrum. The ideal lever does not dissipate or store energy, which there is no friction in the hinge or bending in the beam. This is known as the law of the lever, the mechanical advantage of a lever can be determined by considering the balance of moments or torque, T, about the fulcrum. T1 = F1 a, T2 = F2 b where F1 is the force to the lever. The distances a and b are the distances between the forces and the fulcrum. Since the moments of torque must be balanced, T1 = T2, the mechanical advantage of the lever is the ratio of output force to input force, M A = F2 F1 = a b. Levers are classified by the positions of the fulcrum, effort. It is common to call the force the effort and the output force the load or the resistance
29.
Gyrator
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Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two--port devices which cannot be realized with just the four elements. In particular, gyrators make possible network realizations of isolators and circulators, gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op amps using feedback, tellegen invented a circuit symbol for the gyrator and suggested a number of ways in which a practical gyrator might be built. An important property of a gyrator is that it inverts the current-voltage characteristic of a component or network. In the case of linear elements, the impedance is also inverted, in other words, a gyrator can make a capacitive circuit behave inductively, a series LC circuit behave like a parallel LC circuit, and so on. It is primarily used in filter design and miniaturization. An ideal gyrator is a two port device which couples the current on one port to the voltage on the other. The instantaneous currents and instantaneous voltages are related by v 2 = R i 1 v 1 = − R i 2 where R is the resistance of the gyrator. The gyration resistance has an associated direction indicated by an arrow on the schematic diagram, by convention, the given gyration resistance or conductance relates the voltage on the port at the head of the arrow to the current at its tail. The voltage at the tail of the arrow is related to the current at its head by minus the stated resistance, reversing the arrow is equivalent to negating the gyration resistance, or to reversing the polarity of either port. Although a gyrator is characterized by its value, it is a lossless component. From the governing equations, the power into the gyrator is identically zero. The symbol used to represent a gyrator in one-line diagrams, reflects this one-way phase shift, the gyrator is related to the gyroscope by an analogy in its behaviour. The analogy with the gyroscope is due to the relationship between the torque and angular velocity of the gyroscope on the two axes of rotation, a torque on one axis will produce a proportional change in angular velocity on the other axis and vice versa. A mechanical-electrical analogy of the gyroscope making torque and angular velocity the analogs of voltage, an ideal gyrator is similar to an ideal transformer in being a linear, lossless, passive, memoryless two-port device
30.
Differential equation
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A differential equation is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, because such relations are extremely common, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. In pure mathematics, differential equations are studied from different perspectives. Only the simplest differential equations are solvable by explicit formulas, however, if a self-contained formula for the solution is not available, the solution may be numerically approximated using computers. Differential equations first came into existence with the invention of calculus by Newton, jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is a differential equation of the form y ′ + P y = Q y n for which the following year Leibniz obtained solutions by simplifying it. Historically, the problem of a string such as that of a musical instrument was studied by Jean le Rond dAlembert, Leonhard Euler, Daniel Bernoulli. In 1746, d’Alembert discovered the wave equation, and within ten years Euler discovered the three-dimensional wave equation. The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a particle will fall to a fixed point in a fixed amount of time. Lagrange solved this problem in 1755 and sent the solution to Euler, both further developed Lagranges method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Contained in this book was Fouriers proposal of his heat equation for conductive diffusion of heat and this partial differential equation is now taught to every student of mathematical physics. For example, in mechanics, the motion of a body is described by its position. Newtons laws allow one to express these variables dynamically as an equation for the unknown position of the body as a function of time. In some cases, this equation may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity, the balls acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the balls velocity and this means that the balls acceleration, which is a derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation, Differential equations can be divided into several types
31.
Time derivative
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A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is written as t. A variety of notations are used to denote the time derivative, in addition to the normal notation, d x d t A very common short-hand notation used, especially in physics, is the over-dot. X ˙ Higher time derivatives are used, the second derivative with respect to time is written as d 2 x d t 2 with the corresponding shorthand of x ¨. As a generalization, the derivative of a vector, say. Time derivatives are a key concept in physics, for example, for a changing position x, its time derivative x ˙ is its velocity, and its second derivative with respect to time, x ¨, is its acceleration. Even higher derivatives are also used, the third derivative of position with respect to time is known as the jerk. A large number of equations in physics involve first or second time derivatives of quantities. A common occurrence in physics is the derivative of a vector. In dealing with such a derivative, both magnitude and orientation may depend upon time, for example, consider a particle moving in a circular path. With this form for the displacement, the velocity now is found, the time derivative of the displacement vector is the velocity vector. In general, the derivative of a vector is a made up of components each of which is the derivative of the corresponding component of the original vector. Thus, in case, the velocity vector is, v = d r d t = r = r =. Thus the velocity of the particle is nonzero even though the magnitude of the position is constant, the velocity is directed perpendicular to the displacement, as can be established using the dot product, v ⋅ r = ⋅ = − y x + x y =0. Acceleration is then the time-derivative of velocity, a = d v d t = = − r, the acceleration is directed inward, toward the axis of rotation. It points opposite to the vector and perpendicular to the velocity vector. This inward-directed acceleration is called centripetal acceleration, in economics, many theoretical models of the evolution of various economic variables are constructed in continuous time and therefore employ time derivatives. See for example exogenous growth model and ch, one situation involves a stock variable and its time derivative, a flow variable
32.
20-sim
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20-sim is commercial modeling and simulation program for multidomain dynamic systems, which is developed by Controllab. With 20-sim models can be entered as equations, block diagrams, bond graphs, 20-sim is widely used for modeling complex multi-domain systems and the development of control systems. 20-sim supports four methods for modeling systems, iconic diagrams, block diagrams, bond graphs and equations. The package has advanced support for bond graph modeling, making it known in bond graph communities. For modeling physical systems the package provides libraries for electrical systems, mechanical systems, hydraulics systems, for block diagrams, libraries comparable to those of Simulink, are provided. A feature of the software is the option to create models with differential equations, 20-sim models can be simulated using state of the art numerical integration methods. After checking and processing, models are directly converted into machine code, unlike Simulink, simulation results are shown in 20-sim in a separate window called the Simulator. The simulator is versatile, plots can be displayed horizontally and vertically as time and frequency based plots, 20-sim is self containing, i. e. no additional software is required and all toolboxes are included. Toolboxes are available for building, time domain analysis, frequency domain analysis. To enable scripting it is necessary to install either Matlab, GNU Octave, the last is included as an optional feature in the 20-sim installer. Because of its support of bond graph modeling 20-sim is highly rated in the bond graph community. According to Borutzky only 20-sim, MS1 and Symbols can be categorized as an integrated modeling. Roddeck compares several modeling and simulation tools like Simulink, Labview, the book of J. Ledin gives practical guidelines for modeling and simulation of dynamic systems. An entire chapter is spent on simulation tools and this allows for example, the construction of electrical circuit simulations using standard symbols to represent components, such as op-amps and capacitors. A weak point, according to Ledin is the capability for distributed simulation in 20-sim. 20-sim offers tight integration with 20-sim 4C, any 20-sim model can be exported as C-code to 20-sim 4C where it can be used for deployment on hardware. Typical use is the development of controllers for embedded software and the creation of plants for use in hardware-in-the-loop simulators. 20-sim can be controlled by scripting, allowing task automation and scenario building, scripting is supported in Matlab or GNU Octave, and in Python
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LMS Imagine.Lab Amesim
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LMS Imagine. Lab Amesim is a commercial simulation software for the modeling and analysis of multi-domain systems. It is part of systems engineering domain and falls into the engineering field. The software package is a suite of tools used to model, analyze, models are described using nonlinear time-dependent analytical equations that represent the system’s hydraulic, pneumatic, thermal, electric or mechanical behavior. To create a model for a system, a set of libraries is used. The icons in the system have to be connected and for this purpose each icon has ports, causality is enforced by linking the inputs of one icon to the outputs of another icon. The software runs on most UNIX platforms and on Windows platforms, LMS Imagine. Lab Amesim 15 was released in July 2016. LMS Imagine. Lab Amesim is a part of the Siemens PLM Software Simcenter portfolio and this combines 1D simulation, 3D CAE and physical testing with intelligent reporting and data analytics. The LMS Imagine. Lab Amesim software was developed by Imagine S. A. a company which was acquired in June 2007 by LMS International, the initial engineering project involved the deck elevation of the sinking Ekofisk North Sea petroleum platforms. LMS Imagine. Lab Amesim is used by companies in the automotive, aerospace, in its use, LMS Imagine. Lab Amesim is quite similar to Simulink. LMS Imagine. Lab Amesim is a multi-domain software and it allows to link between different physics domains. It is based on the Bond graph theory, between the submodel and parameter mode, the LMS Amesim model is compiled. Under the Windows platform, LMS Imagine. Lab Amesim works with the free Gcc compiler and it also works with the Microsoft Visual C++ compiler and its free Express edition. Since the version 4.3.0 LMS Amesim uses the Intel compiler on all platforms, the C source code of most of the standard submodels are provided allowing the user to start from this base to fit them to his needs. To create a simulation model in LMS Amesim, components from different physical domains are assembled. The physical libraries have developed through engineering services and partnerships with customers. In version 15, LMS Amesim offered up to 40 libraries to answer various application requirements, hydraulic/pneumatic pipes with wave effects and water-hammer effect, flexible hoses, speed of sound, shocks. It is also the reference framework for various Research projects in Europe, LMS Imagine. Lab Amesim product pages Functional Mock-up Interface LMS Imagine. Lab Amesim videos 1D Simulation Knowledge Base Simcenter 1D videos Several overview presentations
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International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker