In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external forces exerted on it. A rigid body is considered as a continuous distribution of mass. In the study of special relativity, a rigid body does not exist. In quantum mechanics a rigid body is thought of as a collection of point masses. For instance, in quantum mechanics molecules are seen as rigid bodies; the position of a rigid body is the position of all the particles. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles; this makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known.
However a different, mathematically more convenient, but equivalent approach is used. The position of the whole body is represented by: the linear position or position of the body, namely the position of one of the particles of the body chosen as a reference point, together with the angular position of the body. Thus, the position of a rigid body has two components: angular, respectively; the same is true for other kinematic and kinetic quantities describing the motion of a rigid body, such as linear and angular velocity, momentum and kinetic energy. The linear position can be represented by a vector with its tail at an arbitrary reference point in space and its tip at an arbitrary point of interest on the rigid body coinciding with its center of mass or centroid; this reference point may define the origin of a coordinate system fixed to the body. There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix.
All these methods define the orientation of a basis set which has a fixed orientation relative to the body, relative to another basis set, from which the motion of the rigid body is observed. For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of three orthogonal unit vectors b1, b2, b3, such that b1 is parallel to the chord line of the wing and directed forward, b2 is normal to the plane of symmetry and directed rightward, b3 is given by the cross product b 3 = b 1 × b 2. In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as rotation, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position. Velocity and angular velocity are measured with respect to a frame of reference; the linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position.
Thus, it is the velocity of a reference point fixed to the body. During purely translational motion, all points on a rigid body move with the same velocity. However, when motion involves rotation, the instantaneous velocity of any two points on the body will not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a vector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which it is rotating. All points on a rigid body experience the same angular velocity at all times. During purely rotational motion, all points on the body change position except for those lying on the instantaneous axis of rotation; the relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the time rate of change of orientation, because there is no such concept as an orientation vector that can be differentiated to obtain the angular velocity.
The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular velocity of a rigid body D in N and the angular velocity of B with respect to D: N ω B = N ω D + D ω B. In this case, rigid bodies and reference frames are indistinguishable and interchangeable. For any set of three points P, Q, R, the position ve
In physics, motion is the change in position of an object with respect to its surroundings in a given interval of time. Motion is mathematically described in terms of displacement, velocity, acceleration and speed. Motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame. If the position of a body is not changing with respect to a given frame of reference, the body is said to be at rest, immobile, stationary, or to have constant position with reference to its surroundings. An object's motion can not change. Momentum is a quantity, used for measuring the motion of an object. An object's momentum is directly related to the object's mass and velocity, the total momentum of all objects in an isolated system does not change with time, as described by the law of conservation of momentum; as there is no absolute frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving.
Motion applies to objects and matter particles, to radiation, radiation fields and radiation particles, to space, its curvature and space-time. One can speak of motion of shapes and boundaries. So, the term motion, in general, signifies a continuous change in the configuration of a physical system. For example, one can talk about motion of a wave or about motion of a quantum particle, where the configuration consists of probabilities of occupying specific positions. In physics, motion is described through two sets of contradictory laws of mechanics. Motions of all large-scale and familiar objects in the universe are described by classical mechanics. Whereas the motion of small atomic and sub-atomic objects is described by quantum mechanics. Classical mechanics is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets and galaxies, it produces accurate results within these domains, is one of the oldest and largest in science and technology.
Classical mechanics is fundamentally based on Newton's laws of motion. These laws describe the relationship between the forces acting on a body and the motion of that body, they were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, first published on July 5, 1687. Newton's three laws are: A body either is at rest or moves with constant velocity and unless an outer force is applied to it. An object will travel in one direction. Whenever one body exerts a force F onto a second body, the second body exerts the force −F on the first body. F and − F are equal in opposite in sense. So, the body which exerts F will go backwards. Newton's three laws of motion were the first to provide a mathematical model for understanding orbiting bodies in outer space; this explanation unified motion of objects on earth. Classical mechanics was further enhanced by Albert Einstein's special relativity and general relativity. Special relativity is concerned with the motion of objects with a high velocity, approaching the speed of light.
Uniform Motion: When an object moves with a constant speed at a particular direction at regular intervals of time it's known as the uniform motion. For example: a bike moving in a straight line with a constant speed. Equations of Uniform Motion: If v = final velocity, u = initial velocity, a = acceleration, t = time, s = displacement, then: v = u + a t v 2 = u 2 + 2 a s s = u t + a t 2 2 Quantum mechanics is a set of principles describing physical reality at the atomic level of matter and the subatomic particles; these descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation energy as described in the wave–particle duality. In classical mechanics, accurate measurements and predictions of the state of objects can be calculated, such as location and velocity. In the quantum mechanics, due to the Heisenberg uncertainty principle, the complete state of a subatomic particle, such as its location and velocity, cannot be determined. In addition to describing the motion of atomic level phenomena, quantum mechanics is useful in understanding some large-scale phenomenon such as superfluidity, superconductivity, biological systems, including the function of smell receptors and the structures of proteins.
Humans, like all known things in the universe, are in constant motion. Many of these "imperceptible motions" are only perceivable with the help of special tools and careful observation; the larger scales of imperceptible motions are difficult for humans to perceive
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin vibrationem; the oscillations may be periodic, such as the motion of a pendulum—or random, such as the movement of a tire on a gravel road. Vibration can be desirable: for example, the motion of a tuning fork, the reed in a woodwind instrument or harmonica, a mobile phone, or the cone of a loudspeaker. In many cases, vibration is undesirable, wasting energy and creating unwanted sound. For example, the vibrational motions of engines, electric motors, or any mechanical device in operation are unwanted; such vibrations could be caused by imbalances in the rotating parts, uneven friction, or the meshing of gear teeth. Careful designs minimize unwanted vibrations; the studies of sound and vibration are related. Sound, or pressure waves, are generated by vibrating structures. Hence, attempts to reduce noise are related to issues of vibration. Free vibration occurs when a mechanical system is set in motion with an initial input and allowed to vibrate freely.
Examples of this type of vibration are pulling a child back on a swing and letting it go, or hitting a tuning fork and letting it ring. The mechanical system vibrates at one or more of its natural frequencies and damps down to motionlessness. Forced vibration is; the disturbance can be a transient input, or a random input. The periodic input can be a non-harmonic disturbance. Examples of these types of vibration include a washing machine shaking due to an imbalance, transportation vibration caused by an engine or uneven road, or the vibration of a building during an earthquake. For linear systems, the frequency of the steady-state vibration response resulting from the application of a periodic, harmonic input is equal to the frequency of the applied force or motion, with the response magnitude being dependent on the actual mechanical system. Damped vibration: When the energy of a vibrating system is dissipated by friction and other resistances, the vibrations are said to be damped; the vibrations reduce or change in frequency or intensity or cease and the system rests in its equilibrium position.
An example of this type of vibration is the vehicular suspension. Vibration testing is accomplished by introducing a forcing function into a structure with some type of shaker. Alternately, a DUT is attached to the "table" of a shaker. Vibration testing is performed to examine the response of a device under test to a defined vibration environment; the measured response may rattle sound output. Squeak and rattle testing is performed with a special type of quiet shaker that produces low sound levels while under operation. For low frequency forcing, servohydraulic shakers are used. For higher frequencies, electrodynamic shakers are used. One or more "input" or "control" points located on the DUT-side of a fixture is kept at a specified acceleration. Other "response" points experience maximum vibration level or minimum vibration level, it is desirable to achieve anti-resonance to keep a system from becoming too noisy, or to reduce strain on certain parts due to vibration modes caused by specific vibration frequencies.
The most common types of vibration testing services conducted by vibration test labs are Sinusoidal and Random. Sine tests are performed to survey the structural response of the device under test. A random test is considered to more replicate a real world environment, such as road inputs to a moving automobile. Most vibration testing is conducted in a'single DUT axis' at a time though most real-world vibration occurs in various axes simultaneously. MIL-STD-810G, released in late 2008, Test Method 527, calls for multiple exciter testing; the vibration test fixture used to attach the DUT to the shaker table must be designed for the frequency range of the vibration test spectrum. For smaller fixtures and lower frequency ranges, the designer targets a fixture design, free of resonances in the test frequency range; this becomes more difficult as the test frequency increases. In these cases multi-point control strategies can mitigate some of the resonances that may be present in the future. Devices designed to trace or record vibrations are called vibroscopes.
Vibration Analysis, applied in an industrial or maintenance environment aims to reduce maintenance costs and equipment downtime by detecting equipment faults. VA is a key component of a Condition Monitoring program, is referred to as Predictive Maintenance. Most VA is used to detect faults in rotating equipment such as Unbalance, rolling element bearing faults and resonance conditions. VA can use the units of Displacement and Acceleration displayed as a time waveform, but most the spectrum is used, derived from a fast Fourier transform of the TWF; the vibration spectrum provides important frequency information that can pinpoint the faulty component. The fundamentals of vibration analysis can be understood by studying the simple Mass-spring-damper model. Indeed a complex structure such as an automobile body can be modeled as a "summation" of simple mass–spring–damper models. T
In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass i.e. to accelerate. Force can be described intuitively as a push or a pull. A force has both direction, making it a vector quantity, it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, is inversely proportional to the mass of the object. Concepts related to force include: thrust. In an extended body, each part applies forces on the adjacent parts; such internal mechanical stresses cause no acceleration of that body as the forces balance one another. Pressure, the distribution of many 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 causes deformation of solid materials, or flow in fluids. Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part this was due to an incomplete understanding of the sometimes non-obvious force of friction, a inadequate view of the nature of natural motion. A fundamental error was the belief that a force is required to maintain motion at a constant velocity. Most of the previous misunderstandings about motion and force were corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved for nearly three hundred years. By the early 20th century, Einstein developed a theory of relativity that predicted the action of forces on objects with increasing momenta near the speed of light, provided insight into the forces produced by gravitation and inertia.
With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. 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 and gravitational. 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 simple 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 culminated in the work of Archimedes, famous for formulating a treatment of buoyant forces inherent in fluids.
Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed of the elements earth and water, to be in their natural place on the ground and that they will stay that way if left alone, he distinguished between the innate tendency of objects to find their "natural place", which led to "natural motion", unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows; the place where the archer moves the projectile was at the start of the flight, while the projectile sailed through the air, no discernible efficient cause acts on it.
Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation demands a continuum like air for change of place in general. Aristotelian physics began facing criticism in medieval science, first by John Philoponus in the 6th century; the shortcomings of Aristotelian physics would not be corrected until the 17th century work of Galileo Galilei, influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion, he showed that the bodies were accelerated by gravity to an extent, independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction. Sir Isaac Newton described the motion of all objects using the concepts of inertia and force, in doing so he found they obey certain conservation laws.
In 1687, Newton published his thesis Philosophiæ Naturalis Principia Mathematica. In this work Newton set out three laws of motion that to this day are t
In Newtonian mechanics, linear momentum, translational momentum, or momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a direction in three-dimensional space. If m is an object's mass and v is the velocity the momentum is p = m v, In SI units, it is measured in kilogram meters per second. Newton's second law of motion states that a body's rate of change in momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is conserved in special relativity and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, general relativity, it is an expression of one of the fundamental symmetries of time: translational symmetry. Advanced formulations of classical mechanics and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints.
In these systems the conserved quantity is generalized momentum, in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function; the momentum and position operators are related by the Heisenberg uncertainty principle. In continuous systems such as electromagnetic fields and deformable bodies, a momentum density can be defined, a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids. Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension; the vector equations are identical to the scalar equations. The momentum of a particle is conventionally represented by the letter p.
It is the product of two quantities, the particle's mass and its velocity: p = m v. The unit of momentum is the product of the units of velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second the momentum is in kilogram meters per second. In cgs units, if the mass is in grams and the velocity in centimeters per second the momentum is in gram centimeters per second. Being a vector, momentum has 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 with reference to the ground; the momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m1 and m2, velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2; the momenta of more than two particles can be added more with the following: p = ∑ i m i v i. A system of particles has a center of mass, a point determined by the weighted sum of their positions: r cm = m 1 r 1 + m 2 r 2 + ⋯ m 1 + m 2 + ⋯ = ∑ i m i r i ∑ i m i.
If all the particles are moving, the center of mass will 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 Euler's first law. If the net force applied to a particle is a constant F, is applied for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t. In differential form, this is Newton's second law. If the net force experienced by a particle changes as a function of time, F, the change in momentum between times t1 and t2 is Δ p = J = ∫ t 1
In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object. Energy is a conserved quantity; the SI unit of energy is the joule, the energy transferred to an object by the work of moving it a distance of 1 metre against a force of 1 newton. Common forms of energy include the kinetic energy of a moving object, the potential energy stored by an object's position in a force field, the elastic energy stored by stretching solid objects, the chemical energy released when a fuel burns, the radiant energy carried by light, the thermal energy due to an object's temperature. Mass and energy are related. Due to mass–energy equivalence, any object that has mass when stationary has an equivalent amount of energy whose form is called rest energy, any additional energy acquired by the object above that rest energy will increase the object's total mass just as it increases its total energy. For example, after heating an object, its increase in energy could be measured as a small increase in mass, with a sensitive enough scale.
Living organisms require exergy to stay alive, such as the energy. Human civilization requires energy to function, which it gets from energy resources such as fossil fuels, nuclear fuel, or renewable energy; the processes of Earth's climate and ecosystem are driven by the radiant energy Earth receives from the sun and the geothermal energy contained within the earth. The total energy of a system can be subdivided and classified into potential energy, kinetic energy, or combinations of the two in various ways. Kinetic energy is determined by the movement of an object – or the composite motion of the components of an object – and potential energy reflects the potential of an object to have motion, is a function of the position of an object within a field or may be stored in the field itself. While these two categories are sufficient to describe all forms of energy, it is convenient to refer to particular combinations of potential and kinetic energy as its own form. For example, macroscopic mechanical energy is the sum of translational and rotational kinetic and potential energy in a system neglects the kinetic energy due to temperature, nuclear energy which combines utilize potentials from the nuclear force and the weak force), among others.
The word energy derives from the Ancient Greek: translit. Energeia, lit.'activity, operation', which appears for the first time in the work of Aristotle in the 4th century BC. In contrast to the modern definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness and pleasure. In the late 17th century, Gottfried Leibniz proposed the idea of the Latin: vis viva, or living force, which defined as the product of the mass of an object and its velocity squared. To account for slowing due to friction, Leibniz theorized that thermal energy consisted of the random motion of the constituent parts of matter, although it would be more than a century until this was accepted; the modern analog of this property, kinetic energy, differs from vis viva only by a factor of two. In 1807, Thomas Young was 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, in 1853, William Rankine coined the term "potential energy".
The law of conservation of energy was first postulated in the early 19th century, applies to any isolated system. It was argued for some years whether heat was a physical substance, dubbed the caloric, or a physical quantity, such as momentum. In 1845 James Prescott Joule discovered the generation of heat; these developments led to the theory of conservation of energy, formalized by William Thomson as the field of thermodynamics. Thermodynamics aided the rapid development of explanations of chemical processes by Rudolf Clausius, Josiah Willard Gibbs, Walther Nernst, it led to a mathematical formulation of the concept of entropy by Clausius and to the introduction of laws of radiant energy by Jožef Stefan. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time. Thus, since 1918, theorists have understood that the law of conservation of energy is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time.
In 1843, James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. The most famous of them used the "Joule apparatus": a descending weight, attached to a string, caused rotation of a paddle immersed in water insulated from heat transfer, it showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction with the paddle. In the International System of Units, the unit of energy is the joule, named after James Prescott Joule, it is a derived unit. It is equal to the energy expended in applying a force of one newton through a distance of one metre; however energy is expressed in many other units not part of the SI, such as ergs, British Thermal Units, kilowatt-hours and kilocalories, which require a conversion factor when expressed in SI units. The SI unit of energy rate is the watt, a joule per second. Thus, one joule is one watt-second, 3600 joules equal one wa
In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a vector quantity, impulse is a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum in the same direction; the SI unit of impulse is the newton second, the dimensionally equivalent unit of momentum is the kilogram meter per second. The corresponding English engineering units are the slug-foot per second. A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time therefore produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly. J = F average The impulse is the integral of the resultant force with respect to time: J = ∫ F d t Impulse J produced from time t1 to t2 is defined to be J = ∫ t 1 t 2 F d t where F is the resultant force applied from t1 to t2.
From Newton's second law, force is related to momentum p by F = d p d t Therefore, J = ∫ t 1 t 2 d p d t d t = ∫ p 1 p 2 d p = p 2 − p 1 = Δ p where Δp is the change in linear momentum from time t1 to t2. This is called the impulse-momentum theorem; as a result, an impulse may be regarded as the change in momentum of an object to which a resultant force is applied. The impulse may be expressed in a simpler form when the mass is constant: J = ∫ t 1 t 2 F d t = Δ p = m v 2 − m v 1 where F is the resultant force applied, t1 and t2 are times when the impulse begins and ends m is the mass of the object, v2 is the final velocity of the object at the end of the time interval, v1 is the initial velocity of the object when the time interval begins. Impulse has the same dimensions as momentum. In the International System of Units, these are kg⋅m/s = N⋅s. In English engineering units, they are slug⋅ft/s = lbf⋅s; the term "impulse" is used to refer to a fast-acting force or impact. This type of impulse is idealized so that the change in momentum produced by the force happens with no change in time.
This sort of change is a step change, is not physically possible. However, this is a useful model for computing the effects of ideal collisions. Additionally, in rocketry, the term "total impulse" is used and is considered synonymous with the term "impulse"; the application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools for jet- or rocket-propelled vehicles. In the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse; this fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicle's propulsive change in velocity to the engine's specific impulse and the vehicle's propellant-mass ratio. Wave–particle duality defines the impulse of a wave collision; the preservation of momentum in the collision is called phase matching. Applications include: Compton effect Nonlinear optics Acousto-optic modulator Electron phonon scattering Serway, Raymond A..
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