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52-556: The origin of the term "Feder" for these swords is uncertain. The German word Feder means "feather" or "quill", but came to be used of metal springs in the 17th century (i.e. at about the same time as the name of the sparring weapon and possibly influenced by it). The term Fechtfeder itself seems to be connected to the name of the Federfechter , i.e. "feather fencers", a guild or brotherhood of fencers formed in 1570 in Prague . It

104-402: A linear function . Force of fully compressed spring where Zero-length spring is a term for a specially designed coil spring that would exert zero force if it had zero length. That is, in a line graph of the spring's force versus its length, the line passes through the origin. A real coil spring will not contract to zero length because at some point the coils touch each other. "Length" here

156-399: A torque proportional to the angle. A torsion spring's rate is in units of torque divided by angle, such as N·m / rad or ft·lbf /degree. The inverse of spring rate is compliance, that is: if a spring has a rate of 10 N/mm, it has a compliance of 0.1 mm/N. The stiffness (or rate) of springs in parallel is additive , as is the compliance of springs in series. Springs are made from

208-504: A cantilevered beam under a uniform load is given by: δ B = q L 4 8 E I ϕ B = q L 3 6 E I {\displaystyle {\begin{aligned}\delta _{B}&={\frac {qL^{4}}{8EI}}\\[1ex]\phi _{B}&={\frac {qL^{3}}{6EI}}\end{aligned}}} where The deflection at any point, x {\displaystyle x} , along

260-445: A load can be calculated by integrating the function that mathematically describes the slope of the deflected shape of the member under that load. Standard formulas exist for the deflection of common beam configurations and load cases at discrete locations. Otherwise methods such as virtual work , direct integration , Castigliano's method , Macaulay's method or the direct stiffness method are used. The deflection of beam elements

312-431: A non-metallic spring is the bow , made traditionally of flexible yew wood, which when drawn stores energy to propel an arrow . When a conventional spring, without stiffness variability features, is compressed or stretched from its resting position, it exerts an opposing force approximately proportional to its change in length (this approximation breaks down for larger deflections). The rate or spring constant of

364-453: A piece of inelastic material of the proper length so the zero force point would occur at zero length. A zero-length spring can be attached to a mass on a hinged boom in such a way that the force on the mass is almost exactly balanced by the vertical component of the force from the spring, whatever the position of the boom. This creates a horizontal pendulum with very long oscillation period . Long-period pendulums enable seismometers to sense

416-411: A spring is the change in the force it exerts, divided by the change in deflection of the spring. That is, it is the gradient of the force versus deflection curve . An extension or compression spring's rate is expressed in units of force divided by distance, for example or N/m or lbf/in. A torsion spring is a spring that works by twisting; when it is twisted about its axis by an angle, it produces

468-562: A system can be determined through the spring constant k and its displacement x : U = ( 1 2 ) k x 2 {\displaystyle U=\left({\frac {1}{2}}\right)kx^{2}} The kinetic energy K of an object in simple harmonic motion can be found using the mass of the attached object m and the velocity at which the object oscillates v : K = ( 1 2 ) m v 2 {\displaystyle K=\left({\frac {1}{2}}\right)mv^{2}} Since there

520-444: A variable rate. However, a conical spring can be made to have a constant rate by creating the spring with a variable pitch. A larger pitch in the larger-diameter coils and a smaller pitch in the smaller-diameter coils forces the spring to collapse or extend all the coils at the same rate when deformed. Since force is equal to mass, m , times acceleration, a , the force equation for a spring obeying Hooke's law looks like: The mass of

572-481: A variety of elastic materials, the most common being spring steel. Small springs can be wound from pre-hardened stock, while larger ones are made from annealed steel and hardened after manufacture. Some non-ferrous metals are also used, including phosphor bronze and titanium for parts requiring corrosion resistance, and low- resistance beryllium copper for springs carrying electric current . Simple non-coiled springs have been used throughout human history, e.g.

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624-633: A variety of loading and boundary conditions. A number of simple examples are shown below. The formulas expressed are approximations developed for long, slender, homogeneous, prismatic beams with small deflections, and linear elastic properties. Under these restrictions, the approximations should give results within 5% of the actual deflection. Cantilever beams have one end fixed, so that the slope and deflection at that end must be zero. The elastic deflection δ {\displaystyle \delta } and angle of deflection ϕ {\displaystyle \phi } (in radians ) at

676-423: Is a device consisting of an elastic but largely rigid material (typically metal) bent or molded into a form (especially a coil) that can return into shape after being compressed or extended. Springs can store energy when compressed. In everyday use, the term most often refers to coil springs , but there are many different spring designs. Modern springs are typically manufactured from spring steel . An example of

728-506: Is appropriate only in the low-strain region. Hooke's law is a mathematical consequence of the fact that the potential energy of the rod is a minimum when it has its relaxed length. Any smooth function of one variable approximates a quadratic function when examined near enough to its minimum point as can be seen by examining the Taylor series . Therefore, the force – which is the derivative of energy with respect to displacement – approximates

780-412: Is defined as the distance between the axes of the pivots at each end of the spring, regardless of any inelastic portion in-between. Zero-length springs are made by manufacturing a coil spring with built-in tension (A twist is introduced into the wire as it is coiled during manufacture; this works because a coiled spring unwinds as it stretches), so if it could contract further, the equilibrium point of

832-454: Is found by taking the inverse of the period: f = 1 T = ω 2 π = 1 2 π k m {\displaystyle f={\frac {1}{T}}={\frac {\omega }{2\pi }}={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}} In classical physics , a spring can be seen as a device that stores potential energy , specifically elastic potential energy , by straining

884-596: Is known for all x {\displaystyle x} . Where: If the beam is uniform and the deflection at any point is known, this can be calculated without knowing other properties of the beam. The formulas supplied above require the use of a consistent set of units. Most calculations will be made in the International System of Units (SI) or US customary units, although there are many other systems of units. Other units may be used as well, as long as they are self-consistent. For example, sometimes

936-459: Is no energy loss in such a system, energy is always conserved and thus: E = K + U {\displaystyle E=K+U} The angular frequency ω of an object in simple harmonic motion, given in radians per second, is found using the spring constant k and the mass of the oscillating object m : ω = k m {\displaystyle \omega ={\sqrt {\frac {k}{m}}}} The period T ,

988-529: Is possible that the term Feder for the sparring sword arose in the late 16th century at first as a term of derision of the practice weapon used by the Federfechter (who were so called for unrelated reasons, because of a feather or quill used as their heraldic emblem) by their rivals, the Marx Brothers , who would tease the Federfechter as "fencing with quills" as opposed to with real weapons, or as scholars or academics supposedly better at "fighting with

1040-406: Is the degree to which a part of a long structural element (such as beam ) is deformed laterally (in the direction transverse to its longitudinal axis) under a load . It may be quantified in terms of an angle ( angular displacement ) or a distance (linear displacement ). A longitudinal deformation (in the direction of the axis) is called elongation . The deflection distance of a member under

1092-554: Is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory. An example of the use of deflection in this context is in building construction. Architects and engineers select materials for various applications. Beams can vary greatly in their geometry and composition. For instance, a beam may be straight or curved. It may be of constant cross section, or it may taper. It may be made entirely of

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1144-516: The bow (and arrow). In the Bronze Age more sophisticated spring devices were used, as shown by the spread of tweezers in many cultures. Ctesibius of Alexandria developed a method for making springs out of an alloy of bronze with an increased proportion of tin, hardened by hammering after it was cast. Coiled springs appeared early in the 15th century, in door locks. The first spring powered-clocks appeared in that century and evolved into

1196-406: The superposition principle . The change in length Δ L {\displaystyle \Delta L} of the beam, projected along the line of the unloaded beam, can be calculated by integrating the slope θ x {\displaystyle \theta _{x}} function, if the deflection function δ x {\displaystyle \delta _{x}}

1248-604: The 21st century for use as sparring weapons and for competition in the context of the Historical European martial arts revival . Among some HEMA groups, it is believed that certain historical Federn had gradually thinning hilts, though this is not always applied to modern reconstructions of the weapons. Additionally, the Schilden, the blade-catchers, of the modern reconstructions vary from flat squares to double-troughed Parierhaken. Some also have hilts customized into

1300-410: The amount of time for the spring-mass system to complete one full cycle, of such harmonic motion is given by: T = 2 π ω = 2 π m k {\displaystyle T={\frac {2\pi }{\omega }}=2\pi {\sqrt {\frac {m}{k}}}} The frequency f , the number of oscillations per unit time, of something in simple harmonic motion

1352-414: The beam), the δ x {\displaystyle \delta _{x}} and ϕ x {\displaystyle \phi _{x}} equations are identical to the δ B {\displaystyle \delta _{B}} and ϕ B {\displaystyle \phi _{B}} equations above. The deflection, at the free end B, of

1404-417: The bonds between the atoms of an elastic material. Hooke's law of elasticity states that the extension of an elastic rod (its distended length minus its relaxed length) is linearly proportional to its tension , the force used to stretch it. Similarly, the contraction (negative extension) is proportional to the compression (negative tension). This law actually holds only approximately, and only when

1456-603: The closest support and is given by: x 1 = L 2 − a 2 3 {\displaystyle x_{1}={\sqrt {\frac {L^{2}-a^{2}}{3}}}} The elastic deflection (at the midpoint C) on a beam supported by two simple supports, under a uniform load (as pictured) is given by: δ C = 5 q L 4 384 E I {\displaystyle \delta _{C}={\frac {5qL^{4}}{384EI}}} where The deflection at any point, x {\displaystyle x} , along

1508-460: The closest support, is given by: δ max = F a ( L 2 − a 2 ) 3 / 2 9 3 L E I {\displaystyle \delta _{\text{max}}={\frac {Fa\left(L^{2}-a^{2}\right)^{3/2}}{9{\sqrt {3}}LEI}}} where This maximum deflection occurs at a distance x 1 {\displaystyle x_{1}} from

1560-571: The cross-section, and M {\displaystyle M} is the internal bending moment in the beam. If, in addition, the beam is not tapered and is homogeneous , and is acted upon by a distributed load q {\displaystyle q} , the above expression can be written as : E I   d 4 w ( x ) d x 4 = q ( x ) {\displaystyle EI~{\frac {\mathrm {d} ^{4}w(x)}{\mathrm {d} x^{4}}}=q(x)} This equation can be solved for

1612-446: The deformation (extension or contraction) is small compared to the rod's overall length. For deformations beyond the elastic limit , atomic bonds get broken or rearranged, and a spring may snap, buckle, or permanently deform. Many materials have no clearly defined elastic limit, and Hooke's law can not be meaningfully applied to these materials. Moreover, for the superelastic materials, the linear relationship between force and displacement

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1664-471: The first large watches by the 16th century. In 1676 British physicist Robert Hooke postulated Hooke's law , which states that the force a spring exerts is proportional to its extension. On March 8, 1850, John Evans, Founder of John Evans' Sons, Incorporated, opened his business in New Haven, Connecticut, manufacturing flat springs for carriages and other vehicles, as well as the machinery to manufacture

1716-515: The force with which the spring pushes back is linearly proportional to the distance from its equilibrium length: where Most real springs approximately follow Hooke's law if not stretched or compressed beyond their elastic limit . Coil springs and other common springs typically obey Hooke's law. There are useful springs that don't: springs based on beam bending can for example produce forces that vary nonlinearly with displacement. If made with constant pitch (wire thickness), conical springs have

1768-513: The free end in the example image: A (weightless) cantilever beam, with an end load, can be calculated (at the free end B) using: δ B = F L 3 3 E I ϕ B = F L 2 2 E I {\displaystyle {\begin{aligned}\delta _{B}&={\frac {FL^{3}}{3EI}}\\[1ex]\phi _{B}&={\frac {FL^{2}}{2EI}}\end{aligned}}} where Note that if

1820-409: The initial displacement and velocity of the mass. The graph of this function with B = 0 {\displaystyle B=0} (zero initial position with some positive initial velocity) is displayed in the image on the right. In simple harmonic motion of a spring-mass system, energy will fluctuate between kinetic energy and potential energy , but the total energy of the system remains

1872-419: The kilogram-force ( k g f {\displaystyle \mathrm {kgf} } ) unit is used to measure loads. In such a case, the modulus of elasticity must be converted to k g f m 2 {\displaystyle \mathrm {\frac {kgf}{m^{2}}} } . Building codes determine the maximum deflection, usually as a fraction of the span e.g. 1/400 or 1/600. Either

1924-413: The midpoint C of a beam, loaded at its center, supported by two simple supports is then given by: δ C = F L 3 48 E I {\displaystyle \delta _{C}={\frac {FL^{3}}{48EI}}} where The maximum elastic deflection on a beam supported by two simple supports, loaded at a distance a {\displaystyle a} from

1976-408: The quill" than at real fighting (reflecting the different professional backgrounds of the rival fencing guilds). Johann Fischart in his Gargantua (1575) already compares the fencing weapon to a "quill" writing in blood. The recharacterized term Federschwert is modern. The sword consists of a very thin, rounded blade with a large ricasso and a heavy hilt and pommel . Because of this, it has

2028-658: The same material (homogeneous), or it may be composed of different materials (composite). Some of these things make analysis difficult, but many engineering applications involve cases that are not so complicated. Analysis is simplified if: In this case, the equation governing the beam's deflection ( w {\displaystyle w} ) can be approximated as: d 2 w ( x ) d x 2 = M ( x ) E ( x ) I ( x ) {\displaystyle {\frac {\mathrm {d} ^{2}w(x)}{\mathrm {d} x^{2}}}={\frac {M(x)}{E(x)I(x)}}} where

2080-430: The same weight and center of balance as a real sword, and handles identically. This odd construction also has the effect of moving the sword's center of percussion to a theoretical point beyond its tip. The tip of a Federschwert is spatulated and may have been covered with a leather sleeve to make thrusting safer, though no direct historical evidence exists of such use. Production of Fechtfedern has been revived in

2132-399: The same. A spring that obeys Hooke's Law with spring constant k will have a total system energy E of: E = ( 1 2 ) k A 2 {\displaystyle E=\left({\frac {1}{2}}\right)kA^{2}} Here, A is the amplitude of the wave-like motion that is produced by the oscillating behavior of the spring. The potential energy U of such

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2184-450: The second derivative of its deflected shape with respect to x {\displaystyle x} ( x {\displaystyle x} being the horizontal position along the length of the beam) is interpreted as its curvature, E {\displaystyle E} is the Young's modulus , I {\displaystyle I} is the area moment of inertia of

2236-442: The shape of a wayward "S", and others' are extended about two inches. Pommel shapes also vary, between classic spheres , various polyhedrons , arming sword -style disks, or most commonly teardrops or eggs . This article relating to swords is a stub . You can help Misplaced Pages by expanding it . This German history article is a stub . You can help Misplaced Pages by expanding it . Spring (mechanical) A spring

2288-414: The slowest waves from earthquakes. The LaCoste suspension with zero-length springs is also used in gravimeters because it is very sensitive to changes in gravity. Springs for closing doors are often made to have roughly zero length, so that they exert force even when the door is almost closed, so they can hold it closed firmly. Deflection (engineering) In structural engineering , deflection

2340-717: The span doubles, the deflection increases eightfold. The deflection at any point, x {\displaystyle x} , along the span of an end loaded cantilevered beam can be calculated using: δ x = F x 2 6 E I ( 3 L − x ) ϕ x = F x 2 E I ( 2 L − x ) {\displaystyle {\begin{aligned}\delta _{x}&={\frac {Fx^{2}}{6EI}}(3L-x)\\[1ex]\phi _{x}&={\frac {Fx}{2EI}}(2L-x)\end{aligned}}} Note: At x = L {\displaystyle x=L} (the end of

2392-487: The span of a center loaded simply supported beam can be calculated using: δ x = F x 48 E I ( 3 L 2 − 4 x 2 ) {\displaystyle \delta _{x}={\frac {Fx}{48EI}}\left(3L^{2}-4x^{2}\right)} for 0 ≤ x ≤ L 2 {\displaystyle 0\leq x\leq {\frac {L}{2}}} The special case of elastic deflection at

2444-843: The span of a uniformly loaded cantilevered beam can be calculated using: δ x = q x 2 24 E I ( 6 L 2 − 4 L x + x 2 ) ϕ x = q x 6 E I ( 3 L 2 − 3 L x + x 2 ) {\displaystyle {\begin{aligned}\delta _{x}&={\frac {qx^{2}}{24EI}}\left(6L^{2}-4Lx+x^{2}\right)\\[1ex]\phi _{x}&={\frac {qx}{6EI}}\left(3L^{2}-3Lx+x^{2}\right)\end{aligned}}} Simply supported beams have supports under their ends which allow rotation, but not deflection. The deflection at any point, x {\displaystyle x} , along

2496-437: The span of a uniformly loaded simply supported beam can be calculated using: δ x = q x 24 E I ( L 3 − 2 L x 2 + x 3 ) {\displaystyle \delta _{x}={\frac {qx}{24EI}}\left(L^{3}-2Lx^{2}+x^{3}\right)} The deflection of beams with a combination of simple loads can be calculated using

2548-537: The spring is small in comparison to the mass of the attached mass and is ignored. Since acceleration is simply the second derivative of x with respect to time, This is a second order linear differential equation for the displacement x {\displaystyle x} as a function of time. Rearranging: the solution of which is the sum of a sine and cosine : A {\displaystyle A} and B {\displaystyle B} are arbitrary constants that may be found by considering

2600-406: The spring, the point at which its restoring force is zero, occurs at a length of zero. In practice, the manufacture of springs is typically not accurate enough to produce springs with tension consistent enough for applications that use zero length springs, so they are made by combining a negative length spring, made with even more tension so its equilibrium point would be at a negative length, with

2652-521: The springs. Evans was a Welsh blacksmith and springmaker who emigrated to the United States in 1847, John Evans' Sons became "America's oldest springmaker" which continues to operate today. Springs can be classified depending on how the load force is applied to them: They can also be classified based on their shape: The most common types of spring are: Other types include: An ideal spring acts in accordance with Hooke's law, which states that

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2704-427: The strength limit state (allowable stress) or the serviceability limit state (deflection considerations among others) may govern the minimum dimensions of the member required. The deflection must be considered for the purpose of the structure. When designing a steel frame to hold a glazed panel, one allows only minimal deflection to prevent fracture of the glass. The deflected shape of a beam can be represented by

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