formulasearchengine - User contributions [en]

Gliese 436

2015-01-07T09:39:03Z

129.16.192.179: /* Properties */

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Thermodynamic integration

2014-12-18T13:06:08Z

129.16.112.149: /* Derivation */

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Anscombe transform

2014-09-22T11:31:17Z

129.16.187.76:

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Modular group

2014-02-28T12:52:10Z

129.16.126.117: /* Congruence subgroups */

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LaTeX

2014-01-29T14:10:04Z

129.16.106.189:

In [[Measure (mathematics)|measure theory]], the '''Lebesgue measure''', named after [[france|French]] mathematician [[Henri Lebesgue]], is the standard way of assigning a [[measure (mathematics)|measure]] to [[subset]]s of ''n''-dimensional [[Euclidean space]]. For ''n'' = 1, 2, or 3, it coincides with the standard measure of [[length]], [[area]], or [[volume]]. In general, it is also called '''''n''-dimensional volume''', '''''n''-volume''', or simply '''volume'''.<ref>The term ''[[volume]]'' is also used, more strictly, as a [[synonym]] of 3-dimensional volume</ref> It is used throughout [[real analysis]], in particular to define [[Lebesgue integration]]. Sets that can be assigned a Lebesgue measure are called '''Lebesgue measurable'''; the measure of the Lebesgue measurable set ''A'' is denoted by λ(''A'').

Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902.<ref>{{cite journal |author=Henri Lebesgue |title=Intégrale, longueur, aire |year=1902 |publisher=Université de Paris }}</ref>

The Lebesgue measure is often denoted ''dx'', but this should not be confused with the distinct notion of a [[volume form]].

== Definition ==

Given a subset <math>E\subset\mathbb{R}</math>, with the length of an (open, closed, semi-open) interval <math>I = [a,b]</math> given by <math>l(I)=b - a</math>, the Lebesgue outer measure <math>\lambda^*(E)</math> is defined as

:<math>\lambda^*(E) = \text{inf} \left\{\sum_{k=1}^\infty l(I_k) : {(I_k)_{k \in \mathbb N}} \text{ is a sequence of open intervals with } E\subset \bigcup_{k=1}^\infty I_k\right\}</math>.

The Lebesgue measure of E is given by its Lebesgue outer measure <math>\lambda(E)=\lambda^*(E)</math> if, for every <math> A\subset\mathbb{R}</math>,

:<math>\lambda^*(A) = \lambda^*(A \cap E) + \lambda^*(A \cap E^c) </math>.

== Examples ==

* Any [[closed interval]] [''a'', ''b''] of [[real number]]s is Lebesgue measurable, and its Lebesgue measure is the length ''b''−''a''. The [[open interval]] (''a'', ''b'') has the same measure, since the [[set difference|difference]] between the two sets consists only of the end points ''a'' and ''b'' and has [[measure zero]].
* Any [[Cartesian product]] of intervals [''a'', ''b''] and [''c'', ''d''] is Lebesgue measurable, and its Lebesgue measure is (''b''−''a'')(''d''−''c''), the area of the corresponding [[rectangle]].
* The Lebesgue measure of the set of [[rational numbers]] in an interval of the line is 0, although the set is [[Dense set|dense]] in the interval.
* The [[Cantor set]] is an example of an [[uncountable set]] that has Lebesgue measure zero.
* [[Vitali set]]s are examples of sets that are [[non-measurable set|not measurable]] with respect to the Lebesgue measure. Their existence relies on the [[axiom of choice]].

== Properties ==
[[File:Translation of a set.svg|thumb|300px|Translation invariance: The Lebesgue measure of <math>A</math> and <math>A+t</math> are the same.]]
The Lebesgue measure on '''R'''''n'' has the following properties:

# If ''A'' is a [[cartesian product]] of [[interval (mathematics)|intervals]] ''I''1 × ''I''2 × ... × ''I''''n'', then ''A'' is Lebesgue measurable and <math>\lambda (A)=|I_1|\cdot |I_2|\cdots |I_n|.</math> Here, |''I''| denotes the length of the interval ''I''.
# If ''A'' is a [[disjoint union]] of [[countable|countably many]] disjoint Lebesgue measurable sets, then ''A'' is itself Lebesgue measurable and λ(''A'') is equal to the sum (or [[infinite series]]) of the measures of the involved measurable sets.
# If ''A'' is Lebesgue measurable, then so is its [[Complement (set theory)|complement]].
# λ(''A'') ≥ 0 for every Lebesgue measurable set ''A''.
# If ''A'' and ''B'' are Lebesgue measurable and ''A'' is a subset of ''B'', then λ(''A'') ≤ λ(''B''). (A consequence of 2, 3 and 4.)
# Countable [[Union (set theory)|unions]] and [[Intersection (set theory)|intersections]] of Lebesgue measurable sets are Lebesgue measurable. (Not a consequence of 2 and 3, because a family of sets that is closed under complements and disjoint countable unions need not be closed under countable unions: <math>\{\emptyset, \{1,2,3,4\}, \{1,2\}, \{3,4\}, \{1,3\}, \{2,4\}\}</math>.)
# If ''A'' is an [[open set|open]] or [[closed set|closed]] subset of '''R'''''n'' (or even [[Borel set]], see [[metric space]]), then ''A'' is Lebesgue measurable.
# If ''A'' is a Lebesgue measurable set, then it is "approximately open" and "approximately closed" in the sense of Lebesgue measure (see the [[regularity theorem for Lebesgue measure]]).
# Lebesgue measure is both [[Locally finite measure|locally finite]] and [[Inner regular measure|inner regular]], and so it is a [[Radon measure]].
# Lebesgue measure is [[Strictly positive measure|strictly positive]] on non-empty open sets, and so its [[Support (measure theory)|support]] is the whole of '''R'''''n''.
# If ''A'' is a Lebesgue measurable set with λ(''A'') = 0 (a [[null set]]), then every subset of ''A'' is also a null set. [[A fortiori]], every subset of ''A'' is measurable.
# If ''A'' is Lebesgue measurable and ''x'' is an element of '''R'''''n'', then the ''translation of ''A'' by x'', defined by ''A'' + ''x'' = {''a'' + ''x'' : ''a'' ∈ ''A''}, is also Lebesgue measurable and has the same measure as ''A''.
# If ''A'' is Lebesgue measurable and <math>\delta>0</math>, then the ''dilation of <math>A</math> by <math>\delta</math>'' defined by <math>\delta A=\{\delta x:x\in A\}</math> is also Lebesgue measurable and has measure <math>\delta^{n}\lambda\,(A).</math>
# More generally, if ''T'' is a [[linear transformation]] and ''A'' is a measurable subset of '''R'''''n'', then ''T''(''A'') is also Lebesgue measurable and has the measure <math>|\det(T)|\, \lambda\,(A)</math>.

All the above may be succinctly summarized as follows:

: The Lebesgue measurable sets form a [[sigma-algebra|σ-algebra]] containing all products of intervals, and λ is the unique [[Complete measure|complete]] [[translational invariance|translation-invariant]] [[measure (mathematics)|measure]] on that σ-algebra with <math>\lambda([0,1]\times [0, 1]\times \cdots \times [0, 1])=1.</math>

The Lebesgue measure also has the property of being [[Sigma-finite measure|σ-finite]].

== Null sets ==
{{main|Null set}}
A subset of '''R'''''n'' is a ''null set'' if, for every ε > 0, it can be covered with countably many products of ''n'' intervals whose total volume is at most ε. All [[countable]] sets are null sets.

If a subset of '''R'''''n'' has [[Hausdorff dimension]] less than ''n'' then it is a null set with respect to ''n''-dimensional Lebesgue measure. Here Hausdorff dimension is relative to the [[Euclidean metric]] on '''R'''''n'' (or any metric [[Lipschitz]]{{dn|date=May 2012}} equivalent to it). On the other hand a set may have [[topological dimension]] less than ''n'' and have positive ''n''-dimensional Lebesgue measure. An example of this is the [[Smith–Volterra–Cantor set]] which has topological dimension 0 yet has positive 1-dimensional Lebesgue measure.

In order to show that a given set ''A'' is Lebesgue measurable, one usually tries to find a "nicer" set ''B'' which differs from ''A'' only by a null set (in the sense that the [[symmetric difference]] (''A'' − ''B'') <math>\cup</math>(''B'' − ''A'') is a null set) and then show that ''B'' can be generated using countable unions and intersections from open or closed sets.

== Construction of the Lebesgue measure ==
The modern construction of the Lebesgue measure is an application of [[Carathéodory's extension theorem]]. It proceeds as follows.

Fix {{nowrap|''n'' ∈ '''N'''}}. A '''box''' in '''R'''''n'' is a set of the form
:<math>B=\prod_{i=1}^n [a_i,b_i] \, ,</math>
where {{nowrap|''bi'' ≥ ''ai''}}, and the product symbol here represents a Cartesian product. The volume of this box is defined to be
:<math>\operatorname{vol}(B)=\prod_{i=1}^n (b_i-a_i) \, .</math>

For ''any'' subset ''A'' of '''R'''''n'', we can define its [[outer measure]] ''λ''*(''A'') by:

:<math>\lambda^*(A) = \inf \Bigl\{\sum_{B\in \mathcal{C}}\operatorname{vol}(B) : \mathcal{C}\text{ is a countable collection of boxes whose union covers }A\Bigr\} .</math>

We then define the set ''A'' to be Lebesgue measurable if for every subset ''S'' of '''R'''''n'',
:<math>\lambda^*(S) = \lambda^*(S \cap A) + \lambda^*(S - A) \, .</math>

These Lebesgue measurable sets form a [[σ-algebra]], and the Lebesgue measure is defined by {{nowrap|''λ''(''A'') {{=}} ''λ''*(''A'')}} for any Lebesgue measurable set ''A''.

The existence of sets that are not Lebesgue measurable is a consequence of a certain set-theoretical [[axiom]], the [[axiom of choice]], which is independent from many of the conventional systems of axioms for [[set theory]]. The [[Vitali set|Vitali theorem]], which follows from the axiom, states that there exist subsets of '''R''' that are not Lebesgue measurable. Assuming the axiom of choice, [[non-measurable set]]s with many surprising properties have been demonstrated, such as those of the [[Banach–Tarski paradox]].

In 1970, [[Robert M. Solovay]] showed that the existence of sets that are not Lebesgue measurable is not provable within the framework of [[Zermelo–Fraenkel set theory]] in the absence of the axiom of choice (see [[Solovay's model]]).<ref>{{Cite journal |last=Solovay |first=Robert M. |title=A model of set-theory in which every set of reals is Lebesgue measurable |journal=[[Annals of Mathematics]] |jstor=1970696 |series=Second Series |volume=92 |year=1970 |issue=1 |pages=1–56 |doi=10.2307/1970696 }}</ref>

== Relation to other measures ==
The [[Borel measure]] agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets. The Borel measure is translation-invariant, but not [[Complete measure|complete]].

The [[Haar measure]] can be defined on any [[locally compact]] [[topological group|group]] and is a generalization of the Lebesgue measure ('''R'''''n'' with addition is a locally compact group).

The [[Hausdorff measure]] is a generalization of the Lebesgue measure that is useful for measuring the subsets of '''R'''''n'' of lower dimensions than ''n'', like [[submanifold]]s, for example, surfaces or curves in '''R'''³ and [[fractal]] sets. The Hausdorff measure is not to be confused with the notion of [[Hausdorff dimension]].

It can be shown that [[There is no infinite-dimensional Lebesgue measure|there is no infinite-dimensional analogue of Lebesgue measure]].

== See also ==
* [[Lebesgue's density theorem]]
* [[Duffin–Schaeffer conjecture]]

== References ==
{{reflist}}

[[Category:Measures (measure theory)]]

Scale factor (cosmology)

2013-12-10T11:06:26Z

129.16.200.109: The distance between single galaxies in the same cluster are typically not affected by the expansion of the Universe since they are gravitationally bound. It is the distance between the galaxy clusters that increase.

:''"Preload" redirects here; see also [[residual stress]].''

[[Image:Bolted joint.svg|thumb|150px|Bolted joint in vertical [[cutaway drawing|cutaway]]|right]]
[[Image:Bolted joint 2.svg|thumb|150px|Screw joint|right]]
[[Image:Bolted joint 1.svg|thumb|150px|Stud joint|right]]

'''Bolted joints''' are one of the most common elements in [[construction]] and machine design. They consist of [[fastener]]s that capture and join other parts, and are secured with the mating of [[screw thread]]s.

There are two main types of bolted joint designs: tension joints and shear joints.

In the tension joint, the bolt and clamped components of the joint are designed to transfer the external tension load through the joint by way of the clamped components through the design of a proper balance of joint and bolt stiffness. The joint should be designed such that the clamp load is never overcome by the external tension [[force]]s acting to separate the joint (and therefore the joined parts see no relative motion).

The second type of bolted joint transfers the applied load in shear on the bolt shank and relies on the [[shear strength]] of the bolt. Tension loads on such a joint are only incidental. A preload is still applied but is not as critical as in the case where loads are transmitted through the joint in tension. Other such shear joints do not employ a preload on the bolt as they allow rotation of the joint about the bolt, but use other methods of maintaining bolt/joint integrity. This may include [[Clevis pin|clevis]] linkages, joints that can move, and joints that rely on a locking mechanism (like [[lock washer]]s, thread [[adhesive]]s, and lock [[nut (hardware)|nuts]]).

Proper joint design and bolt preload provides useful properties:
* For cyclic tension loads, the fastener is not subjected to the full amplitude of the load; as a result, the fastener's [[fatigue (material)|fatigue]] life is increased or—if the material exhibits an endurance limit its life extends indefinitely.<ref>Collins, p. 481.</ref>
* As long as the external tension loads on a joint do not exceed the clamp load, the fastener is not subjected to motion that would loosen it, obviating the need for locking mechanisms. (Questionable under Vibration Inputs.)
*For the shear joint, a proper clamping force on the joint components prevents relative motion of those components and the [[fretting]] wear of those that would result in fatigue cracks.

In both the tension and shear joint design cases, some level of tension preload in the bolt and resulting compression preload in the clamped components is essential to the joint integrity. The preload target can be achieved by applying a measured [[torque]] to the bolt, measuring bolt extension, heating to expand the bolt then turning the nut down, torquing the bolt to the yield point, testing ultrasonically or by a certain number of degrees of relative rotation of the threaded components. Each method has a range of uncertainties associated with it, some of which are very substantial.

== Theory ==
Typically, a bolt is tensioned (preloaded) by the application of a torque to either the bolt head or the nut. The preload developed in a bolt is due to the applied torque and is a function of the bolt diameter, length, the geometry of the threads and the coefficients of friction that exist in the threads and under the bolt head or nut. The stiffness of the components clamped by the bolt has no relation to the preload that is developed by the torque. The relative stiffness of the bolt and the clamped joint components do, however, determine the fraction of the external tension load that the bolt will carry and that in turn determines preload needed to prevent joint separation and by that means to reduce the range of stress the bolt experiences as the tension load is repeatedly applied. This determines the durability of the bolt when subjected to repeated tension loads. Maintaining a sufficient joint preload also prevents relative slippage of the joint components that would produce fretting wear that could result in a fatigue failure of those parts when subjected to in-plane shearing forces.

[[Image:Bolted joint spring analogy.svg|440px|right]]

The clamp load, also called preload, of a fastener is created when a torque is applied, and so develops a tensile preload that is generally a substantial percentage of the fastener's [[proof strength]]. A fastener is manufactured to various standards that define, among other things, its strength and clamp load. ''Torque charts'' are available to identify the required torque for a fastener based on its ''property class'' (fineness of manufacture and fit) or ''grade'' (tensile strength).

When a fastener is torqued a tension preload develops in the bolt and a compressive preload develops in the parts being fastened. This can be [[Scientific modelling|modeled]] as a spring-like assembly that has some assumed distribution of compressive strain in the clamped joint components. As the tension load is applied it relieves the compressive strains induced by the preload, hence the preload acting on the compressed joint components provides the external tension load a path other than through the bolt. As long as the forces acting on the fastened parts do not exceed the preload, the fastener is not subjected to an increase in its tension load.

This however, is a simplified model that is only valid when the fastened parts are much stiffer than the fastener. In reality, the fastener carries a small fraction of the external load even if that external load does not exceed the clamp load. When the fastened parts are less stiff than the fastener (those that use soft, compressed gaskets for example), this model breaks down and the fastener is subjected to a tension load that is the sum of the tension preload and the external tension load.

In some applications, joints are designed so that the fastener eventually fails before more expensive components. In this case, replacing an existing fastener with a higher strength fastener can result in equipment damage. Thus, it is generally good practice to replace old fasteners with new fasteners of the same grade.
{{clear}}

== Setting the torque ==
Engineered joints require the torque to be chosen to provide the correct preload. Applying the torque to fasteners is commonly achieved using a [[torque wrench]].<ref name="oberg1495">{{harvnb|Oberg|Jones|McCauley|Heald|2004|p=1495}}.</ref> The required torque value for a particular fastener application may be quoted in the published standard document, defined by the manufacturer or calculated.

A common relationship used to calculate the torque for a desired preload takes into account the thread geometry and friction in the threads and under the bolt head or nut. The following assumes standard ISO or National Standard bolts and threads are used:

:<math>T = K P_{pre} d </math>

where
:<math>T </math> is the required torque
:<math>K </math> is the nut factor
:<math>P_{pre} </math> is the desired preload
:<math> d </math> is the bolt diameter

The nut factor K accounts for the thread geometry, friction, pitch. When ISO and Unified National Standard threads are used the nut factor is:<ref name=Shigley/>

:<math>K = \frac{d_{m}}{2 d}\,\left(\frac{ \tan \psi +\mu \sec \alpha} { 1 - \mu \tan \psi \sec \alpha}\right) + 0.625 \mu_{c}</math>

where
:<math> d_{m} </math> = the mean thread diameter, close to pitch diameter.
:<math> d </math> = nominal bolt diameter
:<math> \tan\psi </math> = [[fastener|(thread pitch)]]/(pi * dm)
:Thread Pitch = 1/N where N is the number of threads per inch or mm
:<math> \mu </math> = friction coefficient in the treads
:<math> \alpha </math> = half the thread angle (typically 60°) = 30°
:<math> \mu_{c}</math> = friction coefficient under torqued head or nut

When a value of <math> \mu </math> = <math> \mu_{c}</math> =0.15 is used and the dimensions for any size bolt whether course or fine the nut factor is K ≈ 0.20 and the torque/preload relationship becomes

:<math>T = 0.20 P_{pre} d </math>

A study of the effect of torquing a two samples, one of lubricated and the other unlubricated 1/2 in.- 20 UNF bolts to 800 lb-in, produced the same mean preload of 7700 lbf. The preloads for the unlubricated bolt sample had a standard deviation from the mean value of 1100 lbf, whereas the lubricated sample had a standard deviation of 680 lbf. If the preload value and torques are used in the above relation to solve for the nut factor it is found to be K = 0.208, which is very close to the recommended value of 0.20
<ref name=Shigley>{{cite book
|last=Shigley
|first=Joseph|title=Mechanical Engineering Design|year=1977
|publisher=McGraw-Hill|isbn=0-07-056881-2
|pages=246, 247}}</ref>

{| class="wikitable" cellpadding="5" style="margin:auto;"
|+ '''Accuracy of Bolt Preload Based on Bolt Preload Method'''<ref>{{cite web|last=Brown, Morrow, Durbin, Baca|title=Guideline for Bolted Joint Design and Analysis: Version 1.0|url=http://prod.sandia.gov/techlib/access-control.cgi/2008/080371.pdf|work=Sandia Report, SAND2008-0371|publisher=Sandia National Laboratories for United States Dept. of Energy|accessdate=4 December 2013|page=12}}</ref>
|-
|Method || Accuracy
|-
|Torque wrench on unlubricated bolts || ± 35%
|-
|Torque wrench on cad plated bolts || ± 30%
|-
|Torque wrench on lubricated bolts || ± 25%
|-
|Preload indicating washer || ± 10%
|-
|Strain gauges || ± 1%
|-
|Computer controlled wrench (below yield) || ± 15%
|-
|Computer controlled wrench (yield sensing)|| ± 8%
|-
|Bolt elongation || ± 5%
|-
|Ultrasonic sensing || ± 5%
|}

The preferred bolt preload for structural applications should be at least 75% of the fastener's proof load<ref name="oberg1495"/> for the higher strength fasteners and as high as 90% of the proof load for permanent fasteners. To achieve the benefits of the preloading, the clamping force must be higher than the joint separation load. For some joints, multiple fasteners are required to secure the joint; these are all hand tightened before the final torque is applied to ensure an even joint seating.

The torque value is dependent on the friction produced in the threads and under the torqued bolt head or nut and the fastened material or washer if used. This friction can be affected by the application of a lubricant or any plating (e.g. cadmium or zinc) applied to the threads, and the fastener's standard defines whether the torque value is for dry or lubricated threading, as lubrication can reduce the torque value by 15% to 25%; lubricating a fastener designed to be torqued dry could over-tighten it, which may damage threading or stretch the fastener beyond its [[elastic limit]], thereby reducing its clamping ability.

Either the bolt head or the nut can be torqued. If one has a larger bearing area or coefficient of friction it will require more torque to provide the same target preload.<ref>{{cite web|title=Bolt Science|url=http://www.boltscience.com/pages/faq.htm#3|work=Bolt Science Limited|accessdate=1 December 2013}}</ref> Fasteners should only be torqued if they are fitted in [[wiktionary:clearance hole|clearance hole]]s.

Torque wrenches do not give a direct measurement of the preload in the bolt. Much of the torque applied is lost overcoming friction under the torqued bolt head or nut (50%) and in the threads (40%). The remaining 10% of the applied torque does useful work in stretching the bolt and providing the preload. Initially, as the torque is applied, it must overcome static friction under the head of the bolt or nut (depending on which end is being torqued) and also in the threads. Finally, dynamic friction prevails and the torque is distributed in a 50/40/10 manner as the bolt is tensioned.

More accurate methods for determining the preload rely on defining or measuring the ''screw extension'' from the nut. Alternatively, measurement of the angular rotation of the nut can serve as the basis for defining screw extension based on the fastener's [[thread pitch]].<ref>{{harvnb|Oberg|Jones|McCauley|Heald|2004|p=1499}}.</ref> Measuring the screw extension directly allows the clamping force to be very accurately calculated. This can be achieved using a [[dial test indicator]], reading deflection at the fastener tail, using a [[strain gauge]], or ultrasonic length measurement.

Bolt preload can also be controlled by torquing the bolt to the point of yielding. A skilled operator can feel the drop off of the work required to turn the torque wrench as the material of the bolt begins to yield. At that point the bolt has a preload determined by the bolt area and the proof strength of the bolt or yield strength of the bolt material.

There is no simple method to measure the tension of a fastener already in place other than to tighten it and identify at which point the fastener starts extending. This is known as ''re-torqueing''. An electronic torque wrench can be used on the fastener in question, so that the torque applied can be constantly measured as it is slowly increased in magnitude.

Recent developments enable tensions to be estimated by using ultrasonic testing. Another way to ensure correct tension (mainly in steel erecting) involves the use of crush-washers. These are washers that have been drilled and filled with orange [[Vulcanization#Room Temperature Vulcanization|RTV]]. When the orange rubber strands appear, the tension is correct.

Large-volume users (such as auto makers) frequently use computer controlled [[nut driver]]s. With such machines, the computer in effect plots a graph of the torque exerted. Once the torque reaches a set maximum torque chosen by the designer, the machine stops. Such machines are often used to fit wheelnuts and normally tighten all the wheel nuts simultaneously.

==Thread engagement==
''Thread engagement'' is the length or number of threads that are engaged between the screw and the female threads. Screws are designed so that the bolt shank fails before the threads, but for this to hold true, a minimum thread engagement must be used. The following equation defines this minimum thread engagement:
<ref name="threadengagement">{{Citation
| title = Minimum Thread Engagement Formula and Calculation ISO
| url = http://www.engineersedge.com/thread_strength/thread_minimum_length_engagement.htm
| accessdate = 2010-02-08
| archiveurl =
| archivedate =
| postscript =.}}</ref>

:<math>L_e = \frac{2 \times A_t}{0.5 \pi \left( D - 0.64952 p \right)}</math>

Where Le is the thread engagement length, At is the tensile stress area, D is the major diameter of the screw, and p is the pitch. This equation only holds true if the screw and female thread materials are the same. If they are not the same, then the following equations can be used to determine the additional thread length that is required:<ref name="threadengagement"/>

:<math>J = \frac{\text{tensile strength of external thread material}}{\text{tensile strength of internal thread material}}</math>

:<math>L_{e2} = J \times L_e</math>

Where Le2 is the new required thread engagement.

While these formulas give absolute minimum thread engagement, many industries specify that bolted connections be at least fully engaged. For instance, the [[FAA]] has determined that in general cases, at least one thread must be protruding from any bolted connection. [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgAdvisoryCircular.nsf/0/99c827db9baac81b86256b4500596c4e/$FILE/Chapter%2007.pdf]

== Failure modes ==
The most common mode of [[structural failure|failure]] is overloading: Operating forces of the application produce loads that exceed the clamp load, causing the joint to loosen over time or fail catastrophically.

Overtorquing might cause failure by damaging the threads and deforming the fastener, though this can happen over a very long time. Undertorquing can cause failures by allowing a joint to come loose, and it may also allow the joint to flex and thus fail under fatigue.

[[Brinelling]] may occur with poor quality washers, leading to a loss of clamp load and subsequent failure of the joint.

Other modes of failure include [[corrosion]], [[embedment]], and exceeding the [[shear stress]] limit.

Bolted joints may be used intentionally as [[sacrificial part]]s, which are intended to fail before other parts, as in a [[shear pin]].

== Locking mechanisms ==
[[Image:Car hub cotter pin.jpg|thumb|right|Bolted joints in an automobile wheel. Here the outer fasteners are four studs with three of the four nuts that secure the wheel. The central nut (with locking cover and [[Split pin|cotter pin]]) secures the wheel bearing to the spindle.]]

Locking mechanisms keep bolted joints from coming loose. They are required when [[vibration]] or joint movement will cause loss of [[clamp (tool)|clamp]] [[Structural load|load]] and joint failure, and in equipment where the [[security]] of bolted joints is essential.
* Two nuts, tightened on each other. In this application a thinner nut should be placed adjacent to the joint, and a thicker nut tightened onto it. The thicker nut applies more force to the joint, first relieving the force on the threads of the thinner nut and then applying a force in the opposite direction. In this way the thicker nut presses tightly on the side of the threads away from the joint, while the thinner nut presses on the side of the threads nearest the joint, tightly locking the two nuts against the threads in both directions.<ref>{{cite web|url=http://www.boltscience.com/pages/twonuts.htm|title=The use of two nuts to prevent self loosening|publisher=boltscience.com}}</ref>

==Bolt banging==
{{Expand section|date=September 2008}}

''Bolt banging'' occurs in buildings when bolted joints slip into bearing under load, thus causing a loud and potentially frightening noise resembling a rifle shot that is not, however, of structural significance and does not pose any threat to occupants.
<ref>Carter, C.J.: "Steel Interchange: Banging Bolts", ''[http://www.modernsteel.com/Uploads/Issues/July_1999/071999_Interchange%207-99%20N%20adds.pdf MSC: Modern Steel Construction]'', July 1999.</ref>

== International standards ==
* SA-193/SA-193M: "Specification for alloy-steel and stainless steel bolting materials for high-temperature service"
* SA-194/SA-194M: "Specification for carbon and alloy steel nuts for bolts for high-temperature service"
* SA-320/SA-320M: "Specification for alloy steel bolting materials for low-temperature service"
* EN 1515: "Flanges and their joints - Bolting"
** EN 1515-1: "Flanges and their joints - Bolting - Part 1: Selection of bolting"
** EN 1515-2: "Flanges and their joints — Bolting — Part 2: Classification of bolt materials for steel flanges, PN designated"
** EN 1515-2: "Flanges and their joints — Bolting — Part 3: Classification of bolt materials for steel flanges, class designated"
* ISO 4017: "Hexagon head screws - Product grades A and B"
* ISO 4032: "Hexagon nuts, style 1 - Product grades A and B"
* ISO 4033: "Hexagon nuts, style 2 - Product grades A and B"

== See also ==
* [[Bearing surface]]
* [[Bolt manufacturing process]]
* [[Castellated nut]]/capscrew (common in the aircraft industry)
* [[Quench]]ing and [[tempering (metallurgy)|tempering]] (Q&T)
* [[Rivet]]
* [[Locknut]] ([[prevailing torque nuts]])
** [[Polymer insert nut]]
** [[Oval lock nut]]
* [[washer (hardware)|Lock washer]]
* [[Safety wire|Lock wire]]
* [[Mechanical joint]]
* [[Thread adhesive]]

== References ==
;Notes
{{reflist|2}}

;Bibliography
* {{Citation | last = Collins | first = Jack A. | last2 = Staab | first2 = George H. | last3 = Busby | first3 = Henry R. | title = Mechanical Design of Machine Elements and Machines | publisher = Wiley | year = 2002 | isbn = 0-471-03307-3 | postscript =.}}

*{{citation | last = Oberg | first = Erik | last2 = Jones | first2 = Franklin D. | last3 = McCauley | first3 = Christopher J. | last4 = Heald | first4 = Ricardo M. | year = 2004 | title = [[Machinery's Handbook]] | edition = 27th | publisher = [[Industrial Press]] | isbn = 978-0-8311-2700-8 | postscript =.}}

== External links ==
* [http://www.piping-engineering.com/bolt-length-calculation-method.html Bolt Length Calculation Method]
* [http://www.aisc.org/SearchTaxonomy/TechnicalLibraryResults.aspx?topic=388 Bolts - AISC | Home]
* [http://www.aisc.org/bookstore/itemRedirector.aspx?id=14854 AISC THE BANGING BOLT SYNDROME]
* [http://www.aisc.org/bookstore/itemRedirector.aspx?id=14858 AISC BANGING BOLTS— ANOTHER PERSPECTIVE]
* [http://www.boltscience.com/pages/josteffect.htm Bolt Science - The Jost Effect]
* [http://assist.daps.dla.mil/quicksearch/basic_profile.cfm?ident_number=70455 ''Threaded Fasteners - Tightening to Proper Tension''], US Department of Defense document MIL-HDBK-60, 2.6MB pdf.
* [http://gltrs.grc.nasa.gov/reports/1990/RP-1228.pdf NASA Reference Publication 1228 Fastener Design Manual]
* [http://www.mech.uwa.edu.au/DANotes/threads/mechanics/mechanics.html Mechanics of screws]
* [http://rgl.faa.gov/Regulatory_and_Guidance_Library/rgAdvisoryCircular.nsf/0/99c827db9baac81b86256b4500596c4e/$FILE/Chapter%2007.pdf FAA Advisory Circular 43.13-1B], Paragraph 7-37 "Grip Length"

{{DEFAULTSORT:Bolted Joint}}
[[Category:Threaded fasteners]]
[[Category:Mechanical engineering]]
[[Category:Structural connectors]]

[[vi:Ốc vít#Bu lông]]

Tsallis entropy

2013-10-23T05:56:22Z

129.16.126.117: deleted erroneous category classification: this has nothing to do with "q-analogues"

{{about| mechanical resonance in [[physics]] and [[engineering]]|a general description of resonance| resonance|mechanical resonance of sound including musical instruments|acoustic resonance|the music album by American rock band [[Tesla (band)|Tesla]]|Mechanical Resonance}}

[[Image:Resonance.PNG|thumb|right|Graph showing mechanical [[resonance]] in a mechanical oscillatory system]]

'''Mechanical resonance''' is the tendency of a [[mechanics|mechanical system]] to respond at greater amplitude when the [[frequency]] of its oscillations matches the system's natural frequency of [[vibration]] (its ''[[resonance frequency]]'' or ''resonant frequency'') than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings and airplanes—a phenomenon known as resonance disaster.

Avoiding resonance disasters is a major concern in every building, tower and bridge [[construction]] project. The [[Taipei 101]] building relies on a 660-ton [[pendulum]] — a [[tuned mass damper]] — to modify the response at resonance. Furthermore, the structure is designed to resonate at a frequency which does not typically occur. Buildings in [[seismic]] zones are often constructed to take into account the oscillating frequencies of expected ground motion. In addition, [[engineer]]s designing objects having engines must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other strongly oscillating parts.

Many resonant objects have more than one resonance frequency. It will vibrate easily at those frequencies, and less so at other frequencies. Many [[clock]]s keep time by mechanical resonance in a [[balance wheel]], [[pendulum]], or [[Quartz clock|quartz crystal]].

==Description==
The natural frequency of a simple mechanical system consisting of a weight suspended by a spring is:

:<math>f = {1\over 2 \pi} \sqrt {k\over m} </math>

where ''m'' is the [[mass]] and ''k'' is the [[spring constant]].

A [[swing set]] is a simple example of a resonant system with which most people have practical experience. It is a form of pendulum. If the system is excited (pushed) with a period between pushes equal to the inverse of the pendulum's natural frequency, the swing will swing higher and higher, but if excited at a different frequency, it will be difficult to move. The resonance frequency of a pendulum, the only frequency at which it will vibrate, is given approximately, for small displacements, by the equation:<ref>[http://www.physics.rutgers.edu/~jackph/2005s/PS02.pdf Mechanical resonance]</ref>

:<math>f = {1\over 2 \pi} \sqrt {g\over L} </math>

where ''g'' is the [[standard gravity|acceleration due to gravity]] (about 9.8 m/s2 near the surface of [[Earth]]), and ''L'' is the length from the pivot point to the center of mass.(An [[elliptic integral]] yields a description for any displacement). Note that, in this approximation, the frequency does not depend on [[mass]].

Mechanical resonators work by transferring energy repeatedly from [[kinetic energy|kinetic]] to [[potential energy|potential]] form and back again. In the pendulum, for example, all the energy is stored as [[gravity|gravitational]] energy (a form of potential energy) when the bob is instantaneously motionless at the top of its swing. This energy is proportional to both the mass of the [[Bob (physics)|bob]] and its height above the lowest point. As the bob descends and picks up speed, its potential energy is gradually converted to kinetic energy (energy of movement), which is proportional to the bob's mass and to the square of its speed. When the bob is at the bottom of its travel, it has maximum kinetic energy and minimum potential energy. The same process then happens in reverse as the bob climbs towards the top of its swing.

Some resonant objects have more than one resonance frequency, particularly at harmonics (multiples) of the strongest resonance. It will vibrate easily at those frequencies, and less so at other frequencies. It will "pick out" its resonance frequency from a complex excitation, such as an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies other than its resonance. In the example above, the swing cannot easily be excited by harmonic frequencies, but can be excited by [[subharmonic]]s.

==Examples==
Various examples of mechanical resonance include:
*[[musical instruments]] ([[acoustic resonance]]).
*Most [[clock]]s keep time by mechanical resonance in a [[balance wheel]], [[pendulum]], or [[Quartz clock|quartz crystal]].
*[[tidal resonance]] of the [[Bay of Fundy]].
*[[Orbital resonance]] as in some [[natural satellite|moon]]s of the [[solar system]]'s [[gas giants]].
*The resonance of the [[basilar membrane]] in the [[ear]].
*Making a child's [[Swing (seat)|swing]] swing higher by pushing it at each swing.
*A wineglass breaking when someone sings a loud note at exactly the right pitch.

[[Image:Display 01.jpg|thumbnail|right|''Resonance Rings'' exhibit at [[California Science Center]]]]

Resonance may cause violent swaying motions in improperly constructed structures, such as bridges and buildings. The [[London Millennium Footbridge]] (nicknamed the ''Wobbly Bridge'') exhibited this problem. A faulty bridge can even be destroyed by its resonance (see "[[Angers Bridge]]"); that is why soldiers are trained not to march in [[lockstep marching|lockstep]] across a bridge, although it is suspected to be a myth, see e.g., [[MythBusters (2004 season)#Breakstep Bridge|MythBusters' 'Breakstep Bridge']]. Mechanical systems store potential energy in different forms. For example, a [[spring (device)|spring]]/mass system stores energy as tension in the spring, which is ultimately stored as the energy of bonds between [[atom]]s.

==Resonance disaster==
In mechanics and construction a '''resonance disaster''' describes the destruction of a building or a technical mechanism by induced vibrations at a system's [[resonance]] frequency, which causes it to [[oscillate]]. Periodic excitation optimally transfers to the [[system]] the [[energy]] of the vibration and stores it there. Because of this repeated storage and additional energy input the system swings ever more strongly, until its load limit is exceeded.

===Failure of the original Tacoma Narrows Bridge===
{{main|Tacoma Narrows Bridge (1940)}}
The dramatic, rhythmic twisting that resulted in the 1940 collapse of "Galloping Gertie", the original [[Tacoma Narrows Bridge (1940)|Tacoma Narrows Bridge]], is sometimes characterized in physics textbooks as a classic example of resonance; however, this description is misleading. The catastrophic vibrations that destroyed the bridge were not due to simple mechanical resonance, but to a more complicated oscillation caused by interactions between the bridge and the winds passing through its structure — a phenomenon known as [[aeroelasticity#Flutter|aeroelastic flutter]]. [[Robert H. Scanlan]], father of the field of bridge aerodynamics, wrote an article about this misunderstanding.<ref>K. Billah and R. Scanlan (1991), ''Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks'', [[American Journal of Physics]], 59(2), 118--124 [http://www.ketchum.org/billah/Billah-Scanlan.pdf (PDF)]</ref>

===Other Examples===
* Collapse of [[Broughton Suspension Bridge]] (due to soldiers walking in step)
* Collapse of [[Angers Bridge]]
* Collapse of [[Königs Wusterhausen Central Tower]]
* Resonance of the [[Millennium Bridge (London)#Resonance|Millennium Bridge]]
* [[The_Power_(Snap!_song)#Tremors_at_Techno-Mart|Evacuation of the 39-story TechnoMart]] commercial-residential high-rise in Korea in 2011 due to a class performing [[Tae Bo]] exercises to the song "The Power".

==Applications==
Various method of inducing mechanical resonance in a medium exist. Mechanical waves can be generated in a medium by subjecting an electromechanical element to an alternating electric field having a frequency which induces mechanical resonance and is below any electrical resonance frequency.<ref>Allensworth, et al., United States Patent 4,524,295. June 18, 1985</ref> Such devices can apply mechanical energy from an external source to an element to mechanically stress the element or apply mechanical energy produced by the element to an external load.

The [[United States Patent Office]] classifies devices that tests mechanical resonance under subclass 579, [[resonance]], [[frequency]], or [[amplitude]] study, of Class 73, [[Measuring]] and [[Experiment|test]]ing. This subclass is itself indented under subclass 570, Vibration.<ref>USPTO, [http://www.uspto.gov/go/classification/uspc073/defs073.htm Class 73, Measuring and testing]</ref> Such devices test an article or [[mechanism (technology)|mechanism]] by subjecting it to a vibratory force for determining qualities, characteristics, or conditions thereof, or sensing, studying or making analysis of the vibrations otherwise generated in or existing in the article or mechanism. Devices include methods to cause vibrations at a natural mechanical resonance and measure the [[frequency]] and/or [[amplitude]] the resonance made. Various devices study the amplitude response over a [[frequency range]] is made. This includes [[nodal point]]s, [[wave length]]s, and [[standing wave]] characteristics measured under predetermined vibration conditions.

== See also ==

* [[Resonator]]
* [[Reed switch]]
* [[Transducer]]
* [[Electrical resonance]]
* [[Laser applications]]
* [[Dunkerley's Method]]
* [[String resonance]]

== Notes ==

{{reflist}}

== Further reading ==

* S Spinner, WE Tefft, ''A method for determining mechanical resonance frequencies and for calculating elastic moduli from these frequencies''. American Society for testing and materials.
* CC Jones, ''A mechanical resonance apparatus for undergraduate laboratories''. American Journal of Physics, 1995.

== Patents ==

* {{US patent|1414077}} Method and apparatus for inspecting materials
* {{US patent|1517911}} Apparatus for testing textiles
* {{US patent|1598141}} Apparatus for testing textiles and like materials
* {{US patent|1930267}} Testing and adjusting device
* {{US patent|1990085}} Method and apparatus for testing materials
* {{US patent|2352880}} Article testing machine
* {{US patent|2539954}} Apparatus for determining the behavior of suspended cables
* {{US patent|2729972}} Mechanical resonance detection systems
* {{US patent|2918589}} Vibrating-blade relays with electro-mechanical resonance
* {{US patent|2948861}} Quantum mechanical resonance devices
* {{US patent|3044290}} Mechanical resonance indicator
* {{US patent|3141100}} Piezoelectric resonance device
* {{US patent|3990039}} Tuned ground motion detector utilizing principles of mechanical resonance
* {{US patent|4524295}} Apparatus and method for generating mechanical waves
* {{US patent|4958113}} Method of controlling mechanical resonance hand
* {{US patent|7027897}} Apparatus and method for suppressing mechanical resonance in a mass transit vehicle

[[Category:Mechanical vibrations]]
[[Category:Earthquake engineering]]

[[ru:Резонанс#.D0.9C.D0.B5.D1.85.D0.B0.D0.BD.D0.B8.D0.BA.D0.B0]]
[[sv:Självsvängning]]

Error analysis

2013-10-11T11:04:34Z

129.16.110.19:

{{Expert-subject|pharmacology|date=November 2008}}'''Schild regression analysis''', named for [[Heinz Otto Schild]], is a useful tool for studying the effects of [[agonist]]s and [[Receptor antagonist|antagonist]]s on the cellular response caused by the [[Receptor_(biochemistry)|receptor]] or on ligand-receptor binding.

Using a [[dose-response curve]] or an equivalent curve with concentration and binding %, it is possible to determine the dose ratio, this is a measure of the potency of a drug; it is obtained by dividing the increased [[equilibrium constant]] due to drug inhibition by the equilibrium constant without the drug. A Schild plot is a double logarithmic plot, typically Log(dr-1) as the ordinate and Log[B] as the abscissa. This is because a competitive drug B will have a linear plot with the <math>dr=1+[B]/K_B</math>.
These experiments must be carried out on a very wide range (therefore the logarithmic scale) as the mechanisms differ over a large scale, such as at high concentration of drug.

== Schild regression for ligand binding ==
Although most experiments use cellular response as a measure of the effect, the effect is, in essence, a result of the binding kinetics; so, in order to illustrate the mechanism, [[ligand]] binding is used. A ligand A will bind to a [[receptor (biochemistry)|receptor]] R according to an equilibrium constant :

Kd=k-1/k1

Although the equilibrium constant is more meaningful, texts often mention its inverse, the affinity constant (Kaff = k1/k-1): A better binding means an increase of binding affinity.

The equation for simple ligand binding to a single homogeneous receptor is
<math>[AR]=[R]t[A]/([A]+Kd)</math>

This is the Hill-Langmuir equation, which is practically the [[Hill equation (biochemistry)]] described for the agonist binding. In chemistry, this relationship is called the [[Langmuir equation]], which describes the adsorption of molecules onto sites of a surface (see [[adsorption]]).

[R]total is the total number of binding sites, and when the equation is plotted it is the horizontal asymptote to which the plot tends; more binding sites will be occupied as the ligand concentration increases, but there will never be 100% occupancy. The binding affinity is the concentration needed to occupy 50% of the sites; the lower this value is the easier it is for the ligand to occupy the binding site.

The binding of the ligand to the receptor at equilibrium follows the same kinetics as an enzyme at steady-state ([[Michaelis-Menten equation]]) without the conversion of the bound substrate to product.

Agonists and antagonists can have various effects on ligand binding. They can change the maximum number of binding sites, the affinity of the ligand to the receptor, both effects together or even more bizarre effects when the system being studied is more intact, such as in tissue samples. (Tissue absorption, densitization, and other non equilibrium steady-state can be a problem.)

A surmountable drug changes the binding affinity:
* competitive ligand K'd=Kd (1+[B]/Kb)
* cooperative allosteric ligand K'd=Kd (KB+[B]/(KB+[B]/α))

A nonsurmountable drug changes the maximum binding:
* noncompetitive binding [R]t'=[R]t /(1+[B]/Kb)
* irreversible binding

The Schild regression also can reveal if there are more than one type of receptor and it can show if the experiment was done wrong as the system has not reached equilibrium. 
[[Image:Schild regression bindings.jpg|1200px]]

== Radioligand binding assays ==
The first radio-receptor assay (RRA) was done in 1970 by Lefkowitz et al., using a radiolabeled hormone to determine the binding affinity for its receptor.<ref name="pmid4319388">{{cite journal |author=Lefkowitz RJ, Roth J, Pastan I |title=Radioreceptor assay of adrenocorticotropic hormone: new approach to assay of polypeptide hormones in plasma |journal=Science |volume=170 |issue=3958 |pages=633–5 |date=November 1970 |pmid=4319388 |doi= 10.1126/science.170.3958.633|url=}}</ref>

A radio-receptor assay requires the separation of the bound from the free ligand. This is done by [[filtration]], [[centrifugation]] or [[dialysis]].<ref name="pmid16253574">{{cite journal |author=de Jong LA, Uges DR, Franke JP, Bischoff R |title=Receptor-ligand binding assays: technologies and applications |journal=J. Chromatogr. B Analyt. Technol. Biomed. Life Sci. |volume=829 |issue=1–2 |pages=1–25 |date=December 2005 |pmid=16253574 |doi=10.1016/j.jchromb.2005.10.002 |url=}}</ref>

A method that does not require separation is the scintillation proximity assay that relies on the fact that β-rays from 3H travel extremely short distances. The receptors are bound to beads coated with a polyhydroxy scintillator. Only the bound ligands to be detected.

Today, the fluorescence method is preferred to radioactive materials due to a much lower cost, lower hazard, and the possibility of multiplexing the reactions in a high-throughput manner. One problem is that fluorescent-labeled ligands have to bear a bulky fluorophore that may cause it to hinder the ligand binding. Therefore, the fluorophore used, the length of the linker, and its position must be carefully selected.

An example is by using [[FRET]], where the ligand's fluorophore transfers its energy to the fluorophore of an antibody raised against the receptor.

Other detection methods such as [[surface plasmon resonance]] do not even require fluorophores.

== References ==
''Ligand receptor binding:''
Kenakin T, 1993. Pharmacological analysis of drug-receptor interaction
New York: Raven Press
{{reflist}}

== See also ==
* [[Dose-response relationship]]

==External links==
* [http://www.curvefit.com/schild.htm curvefit.com - Dose-response curves in the presence of antagonists], for a clear explanation.

[[Category:Pharmacology]]
[[Category:Biochemistry methods]]

Lattice (group)

2013-08-30T11:42:33Z

129.16.126.117: /* Symmetry considerations and examples */

{{Economics sidebar}}
In [[economics]], the consumer's preferences, monetary income (money) and prices play an important role in solving the consumer's optimization problem (maximization of their [[utility]] subject to a budget constraint). The '''income effect''' in [[economics]] can be defined as the change in consumption resulting from a change in [[real income]].<ref>{{cite book
| last = Sullivan
| first = arthur
| authorlink = Arthur O' Sullivan
| coauthors = Steven M. Sheffrin
| title = Economics: Principles in action
| publisher = Pearson Prentice Hall
| year = 2003
| location = Upper Saddle River, New Jersey 07458
| pages = 80
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The [[comparative statics]] of consumer behavior investigates the effects of changes in the exogenous or independent variables i.e. prices and money incomes of the consumers on the equilibrium values of the endogenous or dependent variables i.e. the consumer's demand for goods. When the income of the consumer rises with the prices held constant, the optimal bundle chosen by the consumer changes as the feasible set available to him changes. The '''income–consumption curve''' is the set of optimal points of intersection of the points of tangency of the sets of budget constraint lines and indifference curves as income varies, with prices held constant.

==Consumer theory==
[[File:Income consumption curve graph.svg|thumb|300px|300px|left|Figure 1: An increase in the income, with the prices of all goods fixed, causes consumers to alter their choice of market basket.]]

The income effect is a phenomenon observed through changes in purchasing power. It reveals the change in quantity demanded brought by a change in real income ([[utility]]). The figure 1 on the left, shows the consumption patterns of the consumer of two goods X1 and X2, the prices of which are ''p''1 and ''p''2 respectively. The initial bundle X*, is the bundle which is chosen by the consumer on the budget line B1. An increase in the money income of the consumer, with ''p''1 and ''p''2 constant, will shift the budget line outward parallel to itself.

In the figure, this means that the change in the money income of the consumer will shift the budget line B1 outward parallel to itself to B2 where the bundle X' bundle will be chosen. Again, an increase in the money income of the consumer will push the budget line B2 outward parallel to itself to B3 where the bundle X" will be the bundle which will be chosen. Thus, it can be said that, with variations in income of the consumers and with the prices held constant the '''income–consumption curve''' can be traced out as the set of optimal points.
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==Income–consumption curve for different goods==

In the case illustrated with the help of Figure 1 both X1 and X2 are [[normal goods]] in which case, the demand for the good increases as money income rises. However, if the consumer has different preferences, he has the option to choose X0 or X+ on budget line B2. As the income of the consumer rises,and the consumer chooses X0 instead of X' i.e. if the consumer's indifference curve is I4 and not I2, then the demand for X1 would fall . In that case, X1 would be called an [[inferior]] good i.e. demand for good X1 decreases with a rise in income of the consumer. Thus, a rise in income of the consumer may lead his demand for a good to rise, fall or not change at all. It is important to note here that, the knowledge of preferences of the consumer is essential to predict whether a particular good is inferior or normal.

===Normal goods===
[[File:Income consumption curve graph - upward sloping (normal goods).svg|thumb|left|300px|300px|Figure 2: Income-consumption curve for Normal goods]]
In the figure 2 to the left, B1, B2 and B3 are the different budget lines and I1, I2 and I3 are the indifference curves that are available to the consumer. As shown earlier, as the income of the consumer rises, the budget line moves outwards parallel to itself. In this case, from initial bundle X*, with an increase in the income of the consumer the budget line moves from B1 to B2 and the consumer would choose X' bundle and subsequently, with a further rise in consumer's income the budget line moves from B2 to B3 and the consumer would choose X" bundle and so on. And so the consumer would maximize his utility at the points X*, X' and X", and by joining these points, the '''income-consumption curve''' can be obtained.<ref name="oup.com">http://www.oup.com/us/pdf/microecon/ch04ppt.pdf</ref>

The upward sloping income-consumption curve implies that there will be an increase in the demand for both X1 and X2 as the income of the consumer rises and will cause the demand curves of the goods to shift to the right.<ref name=P&R>{{cite book|last=Rubinfeld, Pindyck|first=Daniel, Robert|title=Microeconomics|year=1995|publisher=Tsinghua University Press/ Prentice-Hall|location=Mainland China|isbn=7-302-02494-4|pages=699|url=http://books.google.com/books?id=6AE6AAAACAAJ&dq=microeconomics+rubinfeld+and+pindyck+1995&hl=en&ei=DatsTs7fLoTJrAfe77ypBQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA}}</ref>
When the income-consumption curve has a positive [[slope]] then the income elasticity of demand will be positive. The greater the shifts of the demand curve to the right, the greater the income-elasticity of demand. In such a case, the goods will be normal goods.<ref name=P&R>{{cite book|last=Rubinfeld, Pindyck|first=Daniel, Robert|title=Microeconomics|year=1995|publisher=Tsinghua University Press/ Prentice-Hall|location=Mainland China|isbn=9787302024941|pages=699|url=http://books.google.com/books?id=6AE6AAAACAAJ&dq=microeconomics+rubinfeld+and+pindyck+1995&hl=en&ei=DatsTs7fLoTJrAfe77ypBQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCkQ6AEwAA}}</ref>

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===Inferior goods===

[[File:Income consumption curve graph - downward sloping (inferior goods).svg|thumb|300px|300px|right|Figure 3: with an increase in the income of demand for normal good X2 rises while, demand for inferior good X1 falls.]]

The figure 3 on the right, shows the consumption patterns of the consumer of two goods X1 and X2, the prices of which are ''p''1 and ''p''2 respectively, where B1 and B2 are the budget lines and I1 and I2 are the indifference curves. Figure 3 clearly shows that, with a rise in the income of the consumer, the initial budget line B1 moves outward parallel to itself to B2 and the consumer now chooses X' bundle to the initial bundle X*. The figure shows that, the demand for X2 has risen from X21 to X22 with an outward shift of the budget line from B1 to B2 (caused due to rise in the income of the consumer). This essentially means that, good X2 is a normal good as the demand for X2 rose with an increase in the income of the consumer.

In contrast, it is to be noted from the figure, that the demand for X1 has fallen from X11 to X12 with an outward shift of the budget line from B1 to B2 (caused due to rise in the income of the consumer). This implies that, good X1 is an inferior good as the demand for X1 fell with an increase in the income of the consumer.

The consumer maximizes his utility at points X* and X' and by joining these points, the income–consumption curve can be obtained.<ref name="oup.com"/> In figure 3, the income–consumption curve bends back on itself as with an increase income, the consumer demands more of X2 and less of X1.<ref name=econconcepts>http://economicsconcepts.com/application_of_indifference_curves.htm</ref>
The income–consumption curve in this case is negatively sloped and the income elasticity of demand will be negative.<ref name="P&R"/> Also the price effect for X2 is positive, while it is negative for X1.<ref name="econconcepts"/>

<math>\Delta X_n^1</math> is the change in the demand for good 1 when we change income from <math>m'</math> to <math>m</math>, holding the price of good 1 fixed at <math> p_1</math>:

<math>\Delta X_n^1 = X^1(p_1, m) - X^1(p_1,m').</math>
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===Perfect Substitutes===

[[File:Income consumption curve graph - horizontal (perfect substitutes).svg|thumb|300px|300px||right|Figure4: Income–consumption curve for perfect substitutes]]

The figure on the right depicts the case of two goods X1 and X2 which are [[perfect substitutes]], prices of which are ''p''1 and ''p''2 respectively. Here, I1, I2, I3, I4 and I5 are the straight line indifference curves, B1, B2 and B3 are the budget constraints and X*, X' and X" are the bundles chosen by the consumer. If it is assumed that, ''p''1 < ''p''2 then the consumer would consume only X1 as this would maximize his utility.<ref name=studentclub>[http://www.commerce.usask.ca/studentclubs/finance/exams/Econ211Chapter%204%20SolutionsStLouis0405.pdf]</ref>

In the figure on the right, B1 is the initial budget line and the consumer chooses X* as his optimal bundle and as the money income of the consumer rises, his budget line will shift outward and parallel to itself to B2. At the budget line B2 and indifference curve I4, the consumer will choose X'. Subsequently, as the income rises further, the budget line will again shift outward and parallel to itself to B3, where, the consumer will choose the optimal bundle X".

Thus, it can be said that, the amount of good X1 that the consumer consumes will increase, with an increase in the income of the consumer.<ref name=HRV>{{cite book|last=R Varian|first=Hal|title=Intermediate Microeconomics : A modern approach|year=2006|publisher=W.W. Norton & Co|isbn=0-393-92702-4|pages=754}}</ref> Thus, the income–consumption curve for the perfect substitutes X1 and X2 will be the horizontal axis.<ref name="studentclub"/>

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===Perfect complements===
[[File:Income consumption curve graph - linear (perfect complements).svg|thumb|300px|300px|left|Figure 5: Income–consumption curve for perfect complements]]

In case of perfect complements, the same amount of goods will be consumed by the consumer irrespective of say income, prices etc.<ref name="HRV"/> As the level of consumption remains the same, the income–consumption curve for perfect complements is the diagonal line passing through the origin as shown in Figure 5 on the left.

In the figure on the left, X*, X' and X" are the utility maximization points where the budget constraint lines B1, B2 and B3 touch the kinks of the ''L''-shaped indifference curves I1, I2 and I3 respectively.<ref name="studentclub"/> And by joining these points of utility maximization, the income–consumption curve for perfect substitutes is obtained.

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===Cobb–Douglas preferences===
The demand functions for both good X1 and X2 are linear functions of income and thus, the income-consumption curve will be straight lines through the origin.<ref name=HRV>{{cite book|last=R Varian|first=Hal|title=Intermediate Microeconomics : A modern approach 7th Edition |year=2006|publisher=W.W. Norton & Co|isbn=0-393-92702-4|pages=754}}</ref>

If <math> u(X^1, X^2) = X_1^a X_2^{(1-a)}\,</math>

Then, the demand for good X1 will have the form <math>X_1= am/p_1\,</math><ref name="HRV"/>

If ''p''1 is kept as a fixed value, then this will be a linear function of m. By doubling ''m'', the demand will double. In fact, if ''m'' is multiplied by any positive number say, ''t'', the demand will be multiplied with the same amount.

For good <math> X_2</math>, the demand is given by <math> X_2=(1-a)m/p_2\,</math>.<ref name="HRV"/> This too is clearly linear and thus making the income-consumption curves for both the goods X1 and X2 straight lines passing through the origin.

==Income–consumption curve and Engel curves==
{{Main|Engel Curve}}
At each level of income level, say ''m'', there would an optimal choice for each of the goods. If suppose only the case of a good say, good X1 is taken, then the optimal choice at each set of prices and income or in other words, the demand function for good X1 can be written as:

<math>X^1=X^1(p_1,p_2,m) \,</math> <ref name="HRV"/>

Where,
''p''1 is the price of good X1, ''p''2 is the price of good X2 and ''m'' is the income of the consumer.

If the prices of the goods X1 and X2 are held constant and the changes in demand are observed in relation to changes in income, the [[Engel curve]] can be generated. With all prices held constant, the [[Engel curve]] can be defined as a graph depicting the demand for one good as a function of income.
<ref name="HRV"/>

==See also==
{{Portal|Business and economics}}
*[[Consumer theory#Income effect]]
*[[Price–consumption curve]]
*[[Engel Curve]]

==References==
{{Reflist}}

{{Microeconomics}}

{{DEFAULTSORT:Income Effect}}
[[Category:Economics]]
[[Category:Economics terminology]]
[[Category:Consumer theory]]