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'''Pseudomanifold''' is a special type of [[topological space]].
In the study of [[mechanical network]]s in [[control theory]], an '''inerter''' is a two-terminal device in which the forces applied at the terminals are equal, opposite, and proportional to relative acceleration between the nodes. The generated strength satisfies the equation
It looks like a manifold at most of the points, but may contain singularities.
: <math>F=b(\dot{v}_2-\dot{v}_1)</math>,
For example, the cone of solutions of <math>z^2=x^2+y^2</math> forms a pseudomanifold.
for a suitable constant ''b''.


[[File:Pinched torus.jpg|thumb|<center>A pinched torus</center>]]
[[Malcolm C. Smith]], a [[control theory|control]] [[engineering]] professor at the [[University of Cambridge]], first introduced inerters in a 2002 paper.<ref>{{cite doi|10.1109/TAC.2002.803532}}</ref> However, a similar device was demonstrated in 1998 by Polish engineer Lucianus Lagiewka and used in an automobile bumper application. The principle and design are the same: a kinetic energy absorbing system that can be used as a bumper or vibration absorber.<ref>{{cite web| url= http://www.rexresearch.com/lagiewka/lagiewka.htm|title=Device for transforming kinetic energy}}</ref><ref>{{cite web| url=http://www.youtube.com/watch?feature=endscreen&NR=1&v=4_5oseSVUc4|title=Project EPAR}}</ref> The device was in the public knowledge since 1998. Smith extended the analogy between [[electrical network|electrical]] and [[mechanical network]]s (the [[mobility analogy]]). He observed that the analogy was incomplete, since it was missing a mechanical device playing the same role as an electrical [[capacitor]]. It was found that it is possible to construct such a device using gears.
A pseudomanifold can be considered as a [[combinatorial]] realisation of the general idea of a [[manifold]] with [[Mathematical singularity|singularities]].  
The concepts of [[orientability]], orientation and [[degree of a mapping]] make sense for pseudomanifolds and moreover, within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts.<ref>{{Citation|first=H.|last=Steifert|first2=W.|last2=Threlfall|title=Textbook of Topology|publisher=Academic Press Inc.|year=1980|ISBN=0-12-634850-2}}</ref><ref>{{Citation|first=H.|last=Spanier|title=Algebraic Topology|publisher=McGraw-Hill Education|year=1966|ISBN=0-07-059883-5}}</ref>


== Definition ==
==Applications==
Shortly after its discovery, the inerter principle was used under the name of '''J-damper''' in the suspension systems of [[Formula 1]] [[racing car]]s. When tuned to the natural oscillation frequencies of the tires, the inerter reduced the mechanical load on the suspension. [[McLaren Mercedes]] began using a J-damper in early 2005, and [[Renault]] shortly thereafter.<ref>{{cite doi|10.1109/MCAS.2008.931738}}</ref>


A topological space ''X'' endowed with a [[Triangulation (topology)|triangulation]] ''K'' is an ''n''-dimensional pseudomanifold if the following conditions hold:<ref name="BRAS">{{cite journal |last1=Brasselet|first1=J. P.|year=1996 |title=Intersection of Algebraic Cycles |journal= Journal of Mathematical Sciences|publisher=Springer New York|volume= 82|issue= 5|pages=3625 − 3632|url=http://www.springerlink.com/content/ju28j2wqm174hx10}}</ref>
J-dampers were at the center of the [[2007 Formula One espionage controversy]] which arose when Phil Mackereth left McLaren for Renault.


# {{nowrap|1=''X'' = {{!}}''K''{{!}}}} is the [[union (set theory)|union]] of all ''n''-[[simplex|simplices]].
==Construction==
# Every {{nowrap|1=(''n'' – 1)-simplex}} is a [[Euclidean_simplex#Faces|face]] of exactly two ''n''-simplices for ''n > 1''.
A linear inerter can be constructed by meshing a [[flywheel]] with a [[rack gear]]. The pivot of the flywheel forms one terminal of the device, and the rack gear forms the other.
# For every pair of ''n''-simplices σ and σ' in ''K'', there is a [[sequence]] of ''n''-simplices {{nowrap|1=σ = σ<sub>0</sub>, σ<sub>1</sub>, &hellip;, σ<sub>''k''</sub> = σ'}} such that the [[intersection (mathematics)|intersection]] {{nowrap|1=σ<sub>''i''</sub> ∩ σ<sub>''i''+1</sub>}} is an {{nowrap|1=(''n'' − 1)-simplex}} for all ''i''.


=== Implications of the definition ===
A rotational inerter can be constructed by meshing a flywheel with the ring gear of a [[differential (mechanical device)|differential]].  The side gears of the differential form the two terminals.


*Condition 2 means that ''X'' is a '''non-branching''' [[simplicial complex]].<ref name="ANO">{{citeweb|url=http://eom.springer.de/p/p075720.htm|author=D. V. Anosov|title=Pseudo-manifold|accessdate=August 6, 2010}}</ref>
==References==
*Condition 3 means that ''X'' is a '''strongly connected''' simplicial complex.<ref name="ANO"/>
{{Reflist}}
 
== Examples ==
*A [[Pinched Torus|pinched torus]] (see figure) is an example of an [[orientable surface|orientable]], [[compact surface|compact]] 2-dimensional pseudomanifold.<ref name="BRAS"/>
 
* Complex [[algebraic varieties]] (even with singularities) are examples of pseudomanifolds.<ref name="ANO"/>
 
* [[Thom space]]s of [[vector bundle]]s over triangulable [[compact manifold]]s are examples of pseudomanifolds.<ref name="ANO"/>


* Triangulable, [[compact space|compact]], [[connected space|connected]], [[homology manifold]]s over '''Z''' are examples of pseudomanifolds.<ref name="ANO"/>
==External links==
* [http://www-control.eng.cam.ac.uk/~mcs/lecture_j.pdf The Inerter Concept and Its Applications], lecture notes
* [http://www.f1technical.net/features/10586 J-dampers in Formula 1]


== References ==
[[Category:Control theory]]
 
{{Reflist}}


[[Category:Topological spaces]]
{{automotive-tech-stub}}

Revision as of 08:18, 18 August 2014

In the study of mechanical networks in control theory, an inerter is a two-terminal device in which the forces applied at the terminals are equal, opposite, and proportional to relative acceleration between the nodes. The generated strength satisfies the equation

,

for a suitable constant b.

Malcolm C. Smith, a control engineering professor at the University of Cambridge, first introduced inerters in a 2002 paper.[1] However, a similar device was demonstrated in 1998 by Polish engineer Lucianus Lagiewka and used in an automobile bumper application. The principle and design are the same: a kinetic energy absorbing system that can be used as a bumper or vibration absorber.[2][3] The device was in the public knowledge since 1998. Smith extended the analogy between electrical and mechanical networks (the mobility analogy). He observed that the analogy was incomplete, since it was missing a mechanical device playing the same role as an electrical capacitor. It was found that it is possible to construct such a device using gears.

Applications

Shortly after its discovery, the inerter principle was used under the name of J-damper in the suspension systems of Formula 1 racing cars. When tuned to the natural oscillation frequencies of the tires, the inerter reduced the mechanical load on the suspension. McLaren Mercedes began using a J-damper in early 2005, and Renault shortly thereafter.[4]

J-dampers were at the center of the 2007 Formula One espionage controversy which arose when Phil Mackereth left McLaren for Renault.

Construction

A linear inerter can be constructed by meshing a flywheel with a rack gear. The pivot of the flywheel forms one terminal of the device, and the rack gear forms the other.

A rotational inerter can be constructed by meshing a flywheel with the ring gear of a differential. The side gears of the differential form the two terminals.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Template:Automotive-tech-stub

  1. ↑ Template:Cite doi
  2. ↑ Template:Cite web
  3. ↑ Template:Cite web
  4. ↑ Template:Cite doi
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