en>Cnilep: disambiguate

2013-10-17T03:57:43Z

disambiguate

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{{multiple image
|direction = vertical
| width = 180
| footer = <center>'''Examples of a moving load.'''</center>
| footer_background=#fffcbf
| background color=#fffcbf
| align = right

| image1 = Cb_pant.png
| alt1 = Pantograph
| caption1 = <center>Pantograph</center>

| image2 = Transrapid.jpg
| alt2 = Train
| caption2 = <center>Train</center>

| image3 = Winchester 1897.jpg
| alt3 = Rifle
| caption3 = <center>Rifle</center>
}}

{{multiple image
| footer = <center>'''Types of a moving load.'''</center>
| footer_background=#fffcbf
| background color=#fffcbf
| align = right

| image1 = force_as_a_load.png
| width1 = 70
| alt1 = Force
| caption1 = <center>Force</center>

| image2 = oscillator_a_s_a_load.png
| width2 = 102
| alt2 = Oscillator
| caption2 = <center>Oscillator</center>

| image3 = mass_as_a_load.png
| width3 = 104
| alt3 = Mass
| caption3 = <center>Mass</center>
}}

In structural dynamics this is the load that changes in time the place to which is applied. Examples: vehicles that pass bridges, trains on the track, guideways, etc. In computational models the load is usually applied as:
* a simple massless force,
* an oscillator,
* an inertial force (mass and a massless force).

Numerous historical reviews concerning the moving load problem exist (for example,<ref name="inglis">[[Charles Inglis (engineer)|C.E. Inglis]]. A mathematical treatise on vibrations in railway bridges. Cambridge University Press, 1934.</ref><ref name="schalle">A. Schallenkamp. Schwingungen von Tragern bei bewegten Lasten. Ingenieur-Archiv, 8, 182-198, 1937.</ref>).
Several publications deal with similar problems.<ref name="bergman">{{cite article|author=A.V. Pesterev, L.A. Bergman, C.A. Tan, T.C. Tsao, B. Yang|title=On asymptotics of the solution of the moving oscillator problem|journal=J. Sound and Vibr.|volume=260|pages=519-536|year=2003|url=http://www.eng.wayne.edu/user_files/258/09_EquivalenceJSV_JournalArticle.pdf}}</ref>

The fundamental monograph is devoted to massless loads.<ref name="fryba">{{cite book|author=L. Fryba|title=Vibrations of solids and structures under moving loads.|publisher=Thomas Telford House|year=1999|url=http://books.google.pl/books/about/Vibration_Of_Solids_And_Structures_Under.html?id=3RP4T4Oc0LUC&redir_esc=y}}</ref> Inertial load in numerical models are described in <ref name ="cb_bd_b">{{cite book|author=C.I. Bajer and B. Dyniewicz|title=Numerical analysis of vibrations of structures under moving inertial load|publisher=Springer|year=2012|url=http://link.springer.com/book/10.1007/978-3-642-29548-5/page/1}}</ref>
Unexpected property of differential equations that govern the motion of the mass particle travelling on the string, [[Timoshenko beam theory|Timoshenko]] beam, and [[Mindlin–Reissner plate theory|Mindlin]] plate is described in.<ref>{{cite article|author=B. Dyniewicz and C.I. Bajer|title=Paradox of the particle's trajectory moving on a string|journal=Arch. Appl. Mech.|volume=79|number=3| pages=213-223|year=2009|url=http://link.springer.com/article/10.1007%2Fs00419-008-0222-9?LI=true}}</ref> It is the discontinuity of the mass trajectory near the end of the span (well visible in string at the speed ''v''=0.5''c''). The moving load significantly increases displacements. The critical velocity, at which the growth of displacements is the maximum, must be taken into account in engineering projects.
Structures that carry moving loads can have finite dimensions or can be infinite and supported periodically or placed on the elastic foundation.

Let us consider simply supported string of the length ''l'', cross-sectional
area ''A'', mass density ρ, tensile force ''N'', subjected to a constant force ''P''
moving with constant velocity ''v''. The motion equation of the string under
the moving force has a form

: <math>
-N\frac{\partial^2w(x,t)}{\partial x^2}+\rho
A\frac{\partial^2w(x,t)}{\partial t^2}=\delta(x-vt)P\ .
</math>

Displacements of any point of the simply supported string is given by
the sinus series
: <math>
w(x,t) = \frac{2P}{\rho
Al}\sum_{j=1}^{\infty}\frac{1}{\omega_{(j)}^2-\omega^2}\left(\sin(\omega
t)-\frac{\omega}{\omega_{(j)}}\sin(\omega_{(j)}t)\right)\ ,
</math>
where
: <math>
\omega=\frac{j\pi v}{l}\ ,
</math>
and the natural circular frequency of the string
: <math>
\omega_{(j)}^2=\frac{j^2\pi^2}{l^2}\frac{N}{\rho A}\ .
</math>
In the case of inertial moving load the analytical solutions are
unknown. The equation of motion is increased by the term related to the
inertia of the moving load. A concentrated mass ''m'' accompanied by a point
force ''P'':
: <math>
-N\frac{\partial^2w(x,t)}{\partial x^2}+\rho
A\frac{\partial^2w(x,t)}{\partial
t^2}=\delta(x-vt)P-\delta(x-vt)m\frac{\mbox{d}^2w(vt,t)}{\mbox{d}t^2}\ .
</math>

[[File:Stru mody kol.png|thumb|240px|Convergence of the solution for different number of terms.]]
The last term, because of complexity of computations, is often neglected by engineers. The load influence is reduced to the massless load term. Sometimes the oscillator is placed in the contact point. Such approaches are acceptable only in low range of the travelling load velocity. In higher ranges both the amplitude and the frequency of vibrations differ significantly in the case of both types of a load.

The differential equation can be solved in a semi-analytical way only for simple problems. The series determining the solution converges well and 2-3 terms are sufficient in practice. More complex problems can be solved by the [[finite element method]] or [[space-time finite element method]].

{| class="wikitable"
!massless load
!inertial load
|-
|width="50%"|
[[File:Wiki01f50.gif|thumb|321px|Vibrations of a string under a moving massless force (''v''=0.1''c''); ''c'' is the wave speed.]]
[[File:Wiki05f50.gif|thumb|321px|Vibrations of a string under a moving massless force (''v''=0.5''c''); ''c'' is the wave speed.]]
|width="50%"|
[[File:Wiki01m50.gif|thumb|321px|Vibrations of a string under a moving inertial force (''v''=0.1''c''); ''c'' is the wave speed.]]
[[File:Wiki05m50.gif|thumb|321px|Vibrations of a string under a moving inertial force (''v''=0.5''c''); ''c'' is the wave speed.]]
|}

The discontinuity of the mass trajectory is also well visible in the Timoshenko beam. High shear stiffness emphasizes the phenomenon.
[[File:Timo04n.png|thumb|320px|Vibrations of the Timoshenko beam: red lines - beam axes in time, black line - mass trajectory (w0- static deflection).]]

<big>'''The Renaudot approach vs. the Yakushev approach'''</big> 
The Renaudot approach
: <math>
\delta(x-vt)\frac{\mbox{d}}{\mbox{d}t}\left[m\frac{\mbox{d}w(vt,t)}{\mbox{d}t}\right]=\delta(x-vt)m\frac{\mbox{d}^2w(vt,t)}{\mbox{d}t^2}\ .
</math>
The Yakushev approach
: <math>
\frac{\mbox{d}}{\mbox{d}t}\left[\delta(x-vt)m\frac{\mbox{d}w(vt,t)}{\mbox{d}t}\right]=-\delta^\prime(x-vt)mv\frac{\mbox{d}w(vt,t)}{\mbox{d}t}+\delta(x-vt)m\frac{\mbox{d}^2w(vt,t)}{\mbox{d}t^2}\ .
</math>

<big>'''Massless string under moving inertial load'''</big> 
Let us consider a massless string, which is a particular case of moving
inertial load problem. The first solve the problem Smith
.<ref name="smith64">{{cite article|author=C.E. Smith|title=Motion of a stretched string carrying a moving mass particle|journal=J. Appl. Mech.|year=1964|volume=31|number=1|pages=29-37}}</ref>
The analysis will follow the solution of Fryba.<ref name="fryba" /> Assuming
{{math|ρ}}=0, the equation of motion of a string under a moving mass can
be put into the following form
: <math>
-N\frac{\partial^2w(x,t)}{\partial x^2}=\delta(x-vt)P-\delta(x-vt)\,m\frac{\mbox{d}^2w(vt,t)}{\mbox{d}t^2}\ .
</math>
We impose simply-supported boundary conditions and zero initial
conditions. To solve this equation we use the convolution property. We
assume dimensionless displacements of the string {{math|y}} and
dimensionless time {{math|τ}} :

[[File:Wiki rozero kol.png|thumb|240px|Massless string and a moving mass - mass trajectory.]]

: <math>
y(\tau)=\frac{w(vt,t)}{w_{st}}\ ,\ \ \ \ \tau\ =\ \frac{vt}{l}\ ,
</math>
where {{math|w}}st is the static deflection in the middle of
the string.
The solution is given by a sum
: <math>
y(\tau)=\frac{4\,\alpha}{\alpha\,-\,1}\,\tau\,(\tau-1)\,\sum_{k=1}^\infty\,\prod_{i=1}^k\frac{(a+i-1)(b+i-1)}{c+i-1}\;\frac{\tau^k}{k!}\ ,
</math>
where {{math|α}} is the dimensionless parameters :
: <math>
\alpha=\frac{Nl}{2mv2}\,>\,0\ \ \ \wedge\ \ \ \alpha\,\neq\,1\ .
</math>
Parameters {{math|a}}, {{math|b}} and {{math|c}} are given below
: <math>
a_{1,2}=\frac{3\,\pm\,\sqrt{1+8\alpha}}{2}\ ,\ \ \ \ \ b_{1,2}=\frac{3\,\mp\,\sqrt{1+8\alpha}}{2}\ ,\ \ \ \ \ c=2\ .
</math>

[[File:Wiki alfa1 kol.png|thumb|240px|Massless string and a moving mass - mass trajectory, α=1.]]

In the case of {{math|α}}=1 the considered problem has a closed solution
: <math>
y(\tau )=\left[\frac{4}{3}\tau (1-\tau) -\frac{4}{3}\tau \left( 1+2
\tau\ln (1-\tau )+2\ln (1-\tau )\right)\right]\ .
</math>

'''References'''
<references />



[[Category:Mechanical vibrations]]

Logical consequence - Revision history

en>Cnilep: disambiguate