{{Short description|Load that changes in time}} {{More citations needed | date = June 2021}} {{multiple image |direction = vertical | width = 180 | footer = '''Examples of a moving load.''' | footer_align = center | footer_background=#fffcbf | background color=#fffcbf | align = right | caption_align = center
| image1 = Cb_pant.png | alt1 = Pantograph | caption1 = Pantograph
| image2 = Transrapid 08.jpg | alt2 = Train | caption2 = Train
| image3 = Winchester 1897.jpg | alt3 = Rifle | caption3 = Rifle }}
{{multiple image | footer = '''Types of a moving load.''' | footer_align = center | footer_background=#fffcbf | background color=#fffcbf | align = right | caption_align = center
| image1 = force_as_a_load.png | width1 = 70 | alt1 = Force | caption1 = Force
| image2 = oscillator_a_s_a_load.png | width2 = 102 | alt2 = Oscillator | caption2 = Oscillator
| image3 = mass_as_a_load.png | width3 = 104 | alt3 = Mass | caption3 = Mass }}
In structural dynamics, a '''moving load''' changes the point at which the load is applied over time.{{Citation needed|date=June 2021}} Examples include a vehicle that travels across a bridge{{Citation needed|date=June 2021}} and a train moving along a track.{{Citation needed|date=June 2021}}
== Properties ==
In computational models, load is usually applied as * a simple massless force,{{Citation needed|date=June 2021}} * an oscillator,{{Citation needed|date=June 2021}} or * an inertial force (mass and a massless force).{{Citation needed|date=June 2021}}
Numerous historical reviews of the moving load problem exist.<ref name="inglis">{{cite book | first1 = C.E. | last1 = Inglis | author-link1 = Charles Inglis (engineer) | title = A Mathematical Treatise on Vibrations in Railway Bridges | publisher = Cambridge University Press | date = 1934}}</ref><ref name="schalle">{{cite journal | first1 = A. | last1 = Schallenkamp | title = Schwingungen von Tragern bei bewegten Lasten | journal = Ingenieur-Archiv | publisher = Stringer Nature | language = German | volume = 8 | pages = 182–98 | year = 1937| issue = 3 | doi = 10.1007/BF02085995 | s2cid = 122387048 }}</ref> Several publications deal with similar problems.<ref name="bergman">{{cite news|author1=A.V. Pesterev|author2=L.A. Bergman|author3=C.A. Tan|author4=T.C. Tsao|author5=B. Yang|title=On Asymptotics of the Solution of the Moving Oscillator Problem|journal=J. Sound Vib.|volume=260|pages=519–36|year=2003|url=http://www.eng.wayne.edu/user_files/258/09_EquivalenceJSV_JournalArticle.pdf|access-date=2012-11-09|archive-url=https://web.archive.org/web/20121018151015/http://www.eng.wayne.edu/user_files/258/09_EquivalenceJSV_JournalArticle.pdf|archive-date=2012-10-18|url-status=dead}}</ref>
The fundamental monograph is devoted to massless loads.<ref name="fryba">{{cite book|first1 = L. | last1 = Fryba | title = Vibrations of Solids and Structures Under Moving Loads | publisher = Thomas Telford House | date = 1999 | url=https://books.google.com/books?id=3RP4T4Oc0LUC|isbn=9780727727411}}</ref> Inertial load in numerical models is described in <ref name ="cb_bd_b">{{cite book | first1 = C.I. | last1 = Bajer | first2 = B. | last2 = Dyniewicz | title = Numerical Analysis of Vibrations of Structures Under Moving Inertial Load | volume = 65 | publisher = Springer | date = 2012| doi = 10.1007/978-3-642-29548-5 | series = Lecture Notes in Applied and Computational Mechanics | isbn=978-3-642-29547-8 }}</ref>
Unexpected property of differential equations that govern the motion of the mass particle travelling on the string, Timoshenko beam, and Mindlin plate is described in.<ref>{{cite journal|author1=B. Dyniewicz |author2=C.I. Bajer |name-list-style=amp | title = Paradox of the Particle's Trajectory Moving on a String | journal = Arch. Appl. Mech. | volume = 79 | number = 3 | pages = 213–23 | year = 2009 | doi = 10.1007/s00419-008-0222-9 |bibcode=2009AAM....79..213D |s2cid=56291972 }}</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'').{{Citation needed|date=June 2021}} The moving load significantly increases displacements.{{Citation needed|date=June 2021}} The critical velocity, at which the growth of displacements is the maximum, must be taken into account in engineering projects.{{Citation needed|date=June 2021}}
Structures that carry moving loads can have finite dimensions or can be infinite and supported periodically or placed on the elastic foundation.{{Citation needed|date=June 2021}}
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{{Citation needed|date=June 2021}}
: <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{{Citation needed|date=June 2021}} : <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)\sin\frac{j\pi x}{l}\ , </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.{{Citation needed|date=June 2021}} 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'':{{Citation needed|date=June 2021}} : <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>
thumb|240px|Convergence of the solution for different number of terms. The last term, because of complexity of computations, is often neglected by engineers.{{Citation needed|date=June 2021}} The load influence is reduced to the massless load term.{{Citation needed|date=June 2021}} Sometimes the oscillator is placed in the contact point.{{Citation needed|date=June 2021}} Such approaches are acceptable only in low range of the travelling load velocity.{{Citation needed|date=June 2021}} In higher ranges both the amplitude and the frequency of vibrations differ significantly in the case of both types of a load.{{Citation needed|date=June 2021}}
The differential equation can be solved in a semi-analytical way only for simple problems.{{Citation needed|date=June 2021}} The series determining the solution converges well and 2-3 terms are sufficient in practice.{{Citation needed|date=June 2021}} More complex problems can be solved by the finite element method{{Citation needed|date=June 2021}} or space-time finite element method.{{Citation needed|date=June 2021}}
{| class="wikitable" !massless load !inertial load |- |width="50%"| thumb|321px|Vibrations of a string under a moving massless force (''v''=0.1''c''); ''c'' is the wave speed. thumb|321px|Vibrations of a string under a moving massless force (''v''=0.5''c''); ''c'' is the wave speed. |width="50%"| thumb|321px|Vibrations of a string under a moving inertial force (''v''=0.1''c''); ''c'' is the wave speed. 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.{{Citation needed|date=June 2021}} High shear stiffness emphasizes the phenomenon.{{Citation needed|date=June 2021}} thumb|320px|Vibrations of the Timoshenko beam: red lines - beam axes in time, black line - mass trajectory (w<sub>0</sub>- static deflection).
==The Renaudot approach vs. the Yakushev approach== ===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>{{Citation needed|date=June 2021}}
===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>{{Citation needed|date=June 2021}}
==Massless string under moving inertial load== Consider a massless string, which is a particular case of moving inertial load problem. The first to solve the problem was Smith.<ref name="smith64">{{cite news|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{{Citation needed|date=June 2021}} : <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.{{Citation needed|date=June 2021}} To solve this equation we use the convolution property.{{Citation needed|date=June 2021}} We assume dimensionless displacements of the string {{math|y}} and dimensionless time {{math|τ}}:{{Citation needed|date=June 2021}}
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}}<sub>st</sub> 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>
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:{{Citation needed|date=June 2021}} <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