moving load

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.{{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}}{{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 }}

Several publications deal with similar problems.{{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}}

The fundamental monograph is devoted to massless loads.{{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}} Inertial load in numerical models is described in {{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 }}

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.{{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 }} It is the discontinuity of the mass trajectory near the end of the span (well visible in string at the speed v=0.5c).{{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}}

: $$

-N\frac{\partial^2w(x,t)}{\partial x^2}+\rho

A\frac{\partial^2w(x,t)}{\partial t^2}=\delta(x-vt)P\ .

Displacements of any point of the simply supported string is given by the sinus series{{Citation needed|date=June 2021}}

: $$

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}\ ,

where

: $$

\omega=\frac{j\pi v}{l}\ ,

and the natural circular frequency of the string

: $$

\omega_{(j)}^2=\frac{j^2\pi^2}{l^2}\frac{N}{\rho A}\ .

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}}

: $$

-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}\ .

File:Stru mody kol.png

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}}

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}}

File:Timo04n.png

: $$

\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}\ .

{{Citation needed|date=June 2021}}

: $$

\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}\ .

{{Citation needed|date=June 2021}}

Consider a massless string, which is a particular case of moving inertial load problem. The first to solve the problem was Smith.{{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}}

The analysis will follow the solution of 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}}

: $$

-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}\ .

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}}

File:Wiki rozero kol.png

: $$

y(\tau)=\frac{w(vt,t)}{w_{st}}\ ,\ \ \ \ \tau\ =\ \frac{vt}{l}\ ,

where {{math|w}}_st is the static deflection in the middle of

the string.

The solution is given by a sum

: $$

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!}\ ,

where {{math|α}} is the dimensionless parameters :

: $$

\alpha=\frac{Nl}{2mv2}\,>\,0\ \ \ \wedge\ \ \ \alpha\,\neq\,1\ .

Parameters {{math|a}}, {{math|b}} and {{math|c}} are given below

: $$

a_{1,2}=\frac{3\,\pm\,\sqrt{1+8\alpha}}{2}\ ,\ \ \ \ \ b_{1,2}=\frac{3\,\mp\,\sqrt{1+8\alpha}}{2}\ ,\ \ \ \ \ c=2\ .

File:Wiki alfa1 kol.png

In the case of {{math|α}}=1, the considered problem has a closed solution:{{Citation needed|date=June 2021}} $$

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]\ .

Category:Mechanical vibrations