Moving load

In structural dynamics, a moving load changes the point at which the load is applied over time.^{[citation needed]} Examples include a vehicle that travels across a bridge^{[citation needed]} and a train moving along a track.^{[citation needed]}

Pantograph

Train

Rifle

Examples of a moving load.

Force

Oscillator

Mass

Types of a moving load.

Properties

In computational models, load is usually applied as

a simple massless force,^{[citation needed]}
an oscillator,^{[citation needed]} or
an inertial force (mass and a massless force).^{[citation needed]}

Numerous historical reviews of the moving load problem exist.^[1]^[2] Several publications deal with similar problems.^[3]

The fundamental monograph is devoted to massless loads.^[4] Inertial load in numerical models is described in ^[5]

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.^[6] 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]} The moving load significantly increases displacements.^{[citation needed]} The critical velocity, at which the growth of displacements is the maximum, must be taken into account in engineering projects.^{[citation needed]}

Structures that carry moving loads can have finite dimensions or can be infinite and supported periodically or placed on the elastic foundation.^{[citation needed]}

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]}

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

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]} 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]}

-N{\frac {\partial ^{2}w(x,t)}{\partial x^{2}}}+\rho A{\frac {\partial ^{2}w(x,t)}{\partial t^{2}}}=\delta (x-vt)P-\delta (x-vt)m{\frac {{\mbox{d}}^{2}w(vt,t)}{{\mbox{d}}t^{2}}}\ .

Convergence of the solution for different number of terms.

The last term, because of complexity of computations, is often neglected by engineers.^{[citation needed]} The load influence is reduced to the massless load term.^{[citation needed]} Sometimes the oscillator is placed in the contact point.^{[citation needed]} Such approaches are acceptable only in low range of the travelling load velocity.^{[citation needed]} In higher ranges both the amplitude and the frequency of vibrations differ significantly in the case of both types of a load.^{[citation needed]}

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

massless load	inertial load
Vibrations of a string under a moving massless force (v=0.1c); c is the wave speed. Vibrations of a string under a moving massless force (v=0.5c); c is the wave speed.	Vibrations of a string under a moving inertial force (v=0.1c); c is the wave speed. Vibrations of a string under a moving inertial force (v=0.5c); c is the wave speed.

The discontinuity of the mass trajectory is also well visible in the Timoshenko beam.^{[citation needed]} High shear stiffness emphasizes the phenomenon.^{[citation needed]}

Vibrations of the Timoshenko beam: red lines - beam axes in time, black line - mass trajectory (w₀- static deflection).

The Renaudot approach vs. the Yakushev approach

Renaudot approach

\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}}^{2}w(vt,t)}{{\mbox{d}}t^{2}}}\ .

^{[citation needed]}

Yakushev approach

{\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}}^{2}w(vt,t)}{{\mbox{d}}t^{2}}}\ .

^{[citation needed]}

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.^[7] The analysis will follow the solution of Fryba.^[4] Assuming $ρ$ =0, the equation of motion of a string under a moving mass can be put into the following form^{[citation needed]}

-N{\frac {\partial ^{2}w(x,t)}{\partial x^{2}}}=\delta (x-vt)P-\delta (x-vt)\,m{\frac {{\mbox{d}}^{2}w(vt,t)}{{\mbox{d}}t^{2}}}\ .

We impose simply-supported boundary conditions and zero initial conditions.^{[citation needed]} To solve this equation we use the convolution property.^{[citation needed]} We assume dimensionless displacements of the string $y$ and dimensionless time $τ$ :^{[citation needed]}

Massless string and a moving mass - mass trajectory.

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

where $w$ _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 $α$ is the dimensionless parameters :

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

Parameters $a$ , $b$ and $c$ 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\ .

Massless string and a moving mass - mass trajectory, α=1.

In the case of $α$ =1, the considered problem has a closed solution:^{[citation needed]} $y(\tau )=\left[{\frac {4}{3}}\tau (1-\tau )-{\frac {4}{3}}\tau \left(1+2\tau \ln(1-\tau )+2\ln(1-\tau )\right)\right]\ .$

References

^ Inglis, C.E. (1934). A Mathematical Treatise on Vibrations in Railway Bridges. Cambridge University Press.
^ Schallenkamp, A. (1937). "Schwingungen von Tragern bei bewegten Lasten". Ingenieur-Archiv (in German). 8 (3). Stringer Nature: 182–98. doi:10.1007/BF02085995. S2CID 122387048.
^ A.V. Pesterev; L.A. Bergman; C.A. Tan; T.C. Tsao; B. Yang (2003). "On Asymptotics of the Solution of the Moving Oscillator Problem" (PDF). J. Sound Vib. Vol. 260. pp. 519–36. Archived from the original (PDF) on 2012-10-18. Retrieved 2012-11-09.
^ ^a ^b Fryba, L. (1999). Vibrations of Solids and Structures Under Moving Loads. Thomas Telford House. ISBN 9780727727411.
^ Bajer, C.I.; Dyniewicz, B. (2012). Numerical Analysis of Vibrations of Structures Under Moving Inertial Load. Lecture Notes in Applied and Computational Mechanics. Vol. 65. Springer. doi:10.1007/978-3-642-29548-5. ISBN 978-3-642-29547-8.
^ B. Dyniewicz & C.I. Bajer (2009). "Paradox of the Particle's Trajectory Moving on a String". Arch. Appl. Mech. 79 (3): 213–23. Bibcode:2009AAM....79..213D. doi:10.1007/s00419-008-0222-9. S2CID 56291972.
^ C.E. Smith (1964). "Motion of a stretched string carrying a moving mass particle". J. Appl. Mech. Vol. 31, no. 1. pp. 29–37.

[inglis-1] Inglis, C.E. (1934). A Mathematical Treatise on Vibrations in Railway Bridges. Cambridge University Press.

[schalle-2] Schallenkamp, A. (1937). "Schwingungen von Tragern bei bewegten Lasten". Ingenieur-Archiv (in German). 8 (3). Stringer Nature: 182–98. doi:10.1007/BF02085995. S2CID 122387048.

[bergman-3] A.V. Pesterev; L.A. Bergman; C.A. Tan; T.C. Tsao; B. Yang (2003). "On Asymptotics of the Solution of the Moving Oscillator Problem" (PDF). J. Sound Vib. Vol. 260. pp. 519–36. Archived from the original (PDF) on 2012-10-18. Retrieved 2012-11-09.

[fryba-4] Fryba, L. (1999). Vibrations of Solids and Structures Under Moving Loads. Thomas Telford House. ISBN 9780727727411.

[cb_bd_b-5] Bajer, C.I.; Dyniewicz, B. (2012). Numerical Analysis of Vibrations of Structures Under Moving Inertial Load. Lecture Notes in Applied and Computational Mechanics. Vol. 65. Springer. doi:10.1007/978-3-642-29548-5. ISBN 978-3-642-29547-8.

[6] B. Dyniewicz & C.I. Bajer (2009). "Paradox of the Particle's Trajectory Moving on a String". Arch. Appl. Mech. 79 (3): 213–23. Bibcode:2009AAM....79..213D. doi:10.1007/s00419-008-0222-9. S2CID 56291972.

[smith64-7] C.E. Smith (1964). "Motion of a stretched string carrying a moving mass particle". J. Appl. Mech. Vol. 31, no. 1. pp. 29–37.

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