Moving load

Pantograph

Train

Rifle

Examples of a moving load.

Force

Oscillator

Mass

Types of a 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 ]}

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 ]}

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

Displacements of any point of the simply supported string is given by the sinus series^{[ citation needed ]}

${\displaystyle w(x,t)={\frac {2P}{\rho Al}}\sum _{j=1}^{\infty }{\frac {1}{\omega _{(j)}^{2}-\omega ^{2}}}\left(\sin(\omega t)-{\frac {\omega }{\omega _{(j)}}}\sin(\omega _{(j)}t)\right)\sin {\frac {j\pi x}{l}}\ ,}$

where

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

and the natural circular frequency of the string

${\displaystyle \omega _{(j)}^{2}={\frac {j^{2}\pi ^{2}}{l^{2}}}{\frac {N}{\rho A}}\ .}$

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 ]}

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

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

${\displaystyle \delta (x-vt){\frac {\mbox{d}}{{\mbox{d}}t}}\left[m{\frac {{\mbox{d}}w(vt,t)}{{\mbox{d}}t}}\right]=\delta (x-vt)m{\frac {{\mbox{d}}^{2}w(vt,t)}{{\mbox{d}}t^{2}}}\ .}$^{[ citation needed ]}

Yakushev approach

${\displaystyle {\frac {\mbox{d}}{{\mbox{d}}t}}\left[\delta (x-vt)m{\frac {{\mbox{d}}w(vt,t)}{{\mbox{d}}t}}\right]=-\delta ^{\prime }(x-vt)mv{\frac {{\mbox{d}}w(vt,t)}{{\mbox{d}}t}}+\delta (x-vt)m{\frac {{\mbox{d}}^{2}w(vt,t)}{{\mbox{d}}t^{2}}}\ .}$^{[ 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 ]}

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

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.

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

where w_st is the static deflection in the middle of the string. The solution is given by a sum

${\displaystyle y(\tau )={\frac {4\,\alpha }{\alpha \,-\,1}}\,\tau \,(\tau -1)\,\sum _{k=1}^{\infty }\,\prod _{i=1}^{k}{\frac {(a+i-1)(b+i-1)}{c+i-1}}\;{\frac {\tau ^{k}}{k!}}\ ,}$

where α is the dimensionless parameters :

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

Parameters a, b and c are given below

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

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 ]}${\displaystyle y(\tau )=\left[{\frac {4}{3}}\tau (1-\tau )-{\frac {4}{3}}\tau \left(1+2\tau \ln(1-\tau )+2\ln(1-\tau )\right)\right]\ .}$

References

1. Inglis, C.E. (1934). A Mathematical Treatise on Vibrations in Railway Bridges. Cambridge University Press.
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.
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.
4. Fryba, L. (1999). Vibrations of Solids and Structures Under Moving Loads. Thomas Telford House. ISBN   9780727727411.
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.
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.