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
Pantograph
Train
Train
Rifle
Rifle
Examples of a moving load.
Force
Force
Oscillator
Oscillator
Mass
Mass
Types of a moving load.

Properties

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In computational models, load is usually applied as

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]

 

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

 

where

 

and the natural circular frequency of the string

 

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]

 
 
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

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Renaudot approach

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 [citation needed]

Yakushev approach

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 [citation needed]

Massless string under moving inertial load

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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]

 

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.
 

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

 

where α is the dimensionless parameters :

 

Parameters a, b and c are given below

 
 
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]  

References

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  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. ^ a b 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.