In structural dynamics, a moving load changes the point at which the load is applied over time. Examples include a vehicle that travels across a bridge and a train moving along a track.
Properties In computational models, load is usually applied as
a simple massless force, an oscillator, or an inertial force (mass and a massless force). Numerous historical reviews of the moving load problem exist. Several publications deal with similar problems. The fundamental monograph is devoted to massless loads. Inertial load in numerical models is described in 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. It is the discontinuity of the mass trajectory near the end of the span (well visible in string at the speed v=0.5c). The moving load significantly increases displacements. The critical velocity, at which the growth of displacements is the maximum, must be taken into account in engineering projects. Structures that carry moving loads can have finite dimensions or can be infinite and supported periodically or placed on the elastic foundation. 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
− N ∂ 2 w ( x , t ) ∂ x 2 + ρ A ∂ 2 w ( x , t ) ∂ t 2 = δ ( x − v t ) P . {\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
w ( x , t ) = 2 P ρ A l ∑ j = 1 ∞ 1 ω ( j ) 2 − ω 2 ( sin ( ω t ) − ω ω ( j ) sin ( ω ( j ) t ) ) sin j π x l , {\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
ω = j π v l , {\displaystyle \omega ={\frac {j\pi v}{l}}\ ,}
and the natural circular frequency of the string
ω ( j ) 2 = j 2 π 2 l 2 N ρ A . {\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. 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:
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