zenduck.me: Moving load Wikipedia

Load that changes in time

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[edit]

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]}

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

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

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

where

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

and the natural circular frequency of the string

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

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]}

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

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

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

The Renaudot approach vs. the Yakushev approach[edit]

Renaudot approach[edit]

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

Yakushev approach[edit]

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

Massless string under moving inertial load[edit]

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]}

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

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]}

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

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

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

where α is the dimensionless parameters :

$\alpha =\frac{Nl}{2mv2}>0\wedge \alpha \ne 1.$

Parameters a, b and c are given below

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

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

References[edit]