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Load that changes in time
Examples 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[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]
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]
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]
- [citation needed]
Yakushev approach[edit]
- [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]
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]
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
In the case of α=1, the considered problem has a closed solution:[citation needed]