Many researchers have paid considerable attention to the dynamics of mechanical structures due to rain, wind, earthquake, machines and traffic-induced vibrations. These vibrations in mechanical structures are of great importance because of their impact on our life. The Tacoma Narrow suspension bridge is a well-known example of a structural collapse because of wind. The motion of mechanical structures, such as the vibrations of suspension bridges, overhead transmission lines, dynamically loaded helocal springs, conveyor belts and elevator cables, can be expressed as initial-boundary value problems for wave or beam equations. The main objective of this poster is to study boundary reflection and damping properties of waves in semi-infinite strings.
Research focus: understanding of how waves are damped and reflected by these boundaries, and how much energy is dissipated at the boundary.
Reflection Properties and Damping Properties
for Semi-Infinite Wave and Beam Equations
Tugce Akkaya, Wim van Horssen
Department of Mathematical Physics, Delft Institute of Applied Mathematics,
Delft University of Technology, Delft, The Netherlands
Title
Introduction
Reference
T. Akkaya, W.T. van Horssen (2015): Reflection and damping properties for semi-infinite string equations with non-classical boundary conditions. Submitted to Journal of Sound and Vibration.
Copyright © 2014 Tugce Akkaya - T.Akkaya@tudelft.nl
Title
Mathematical Model
Title
Methods
Conclusions
Title
Results
The D’Alembert method is used to construct explicit solutions of the boundary value problem for the one-dimensional wave equation on the semi-infinite domain. The general solution of the 1D-wave equation is
where
Then,
•
It is found that the solution is bounded by using an energy integral.•
Figure 2(a) shows some reflected waves for different cases.•
It can be seen that at the incident wave hits the boundary, and from that time the energy is dissipated. However, the energy is conserved when the damping coefficient is equal to zero.•
Before starting with more complicated situations, such as initial-boundary value problems for fourth order linear partial differential equations, it is good to examine a one-dimensional string example in some detail first to have some of the procedures and concepts in their simplest form.Euler Bernoulli Beam Equation:
The perfectly flexible string is attached to a mass , spring and dashpot at . is the vertical displacement of the string, where is the position
along the string, and is the time (Figure 1). Let us assume that gravity and other external forces can be neglected. The initial boundary value problem is given by,
where the wave speed , is the tension and is the mass density of the string. Here, and represent the initial displacement and initial velocity of the string, respectively. In order to put the equation in a non-dimensional form the following dimensionless quantities are used:
where and time are some characteristic quantities for the length and the time, and we obtain
u x k α m
