A Theoretical Study on the Generation and Propagation of Traveling Waves in Strings
The goal of this work is to understand the generation and propagation of one-dimensional steady state traveling waves in a finite medium with a two-force excitation methodology. The solution to the second order partial differential equation describing the
- PDF / 690,451 Bytes
- 6 Pages / 595.516 x 790.987 pts Page_size
- 87 Downloads / 203 Views
A Theoretical Study on the Generation and Propagation of Traveling Waves in Strings Isil Anakok, V. V. N. Sriram Malladi, and Pablo A. Tarazaga
Abstract The goal of this work is to understand the generation and propagation of one-dimensional steady state traveling waves in a finite medium with a two-force excitation methodology. The solution to the second order partial differential equation describing the equation of motion for a string is theoretically solved considering a fixed-fixed boundary condition. The parameters that affect the generation and propagation of waves should be well understood to control and manipulate the desired system’s response. The string equation is solved by rearranging it based on linear wave components and phase difference components needed to generate steady-state traveling waves in a string. Two excitation forces are applied to a string near the boundaries to understand the generation and propagation of traveling waves at various frequencies. Determining the quality of the traveling waves and understanding the parameters on the wave propagation of a string can lead to further understand and leverage various engineering disciplines such as mechanical actuation mechanisms, propulsion of flagella, and the basilar membrane in the ear’s cochlea. Keywords Traveling waves · Vibration · Cost function · Strings · Two force excitations
32.1 Introduction In previous works, Gabai et al. studied how multiple forces can be applied to one-dimensional structures as to develop and characterize traveling waves [1, 2]. Malladi et al. studied non-reflective waves in beams analytically and experimentally and in such studies a cost function (CF) is introduced to understand the quality of the traveling waves generated in a beam with free-free boundary condition [3–7]. The theoretical study of generating traveling waves for one-dimensional beams and higher-dimensional structures were further developed to great detail in [8]. Earlier work by the authors studied applied a two-force excitation and a cost function approach to generate and propagate steady-state travelling, standing, and hybrid waves in a string under fixed-fixed boundary conditions in [9]. The work was later extended by studying a string at different boundary conditions and adapting their cost functions to generate traveling waves at various frequencies [10]. Furthermore, the analytical study of a fixed-fixed string was validated by an experiment in [10]. This work analyzes and widens the simplified theory of generating traveling waves in strings. The solution to the second order partial differential equation describing the motion for a string is studied analytically and the direction of generation of traveling waves is discussed based on the effect of phase difference between two applied forces.
32.2 Analysis of the One-Dimensional Equation of Motion for Strings The homogeneous solution of the second order partial differential equation describing a string can be found in [9]. A fixedfixed string of length L excited by two forces (F1 = cos (ω
Data Loading...
