![]() ![]() The significant parameters in this formulation are L, EI, and ρ A. Relationship between angular frequency and wave number. Applying the method of separation of variables, a general solution to the equation in figure 8 with a pinned-pinned end condition can be derived and takes the form of the equation in figure 4, with ω n and k n being interrelated as follows:įigure 10. Note that the equation in figure 8 is not of the wave equation form and that a does not have the dimension of velocity. Vibration parameter for a classical beam. Where a is defined as the vibration parameter for classical beam, which can be solved as follows:įigure 9. ![]() EOM for a classical beam, rewritten with vibration parameter. The equation in figure 7 may be rewritten as follows:įigure 8. #TRANSVERSE VIBRATION OF STRINGS THEORY FREE#The governing equation for the free transverse vibration of a Bernoulli-Euler beam is given by the following: (4)įigure 7. Also note that from the equation in figure 5, ω n is proportional to the mode number, n.įrom the equations in figure 3 and figure 5, the cable tension H can be determined from the fundamental natural frequency, f, as follows:į, in Hz is related to the angular frequency ω such that f = ω/2π. ![]() Note that the only significant parameters in figure 1 through figure 5 are L, H, and ρ A. The equation in figure 1 is a linearized EOM in which nonlinearities arising from finite sag are ignored. The natural frequencies, ω n, are the eigenvalues representing the discrete frequencies at which the system is capable of undergoing harmonic motion. The equation in figure 4 indicates that the motion of the string is represented by a superposition of standing waves with mode shapes of sin k nx and time-varying amplitudes of C ncos(ω n t - α n). The angular frequencies and wave numbers are not independent of each other but are interrelated as follows:įigure 5. Α n = Phase angle of time-dependent part of transverse in-plane displacement due to vibration. K n = Wave number of the nth mode of vibration.Ĭ n = Amplitude of in-plane displacement due to vibration. Ω n = Natural angular frequency of the nth mode of vibration. General solution of EOM of a taut string. Where c is the phase velocity, which is defined as follows:Īpplying the method of separation of variables, a general solution to the equation from figure 1 with a fixed-fixed end condition may readily be derived as follows:įigure 4. The equation in figure 1 may be rewritten as follows:įigure 2. Y = Transverse in-plane displacement due to vibration.Ī = Cross-sectional area of the string, beam, or cable. H = Axial tension force in a string or cable. Equation of motion (EOM) for a taut string. The governing equation for the free transverse vibration of a taut string is as follows: (4)įigure 1. Mitigation of Wind-Induced Vibration of Stay Cables: Numerical Simulations and Evaluations ![]()
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