Multi-harmonic forced vibration and resonance of simple beams to moving vehicles

Abstract

This study modeled the moving-vehicle-induced forcing excitation on a single-span prismatic bridge as a multiple frequency-multiplication harmonic load on the modal coordinates of a linear elastic simple Euler–Bernoulli beam, and investigated the forced modal oscillation and resonance behavior of this type of dynamic system. The forced modal responses consist of multiple frequency-multiplication steady-state harmonics and one damped mono-frequency complementary harmonic. The analysis revealed that a moving load induces high-harmonic forced resonance amplification when the moving speed is low. To verify the occurrence of high-harmonic forced resonance, numerical tests were conducted on single-span simple beams based on structural modeling using the finite element method (FEM) and a moving sprung-mass oscillator vehicle model. The forced resonance amplification characteristics of the fundamental mode for beam response estimation are presented with consideration to different end restraint conditions. The results reveal that the high-harmonic forced resonance may be significant for the investigated beams subjected to vehicle loads moving at specific low speeds. For the investigated single-span simple beams, the moving vehicle carriage heaving oscillation modulates the beam modal frequency, but does not induce notable variation of the modal oscillation harmonic structure for the cases that vehicle of small mass moves in low speed.

Multi-beam Line-finding Problem Based on Geometric Model

“…Establish the geometric model of the measuring ship, the path of the acoustic signal emitted by the transducer of the measuring ship [8], the seafloor slope and the plane perpendicular to the direction of the measurement line, first calculate the coverage width W, then calculate the depth of seawater at the distance from the center D, and finally calculate the overlap rate between adjacent strips [9,10].…”

“…Establish the geometrical model of the measuring ship at point A' and point A, the path of the acoustic signal emitted by the transducer of the measuring ship, the seabed slope and the plane perpendicular to the direction of the measuring line, as in Fig. 2, make B'N perpendicular to the AE at point N over point B', and extend A'E' to intersect B'N at point M. In ∆B'E'M, the𝐵 𝑀 𝑊′ 𝑐𝑜𝑠𝛼 𝐸 𝑀 𝑊′ 𝑠𝑖𝑛𝛼 (8) Assuming A'A=d,so MN=d, from similar triangles ΔB'EN and ΔB'EM we get (9) i.e. (10) solve for…”