“…The longitudinal and bending forces on the particle of the inner wall of the flow tube can be defined as follows 52 : where is the longitudinal force, is the bending force, is the excitation frequency, is the mass of the particle, is the amplitude of excitation force, is the tilt angle of the excitation force, is the phase of the longitudinal vibration, is the phase of the bending vibration, is the longitudinal vibration mode at the horizontal coordinate point , is the bending vibration mode at the horizontal coordinate point , is the action point of the excitation force in the horizontal coordinate, is the ratio of longitudinal vibration excitation frequency to natural frequency, is the ratio of longitudinal vibration excitation frequency to natural frequency bending vibration, is the modal stiffness of longitudinal vibration, is the relative damping coefficient of longitudinal vibration, and is the relative damping coefficient of bending vibration.…”