Acoustic Noise Suppression: Gradient Self-Help?

Recently, Tomasi and Ernst introduced a simple string model for understanding the vibrations of MRI gradient coils. [1] We build upon this simple foundation to provide a window into a particular method for active noise cancellation, where the pulse sequences that caused the problem may be designed to solve the problem. Previous work has focused on active and passive shielding to control eddy currents [2, 3] and on softening the gradient pulses [4]. More et al. [5] have noted that, because of the predictable harmonic behavior, the MR environment is ideal for general active noise control methods, such as earlier suggestions involving external microphone-controlled mounted plates [6] and active noise control focused on measuring and generating “anti-sound” [7]. Here, we consider the alternative possibility of augmenting the gradient pulse sequence itself to provide force/anti-force cancellations. We model the steady-state damped vibrations induced by the periodic application of one force followed by a second force. Motivated by this result, we investigate simple gradient-pulse steady-state experiments to search for sound suppression. Theory For a string of length L fixed at each end with tension T and mass density ρ, one can write the following expression for motion transverse to the length of the string 2 2 2 y y x υ ∂ = ∂ && (1.1) where T υ ρ = . Solutions to this equation are of the form y = Asin(k x)sin(ω t) . We can straightforwardly show that for a particular string disturbance at an initial time, a follow-up disturbance that is either a half-period later or otherwise 180 out of phase, will completely cancel and quiet the string down for the mode of interest. For illustrative purposes consider the following steady-state problem, a one-dimensional driven damped simple harmonic oscillator can be used to model vibrational motion. The closed form solutions for such a system are easily found, and when the force is considered to be instantaneous and the time constant due to the damping small compared with the natural period of the spring resonance (that is to say, we are under-damped) the solution has the following form 2 ( ) sin b t m k b x t e t m m − ⎡ ⎤