Vibrational response of a MRI gradient coil cylinder to time-harmonic Lorentz-force excitations: An exact linear elastodynamic model for shielded longitudinal gradient coils
Applied Mathematical Modelling
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© 2019 Elsevier Inc. The reduction of gradient coil (GC) vibration continues to be a challenging problem in the optimization of magnetic resonance imaging systems. A key deficiency for passive reduction strategies is that, under realistic thick-shell conditions, there are no existing mathematical models that can provide reliable theoretical predictions about the parametrical behaviors of the vibration response. In this paper, we introduce a simple linear elastodynamic model of a shielded longitudinal GC that can serve as a baseline theoretical model for studying the steady-state linear vibration response of an undamped GC cylinder under the condition that only the Z-coil windings are excited by Lorentz forces. The exact three-dimensional theory of linear elasticity is used to formulate the model, and hence, there are no built-in geometrical constraints. An exact closed-form solution for the steady-state displacement field of the GC cylinder is given, and the solution is then used to numerically investigate the frequency response of a typical whole-body GC cylinder. A core prediction of the model is that the frequency response is essentially governed by the Fourier decomposition of a dimensionless “profile function” that specifies how the current density varies along the GC cylinder axis. An interesting corollary is that generally the same set of resonant modes are excited independent of how the currents are spatially distributed. The model also predicts that the widths of all resonances are substantially decreased when shielding currents are present. Numerical results obtained using generic profile functions are found to be remarkably consistent with available in-situ experimental data and existing numerical models. The model is therefore well-suited for parametric studies of the steady-state linear vibration response assuming that Z-coil windings are excited exclusively.