Explicit solution techniques in structural dynamics are hampered by repeatedly solving linear systems with the mass matrix at each time step. For this reason, practitioners tend to substitute the consistent mass matrix with an ad hoc (diagonal) approximation, a practice widely known as mass lumping. In immersogeometric analysis, mass lumping techniques such as the row-sum [1] ensure that the critical time step remains bounded, independently of how small trimmed elements are, provided the discretization is sufficiently smooth [2]. This apparently beneficial effect of smoothness has drawn much enthusiasm and admiration but has unfortunately overshadowed another much more subtle effect: while the largest discrete eigenvalues constraining the critical time step may remain bounded, the smallest ones instead converge to zero as the trimmed elements get smaller and the polynomial order increases [3, 4]. Those inaccurate eigenvalues bring in spurious modes in the low-frequency spectrum and their impact on the discrete solution has never been fully understood. In this talk, I show how spurious low-frequency modes may disastrously impact the discrete solution for problems in one and two space dimensions and later propose a stabilization technique based on polynomial extensions. Our strategy consists in modifying the discrete formulation prior to mass lumping and restores a level of accuracy comparable to boundary-fitted discretizations.