  In this article, we study the effect of small-cut elements on the critical
time-step size in an immersogeometric context. We analyze different
formulations for second-order (membrane) and fourth-order (shell-type)
equations, and derive scaling relations between the critical time-step size and
the cut-element size for various types of cuts. In particular, we focus on
different approaches for the weak imposition of Dirichlet conditions: by
penalty enforcement and with Nitsche's method. The stability requirement for
Nitsche's method necessitates either a cut-size dependent penalty parameter, or
an additional ghost-penalty stabilization term is necessary. Our findings show
that both techniques suffer from cut-size dependent critical time-step sizes,
but the addition of a ghost-penalty term to the mass matrix serves to mitigate
this issue. We confirm that this form of `mass-scaling' does not adversely
affect error and convergence characteristics for a transient membrane example,
and has the potential to increase the critical time-step size by orders of
magnitude. Finally, for a prototypical simulation of a Kirchhoff-Love shell,
our stabilized Nitsche formulation reduces the solution error by well over an
order of magnitude compared to a penalty formulation at equal time-step size.
