<p>In this paper, we first introduce a Fourier–Legendre spectral collocation method to solve the two-dimensional static Cauchy–Navier equations with variable coefficients in irregular annular domains. We then present a space-time Fourier–Legendre spectral collocation method for time-dependent Cauchy–Navier equations in such domains. The process begins by applying a polar coordinate transformation to map the irregular annular domain onto a regular one, followed by a linear transformation to map this domain onto the reference element. Classical spectral collocation methods are then employed for numerical simulation on the reference element. The numerical results demonstrate that the proposed method achieves high accuracy.</p>