Measurements of the ac response represent a widely-used method for probing the properties of superconductors. In the surface superconducting state (SSS), increase of the current beyond the surface critical current $I_c$ leads to breakdown of SSS and penetration of external magnetic field into the sample bulk. An interesting free-of-bulk system in SSS is offered by thin-walled superconducting cylinders. The critical state model (CSM) asserts the ac susceptibility $chi$ to exhibit jumps as a function of the external ac field amplitude $H_{ac}$, because of the periodic destruction and restoration of SSS in the cylinder wall. Here, we investigate experimentally the low-frequency (128-8192,Hz) ac response of thin-walled superconducting cylinders in superimposed dc and ac magnetic fields applied parallel to the cylinder axis. Distinct from the CSM predictions, experiments reveal that $chi$ is a smooth function of $H_{ac}$. For the explanation of our observations we propose a phenomenological model of partial penetration of magnetic flux (PPMF). The PPMF model implies that after a restoration of the superconducting state, the magnetic fields inside and outside the cylinder are not equal, and the value of the penetrating flux is random for each penetration. This model fits very well to the experimental data on the temperature dependence of the first-harmonic $chi_1$ at any $H_{ac}$ and dc field magnitude. However, in a certain temperature range the values of physical parameters deduced within the framework of the PPMF model are questionable.
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