  This paper investigates the quasinormal mode (QNM) vibrations of a rotating
cylindrical black hole (or black string) spacetime that is surrounded by a thin
shell rotating synchronously with the black string's axis. The existence of the
thin shell leads to a piecewise metric of the black hole spacetime beyond the
horizon, which is divided into two stationary spacetime parts by the radius of
the thin shell. As a result, the potential function $V(r)$ of the QNM equation
is also discontinuous. To solve the QNM equation with the discontinuous
potential function, we propose two methods, matrix method and generalized
Horowitz-Hubeny Method. We find that the influence of the thin shell can reduce
the QNM frequency of the black string while alleviating their amplitude decay
rate. Our suggested method can be easily applied to other QNM calculations of
black hole spacetime with discontinuous potential function, thus facilitating
investigations into more intricate and realistic black hole spacetimes, such as
those with accretion disks. Additionally, the finite difference method is
employed to investigate the spacetime too. This analysis discloses a
substantial gap in the potential function when the thin shell's mass and charge
achieve sufficiently high values, resulting in the outer spacetime nearing
gravitational collapse and extreme black hole scenarios. Within this gap, the
QNM wave displays oscillations, producing an echo effect. Moreover, it is
established that the closeness of the spacetime to the collapse threshold and
charge extremality have positive correlation with the beat interval of this
echo.
