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We study free vibrations of a crystal plate of AT-cut quartz carrying a thin mass layer operating as a quartz crystal microbalance for mass sensing. The mass layer is imperfectly bonded to the crystal plate with its interface described by the so-called shear-slip model that allows a discontinuity of the interface displacement. The effect of mass layer in-plane shear stiffness is also considered. The equations of anisotropic elasticity are used for the crystal plate with the omission of the small elastic constant c56. The mass layer is governed by the plane-stress equations of elasticity. An analytical solution is obtained using Fourier series, from which the resonant frequencies and vibration mode shapes are calculated. The effects of the mass layer in-plane shear stiffness, imperfect interface bonding on resonant frequencies and energy trapping are examined.