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In the present article, dynamic response of a thick sandwich truncated conical shells with a transversely flexible/inflexible core and nanocomposite face sheets subjected to low-velocity impact was studied. The face sheets are reinforced with single-walled carbon nanotubes where the agglomeration effects are considered based on Mori–Tanaka model. A new equivalent three-degree-of-freedom spring-mass-damper model is utilized to describe the contact force between impactor and sandwich truncated conical shells. Based on an improved higher order sandwich panel theory, the equations of motion are derived by Hamilton’s principal incorporating the curvature, in-plane stress of the core and the structural damping effects. Differential quadrature method is applied for obtaining the contact force and displacement histories. After validity of the present study, the effects of the single-walled carbon nanotubes volume fraction, single-walled carbon nanotubes agglomeration, number of the layers of the face sheets, boundary conditions, semi-vertex angle of the cone, impact velocity, and mass of impactor on the low-velocity impact response of the nanocomposite structure are studied in details. Numerical results show that increasing the volume fraction of single-walled carbon nanotubes can reduce the amplitude of the dynamic response of the nanocomposite structure