Abstract Continuum mechanics and an elastic beam model were employed in the nonlinear force vibrational analysis of an embedded, curved, single-walled carbon nanotube. The analysis considered the effects of the curvature or waviness and midplane stretching of the nanotube on the nonlinear frequency. By utilizing He’s Energy Balance Method (HEBM), the relationships of the nonlinear amplitude and frequency were expressed for a curved, single-walled carbon nanotube. The amplitude frequency response curves of the nonlinear free vibration were obtained for a curved, single-walled carbon nanotube embedded in a Pasternak elastic foundation. Finally, the influence of the amplitude of the waviness, midplane stretching nonlinearity, shear foundation modulus, surrounding elastic medium, radius, and length of the curved carbon nanotube on the amplitude frequency response characteristics are discussed. As a result, the combination effects of waviness and stretching nonlinearity on the nonlinear frequency of the curved SWCNT with a small outer radius were larger than the straight one.