The non-resonant magnetic braking effect induced by an non-axisymmetric magnetic perturbation is investigated on JET and TEXTOR. The collionality dependence of the torque induced by the n = 1, where n is the toroidal mode number, magnetic perturbation generated by the error field correction coils on JET is observed. The observed torque is located mainly in the plasma core (normalized radius r < 0.4) and increases with decreasing collisionality. The neoclassical toroidal plasma viscosity torque in collisionless regime is modeled by using the numerical solution of the bounce averaged drift kinetic equation. The calculated collisionality dependence of the neoclassical toroidal plasma viscosity torque is in good agreement with the experimental observation on JET. The reason for this collisionality dependence is that the torque in the plasma core on JET mainly comes from the flux of the trapped electrons, which are still mainly in the 1/v regime. The strongest neoclassical toroidal plasma viscosity torque torque on JET is also located near the plasma core. The magnitude of the neoclassical toroidal plasma viscosity torque strongly depends on the plasma response, which is also discussed in this paper. There is no obvious braking effect with n = 2 magnetic perturbation generated by the dynamic ergodic divertor on TEXTOR, which is consistent with the neoclassical toroidal plasma viscosity modeling.