The non-resonant magnetic braking effect induced by an Non- Axisymmetric Magnetic Perturbation (NAMP) is investigated on JET and TEXTOR. The collionality dependence of the torque induced by the n = 1 NAMP field is obtained on JET. The observed torque is located mainly in the plasma core (normalized r < 0.4). It increases with decreasing collisionality. The calculation shows that it is close to the transition between the v- v and the superbanana plateau regimes in the plasma core. The NTV torque is modeled by using the smoothly connnected formula in the collisionless regimes. The calculated collisionality dependence shows the same tendency as the experimental observation due to the resonant particle effect. The strongest NTV torque is also located in the plasma core because of the resonant particle effect. However, the magnitudes of the NTV torque is still about 1-2 orders smaller than the observed ones on JET. There is no obvious braking effect with m/n = 6/2 NAMP generated by the Dynamic Ergodic Divertor (DED) on TEXTOR. The calculated NTV torque on TEXTOR is also very small.