From the results focus is on how these parameters and their variations affect the overall performance of the individual models both for the pre-sliding and gross-sliding friction
(a)
(b)
Fig. 3.17 The friction force against (a) displacement, and (b) velocity, for the GMS model for various values of the parameter αi
(a)
(b)
Fig. 3.18 The friction force against (a) displacement, and (b) velocity, for the GMS model for various values of the parameter Ci
(a)
(b)
Fig. 3.19 The friction force against (a) displacement, and (b) velocity, for the GMS model for various values of the parameter σ0i
(a)
(b)
Fig. 3.20 The friction force against (a) displacement, and (b) velocity, for the GMS model for various values of the parameter σ1i
(a)
(b)
Fig. 3.21 The friction force against (a) displacement, and (b) velocity, for the GMS model for various values of the parameter vs
(a)
(b)
Fig. 3.22 The friction force against (a) displacement, and (b) velocity, for the Xiong model for various values of the parameter Ki
(a)
(b)
Fig. 3.23 The friction force against (a) displacement, and (b) velocity, for the Xiong model for various values of the parameter σ1i
(a)
(b)
Fig. 3.24 The friction force against (a) displacement, and (b) velocity, for the Xiong model for various values of the Stribeck parameter vs
(a)
(b)
Fig. 3.25 The friction force against (a) displacement, and (b) velocity, for the Xiong model for various values of the parameter λi
(a)
(b)
Fig. 3.26 The friction force against (a) displacement, and (b) velocity, for the Xiong model for various values of the parameter τ
regimes and their transitions. LuGre model:
Increasing the stiffness increased the stiction slightly, and reduced the pre-sliding displace- ment with smoother transition to gross-sliding, while in the gross sliding regime increments in the stiffness parameter shrink the frictional lag loop and reduced the possibility of loop crossing. Loop crossing simply means a situation whereby the acceleration friction force is lower than the deceleration value at some point as the velocity goes through periodic motion, see figure 2.5. The micro-damping parameter introduced possible undershoot as its value is increased in the pre-sliding regime and decreased the frictional lag loop with the possibility of overlap as it is increased in the gross-sliding regime. This behaviour is why it may not be wise to arbitrarily increase this parameter in the LuGre model. A preferred approach adopted by the authors to model this parameter was to make it a decreasing function of the relative velocity. Increasing the Stribeck parameter also increased the stiction and displacement in the pre-sliding regime with undershoot and loop crossing in the gross-sliding for increments in the parameter. Thus from the above, the LuGre model exhibits appreciable sensitivity to parameter variations.
Proposed model:
The micro-damping parameter increments resulted in the increase of the stiction and re- duction of the pre-sliding displacement in the pre-sliding regime while this did not affect the frictional lag shape in any way in the gross-sliding regime. The stiction force and the pre-sliding displacement were seen to increase in response to increased Stribeck velocity during pre-sliding and increased the frictional lag while preserving the loop shape (loop shift). For the pre-sliding displacement parameter increments resulted in the lowering of the stiction, and no effect on the frictional lag in the gross sliding-regime. The parameter τ did not affect the pre-sliding regime, but was seen to increase the loop shape of the frictional lag. It should be noted that the introduction of this parameter was to improve the frictional lag feature of the new model. From the observations the proposed new model exhibits a degree of robustness to parameter changes and for most variations the general shape and features of the friction phenomenon is preserved both for the pre-sliding and gross-sliding regimes. From the above, the proposed new model was shown to exhibit higher degree of robustness to parameter variations for the simulation tests performed and parameters values used. GMS model:
Increasing the stiffness led to increase in the stiction and breakaway displacement reduction in the pre-sliding while in the gross-sliding regime, the frictional lag shape was reduced. The micro-damping had little to no effect on the stiction and breakaway displacement values as it increased during pre-sliding and general reduction in the frictional lag shape in the gross
sliding regime was observed. The static friction and breakaway displacement increased as the Stribeck velocity increased for the pre-slide regime, while for the gross-slide regime the frictional lag shape increased also. Parameter C increments resulted in decreasing stiction while breakaway displacement remain unchanged, and a reduction in the frictional lag in gross-sliding. The parameter α increased the stiction and breakaway displacement in the pre-slide regime, it also increased the frictional lag shape in the gross-slide regime as it was increased. From these one sees that the GMS model is sensitive to parameter variations though to a less degree compared to the LuGre model.
Xiong model:
Incrementing the stiffness increased the stiction while lowering the breakaway displacement, improving transitions from pre-sliding to gross-sliding. It increased the frictional lag shape in the sliding regime. The micro-damping had little effect on the static friction and none on the breakaway displacement and frictional lag. The Stribeck velocity increased the stic- tion and breakaway displacement as it increased, while in the gross sliding it increased the frictional lag shape (loop shift). As the scaling parameter increased the stiction, breakaway displacement, and the frictional lag were shown to increase. Increasing the τ value smooths transition from pre-slide to gross-slide, it also increased the breakaway displacement while in the gross-sliding the frictional lag shape increased. Note as in the proposed new model the τ parameter functions in similar manner to improve the frictional lag prediction of the model. The Xiong model is thus sensitive to variations to its parameters except for the micro-damping with least effect on the model.
Hence from the above simulations and results, the proposed new model of friction exhib- ited more robustness to variations in it’s parameters more than the other friction models investigated.