41.2.3.2. Longitudinal force
The longitudinal force depends on the:
Vertical force (\({{F}_{z}}\))
Current coefficient of friction (\(\mu\))
Longitudinal slip ratio (\({{S}_{s}}\))
Slip angle (\(\alpha\))
The comprehensive slip ratio (\({{S}_{s\alpha }}\)):
(41.27)\[{{S}_{s\alpha }}=\sqrt{\left( {{S}_{s}}^{2}+{{\tan }^{2}}\alpha \right)}\]
The current value coefficient of friction (\(\mu\)):
(41.28)\[\mu ={{\mu }_{s}}-({{\mu }_{s}}-{{\mu }_{d}})\times {{S}_{s\alpha }}\]
A critical longitudinal slip (\({{S}^{*}}_{s}\))
(41.29)\[{{S}^{*}}_{s}=\left| \frac{\mu \times {{F}_{z}}}{2\times {{K}_{s}}} \right|\]
Elastic Deformation State: \(\left| {{S}_{s}} \right|<{{S}^{*}}_{s}\)
(41.30)\[F_x=-K_s \times S_s\]\({{K}_{s}}\): Partial derivative of longitudinal force (\({{F}_{x}}\)) with respect to longitudinal slip ratio (\({{S}_{s}}\)) at zero longitudinal slip
Complete Sliding State: \(\left| {{S}_{s}} \right|>{{S}^{*}}_{s}\)
(41.31)\[{{F}_{x}}=-sign{({{S}_{s}})({{F}_{x1}}-{{F}_{x2}})}\]- where,
- \({{F}_{x1}}=\mu \times \left| {{F}_{z}} \right|\)\({{F}_{x2}}=\left| \frac{{{\left( \mu \times {{F}_{z}} \right)}^{2}}}{\left( 4\times \left| {{S}_{s}} \right|\times \xi \right)} \right|\)