41.2.4.1. Force Analysis

To compute longitudinal force, lateral force, and self-aligning torque in the contact coordinate system, the user must perform a test to determine the precise operation conditions.

The conditions of interest are:

• Case I: \(\alpha \gamma <0\)

• Case II: \(\alpha \gamma \ge 0\) and \({{K}_{\alpha }}{{S}_{\alpha }}-{{K}_{\gamma }}{{S}_{\gamma }}\ge 0\)

• Case III: \(\alpha \gamma \ge 0\) and \({{K}_{\alpha }}{{S}_{\alpha }}-{{K}_{\gamma }}{{S}_{\gamma }}<0\)

  \({{K}_{\alpha }}\): Lateral Tire Stiffness by Slip Angle

  \({{K}_{\gamma }}\): Lateral Tire Stiffness by Camber Angle

The lateral force \({{F}_{\eta }}\) can be decomposed into two components: \({{F}_{\eta \alpha }}\) and \({{F}_{\eta \gamma }}\). The two components are in the same direction if \(\alpha \gamma <0\) and in opposite direction if \(\alpha \gamma \ge 0\).

41.2.4.1.1. Case I.

Case I: \(\alpha \gamma <0\)

A slip ratio due to the critical inclination angle is denoted by \({{S}^{*}}_{\gamma }\), then it can be defined as:

(41.37)\[{{S}^{*}}_{\gamma }=\mu \frac{\left| {{F}_{z}} \right|}{{{K}_{\gamma }}}\]

If \({{S}^{*}}_{s}\) represents a slip ratio due to the critical Longitudinal slip ratio, then it can be defined as:

(41.38)\[{{S}^{*}}_{s}=3\mu \frac{\left| {{F}_{z}} \right|}{{{K}_{s}}}\]

If a slip ratio due to the critical slip angle is denoted by \({{S}^{*}}_{\alpha }\), then it can be determined as

(41.39)\[{{S}^{*}}_{\alpha }=\frac{{{K}_{s}}}{{{K}_{\alpha }}}\sqrt{{{S}^{*}}{{_{s}}^{2}}-{{S}_{s}}^{2}}-3{{K}_{\gamma }}\frac{{{S}_{\gamma }}}{{{K}_{\alpha }}}\]

when, \({{S}_{s}}\le {{S}^{*}}_{s}\)

• The term “critical” stands for the maximum value which allows an elastic deformation of a tire during pure slip due to pure slip ratio slip angle, or inclination angle.

• Whenever any slip ratio becomes greater than its corresponding critical value, an elastic deformation no longer exists, but instead complete sliding state represents the contact condition between the tire tread base and the contact surface.

A nondimensional slip ratio \({{S}_{n}}\) is determined as:

(41.40)\[{{S}_{n}}=\frac{{{B}_{2}}+\sqrt{{{B}_{2}}^{2}-{{B}_{1}}{{B}_{3}}}}{{{B}_{1}}}\]

where,

\({{B}_{1}}={{(3\mu {{F}_{z}})}^{2}}-{{(3{{K}_{\gamma }}{{S}_{\gamma }})}^{2}}\)

\({{B}_{2}}=3{{K}_{\alpha }}{{S}_{\alpha }}{{K}_{\gamma }}{{S}_{\gamma }}\)

\({{B}_{3}}=-\left| {{({{K}_{s}}{{S}_{s}})}^{2}}+{{({{K}_{\alpha }}{{S}_{\alpha }})}^{2}} \right|\)

A nondimensional contact patch length is determined as:

\({{L}_{n}}=1-{{S}_{n}}\)

A modified later friction coefficient \({{\mu }_{ym}}\) is evaluated as:

(41.41)\[{{\mu }_{ym}}={{\mu }_{y}}-\left( \frac{{{K}_{\gamma }}{{S}_{\gamma }}}{\left| {{F}_{z}} \right|} \right)\]

where, \({{\mu }_{y}}=\mu \sin \theta\) is the available friction as determined by the friction circle.

• To determine the longitudinal force, the lateral force, and the self-aligning torque, considers two sub-cases separately.

• The first case is for the elastic deformation state, while the other is for the complete sliding state without any elastic deformation of a tire.

• Specifically, if all of slip ratios are smaller than those of their corresponding critical values, then there exists an elastic deformation state, otherwise there exists only complete sliding state between the tire tread base and the terrain surface.

1. Elastic Deformation State: \({{S}_{\gamma }}<{{S}^{*}}_{\gamma }\), \({{S}_{s}}<{{S}^{*}}_{s}\) and \({{S}_{\alpha }}<{{S}^{*}}_{\alpha }\)

   In the elastic deformation state, the longitudinal force \({{F}_{\xi }}\), the lateral force \({{F}_{\eta }}\), and three components of the self-aligning torque are written as functions of the elastic stiffness and the slip ratio as well as the normal force and the friction coefficients, such as:

   (41.42)\[{{F}_{\xi }}={{K}_{s}}{{S}_{s}}{{L}_{n}}^{2}+{{\mu }_{x}}\left| {{F}_{z}} \right|\left( 1-3{{L}_{n}}^{2}+2{{L}_{n}}^{3} \right)\]
   (41.43)\[{{F}_{\eta }}={{K}_{\alpha }}{{S}_{\alpha }}{{L}_{n}}^{2}+{{\mu }_{ym}}\left| {{F}_{z}} \right|(1-3{{L}_{n}}^{2}+2{{L}_{n}}^{3})+{{K}_{\gamma }}{{S}_{\gamma }}\]
   (41.44)\[{{T}_{z\alpha }}=\left[ {{K}_{\alpha }}{{S}_{\alpha }}\left( -\frac{1}{2}+\frac{2}{3}{{L}_{n}} \right)+\frac{3}{2}{{\mu }_{ym}}\left| {{F}_{z}} \right|{{S}_{n}}^{2} \right]\ell {{L}_{n}}^{2}\]
   (41.45)\[{{T}_{zs\alpha }}=\frac{2}{3}{{K}_{s}}{{S}_{s}}{{S}_{\alpha }}\ell {{L}_{n}}^{3}+\frac{3{{\mu }_{x}}{{\mu }_{y}}{{F}_{z}}^{2}\ell }{5{{K}_{\alpha }}}\left( 1-10{{L}_{n}}^{3}+15{{L}_{n}}^{4}-6{{L}_{n}}^{5} \right)\]
   (41.46)\[{{T}_{zs\gamma }}=\left| \eta \right|{{F}_{\xi }}\]

   where,

   \(\left| \eta \right|={{S}_{\gamma }}\sqrt{{{r}_{1}}^{2}-{{\ell }^{2}}/4}\) is the offset between the wheel plane center and the tire tread base, \({{r}_{1}}^{2}-{{\ell }^{2}}/4\) is set to zero if it is negative. \({{T}_{z\alpha }}\) is the portion of the self-aligning torque generated by the slip angle \(\alpha\). \({{T}_{zs\alpha }}\) and \({{T}_{zs\gamma }}\) are other components of the self-aligning torque produced by the longitudinal force, which has an offset between the wheel center plane and the tire tread base, due to the slip angle \(\alpha\) and the inclination angle \(\gamma\), respectively. The self-aligning torque \({{T}_{z}}\) is determined as combinations of \({{T}_{z\alpha }}\), \({{T}_{zs\alpha }}\) and \({{T}_{zs\gamma }}\).

2. Complete Sliding State: \({{S}_{\gamma }}\ge {{S}^{*}}_{\gamma }\), \({{S}_{s}}\ge {{S}^{*}}_{s}\) and \({{S}_{\alpha }}\ge {{S}^{*}}_{\alpha }\)

   In the complete sliding state, the longitudinal force, the later force, and three components of the self-aligning torque are determined as functions of the normal force and the friction coefficients without any elastic stiffness and slip ratio as:

   (41.47)\[{{F}_{\xi }}={{\mu }_{x}}{{F}_{z}}\]
   (41.48)\[{{F}_{\eta }}={{\mu }_{y}}{{F}_{z}}\]
   (41.49)\[{{T}_{z\alpha }}=0\]
   (41.50)\[{{T}_{z\alpha s}}=\frac{3{{\mu }_{x}}{{\mu }_{y}}{{F}_{z}}^{2}\ell }{5{{K}_{\alpha }}}\]
   (41.51)\[{{T}_{zs\gamma }}=\left| \eta \right|{{F}_{\xi }}\]

41.2.4.1.2. Case II.

Case II. \(\alpha \gamma \ge 0\) and \({{K}_{\alpha }}{{S}_{\alpha }}-{{K}_{\gamma }}{{S}_{\gamma }}\ge 0\)

Same as in Case II, a slip ratio due to the critical value of the slip ratio can be obtained as:

(41.52)\[{{S}^{*}}_{s}=3\mu \frac{\left| {{F}_{z}} \right|}{{{K}_{s}}}\]

A slip ratio due to the critical value of the slip angle can be found as:

(41.53)\[{{S}^{*}}_{\alpha }=\frac{{{K}_{s}}}{{{K}_{\alpha }}}\sqrt{{{S}^{*}}{{_{s}}^{2}}-{{S}_{s}}^{2}}+3{{K}_{\gamma }}\frac{{{S}_{\gamma }}}{{{K}_{\alpha }}}\]

When \({{S}_{s}}\le {{S}^{*}}_{s}\)

The nondimensional slip ratio \({{S}_{n}}\), is determined as:

(41.54)\[S{}_{n}=\frac{{{B}_{2}}+\sqrt{{{B}_{2}}^{2}-{{B}_{1}}{{B}_{3}}}}{{{B}_{1}}}\]

where,

\(B_1=(3\mu{F_z})^2-(3{K_\gamma}{S_\gamma})^2\)

\(B_2=-3{K_\alpha}{S_\alpha}{K_\gamma}{S_\gamma}\)

\(B_3=|{k_s}{S_s})^2+({K_\alpha}{S_\alpha})^2|\)

The nondimensional contact patch length \({{L}_{n}}\) is found from the equation \({{L}_{n}}=1-{{S}_{n}}\), and the modified lateral friction coefficient \({{\mu }_{ym}}\) is expressed as:

(41.55)\[{{\mu }_{ym}}={{\mu }_{y}}+\frac{{{K}_{\gamma }}{{S}_{\gamma }}}{{{F}_{z}}}\]

For the longitudinal force, the lateral force, and the self-aligning torque two sub-cases should also be considered separately. A slip ratio due to the critical value of the inclination angle is not needed here since the required condition for Case II, \({{K}_{\alpha }}{{S}_{\alpha }}-{{K}_{\gamma }}{{S}_{\gamma }}\ge 0\), replaces the critical condition of the inclination angle.

1. Elastic Deformation State: \({{S}_{s}}<{{S}^{*}}_{s}\) and \({{S}_{\alpha }}<{{S}^{*}}_{\alpha }\)

   In the elastic deformation state:

   (41.56)\[{{F}_{\xi }}={{K}_{s}}{{S}_{s}}{{L}_{n}}^{2}+{{\mu }_{x}}\left| {{F}_{z}} \right|(1-3{{L}_{n}}^{2}+2{{L}_{n}}^{3})\]
   (41.57)\[{{F}_{\eta }}={{K}_{\alpha }}{{S}_{\alpha }}{{L}_{n}}^{2}+{{\mu }_{ym}}\left| {{F}_{z}} \right|(1-3{{L}_{n}}^{2}+2{{L}_{n}}^{3})-{{K}_{\gamma }}{{S}_{\gamma }}\]
   (41.58)\[{{T}_{z\alpha }}=\left[ {{K}_{\alpha }}{{S}_{\alpha }}\left( -\frac{1}{2}+\frac{2}{3}{{L}_{n}} \right)+\frac{3}{2}{{\mu }_{ym}}\left| {{F}_{z}} \right|{{S}_{n}}^{2} \right]\ell {{L}_{n}}^{2}\]
   (41.59)\[{{T}_{zs\alpha }}=\frac{2}{3}{{K}_{s}}{{S}_{s}}{{S}_{\alpha }}\ell {{L}_{n}}^{3}+\frac{3{{\mu }_{x}}{{\mu }_{y}}{{F}_{z}}^{2}\ell }{5{{K}_{\alpha }}}(1-10{{L}_{n}}^{3}+15{{L}_{n}}^{4}-6{{L}_{n}}^{5})\]
   (41.60)\[{{T}_{zs\gamma }}=\left| \eta \right|{{F}_{\xi }}\]

2. Complete sliding state: \({{S}_{s}}\ge {{S}^{*}}_{s}\) or \({{S}_{\alpha }}\ge {{S}^{*}}_{\alpha }\)

   (41.61)\[{{F}_{\xi }}={{\mu }_{x}}{{F}_{z}}\]

   (41.62)\[{{F}_{\eta }}={{\mu }_{y}}{{F}_{z}}\]

   (41.63)\[{{T}_{z\alpha }}=0\]

   (41.64)\[{{T}_{z\alpha s}}=\frac{3{{\mu }_{x}}{{\mu }_{y}}{{F}_{z}}^{2}\ell }{5{{K}_{\alpha }}}\]

   (41.65)\[{{T}_{zs\gamma }}=\left| \eta \right|{{F}_{\xi }}\]

41.2.4.1.3. Case III.

Case III: \(\alpha \gamma \ge 0\) and \({{K}_{\alpha }}{{S}_{\alpha }}-{{K}_{\gamma }}{{S}_{\gamma }}<0\)

Similar to Case I and II, slop ratios due to the critical values of the inclination angle and the slip ratio are obtained as:

(41.66)\[{{S}^{*}}_{\gamma }=\frac{3\mu \left| {{F}_{z}} \right|+{{K}_{\alpha }}{{S}_{\alpha }}}{3{{K}_{\gamma }}}\]

(41.67)\[{{S}^{*}}_{c}=\frac{1}{{{K}_{s}}}\sqrt{{{\left( 3\mu {{F}_{z}} \right)}^{2}}-{{\left( {{K}_{\alpha }}{{S}_{\alpha }}-3{{K}_{\gamma }}{{S}_{\gamma }} \right)}^{2}}}\]

The nondimensional slip ratio \({{S}_{n}}\), is expressed as:

(41.68)\[{{S}_{n}}=\frac{{{B}_{2}}+\sqrt{{{B}_{2}}^{2}-{{B}_{1}}{{B}_{3}}}}{{{B}_{1}}}\]

where,

\({{B}_{1}}={{\left( 3\mu {{F}_{z}} \right)}^{2}}-{{\left( 3{{K}_{\gamma }}{{S}_{\gamma }} \right)}^{2}}\)

\({{B}_{2}}=-3{{K}_{\alpha }}{{S}_{\alpha }}{{K}_{\gamma }}{{S}_{\gamma }}\)

\({{B}_{3}}=-\left[ {{\left( {{K}_{s}}{{S}_{s}} \right)}^{2}}+{{\left( {{K}_{\alpha }}{{S}_{\alpha }} \right)}^{2}} \right]\)

For the longitudinal force, the lateral force, and the self-aligning torque, two sub-cases should also be considered similar to Cases I and II. A slip ratio due to the critical value of the slip angle is not needed here since the required condition for Case III, \({{K}_{\alpha }}{{S}_{\alpha }}-{{K}_{\gamma }}{{S}_{\gamma }}<0\), replaces the critical condition of the slip angle.

1. Elastic Deformation State: \({{S}_{\gamma }}<{{S}^{*}}_{\gamma }\) and \({{S}_{s}}<{{S}^{*}}_{s}\)

   In the elastic deformation state, \({{F}_{\eta }}\) and \({{T}_{z\alpha }}\) may be written:

   (41.69)\[{{F}_{\xi }}={{K}_{s}}{{S}_{s}}{{L}_{n}}^{2}+{{\mu }_{x}}\left| {{F}_{z}} \right|\left( 1-3{{L}_{n}}^{2}+2{{L}_{n}}^{3} \right)\]
   (41.70)\[{{F}_{\eta }}={{K}_{\gamma }}{{S}_{\gamma }}\left( 3{{L}_{n}}^{2}-2{{L}_{n}}^{3} \right)-{{K}_{\alpha }}{{S}_{\alpha }}{{L}_{n}}^{2}+{{\mu }_{y}}\left| {{F}_{z}} \right|\left( 1-3{{L}_{n}}^{2}+2{{L}_{n}}^{3} \right)\]
   (41.71)\[{{T}_{z\alpha }}=\frac{{{K}_{\alpha }}{{S}_{\alpha }}\ell {{L}_{n}}}{6}\]
   (41.72)\[{{T}_{zs\alpha }}=\frac{2}{3}{{K}_{s}}{{S}_{s}}{{S}_{\alpha }}\ell {{L}_{n}}^{3}+\frac{3{{\mu }_{x}}{{\mu }_{y}}{{F}_{z}}^{2}\ell }{5{{K}_{\alpha }}}\left( 1-10{{L}_{n}}^{3}+15{{L}_{n}}^{4}-6{{L}_{n}}^{5} \right)\]
   (41.73)\[{{T}_{zs\gamma }}=\left| \eta \right|{{F}_{\xi }}\]

2. Complete Sliding State: \({{S}_{\gamma }}\ge {{S}^{*}}_{\gamma }\) or \({{S}_{s}}\ge {{S}^{*}}_{s}\)

   (41.74)\[{{F}_{\xi }}={{\mu }_{x}}{{F}_{z}}\]
   (41.75)\[{{F}_{\eta }}={{\mu }_{y}}{{F}_{z}}\]
   (41.76)\[{{T}_{z\alpha }}=0\]
   (41.77)\[{{T}_{z\alpha s}}=\frac{3{{\mu }_{x}}{{\mu }_{y}}{{F}_{z}}^{2}\ell }{5{{K}_{\alpha }}}\]
   (41.78)\[{{T}_{zs\gamma }}=\left| \eta \right|{{F}_{\xi }}\]

   Respectively, the Longitudinal force \({{F}_{\xi }}\), the lateral force \({{F}_{\eta }}\), and three components of the self-aligning torques \({{T}_{z\alpha }}\), \({{T}_{zs\alpha }}\), and \({{T}_{zs\gamma }}\) always have positive values, but they can be transformed to have positive or negative values depending on the slip ratio \(S\), the slip angle \(\alpha\), and the inclination angle \(\gamma\) in the Contact coordinate system.