Adhesion¶

Functions¶

Name	Description
`normal_adhesion_potential()`	The normal adhesion potential.
`normal_adhesion_potential_first_derivative()`	The first derivative of the normal adhesion potential wrt d.
`normal_adhesion_potential_second_derivative()`	The second derivative of the normal adhesion potential wrt d.
`max_normal_adhesion_force_magnitude()`	The maximum normal adhesion force magnitude.
`tangential_adhesion_f0()`	The tangential adhesion mollifier function.
`tangential_adhesion_f1()`	The first derivative of the tangential adhesion mollifier function.
`tangential_adhesion_f2()`	The second derivative of the tangential adhesion mollifier function.
`tangential_adhesion_f1_over_x()`	The first derivative of the tangential adhesion mollifier function divided by y.
`tangential_adhesion_f2_x_minus_f1_over_x3()`	The second derivative of the tangential adhesion mollifier function times y minus the first derivative all divided by y cubed.
`smooth_mu_a0()`	Compute the value of the ∫ μ(y) a₁(y) dy, where a₁ is the first derivative of the smooth tangential adhesion mollifier.
`smooth_mu_a1()`	Compute the value of the μ(y) a₁(y), where a₁ is the first derivative of the smooth tangential adhesion mollifier.
`smooth_mu_a2()`	Compute the value of d/dy (μ(y) a₁(y)), where a₁ is the first derivative of the smooth tangential adhesion mollifier.
`smooth_mu_a1_over_x()`	Compute the value of the μ(y) a₁(y) / y, where a₁ is the first derivative of the smooth tangential adhesion mollifier.
`smooth_mu_a2_x_minus_mu_a1_over_x3()`	Compute the value of the [(d/dy μ(y) a₁(y)) ⋅ y - μ(y) a₁(y)] / y³, where a₁ and a₂ are the first and second derivatives of the smooth tangential adhesion mollifier.

Normal Adhesion Potential¶

double normal_adhesion_potential(const double d, const double dhat_p, const double dhat_a, const double a2);¶

The normal adhesion potential.

Parameters:¶

const double d¶: distance
const double dhat_p¶: distance of largest adhesion force ( \(\hat{d}_p\)) where \(0 < \hat{d}_p < \hat{d}_a\)
const double dhat_a¶: adhesion activation distance ( \(\hat{d}_a\))
const double a2¶: adjustable parameter relating to the maximum derivative of a ( \(a_2\))

Returns:¶

The normal adhesion potential.

double normal_adhesion_potential_first_derivative(const double d, const double dhat_p, const double dhat_a, const double a2);¶

The first derivative of the normal adhesion potential wrt d.

Parameters:¶

const double d¶: distance
const double dhat_p¶: distance of largest adhesion force ( \(\hat{d}_p\)) where \(0 < \hat{d}_p < \hat{d}_a\)
const double dhat_a¶: adhesion activation distance ( \(\hat{d}_a\))
const double a2¶: adjustable parameter relating to the maximum derivative of a ( \(a_2\))

Returns:¶

The first derivative of the normal adhesion potential wrt d.

double normal_adhesion_potential_second_derivative(const double d, const double dhat_p, const double dhat_a, const double a2);¶

The second derivative of the normal adhesion potential wrt d.

Parameters:¶

const double d¶: distance
const double dhat_p¶: distance of largest adhesion force ( \(\hat{d}_p\)) where \(0 < \hat{d}_p < \hat{d}_a\)
const double dhat_a¶: adhesion activation distance ( \(\hat{d}_a\))
const double a2¶: adjustable parameter relating to the maximum derivative of a ( \(a_2\))

Returns:¶

The second derivative of the normal adhesion potential wrt d.

double max_normal_adhesion_force_magnitude( const double dhat_p, const double dhat_a, const double a2);¶

The maximum normal adhesion force magnitude.

Parameters:¶

const double dhat_p¶: distance of largest adhesion force ( \(\hat{d}_p\)) where \(0 < \hat{d}_p < \hat{d}_a\)
const double dhat_a¶: adhesion activation distance ( \(\hat{d}_a\))
const double a2¶: adjustable parameter relating to the maximum derivative of a ( \(a_2\))

Returns:¶

The maximum normal adhesion force magnitude.

Tangential Adhesion Potential¶

double tangential_adhesion_f0(const double y, const double eps_a);¶

The tangential adhesion mollifier function.

Parameters:¶

const double y¶: The tangential relative speed.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The tangential adhesion mollifier function at y.

double tangential_adhesion_f1(const double y, const double eps_a);¶

The first derivative of the tangential adhesion mollifier function.

Parameters:¶

const double y¶: The tangential relative speed.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The first derivative of the tangential adhesion mollifier function at y.

double tangential_adhesion_f2(const double y, const double eps_a);¶

The second derivative of the tangential adhesion mollifier function.

Parameters:¶

const double y¶: The tangential relative speed.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The second derivative of the tangential adhesion mollifier function at y.

double tangential_adhesion_f1_over_x( const double y, const double eps_a);¶

The first derivative of the tangential adhesion mollifier function divided by y.

Parameters:¶

const double y¶: The tangential relative speed.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The first derivative of the tangential adhesion mollifier function divided by y.

double tangential_adhesion_f2_x_minus_f1_over_x3( const double y, const double eps_a);¶

The second derivative of the tangential adhesion mollifier function times y minus the first derivative all divided by y cubed.

Parameters:¶

const double y¶: The tangential relative speed.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The second derivative of the tangential adhesion mollifier function times y minus the first derivative all divided by y cubed.

double smooth_mu_a0(const double y, const double mu_s, const double mu_k, const double eps_a);¶

Compute the value of the ∫ μ(y) a₁(y) dy, where a₁ is the first derivative of the smooth tangential adhesion mollifier.

Note

The a0/a1 are unrelated to the a0/a1 in the normal adhesion.

Parameters:¶

const double y¶: The tangential relative speed.
const double mu_s¶: Coefficient of static adhesion.
const double mu_k¶: Coefficient of kinetic adhesion.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the integral at y.

double smooth_mu_a1(const double y, const double mu_s, const double mu_k, const double eps_a);¶

Compute the value of the μ(y) a₁(y), where a₁ is the first derivative of the smooth tangential adhesion mollifier.

Note

The a1 is unrelated to the a1 in the normal adhesion.

Parameters:¶

const double y¶: The tangential relative speed.
const double mu_s¶: Coefficient of static adhesion.
const double mu_k¶: Coefficient of kinetic adhesion.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the product at y.

double smooth_mu_a2(const double y, const double mu_s, const double mu_k, const double eps_a);¶

Compute the value of d/dy (μ(y) a₁(y)), where a₁ is the first derivative of the smooth tangential adhesion mollifier.

Note

The a1/a2 are unrelated to the a1/a2 in the normal adhesion.

Parameters:¶

const double y¶: The tangential relative speed.
const double mu_s¶: Coefficient of static adhesion.
const double mu_k¶: Coefficient of kinetic adhesion.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the derivative at y.

double smooth_mu_a1_over_x(const double y, const double mu_s, const double mu_k, const double eps_a);¶

Compute the value of the μ(y) a₁(y) / y, where a₁ is the first derivative of the smooth tangential adhesion mollifier.

Note

The x in the function name refers to the parameter y.

Note

The a1 is unrelated to the a1 in the normal adhesion.

Parameters:¶

const double y¶: The tangential relative speed.
const double mu_s¶: Coefficient of static adhesion.
const double mu_k¶: Coefficient of kinetic adhesion.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the product at y.

double smooth_mu_a2_x_minus_mu_a1_over_x3(const double y, const double mu_s, const double mu_k, const double eps_a);¶

Compute the value of the [(d/dy μ(y) a₁(y)) ⋅ y - μ(y) a₁(y)] / y³, where a₁ and a₂ are the first and second derivatives of the smooth tangential adhesion mollifier.

Note

The x in the function name refers to the parameter y.

Note

The a1/a2 are unrelated to the a1/a2 in the normal adhesion.

Parameters:¶

const double y¶: The tangential relative speed.
const double mu_s¶: Coefficient of static adhesion.
const double mu_k¶: Coefficient of kinetic adhesion.
const double eps_a¶: Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the expression at y.