Adhesion¶

Normal Adhesion Potential¶

ipctk.normal_adhesion_potential(d: float, dhat_p: float, dhat_a: float, a2: float) → float¶

The normal adhesion potential.

Parameters:¶

d: float¶

distance

dhat_p: float¶

distance of largest adhesion force (\(\hat{d}_p\)) where \(0 < \hat{d}_p < \hat{d}_a\)

dhat_a: float¶

adhesion activation distance (\(\hat{d}_a\))

a2: float¶

adjustable parameter relating to the maximum derivative of a (\(a_2\))

Returns:¶

The normal adhesion potential.

ipctk.normal_adhesion_potential_first_derivative(d: float, dhat_p: float, dhat_a: float, a2: float) → float¶

The first derivative of the normal adhesion potential wrt d.

Parameters:¶

d: float¶

distance

dhat_p: float¶

distance of largest adhesion force (\(\hat{d}_p\)) where \(0 < \hat{d}_p < \hat{d}_a\)

dhat_a: float¶

adhesion activation distance (\(\hat{d}_a\))

a2: float¶

adjustable parameter relating to the maximum derivative of a (\(a_2\))

Returns:¶

The first derivative of the normal adhesion potential wrt d.

ipctk.normal_adhesion_potential_second_derivative(d: float, dhat_p: float, dhat_a: float, a2: float) → float¶

The second derivative of the normal adhesion potential wrt d.

Parameters:¶

d: float¶

distance

dhat_p: float¶

distance of largest adhesion force (\(\hat{d}_p\)) where \(0 < \hat{d}_p < \hat{d}_a\)

dhat_a: float¶

adhesion activation distance (\(\hat{d}_a\))

a2: float¶

adjustable parameter relating to the maximum derivative of a (\(a_2\))

Returns:¶

The second derivative of the normal adhesion potential wrt d.

ipctk.max_normal_adhesion_force_magnitude(dhat_p: float, dhat_a: float, a2: float) → float¶

The maximum normal adhesion force magnitude.

Parameters:¶

dhat_p: float¶

distance of largest adhesion force (\(\hat{d}_p\)) where \(0 < \hat{d}_p < \hat{d}_a\)

dhat_a: float¶

adhesion activation distance (\(\hat{d}_a\))

a2: float¶

adjustable parameter relating to the maximum derivative of a (\(a_2\))

Returns:¶

The maximum normal adhesion force magnitude.

Tangential Adhesion Potential¶

ipctk.tangential_adhesion_f0(y: float, eps_a: float) → float¶

The tangential adhesion mollifier function.

Parameters:¶

y: float¶

The tangential relative speed.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

The tangential adhesion mollifier function at y.

ipctk.tangential_adhesion_f1(y: float, eps_a: float) → float¶

The first derivative of the tangential adhesion mollifier function.

Parameters:¶

y: float¶

The tangential relative speed.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

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

ipctk.tangential_adhesion_f2(y: float, eps_a: float) → float¶

The second derivative of the tangential adhesion mollifier function.

Parameters:¶

y: float¶

The tangential relative speed.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

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

ipctk.tangential_adhesion_f1_over_x(y: float, eps_a: float) → float¶

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

Parameters:¶

y: float¶

The tangential relative speed.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

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

ipctk.tangential_adhesion_f2_x_minus_f1_over_x3(y: float, eps_a: float) → float¶

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

Parameters:¶

y: float¶

The tangential relative speed.

eps_a: float¶

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.

Smooth \(\mu\)¶

ipctk.smooth_mu_a0(y: float, mu_s: float, mu_k: float, eps_a: float) → float¶

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:¶

y: float¶

The tangential relative speed.

mu_s: float¶

Coefficient of static adhesion.

mu_k: float¶

Coefficient of kinetic adhesion.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the integral at y.

ipctk.smooth_mu_a1(y: float, mu_s: float, mu_k: float, eps_a: float) → float¶

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:¶

y: float¶

The tangential relative speed.

mu_s: float¶

Coefficient of static adhesion.

mu_k: float¶

Coefficient of kinetic adhesion.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the product at y.

ipctk.smooth_mu_a2(y: float, mu_s: float, mu_k: float, eps_a: float) → float¶

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:¶

y: float¶

The tangential relative speed.

mu_s: float¶

Coefficient of static adhesion.

mu_k: float¶

Coefficient of kinetic adhesion.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the derivative at y.

ipctk.smooth_mu_a1_over_x(y: float, mu_s: float, mu_k: float, eps_a: float) → float¶

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

Notes

The x in the function name refers to the parameter y. The a1 is unrelated to the a1 in the normal adhesion.

Parameters:¶

y: float¶

The tangential relative speed.

mu_s: float¶

Coefficient of static adhesion.

mu_k: float¶

Coefficient of kinetic adhesion.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the product at y.

ipctk.smooth_mu_a2_x_minus_mu_a1_over_x3(y: float, mu_s: float, mu_k: float, eps_a: float) → float¶

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.

Notes

The x in the function name refers to the parameter y. The a1/a2 are unrelated to the a1/a2 in the normal adhesion.

Parameters:¶

y: float¶

The tangential relative speed.

mu_s: float¶

Coefficient of static adhesion.

mu_k: float¶

Coefficient of kinetic adhesion.

eps_a: float¶

Velocity threshold below which static adhesion force is applied.

Returns:¶

The value of the expression at y.