physics/mechanics questions

[James Milam]

These statements are found in the Kane's method and Lagrange's method docs and are seemingly contradictory

"In mechanics we are assuming there are 5 basic sets of equations needed to describe a system."

"In mechanics we are assuming there are 3 basic sets of equations needed to describe a system."

I'm guessing some from the 5 sets can be rearranged to match the 3 sets?

Also what does m represent in the 3 sets (on Lagrane's method page). I'd guess mass but it is a function of the generalized coordinates and time and has a time derivative.

Same basic question, what does k represent in the 5 sets (on Kane's method page).

I'm assuming that the f's are different types of forces on both pages.

Thanks for the help,

Brandon

[Jason Moore]

These descriptions reflect the differences in what Langrange's and Kane's method produces.

Lagrange's method produces a kinematical equation (linear in q'), a dynamical equation (linear in q''), and a constraint equation that includes Langrange multipliers (if there are constraints). The m_c, m_dc, and m_d are simply the matrices that contain the coefficients to the linear terms in the equations. m_dc is often called the mass matrix (simple example: m x'' = -c x' - k x - f ). In Lagrange's method m_c is always identity because you simply sub in u = q' to put the equations in first order form.

Kane's method produces a different, but equivalent, set of equations.

1. holonomic constraint equation (non-linear kinematic loop constraints) [ this can also be in Langrange's method as a 4th equation but we currently don't have it explicitly defined]

2. kinematical differential equations (linear in the generalized speeds). This is typically u = q' like in Lagrange's method but Kane's method allows you to define these any way you want so that you can get simpler equations in the end.

3. non-holonomic constraints (linear in the generalized speeds and defines the relationships between the independent speeds and the dependent speeds)

4. dynamical equations of motion (linear in independent speeds)

5. dynamical equations of motion (linear in dependent generalized speeds)

3, 4, and 5 serve to describe the same thing that the last two Lagrange equations describe.

The k terms are simply the linear coefficient matrices in those equations.

Theoretically you should be able to transform the results of Lagrange's method to those of Kane's. I'm not sure if that procedure is laid out anywhere in the literature. But it may have some complications.

The f's are simply vector equations of those variables that are non-linear. They will hold all external forces, but also terms that are non-linear like the Coriolis forces, etc.

Jason
moorepants.info
+01 530-601-9791

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[Jason Moore]

Correction on what Kane's method class produces, in order as presented in the docs:

1. holonomic constraint equation (non-linear in the coordinates, e.g. kinematic loop constraints) [ this can also be in Langrange's method as a 4th equation but we currently don't have it explicitly defined]

2. non-holonomic constraints (linear in the generalized speeds, but the number of equations is less than the number of generalized speeds so you solve these equations for the dependent speeds)
3. kinematical differential equations (linear in the generalized speeds and derivatives of the generalized coordinates). This is typically u = q' like in Lagrange's method but Kane's method allows you to define these any way you want so that you can get simpler equations of motion than other methods.

4. dynamical equations of motion (linear in derivatives of the independent speeds, k_d is the so called mass matrix)

5. differentiated non-holonomic constraints (linear in the derivatives of the dependent speeds, this is non-essential but allows you to integrate the dependent speeds)

Jason
moorepants.info
+01 530-601-9791

[James Milam]

"Lagrange's method produces a kinematical equation (linear in q'), a dynamical equation (linear in q''), and a constraint equation that includes Langrange multipliers (if there are constraints). The m_c, m_dc, and m_d are simply the matrices that contain the coefficients to the linear terms in the equations. m_dc is often called the mass matrix (simple example: m x'' = -c x' - k x - f ). In Lagrange's method m_c is always identity because you simply sub in u = q' to put the equations in first order form."

Isn't the third equation (the one with the lagrange multipliers) the dynamical equation? If so would that mean m_d is the mass matrix and not m_dc? 

In the code the mass matrix represents M in M*udot=F and forcing is F. For mass_matrix_full and forcing_full is it M*[u, udot]^T = F? 

Last question, in Lagrange's method it appears a third "block" is added to mass matrix full which consists of the derivative of the constraint information. Is this just supplying equations for dependent udots? KanesMEthod does not have this third "block" in it's mass_matrix_full. Is this constraint information covered by the derivative of the non-holonomic constraints that it puts in its mass matrix (return Matrix([self._k_d, self._k_dnh]))? If so why are the constraints included in mass_matrix in KanesMethod but held off for mass_matrix_full in LagrangesMethod? Couldn't this cause code that uses mass matrix from Kane's method to not also work using LagrangesMethod mass matrix?