“Analytical Dynamic” in a Nutshell

Abolfazl Mohammadijoo

Abolfazl Mohammadijoo

I am a freelance "Full-Stack Developer" and "Full-Stack Engineer". I have Bachelor and Master Degrees in Mechanical Engineering (Control & Robotics) from best Universities in Iran, and have a great knowledge in Artificial Intelligence, Computer engineering and Electrical Engineering.

Classic mechanic or “Classic Dynamic” is all about applying the Newton’s laws on a dynamic system and deriving the dynamic equations of motions of system. Usually, you first draw Free Body Diagram of system and apply the forces on parts and derive the equations of motion. Therefore, we could say Classic Dynamic is “Force-Based” approach.

In “Analytical Dynamic”, we use another approach to derive “equations of motion” or “Dynamic Equations of System”. In this approach we first calculate the energy of system (kinetic and potential energies) and then derive the dynamic equations from energy of the system, thus, the Analytical Dynamic is “Energy-Based” approach.

The 2 main approaches in Analytical Dynamic are “Hamilton Principle” and “Lagrange Equations”.

In Hamiltonian approach, you need to apply a virtual displacement to generalized coordinates and then calculate virtual work and virtual energies. Then by solving below equation, the dynamic system equations will be derived:

      \[ \int_{t_1}^{t_2} [\delta W_e + \delta T - \delta U] \,dt = 0 \]

In above equation \delta W_e is virtual work by external forces, \delta T is virtual kinetic energy and \delta U is virtual elastic energy. For conservative system, we can substitute the virtual work by external forces with a potential energy and the above equation, will be changed as bellow which is known as Hamilton Principle:

     \[ \delta  \int_{t_1}^{t_2} [ T - (U + V)] \,dt = 0 \]

In “Quantum Mechanics”, there is a term as “Hamiltonian” that is summation of kinetic and potential energy, and sometimes could be mistaken with “Hamilton Principle”. Hamiltonian operator in quantum physics, which is used in deriving the Schrödinger equation, is as below:

  \hat{H} = \hat{T} + \hat{V}

Second approach of Analytical Dynamic, is Lagrange equations of motion. In this approach, you first need to calculate kinetic and potential energies. Then the Lagrangian would be:  L = T – V

By solving below equation which is known as “Lagrange Equation of Motion”, the dynamic equation of system will be derived:

     \[ {\frac{d}{dt}}  ({\frac{\partial L}{\partial  \dot{q_i}}}) - {\frac{\partial L}{\partial q_i}}  = Q_i \]

In above equation, L is lagrangian, q is generalized coordinates and Q is generalized external forces applied in each generalize coordinates.

Sometimes, there are some constraints on the system and you need to solve Lagrange equations, considering system constraints.

If the constraints on system is only function of generalized coordinates, it is called “Holonomic” constraints and is in below form:

     \[ f_j(t, q_1, q_2, ...., q_m) = 0  \quad  \quad   j = 1,2,...,r  \]

But if the constraints are also function of derivative of generalized coordinates, it is called “Non-Holonomic” constraints and is in below form:

     \[ \sum_{i=1}^{m} a_{ji}  \dot{q_i} + b_j = 0   \quad \quad    j = 1,2,...,r  \]

     \[ a_{ji} = a_{ji} (t,q_i) \]

     \[ b_j = b_j(t,q_i)  \]

However, even if are constraints are holonomic, we can take derivative from them and convert them to non-holonomic constraints and Lagrange equation for systems with constraints, is as below:

     \[ {\frac{d}{dt}}  ({\frac{\partial L}{\partial  \dot{q_i}}}) - {\frac{\partial L}{\partial q_i}}  = Q_i  + \sum_{j=1}^{r} \lambda_j a_{ji}    \quad \quad    j = 1,2,...,r  \]

In above equation, \lambda_j are Lagrange coefficients which should be solved and derived. There are 4 techniques to solve the Lagrange Equation with constraints. These techniques are: “Integrated Multiplier Method (IMM)”, “Augmented Method (AM)”, “Elimination Method (EM)” and “Greenwood Method (GM)”. The detail of each approach could be found in any analytical dynamic text books.

There is another approach in Analytical Dynamic which is called “Quaternion”. Working with Euler angles and deriving the rotational matrix, is sometimes complicated. Therefore, in this approach another parameter will be added to 3 Euler angles and these parameters are called “Quaternion” as shown in below formula:

     \[ e_0 = cos({\frac{\alpha}{2}})  \quad   e_1 = cos(\gamma_1) sin({\frac{\alpha}{2}})   \]

     \[ e_2 = cos(\gamma_2) sin({\frac{\alpha}{2}})   \quad   e_3 = cos(\gamma_3) sin({\frac{\alpha}{2}})  \]

     \[  e_0^2 + e_1^2 + e_2^2 + e_3^2 = 1  \]

In above equation, \gamma_1 and \gamma_2 and \gamma_3 are 3 principle rotation angles around 3 coordinates and \alpha is rotation angle around principle line. The rest of problem is like before, the only difference is, instead of having 3 Euler angles (and their derivatives) as generalized coordinates using in Lagrange equation, you have 4 quaternion (and 4 derivatives) to use in Lagrange equation. The formula of converting quaternions to Euler angles and vice-versa and calculating derivative of quaternions, could be found in analytical dynamic text books.

Sometimes, we could ignore some coordinates when we are solving the Lagrange equation and when the number of coordinates are a lot, this will help to speed up the calculation. Actually, the coordinates which satisfy both of two below conditions, are ignorable coordinates:

     \[  {\frac{\partial L}{\partial q_i} = 0} \quad and \quad  Q_i=0 \]

When we found the “Ignorable Coordinates”, then we can use “Routh Method” to solve Lagrange equation with ignorable coordinates. This approach also could be found in any analytical dynamic books with detail.

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