{"id":444,"date":"2023-03-05T04:31:35","date_gmt":"2023-03-05T01:01:35","guid":{"rendered":"https:\/\/mohammadijoo.com\/blog\/?p=444"},"modified":"2023-04-10T00:35:42","modified_gmt":"2023-04-09T21:05:42","slug":"analytical-dynamic-in-a-nutshell","status":"publish","type":"post","link":"https:\/\/mohammadijoo.ir\/en\/blog\/analytical-dynamic-in-a-nutshell\/","title":{"rendered":"“Analytical Dynamic” in a Nutshell"},"content":{"rendered":"\n

Classic mechanic or \u201cClassic Dynamic\u201d is all about applying the Newton\u2019s 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 \u201cForce-Based\u201d approach.<\/p>\n\n\n\n

In \u201cAnalytical Dynamic\u201d, we use another approach to derive \u201cequations of motion\u201d or \u201cDynamic Equations of System\u201d. 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 \u201cEnergy-Based\u201d approach.<\/p>\n\n\n\n\n\n\n\n

The 2 main approaches in Analytical Dynamic are \u201cHamilton Principle\u201d and \u201cLagrange Equations\u201d.<\/p>\n\n\n\n

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:<\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ \int_{t_1}^{t_2} [\delta W_e + \delta T - \delta U] \,dt = 0 \] \"<\/p><\/p>\n\n\n\n

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:<\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ \delta \int_{t_1}^{t_2} [ T - (U + V)] \,dt = 0 \] \"<\/p><\/p>\n\n\n\n

In \u201cQuantum Mechanics\u201d, there is a term as \u201cHamiltonian\u201d that is summation of kinetic and potential energy, and sometimes could be mistaken with \u201cHamilton Principle\u201d. Hamiltonian operator in quantum physics, which is used in deriving the Schr\u00f6dinger equation, is as below:<\/p>\n\n\n\n

\" \hat{H} = \hat{T} + \hat{V} \"<\/p>\n\n\n\n

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 \u2013 V<\/p>\n\n\n\n

By solving below equation which is known as \u201cLagrange Equation of Motion\u201d, the dynamic equation of system will be derived:<\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ {\frac{d}{dt}} ({\frac{\partial L}{\partial \dot{q_i}}}) - {\frac{\partial L}{\partial q_i}} = Q_i \] \"<\/p> <\/p>\n\n\n\n

In above equation, L is lagrangian, q is generalized coordinates and Q is generalized external forces applied in each generalize coordinates.<\/p>\n\n\n\n

Sometimes, there are some constraints on the system and you need to solve Lagrange equations, considering system constraints.<\/p>\n\n\n\n

If the constraints on system is only function of generalized coordinates, it is called \u201cHolonomic\u201d constraints and is in below form:<\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ f_j(t, q_1, q_2, ...., q_m) = 0 \quad \quad j = 1,2,...,r \] \"<\/p><\/p>\n\n\n\n

But if the constraints are also function of derivative of generalized coordinates, it is called \u201cNon-Holonomic\u201d constraints and is in below form:<\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ \sum_{i=1}^{m} a_{ji} \dot{q_i} + b_j = 0 \quad \quad j = 1,2,...,r \] \"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ a_{ji} = a_{ji} (t,q_i) \] \"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ b_j = b_j(t,q_i) \] \"<\/p><\/p>\n\n\n\n

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:<\/p>\n\n\n\n

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

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: \u201cIntegrated Multiplier Method (IMM)\u201d, \u201cAugmented Method (AM)\u201d, \u201cElimination Method (EM)\u201d and \u201cGreenwood Method (GM)\u201d. The detail of each approach could be found in any analytical dynamic text books.<\/p>\n\n\n\n

There is another approach in Analytical Dynamic which is called \u201cQuaternion\u201d. 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 \u201cQuaternion\u201d as shown in below formula:<\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ e_0 = cos({\frac{\alpha}{2}}) \quad e_1 = cos(\gamma_1) sin({\frac{\alpha}{2}}) \] \"<\/p> <\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ e_2 = cos(\gamma_2) sin({\frac{\alpha}{2}}) \quad e_3 = cos(\gamma_3) sin({\frac{\alpha}{2}}) \] \"<\/p><\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ e_0^2 + e_1^2 + e_2^2 + e_3^2 = 1 \] \"<\/p><\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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:<\/p>\n\n\n\n

  <\/span>   <\/span>\" \[ {\frac{\partial L}{\partial q_i} = 0} \quad and \quad Q_i=0 \] \"<\/p><\/p>\n\n\n\n

When we found the \u201cIgnorable Coordinates\u201d, then we can use \u201cRouth Method\u201d to solve Lagrange equation with ignorable coordinates. This approach also could be found in any analytical dynamic books with detail.<\/p>\n","protected":false},"excerpt":{"rendered":"

Classic mechanic or \u201cClassic Dynamic\u201d is all about applying the Newton\u2019s 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 \u201cForce-Based\u201d approach. […]<\/p>\n","protected":false},"author":1,"featured_media":445,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[7,6],"tags":[119,120,121],"class_list":["post-444","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mechanical-engineering","category-robotics-and-control-engineering","tag-analytical-dynamic","tag-hamilton-principle","tag-lagrange-equations"],"yoast_head":"\n"Analytical Dynamic" in a Nutshell - Abolfazl Mohammadijoo's Blog<\/title>\n<meta name=\"description\" content=\"Classic mechanic or \u201cClassic Dynamic\u201d is all about applying the Newton\u2019s laws on a dynamic system and deriving the dynamic equations of motions of system.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mohammadijoo.ir\/en\/blog\/analytical-dynamic-in-a-nutshell\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\""Analytical Dynamic" in a Nutshell - 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