Generalized forces

Generalized forces

Generalized forces are defined via coordinate transformation of applied forces, mathbf{F}_i, on a system of n particles, i. The concept finds use in Lagrangian mechanics, where it plays a conjugate role to generalized coordinates.

A convenient equation from which to derive the expression for generalized forces is that of the virtual work, delta W_a, caused by applied forces, as seen in D'Alembert's principle in accelerating systems and the principle of virtual work for applied forces in static systems. The subscript a is used here to indicate that this virtual work only accounts for the applied forces, a distinction which is important in dynamic systems.cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |language=English |isbn=0-03-063366-4 |chapter=Energy Methods] rp|265

:delta W_a = sum_{i=1}^n mathbf {F}_{i} cdot delta mathbf r_i::delta mathbf r_i is the virtual displacement of the system, which does not have to be consistent with the constraints (in this development)

Substitute the definition for the virtual displacement (differential):rp|265

:delta mathbf{r}_i = sum_{j=1}^m frac {partial mathbf {r}_i} {partial q_j} delta q_j:delta W_a = sum_{i=1}^n mathbf {F}_{i} cdot sum_{j=1}^m frac {partial mathbf {r}_i} {partial q_j} delta q_j

Using the distributive property of multiplication over addition and the associative property of addition, we haverp|265

:delta W_a = sum_{j=1}^m sum_{i=1}^n mathbf {F}_{i} cdot frac {partial mathbf {r}_i} {partial q_j} delta q_j.

From this form, we can see that the generalized applied forces are then defined byrp|265

:Q_j = sum_{i=1}^n mathbf {F}_{i} cdot frac {partial mathbf {r}_i} {partial q_j}.

Thus, the virtual work due to the applied forces isrp|265

:delta W_a = sum_{j=1}^m Q_j delta q_j.

References


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Generalized coordinates — By deriving equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any coordinate system that is ultimately specified. cite book |last=Torby |first=Bruce |title=Advanced Dynamics for… …   Wikipedia

  • Contrôle des naissances, stérilisations et avortements forcés au Tibet — Le contrôle des naissances au Tibet historique a été introduit dans la Région autonome du Tibet (RAT) en 1985 puis a été renforcé en 1992[1]. La population rurale tibétaine de la RAT n avait aucune contrainte avant 1992[réf. nécessaire].… …   Wikipédia en Français

  • Mechanics of planar particle motion — Classical mechanics Newton s Second Law History of classical mechanics  …   Wikipedia

  • Centrifugal force (planar motion) — In classical mechanics, centrifugal force (from Latin centrum center and fugere to flee ) is one of the three so called inertial forces or fictitious forces that enter the equations of motion when Newton s laws are formulated in a non inertial… …   Wikipedia

  • Centrifugal force — Not to be confused with Centripetal force. Classical mechanics Newton s Second Law …   Wikipedia

  • Lagrangian mechanics — is a re formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by Italian mathematician Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is… …   Wikipedia

  • Conjugate variables (thermodynamics) — For a more general mathematical discussion, see Conjugate variables. Thermodynamics …   Wikipedia

  • Frame of reference — A frame of reference in physics, may refer to a coordinate system or set of axes within which to measure the position, orientation, and other properties of objects in it, or it may refer to an observational reference frame tied to the state of… …   Wikipedia

  • Constraint algorithm — In mechanics, a constraint algorithm is a method for satisfying constraints for bodies that obey Newton s equations of motion. There are three basic approaches to satisfying such constraints: choosing novel unconstrained coordinates ( internal… …   Wikipedia

  • First law of thermodynamics — In thermodynamics, the first law of thermodynamics is an expression of the more universal physical law of the conservation of energy. Succinctly, the first law of thermodynamics states: DescriptionThe first law of thermodynamics basically states… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”