Concept in Lagrangian mechanics
In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces Fi, i = 1, …, n, acting on a system that has its configuration defined in terms of generalized coordinates. In the formulation of virtual work, each generalized force is the coefficient of the variation of a generalized coordinate.
Virtual work
Generalized forces can be obtained from the computation of the virtual work, δW, of the applied forces.[1]: 265
The virtual work of the forces, Fi, acting on the particles Pi, i = 1, ..., n, is given by
![{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8861e2a18fec11184acdf38c4d9c31f97131ee6b)
where δri is the virtual displacement of the particle Pi.
Generalized coordinates
Let the position vectors of each of the particles, ri, be a function of the generalized coordinates, qj, j = 1, ..., m. Then the virtual displacements δri are given by
![{\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j},\quad i=1,\ldots ,n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/755ca1be73e83b731fe1b2ab166e0f662b469713)
where δqj is the virtual displacement of the generalized coordinate qj.
The virtual work for the system of particles becomes
![{\displaystyle \delta W=\mathbf {F} _{1}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{1}}{\partial q_{j}}}\delta q_{j}+\ldots +\mathbf {F} _{n}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{n}}{\partial q_{j}}}\delta q_{j}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53ff8e190ab2fc599092163724770a01c60332e)
Collect the coefficients of δqj so that
![{\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{1}}}\delta q_{1}+\ldots +\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{m}}}\delta q_{m}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc9d21109c887a8a12ce098521ff2462c5066823)
Generalized forces
The virtual work of a system of particles can be written in the form
![{\displaystyle \delta W=Q_{1}\delta q_{1}+\ldots +Q_{m}\delta q_{m},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca828023cb817917d10be78c407ae2ddeedac6b7)
where
![{\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}},\quad j=1,\ldots ,m,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6faf2b9a2cf6ac396b113216b8a1e4fa23ad9085)
are called the generalized forces associated with the generalized coordinates qj, j = 1, ..., m.
In the application of the principle of virtual work it is often convenient to obtain virtual displacements from the velocities of the system. For the n particle system, let the velocity of each particle Pi be Vi, then the virtual displacement δri can also be written in the form[2]
![{\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}\delta q_{j},\quad i=1,\ldots ,n.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/244bc66ce82ad848a6ffc95e2617e912efe1ab36)
This means that the generalized force, Qj, can also be determined as
![{\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31f75c4c76285bff1bdf10808ab635bf7983ad41)
D'Alembert's principle
D'Alembert formulated the dynamics of a particle as the equilibrium of the applied forces with an inertia force (apparent force), called D'Alembert's principle. The inertia force of a particle, Pi, of mass mi is
![{\displaystyle \mathbf {F} _{i}^{*}=-m_{i}\mathbf {A} _{i},\quad i=1,\ldots ,n,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afe4fb8b4b890e2fcac4af4b7ebb8d2676cfd864)
where Ai is the acceleration of the particle.
If the configuration of the particle system depends on the generalized coordinates qj, j = 1, ..., m, then the generalized inertia force is given by
![{\displaystyle Q_{j}^{*}=\sum _{i=1}^{n}\mathbf {F} _{i}^{*}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b0262d312c695b5f1f0bb6e08df742e01e1810f)
D'Alembert's form of the principle of virtual work yields
![{\displaystyle \delta W=(Q_{1}+Q_{1}^{*})\delta q_{1}+\ldots +(Q_{m}+Q_{m}^{*})\delta q_{m}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b632b0a9c2dfeb67e9c0bb370309d426169ae81)
References
- ^ Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4.
- ^ T. R. Kane and D. A. Levinson, Dynamics, Theory and Applications, McGraw-Hill, NY, 2005.
See also
- Lagrangian mechanics
- Generalized coordinates
- Degrees of freedom (physics and chemistry)
- Virtual work