Physics:Generalized forces
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
- [math]\displaystyle{ \delta W = \sum_{i=1}^n \mathbf F_i \cdot \delta \mathbf r_i }[/math]
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
- [math]\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, }[/math]
where δqj is the virtual displacement of the generalized coordinate qj.
The virtual work for the system of particles becomes
- [math]\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. }[/math]
Collect the coefficients of δqj so that
- [math]\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. }[/math]
Generalized forces
The virtual work of a system of particles can be written in the form
- [math]\displaystyle{ \delta W = Q_1\delta q_1 + \ldots + Q_m\delta q_m, }[/math]
where
- [math]\displaystyle{ Q_j = \sum_{i=1}^n \mathbf F_i \cdot \frac {\partial \mathbf r_i} {\partial q_j},\quad j=1,\ldots, m, }[/math]
are called the generalized forces associated with the generalized coordinates qj, j = 1, ..., m.
Velocity formulation
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]
- [math]\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. }[/math]
This means that the generalized force, Qj, can also be determined as
- [math]\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. }[/math]
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
- [math]\displaystyle{ \mathbf F_i^*=-m_i\mathbf A_i,\quad i=1,\ldots, n, }[/math]
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
- [math]\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. }[/math]
D'Alembert's form of the principle of virtual work yields
- [math]\displaystyle{ \delta W = (Q_1+Q^*_1)\delta q_1 + \ldots + (Q_m+Q^*_m)\delta q_m. }[/math]
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
![]() | Original source: https://en.wikipedia.org/wiki/Generalized forces.
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