Generalized forces

Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, F_i, 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.

The virtual work of the forces, F_i, acting on the particles P_i, i=1,..., n, is given by

δ W = ∑ i = 1 n F i ⋅ δ r i

where δr_i is the virtual displacement of the particle P_i.

Generalized coordinates

Let the position vectors of each of the particles, r_i, be a function of the generalized coordinates, q_j, j=1,...,m. Then the virtual displacements δr_i are given by

δ r i = ∑ j = 1 m ∂ r i ∂ q j δ q j , i = 1 , … , n ,

where δq_j is the virtual displacement of the generalized coordinate q_j.

The virtual work for the system of particles becomes

δ W = F 1 ⋅ ∑ j = 1 m ∂ r 1 ∂ q j δ q j + … + F n ⋅ ∑ j = 1 m ∂ r n ∂ q j δ q j .

Collect the coefficients of δq_j so that

δ W = ∑ i = 1 n F i ⋅ ∂ r i ∂ q 1 δ q 1 + … + ∑ i = 1 n F i ⋅ ∂ r i ∂ q m δ q m .

The virtual work of a system of particles can be written in the form

δ W = Q 1 δ q 1 + … + Q m δ q m ,

where

Q j = ∑ i = 1 n F i ⋅ ∂ r i ∂ q j , j = 1 , … , m ,

are called the generalized forces associated with the generalized coordinates q_j, 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 P_i be V_i, then the virtual displacement δr_i can also be written in the form

δ r i = ∑ j = 1 m ∂ V i ∂ q ˙ j δ q j , i = 1 , … , n .

This means that the generalized force, Q_j, can also be determined as

Q j = ∑ i = 1 n F i ⋅ ∂ V i ∂ q ˙ j , j = 1 , … , m .

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, P_i, of mass m_i is

F i ∗ = − m i A i , i = 1 , … , n ,

where A_i is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates q_j, j=1,...,m, then the generalized inertia force is given by

Q j ∗ = ∑ i = 1 n F i ∗ ⋅ ∂ V i ∂ q ˙ j , j = 1 , … , m .

D'Alembert's form of the principle of virtual work yields

δ W = ( Q 1 + Q 1 ∗ ) δ q 1 + … + ( Q m + Q m ∗ ) δ q m .

References

Generalized forces Wikipedia

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Contents

Virtual work

Generalized coordinates