Generalized forces#D'Alembert's principle

Generalized forces can be obtained from the computation of the virtual work, {{mvar|δW}}, of the applied forces.{{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 |isbn=0-03-063366-4 |chapter=Energy Methods}}{{rp|265}}

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

$$\backslash delta\; W\; =\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i$$

where {{math|δr_i}} is the virtual displacement of the particle {{mvar|P_i}}.

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

$$\backslash delta\; \backslash mathbf\{r\}\_i\; =\; \backslash sum\_\{j=1\}^m\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; \{r\}\_i\}\; \{\backslash partial\; q\_j\}\; \backslash delta\; q\_j,\backslash quad\; i=1,\backslash ldots,\; n,$$

where {{mvar|δq_j}} is the virtual displacement of the generalized coordinate {{mvar|q_j}}.

The virtual work for the system of particles becomes

$$\backslash delta\; W\; =\; \backslash mathbf\; F\_1\; \backslash cdot\; \backslash sum\_\{j=1\}^m\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; r\_1\}\; \{\backslash partial\; q\_j\}\; \backslash delta\; q\_j\; +\; \backslash dots\; +\; \backslash mathbf\; F\_n\; \backslash cdot\; \backslash sum\_\{j=1\}^m\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; r\_n\}\; \{\backslash partial\; q\_j\}\; \backslash delta\; q\_j.$$

Collect the coefficients of {{mvar|δq_j}} so that

$$\backslash delta\; W\; =\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; r\_i\}\; \{\backslash partial\; q\_1\}\; \backslash delta\; q\_1\; +\; \backslash dots\; +\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; r\_i\}\; \{\backslash partial\; q\_m\}\; \backslash delta\; q\_m.$$

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

$$\backslash delta\; W\; =\; Q\_1\backslash delta\; q\_1\; +\; \backslash dots\; +\; Q\_m\backslash delta\; q\_m,$$

where

$$Q\_j\; =\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; r\_i\}\; \{\backslash partial\; q\_j\},\backslash quad\; j=1,\backslash ldots,\; m,$$

are called the generalized forces associated with the generalized coordinates {{math|1=q_j, 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 P_i be {{math|V_i}}, then the virtual displacement {{math|δr_i}} can also be written in the formT. R. Kane and D. A. Levinson, [https://www.amazon.com/Dynamics-Theory-Applications-Mechanical-Engineering/dp/0070378460 Dynamics, Theory and Applications], McGraw-Hill, NY, 2005.

$$\backslash delta\; \backslash mathbf\; r\_i\; =\; \backslash sum\_\{j=1\}^m\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; V\_i\}\; \{\backslash partial\; \backslash dot\; q\_j\}\; \backslash delta\; q\_j,\backslash quad\; i=1,\backslash ldots,\; n.$$

This means that the generalized force, {{mvar|Q_j}}, can also be determined as

$$Q\_j\; =\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\; F\_i\; \backslash cdot\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; V\_i\}\; \{\backslash partial\; \backslash dot\{q\}\_j\},\; \backslash quad\; j=1,\backslash ldots,\; m.$$

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, {{mvar|P_i}}, of mass {{mvar|m_i}} is

$$\backslash mathbf\; F\_i^*=-m\_i\backslash mathbf\; A\_i,\backslash quad\; i=1,\backslash ldots,\; n,$$

where {{math|A_i}} is the acceleration of the particle.

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

$$Q^*\_j\; =\; \backslash sum\_\{i=1\}^n\; \backslash mathbf\; F^*\_\{i\}\; \backslash cdot\; \backslash frac\; \{\backslash partial\; \backslash mathbf\; V\_i\}\; \{\backslash partial\; \backslash dot\; q\_j\},\backslash quad\; j=1,\backslash ldots,\; m.$$

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

$$\backslash delta\; W\; =\; (Q\_1\; +\; Q^*\_1)\backslash delta\; q\_1\; +\; \backslash dots\; +\; (Q\_m\; +\; Q^*\_m)\backslash delta\; q\_m.$$

• Lagrangian mechanics
• Generalized coordinates
• Degrees of freedom (physics and chemistry)
• Virtual work

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Category:Mechanical quantities

Category:Classical mechanics

Category:Lagrangian mechanics