D'Alembert's Principle for a Particle and a System

D'Alembert's principle rewrites equations of dynamics in a formally equilibrium-like form by adding inertia forces to the applied forces and constraint reactions.

Inertia force of a particle

For a particle of mass $m$ with acceleration $\vec a$, define the inertia force:

$$\vec F^{in}=-m\vec a.$$

This is a computational construct, not an additional physical interaction with another body.

D'Alembert's principle for a particle

The equation $m\vec a=\sum\vec F$ may be rewritten as:

$$\sum\vec F+\vec F^{in}=0.$$

The resulting form resembles static equilibrium even though the particle may be accelerating.

Mechanical system

For every particle of a system, introduce $\vec F_i^{in}=-m_i\vec a_i$. The applied forces, constraint reactions, and inertia forces then form a formally balanced system in the sense of D'Alembert's principle.

Resultant inertia force

For a system of constant mass:

$$\vec R^{in}=\sum\vec F_i^{in}=-M\vec a_C.$$

Thus the resultant inertia force is determined by the acceleration of the center of mass.

Resultant moment of inertia forces

About a selected point $O$:

$$\vec M_O^{in}=\sum\vec r_i\times\vec F_i^{in}.$$

Together with the resultant inertia force, it is useful in rigid-body dynamics and in determining support reactions.

Practical use

The method is convenient when constraint reactions must be found for a system whose motion is known. After introducing inertia forces, equilibrium-style equations may be used, provided all required inertial terms are included correctly.

Example

A 5 kg body translates with acceleration 3 m/s² to the right. Its inertia force has magnitude 15 N and points to the left. In D'Alembert's equation it is included together with the real external forces.

Common mistakes

• treating inertia force as an ordinary interaction force;
• directing $\vec F^{in}$ along acceleration instead of opposite to it;
• omitting moments of inertia forces in rotational motion;
• using static equilibrium equations without all required inertial terms.