D'Alembert's principle

Jean d'Alembert

D'Alembert's principle, also known as the Lagrange-D'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. The principle states that the sum of the differences between the forces acting on a system and the time derivatives of the momenta of the system itself along a virtual displacement consistent with the constraints of the system, is zero. Thus, in symbols d'Alembert's principle is,

$\sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i})\cdot \delta \mathbf {r} _{i}=0$.

$\mathbf {F} _{i}$ are the applied forces

$\delta \mathbf {r} _{i}$ is the virtual displacement of the system, consistent with the constraints

$m_{i}$ are the masses of the particles in the system

$\mathbf {a} _{i}$ are the accelerations of the particles in the system

$m_{i}\mathbf {a} _{i}$ together as products represent the time derivatives of the system momenta

$i$ is an integer used to indicate (via subscript) a variable corresponding to a particular particle

It is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamilton's principle, avoiding restriction to holonomic systems. If the negative terms in accelerations are recognized as inertial forces, the statement of d'Alembert's principle becomes The total virtual work of the impressed forces plus the inertial forces vanishes for reversible displacements.

This above equation is often called d'Alembert's principle, but it was first written in this variational form by Joseph Louis Lagrange. D'Alembert's contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the generalized forces ${\mathbf {Q} }_{j}$ need not include constraint forces.

Derivation

Consider Newton's law for a system of particles, i. The total force on each particle is

$\mathbf {F} _{i}^{(T)}=m_{i}\mathbf {a} _{i}$.

$\mathbf {F} _{i}^{(T)}$ are the total forces acting on the system's particles

$m_{i}\mathbf {a} _{i}$ are the inertial forces that result from the total forces

Moving the inertial forces to the left gives an expression that can be considered to represent quasi-static equilibrium, but which is really just a small algebraic manipulation of Newton's law:

$\mathbf {F} _{i}^{(T)}-m_{i}\mathbf {a} _{i}=\mathbf {0}$.

Considering the virtual work, $\delta W$, done by the total and inertial forces together through an arbitrary virtual displacement, $\delta \mathbf {r} _{i}$, of the system leads to a zero identity, since the forces involved sum to zero for each particle.

$\delta W=\sum _{i}\mathbf {F} _{i}^{(T)}\cdot \delta \mathbf {r} _{i}-\sum _{i}m_{i}\mathbf {a} _{i}\cdot \delta \mathbf {r} _{i}=0$

At this point it should be noted that the original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements. Separating the total forces into applied forces, $\mathbf {F} _{i}$, and constraint forces, $\mathbf {C} _{i}$, yields

$\delta W=\sum _{i}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}+\sum _{i}\mathbf {C} _{i}\cdot \delta \mathbf {r} _{i}-\sum _{i}m_{i}\mathbf {a} _{i}\cdot \delta \mathbf {r} _{i}=0$

If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, the constraint forces do no work. Such displacements are said to be consistent with the constraints. This leads to the formulation of d'Alembert's principle, which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work:

$\delta W=\sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i})\cdot \delta \mathbf {r} _{i}=0$.

There is also a corresponding principle for static systems called the principle of virtual work for applied forces.

D'Alembert's principle of inertial forces

D'Alembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called "inertial force" and "inertial torque" or moment. The inertial force must act through the center of mass and the inertial torque can act anywhere. The system can then be analyzed exactly as a static system subjected to this "inertial force and moment" and the external forces. The advantage is that, in the equivalent static system' one can take moments about any point (not just the center of mass). This often leads to simpler calculations because any force (in turn) can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation (sum of moments = zero). In textbooks of engineering dynamics this is sometimes referred to as d'Alembert's principle.

Example for plane 2D motion of a rigid body

For a planar rigid body, moving in the plane of the body (the x-y plane), and subjected to forces and torques causing rotation only in this plane, the inertial force is

$\mathbf {F} _{i}=-m{\ddot {\mathbf {r} _{c}}}$

where $\mathbf {r} _{c}$ is the position vector of the centre of mass of the body, and $m$ is the mass of the body. The inertial torque (or moment) is

$T_{i}=-I{\ddot {\theta }}$

where $I$ is the moment of inertia of the body. If, in addition to the external forces and torques acting on the body, the inertia force acting through the center of mass is added and the inertial torque is added (acting around the centre of mass is as good as anywhere) the system is equivalent to one in static equilibrium. Thus the equations of static equilibrium

$\sum F_{x}=0$

$\sum F_{y}=0$

$\sum T=0$

hold. The important thing is that $\sum T$ is the sum of torques (or moments, including the inertial moment and the moment of the inertial force) taken about any point. The direct application of Newton's laws requires that the angular acceleration equation be applied only about the center of mass.