D'Alembert's principle explained

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, and Italian-French mathematician Joseph Louis Lagrange. D'Alembert's principle generalizes the principle of virtual work from static to dynamical systems by introducing forces of inertia which, when added to the applied forces in a system, result in dynamic equilibrium.^{[1] [2]}

D'Alembert's principle can be applied in cases of kinematic constraints that depend on velocities.^[1] The principle does not apply for irreversible displacements, such as sliding friction, and more general specification of the irreversibility is required.^{[3] [4]}

Statement of the principle

The principle states that the sum of the differences between the forces acting on a system of massive particles and the time derivatives of the momenta of the system itself projected onto any virtual displacement consistent with the constraints of the system is zero. Thus, in mathematical notation, d'Alembert's principle is written as follows,$$\backslash sum\_i\; (\backslash mathbf\; F\_i\; -\; m\_i\; \backslash dot\backslash mathbf\_i\; -\; \backslash dot\_i\backslash mathbf\_i)\backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; =\; 0,$$

where:

i is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,F _{i} is the total applied force (excluding constraint forces) on the i -th particle,m _{i} is the mass of the i -th particle,v _{i} is the velocity of the i -th particle,\delta r _{i} is the virtual displacement of the i -th particle, consistent with the constraints.Q _{j} need not include constraint forces. It is equivalent to the somewhat more cumbersome Gauss's principle of least constraint.

Derivations

General case with variable mass

The general statement of d'Alembert's principle mentions "the time derivatives of the momenta of the system." By Newton's second law, the first time derivative of momentum is the force. The momentum of the

i -th mass is the product of its mass and velocity:$$\backslash mathbf\; p\_i\; =\; m\_i\; \backslash mathbf\; v\_i$$and its time derivative is$$\backslash dot\_i\; =\; \backslash dot\_i\; \backslash mathbf\_i\; +\; m\_i\; \backslash dot\_i.$$

In many applications, the masses are constant and this equation reduces to$$\backslash dot\_i\; =\; m\_i\; \backslash dot\_i\; =\; m\_i\; \backslash mathbf\_i.$$

However, some applications involve changing masses (for example, chains being rolled up or being unrolled) and in those cases both terms

• m _{i} v _{i} and m _{i} • v _{i} have to remain present, giving$$\backslash sum\_\; (\backslash mathbf\; \_\; -\; m\_i\; \backslash mathbf\_i\; -\; \backslash dot\_i\; \backslash mathbf\_i)\backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; =\; 0.$$

Special case with constant mass

Consider Newton's law for a system of particles of constant mass,

i . The total force on each particle is^[6] $$\backslash mathbf\; \_^\; =\; m\_i\; \backslash mathbf\; \_i,$$whereF ^{(T)} _{i} are the total forces acting on the system's particles,m _{i} 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:^[6] $$\backslash mathbf\; \_^\; -\; m\_i\; \backslash mathbf\; \_i\; =\; \backslash mathbf\; 0.$$

Considering the virtual work,

\delta W , done by the total and inertial forces together through an arbitrary virtual displacement, \delta r _{i} , of the system leads to a zero identity, since the forces involved sum to zero for each particle.^[6]

$$\backslash delta\; W\; =\; \backslash sum\_\; \backslash mathbf\; \_^\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; -\; \backslash sum\_\; m\_i\; \backslash mathbf\_i\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; =\; 0$$

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,

F _{i} , and constraint forces, C _{i} , yields^[6] $$\backslash delta\; W\; =\; \backslash sum\_\; \backslash mathbf\; \_\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; +\; \backslash sum\_\; \backslash mathbf\; \_\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; -\; \backslash sum\_\; m\_i\; \backslash mathbf\_i\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; =\; 0.$$

If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces (which is not usually the case, so this derivation works only for special cases), the constraint forces don't do any work, $\backslash sum\_\; \backslash mathbf\; \_\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; =\; 0$. Such displacements are said to be consistent with the constraints.^[7] 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:^[6] $$\backslash delta\; W\; =\; \backslash sum\_\; (\backslash mathbf\; \_\; -\; m\_i\; \backslash mathbf\_i)\backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; =\; 0.$$

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). Even in the course of Fundamentals of Dynamics and Kinematics of machines, this principle helps in analyzing the forces that act on a link of a mechanism when it is in motion. In textbooks of engineering dynamics, this is sometimes referred to as d'Alembert's principle.

Some educators caution that attempts to use d'Alembert inertial mechanics lead students to make frequent sign errors.^[8] A potential cause for these errors is the sign of the inertial forces. Inertial forces can be used to describe an apparent force in a non-inertial reference frame that has an acceleration

a with respect to an inertial reference frame. In such a non-inertial reference frame, a mass that is at rest and has zero acceleration in an inertial reference system, because no forces are acting on it, will still have an acceleration - a and an apparent inertial, or pseudo or fictitious force - m a will seem to act on it: in this situation the inertial force has a minus sign.

Dynamic equilibrium

D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of

n rigid bodies with m generalized coordinates requires$$\backslash delta\; W\; =\; \backslash left(Q\_1\; +\; Q\_1^*\backslash right)\; \backslash delta\; q\_1\; +\; \backslash dots\; +\; \backslash left(Q\_m\; +\; Q\_m^*\backslash right)\; \backslash delta\; q\_m\; =\; 0,$$for any set of virtual displacements \delta q _{j} with Q _{j} being a generalized applied force and Q ^{*} _{j} being a generalized inertia force. This condition yields m equations,$$Q\_j\; +\; Q^*\_j\; =\; 0,\; \backslash quad\; j=1,\; \backslash ldots,\; m,$$ which can also be written as$$\backslash frac\; \backslash frac\; -\backslash frac\; =\; Q\_j,\; \backslash quad\; j=1,\backslash ldots,m.$$The result is a set of m equations of motion that define the dynamics of the rigid body system.

Formulation using the Lagrangian

D'Alembert's principle can be rewritten in terms of the Lagrangian L=T-V of the system as a generalized version of Hamilton's principle as follows,$$\backslash delta\; \backslash int\_^\; L(\backslash mathbf,\; \backslash dot,t)\; dt\; +\; \backslash sum\_i\backslash int\_^\; \backslash mathbf\_i\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; dt=\; 0,$$where:

r = ( r _{1,...,} r _{N)} F _{i} are the applied forces\delta r _{i} is the virtual displacement of the i -th particle, consistent with the constraints \sum _{iC} _{i} ⋅ \delta r _{i=0}

• the critical curve satisfies the constraints

\sum _{iC} _{i} ⋅ • r _{i=0} With the Lagrangian$$L(\backslash mathbf,\; \backslash dot,t)\; =\; \backslash sum\_i\; \backslash frac\; m\_i\; \backslash dot\; \_i^2,$$the previous statement of d'Alembert principle is recovered.

Generalization for thermodynamics

An extension of d'Alembert's principle can be used in thermodynamics.^[4] For instance, for an adiabatically closed thermodynamic system described by a Lagrangian depending on a single entropy S and with constant masses

m _{i} , such as$$L(\backslash mathbf,\; \backslash dot,S,t)\; =\; \backslash sum\_i\; \backslash frac\; m\_i\; \backslash dot\; \_i^2\; -\; V(\backslash mathbf,S),$$it is written as follows$$\backslash delta\; \backslash int\_^\; L(\backslash mathbf,\; \backslash dot,S,t)\; dt\; +\; \backslash sum\_i\backslash int\_^\; \backslash mathbf\_i\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i\; dt=\; 0,$$where the previous constraints $\backslash sum\_i\backslash mathbf\_i\; \backslash cdot\; \backslash delta\; \backslash mathbf\; \_i=0$ and $\backslash sum\_i\backslash mathbf\_i\; \backslash cdot\; \backslash dot\; \backslash mathbf\; \_i=0$ are generalized to involve the entropy as:\sum _{iC} _{i} ⋅ \delta r _{i+T} \delta S =0 \sum _{iC} _{i} ⋅ • r _{i+T} • S =0 . Here T = \partial V/\partial S is the temperature of the system, F _{i} are the external forces, C _{i} are the internal dissipative forces. It results in the mechanical and thermal balance equations:^[4] $$m\_i\backslash mathbf\_i=-\; \backslash frac+\; \backslash mathbf\_i+\backslash mathbf\_i,\; \backslash ;\backslash ;i=1,...,N\; \backslash qquad\; \backslash qquad\; T\; \backslash dot\; S\; =\; -\backslash sum\_i\backslash mathbf\_i\; \backslash cdot\; \backslash dot\; \backslash mathbf\; \_i.$$Typical applications of the principle include thermo-mechanical systems, membrane transport, and chemical reactions.

For

\delta S = • S =0 the classical d'Alembert principle and equations are recovered.

References

1. Book: Lanczos, Cornelius. 1964. Variational principles of mechanics. 92. Toronto, University of Toronto Press .
2. Book: d'Alembert, Jean le Rond. 1743. Traité de dynamique. 50–51.
3. Udwadia . F. E. . Kalaba . R. E. . On the Foundations of Analytical Dynamics . Intl. Journ. Nonlinear Mechanics . 37 . 6 . 1079–1090 . 2002 . 10.1016/S0020-7462(01)00033-6 . 2002IJNLM..37.1079U . dead . https://web.archive.org/web/20100613031338/http://ae-www.usc.edu/bio/udwadia/papers/On_foundation_of_analytical_dynamics.pdf . 2010-06-13 . 10.1.1.174.5726 .
4. Gay-Balmaz . François . Yoshimura . Hiroaki . 2018 . From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective . Entropy . en . 21 . 1 . 8 . 10.3390/e21010008 . 33266724 . 7514189 . 1904.03738 . 2018Entrp..21....8G . 1099-4300 . free .
5. [Arnold Sommerfeld]
6. Book: Torby, Bruce . Advanced Dynamics for Engineers . HRW Series in Mechanical Engineering . 1984 . CBS College Publishing . United States of America . 978-0-03-063366-9 . Energy Methods.
7. Teaching Students Work and Virtual Work Method in Statics:A Guiding Strategy with Illustrative Examples . Ing-Chang. Jong. 2005 American Society for Engineering Education Annual Conference & Exposition. 2005. June 24, 2014. Improving Mechanics of Materials.
8. https://www.engineering.cornell.edu/faculty-directory/andy-ruina Ruina, Andy L.