D’Alembert’s Principle

D’Alembert’s principle states that for a system of mass of particles, the sum of difference of the force acting on the system and the time derivatives of the momenta is zero when projected onto any virtual displacement. It is also known as Lagrange-d’Alembert principle, named after French mathematician and physicist Jean le Rond d’Alembert.

D’Alembert’s principle mathematical representation

D’Alembert’s principle can be explained mathematically in following way:

\(\sum_{i}(F_{i}-m_{i}a_{i})\delta r_{i}=0\)

Where,

i: integral used for the identification of variable corresponding to particular particle in the system

Fi: total applied force on the ith place

mi: mass of the ith particles

ai: acceleration of ith particles

miai: time derivative representation

𝜹ri: virtual displacement of ith particle

D’Alembert derivation for special cases

Using D’Alembert’s mathematical formula, virtual work can be shown equal to D’Alembert’s principle which is equal to zero.

Following is the derivation:

\(F_{i}^{(T)} = m_{i}a_{i}\) (total force on each particle)

\(F_{i}^{(T)}-m_{i}a_{i}=0\) (inertial force is moved left, representing quasi-static equilibrium)

\(\delta W = \sum_{i}F_{i}^{(T)}.\delta r_{i}-\sum_{i}m_{i}a_{i}.\delta r_{i}=0\) (equated to virtual work )

\(\delta W = \sum_{i}F_{i}.\delta r_{i}+\sum_{i}C_{i}.\delta r_{i}-\sum_{i}m_{i}a_{i}.\delta r_{i}=0\) (separation of applied force and constraint force)

\(\delta W = \sum_{i}(F_{i}-m_{i}a_{i}).\delta r_{i}=0\) (final equation)

D’Alembert principle examples using inertial forces

  • 1D motion of rigid body: T – W = ma or T = W + ma where T is tension force of wire, W is weight of sample model and ma is acceleration force.
  • 2D motion of rigid body: For an object moving in x-y plane following is the mathematical representation: Fi= -mrc where Fi is the total force applied on the ith place, m mass of the body and rc is position vector of centre of mass of the body.

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