Question

In regards to the subject of Vibrations: Equations of Motion may be formulated using Newton’s Second Law force and moment balance or Lagrange’s Equations. Briefly discuss how to decide which method to use.

Answer 1

Equations of Motion using Newton's Law of Motion

In deriving the equations of motion using Newtonian Mechanics, the constraint forces must be known explicitly to derive the accleration, i.e for a single degree of freedom system

Equation of motion using Lagrange's Equations of Motion

In Lagrange approach, the constraint forces need not be known explicitly. Instead, a quantity called Lagrangian(L) is evaluated based on the path a body is constrained to follow.

Where

is the total kinetic energy of the system

V is the total potential energy of the system considering the conservative forces(e.g gravity).

The equations of motion are derived using the following relation.

For a single degree of freedom system

Advantages of Lagrange Equations of Motion

In Newtonian Mechanics, the constraint forces are vectors which need to be considered for equations of motion. For a multibody systems this may give rise to complexity. However, in Lagrange approach, the Lagrangial L is a scaler and is more convenient to use in a multibody system.

However, with the Newtonian approach one can take care of non conservative forces like friction forces. But introduction of nonconservative forces in Lagrangian approach makes it more complex(also limited ).

Also Newtonian approach is more convenient in cartesian coordinate system, but becomes complex in other coordinate systems.

Rules to follow for selection

1. For motion in cartesian coordinate system and with lower number of bodies (typically 2 or less) use Newtonian approach.

2. For motion in polar or spherical coordinate system or with about 3 or more number of bodies use Lagrangian approach.