Spring2024

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Degree of freedom

Degrees of Freedom The minimum number of real numbers we need to specify in oder to completely know the configuration of the mechanical system.

Remark : The DoF are also called co-ordinates.

  1. 1 particle in $\mathbb{R}^2$ (2 DoF)
  2. 1 particle in $\mathbb{R}^3$ (3 DoF)
  3. A particle in a hoop. The hoop lies in $\mathbb{R}^2$ (1 DoF)
    1. A particle in a hoop. The hoop lies in in $\mathbb{R}^3$. (1 DoF)
  4. Two masses connected by a rigid link. The center point is fixed. The whole assembly lies in 2D (1 DoF).
    1. Now the center point is free (3 Dof)
    2. The whole assembly lies in 3D, with the center point fixed. (3 DoF)
    3. The whole assembly lies in 3D, with the center point free to two move. (6 DoF).
  5. A mass $M$ is constrained to lie in a slot. It is connected by a spring to the slot. There is a mass $m$ connected by a rigid link to the $M$ through a hinge (2D) joint. (2 DoF)
  6. A pulley with two hanging masses (1 DoF)
  7. A rigid disk in 2D (3 DoF)
  8. A rigid disk in 2D with center fixed. 1Dof
  9. A disk rolling down the wedge, the wedge is fixed. (1 DoF)

Alt text

Generalized co-ordinates: A set of generalized co-ordinates is a set of real numbers (parameters) that when provided enable us to reconstruct the configuration of the mechanical system, and all configurations can be described this way. The generalized co-ordinates are customerily written as $(q_1,q_2, \ldots, q_n)$, or simply as $q_j$

Proper set of generalized