# Equations of Motion

## Single Degree of Freedom (SDOF) Systems

Writing Newton's Second Law (force equals mass times acceleration) for this system at time (t) results in:

F(t) - ku(t) = mu^{..}(t)

or:

mu^{..}(t) + ku(t) = F(t)

where:

u^{..}(t) is the acceleration of the mass at time (t) and it is equal to the second derivative of u with respect to time

k = is the stiffness of the spring

Considering damping, the above equation becomes:

mu^{..}(t) + cu^{.}(t) + ku(t) = F(t)

where:

u^{.}(t) is the velocity of the mass at time (t), and it is equal to the first derivative of u with respect to time

## Multi Degree of Freedom (MDOF) Systems

For a Multi-Degree-of-Freedom (MDOF) system, m, c, and k become matrices instead of single values and the equations of motion are expressed as:

where:

[M]: mass matrix

[K]: stiffness matrix

[C]: damping matrix

{u(t)}: displacement vector at time t (displacement components of every node)

: acceleration vector at time t (acceleration components of every node)

: velocity vector at time t (velocity components of every node)

{f(t)}: time varying load vector (force components of every node)

**Parent topic**Dynamic Analysis