# Single degree of freedom system

Single degree of freedom system

• Single degree freedom system consists of a mass m concentrated at the roof level, a massless frame that provides stiffness to the system
• And a viscous damper (also known as a dashpot) that dissipates vibrational energy of the system
• The beam and columns are assumed to be inextensible axially
• This system may be considered as an idealization of a one-story structure.
• Each structural member (beam, column, wall, etc.) of the actual structure contributes to the inertial (mass)
• Elastic (stiffness or flexibility), and energy dissipation (damping) properties of the structure.
• In the idealized system, however, each of these properties is concentrated in three separate, pure components: mass component, stiffness component, and damping component
• The number of independent displacements required to define the displaced positions of all the masses relative to their original position is called the number of degrees of freedom (DOFs) for dynamic analysis
• More DOFs are typically necessary to define the stiffness properties of a structure compared to the DOFs necessary for representing inertial properties
• Consider the one-story frame of Fig. constrained to move only in the direction of the excitation
• The static analysis problem has to be formulated with three DOFs lateral displacement.
• And two joint rotations to determine the lateral stiffness of the frame.
• In contrast, the structure has only one DOF lateral displacement for dynamic analysis if it is idealized with mass concentrated at one location, typically the roof level.
• Thus we call this a single-degree-of-freedom (SDF) system.

Below Single-degree-of-freedom system can understand  by given Fig. due to applied force P(t) in lateral direction and its displacement by earthquake load Two types of dynamic excitation will be considered:

1.  external force p(t) in the lateral direction (Fig. a)
2. Earthquake-induced ground motion ug(t) (Fig. b)

In both cases u denotes the relative displacement between the mass and the base of the structure.