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PART I: SINGLE-DEGREE-FREEDOM SYSTEMChapter I: Equation of Motion, Problem Statement and Solution Methods

Terms:Viscous Damper-Commonly used Damping ElementLateral- HorizontalExcitation-Applied Force/Ground MotionAmplitude- Measurement that indicates the movement or vibration of something.Damping- Process by vibration steadily diminishes in amplitude

Joint Rotation-Vibration-Inertia Force- a force equal to the product of mass times its acceleration and acting in a direction opposite to the acceleration.Earthquake- A shaking, trembling, or concussion of the earth, due to subterranean causes, often accompanied by a rumbling noise.

SIMPLE STRUCTURE Idealized as a concentrated or lumped mass m supported by a massless structure with stiffness k in the lateral direction.

IDEALIZED STRUCTURE

Degrees of Freedom (DOF) The number of independent displacements required to dene the displaced positions of all the masses relative to their original position.

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.

SINGLE-DEGREE-OF-FREEDOM SYSTEM

Each structural member (beam, column, wall, etc.) of the actual structure contributes to the inertial (mass), elastic (stiffness or exibility), 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.

FORCEDISPLACEMENT RELATION

Linearly Elastic Systems:

Inelastic SystemsThe initial loading curve is nonlinear at the larger amplitudes of deformation, and the unloading and reloading curves differ from the initial loading branch; such a system is said to be inelastic.

This implies that the forcedeformation relation is path dependent, i.e., it depends on whether the deformation is increasing or decreasing. Thus the resisting force is an implicit function of deformation:

DAMPING FORCE The energy of the vibrating system is dissipated by various mechanisms, and often more than one mechanism. In a vibrating building these include friction at steel connections, opening and closing of microcracks in concrete, and friction between the structure itself and nonstructural elements such as partition walls.

EQUATION OF MOTION: EXTERNAL FORCE

Using Newtons Second Law of Motion

LINEARLY ELASTIC SYSTEMINELASTIC SYSTEMOne of Jean Le Rond dAlembert work.It states that with inertia forces included, a system is in equilibrium at each time instant. Thus a freebody diagram of a moving mass can be drawn, and principles of statics can be used to develop the equation of motion. Dynamic Equilibrium

Stiffness, Damping, and Mass Components

MASS SPRING DUMPER SYSTEM

EQUATION OF MOTION: EARTHQUAKE EXCITATION

= total (or absolute) displacement of the mass = displacement of the ground is denoted= relative displacement between the mass and ground

Subjected to Earthquake ExcitationEquation Dynamic Equilibrium:

Linearly Elastic StructureInelastic Structure