Discrete and Continuous Systems

• CE 137 - Structural Dynamics and Earthquake Engineering. Chapter 1 is an introduction to structural dynamics

Learning Objectives

• Differentiate between static and dynamic loads.
• The different types of dynamic loads should be understood.
• Define structural dynamics.
• Structural dynamics is related to structural analysis.
• Explain the dynamic response of typical structures.
• Define vibration.
• Identify the types of vibratory motion.
• Explain the functional elements of single degree of freedom (SDOF) systems.
• Derive the equations of motion.

Structural Dynamics

• Branch of structural engineering that studies the dynamic properties of structures, including natural frequency and damping mechanism.
• Examines the effects of dynamic loads or excitations on structures in terms of stresses and deformations, velocities, or acceleration.
• The analysis can be linear or nonlinear.
• Linear analysis means material stresses are within the elastic limit.
• Nonlinear analysis means the material exhibits permanent sets.

Static vs. Dynamic Load

• Static loads do not accelerate.
• The sum of forces equals 0.
• Static loads can be moving, but velocity is constant.
• Static loads do not vary with time.
• Dynamic loads accelerate and produce vibration.
• The sum of forces is equal to mass times acceleration.
• Dynamic loads are time-dependent.

Types of dynamic loads

• Live loads
• Blast
• Impact
• Earthquake
• Wind

Overview of structural dynamics

• EXCITATION: Live Load, Wind, Seismic, Blast/Impact, and Ocean
• STRUCTURE: Bridge, Building, Dam, Pipelines, Etc.
• RESPONSE: Body deforms or Body does not deform

Dynamic Response

• Deformations (displacement, velocity, and acceleration)
• Element Forces (stresses, strains)

Relation of Structural Dynamics to Structural Analysis

• Static Linear: ES 11, ES 13, CE 131, CE132
• Static Nonlinear: Undergraduate Thesis, Master Course
• Dynamic Linear: CE 137
• Dynamic Nonlinear: Undergraduate Thesis, Master Course

Definition of Vibration

• Vibration is a type of dynamic behavior where a system oscillates about an equilibrium position
• "Oscillate" means forward and backward translation or rotation of a structure around a certain position.
• "Dynamic" may be defined as time-varying, expressed as y = uo sin (ωnt + θ)

Types of Vibration

• Periodic Motion: motion which repeats itself at regular intervals of time
• Non-periodic Motion: does not repeat itself at constant intervals

Simple Harmonic Motion

• Motion characterized by sinusoidal motion at a constant frequency
• Response is defined as the projection of P on the y-axis

Simple Harmonic Motion Formulae

• T = 1/f = 2π√(m/k)
• Angular frequency (ω): Rate at which the motion oscillates in one complete revolution (rad/s)
• Frequency (f): Number of cycles per unit time (cps or Hz)
• Amplitude (A): Maximum response of a periodic motion
• Period (T): Time it takes to complete one cycle of motion.

Types of Vibrating System

• Rigid systems have no strains and all points move in phase.
• Compliant systems have points that move differently and are characterized by distributed mass.

Compliant system

• Discrete: mass is considered to be concentrated at a finite number of locations
• Continuous/Distributed: mass is distributed throughout the system

Discrete systems

• Single Degree of Freedom Systems (SDOF)
• Multi-Degree of Freedom Systems (SDOF)

Elements of Discrete Systems

• Spring element (k): The stiffness of the structure
• Damping element (c): The energy dissipator of the structure
• Mass element (m): The inertial characteristics of the structure

Inertia Force and Mass element

• Inertia: "object's amount of resistance to a change in velocity (which is quantified by its mass), or sometimes its momentum" (Newton's first law of motion)
• Mass element represents the mass and inertial characteristics of the structure.
• f₁ = mü, where m = mass, ü = acceleration

Structural resisting or Spring Force

• Structural resisting or Spring Force is represented as: fs = ku

Spring Element

• Represents the restoring force and the potential energy capacity of a structure

Damping

• When allowed to freely oscillate, a pendulum will come to a stop because of damping
• With damping, the amplitudes decrease

Damping Element

• Represents the energy dissipation in a structure
• Most of the energy dissipation comes from thermal effects of repeated elastic straining and internal friction when a solid is deformed.
• In actual building structures, mechanisms of energy dissipation include friction at steel connections, opening and closing of microcracks in concrete, and friction between structure and non-structural elements such as partition walls. These mechanisms are very difficult to quantify for an actual structural system.

Damping Force

• Damping Force is represented as: fp = cù
• c = viscous damping coefficient (force x time/length)
• ú = velocity

Equations of Motion

• Can be formulated using:
  • Dynamic Equilibrium (D' Alembert's principle)
  • Principle of Virtual Work
  • Hamilton's Principle (Lagrange's equation)

Equations of Motion - General

• ma = ΣF
• mü = p(t) - fs - fD
• mü = p(t) - ku - cù
• Equation of motion for an SDOF system subjected to an external force that varies with time, t:
  • mü + cù + ku = p(t)

Principle of Virtual Work

• The equations of motion are obtained by introducing virtual displacements corresponding to each degree of freedom and equating the resulting work done to zero.
• For a system that is in equilibrium, the work done by all the forces during an assumed displacement (virtual displacement) which is compatible with the system constraints is equal to zero, ΣδW = 0.

Ground Motion Excitation

• No applied force
• Ground itself is actually moving
• Total structural displacement, ut:
  • u = relative displacement between the structure and the ground
  • ug = ground displacement
• Ground Motion Excitation:
  • ut = ug + u
  • üt = üg +ü
  • ma = ΣF (Newton's 2nd law of motion)
  • m[üg + ü] = −ku – củ
  • mü + ku + cù =-müg

Summary

• Static loads compared to dynamic loads
• Types of dynamic loads (live, wind, earthquake, blast and impact)
• Structural dynamics are the properties of a structure, response of structures under excitation.
• Structural dynamics focus on dynamic linear behavior.
• The dynamic response of typical structures includes deformation and stress.
• Vibration: dynamic behavior, oscillation with respect to equilibrium position.
• Types of vibratory motion: periodic vs. nonperiodic, simple harmonic motion.
• The 3 elements of discrete systems are mass, spring, and damping in an SDOF system.
• Equations of motion can be general and with ground motion excitation.

Explore the key differences between discrete and continuous systems in compliant systems. Learn about multi-degree of freedom systems and the two main categories of discrete systems. Understand structural dynamics, vibratory motion, and the components of SDOF systems.