Chapter 1

Basic Concepts of Structural Dynamics

1.1 The Dynamic Environment

Structural engineers are familiar with the analysis of structures for static loads in which a load is applied to the structure and a single solution is obtained for the resulting displacements and member forces. When considering the analysis of structures for dynamic loads, the term dynamic simply means “time-varying.” Hence, the loading and all aspects of the response vary with time. This results in possible solutions at each instant during the time interval under consideration. From an engineering standpoint, the maximum values of the structural response are usually the ones of particular interest, especially when considering the case of structural design.

Two different approaches, which are characterized as either deterministic or nondeterministic, can be used for evaluating the structural response to dynamic loads. If the time variation of the loading is fully known, the analysis of the structural response is referred to as a deterministic analysis. This is the case even if the loading is highly oscillatory or irregular in character. The analysis leads to a time history of the displacements in the structure corresponding to the prescribed time history of the loading. Other response parameters such as member forces and relative member displacements are then determined from the displacement history.

If the time variation of the dynamic load is not completely known but can be defined in a statistical sense, the loading is referred to as a random dynamic loading, and the analysis is referred to as nondeterministic. The nondeterministic analysis provides information about the displacements in a statistical sense, which results from the statistically defined loading. Hence, the time variation of the displacements is not determined, and other response parameters must be evaluated directly from an independent nondeterministic analysis rather than from the displacement results. Methods for nondeterministic analysis are described in books on random vibration. In this text, we only discuss methods for deterministic analysis.

1.2 Types of Dynamic Loading

Most structural systems will be subjected to some form of dynamic loading during their lifetime. The sources of these loads are many and varied. The ones that have the most effect on structures can be classified as environmental loads that arise from winds, waves, and earthquakes. A second group of dynamic loads occurs as a result of equipment motions that arise in reciprocating and rotating machines, turbines, and conveyor systems. A third group is caused by the passage of vehicles and trucks over a bridge. Blast-induced loads can arise as the result of chemical explosions or breaks in pressure vessels or pressurized transmission lines.

For the dynamic analysis of structures, deterministic loads can be divided into two categories: periodic and nonperiodic. Periodic loads have the same time variation for a large number of successive cycles. The basic periodic loading is termed simple harmonic and has a sinusoidal variation. Other forms of periodic loading are often more complex and nonharmonic. However, these can be represented by summing a sufficient number of harmonic components in a Fourier series analysis. Nonperiodic loading varies from very short duration loads (air blasts) to long-duration loads (winds or waves). An air blast caused by some form of chemical explosion generally results in a high-pressure force having a very short duration (milliseconds). Special simplified forms of analysis may be used under certain conditions for this loading, particularly for design. Earthquake loads that develop in structures as a result of ground motions at the base can have a duration that varies from a few seconds to a few minutes. In this case, general dynamic analysis procedures must be applied. Wind loads are a function of the wind velocity and the height, shape, and stiffness of the structure. These characteristics give rise to aerodynamic forces that can be either calculated or obtained from wind tunnel tests. They are usually represented as equivalent static pressures acting on the surface of the structure.

1.3 Basic Principles

The fundamental physical laws that form the basis of structural dynamics were postulated by Sir Isaac Newton in the Principia (1687).1 These laws are also known as Newton's laws of motion and can be summarized as follows:

Newton referred to the product of the mass, m, and the velocity, dv/dt, as the quantity of motion that we now identify as the momentum. Then Newton's second law of linear momentum becomes

1.1

where both the momentum, m(dv/dt), and the driving force, f, are functions of time. In most problems of structural dynamics, the mass remains constant, and Equation (1.1) becomes

1.2

An exception occurs in rocket propulsion in which the vehicle is losing mass as it ascends. In the remainder of this text, time derivatives will be denoted by dots over a variable. In this notation, Equation (1.2) becomes .

Newton's second law can also be applied to rotational motion, as shown in Figure 1.1. The angular momentum, or moment of momentum, about an origin O can be expressed as

1.3

where

L = the angular momentum

r = the distance from the origin to the mass, m

= the velocity of the mass

Figure 1.1 Rotation of a mass about a fixed point (F. Naeim, The Seismic Design Handbook, 2nd ed. (Dordrecht, Netherlands: Springer, 2001), reproduced with kind permission from Springer Science+Business Media B.V.)

When the mass is moving in a circular arc about the origin, the angular speed is , and the velocity of the mass is . Hence, the angular momentum becomes

1.4

The time rate of change of the angular momentum equals the torque:

1.5

If the quantity mr2 is defined as the moment of inertia, I, of the mass about the axis of rotation (mass moment of inertia), the torque can be expressed as

1.6

where denotes the angular acceleration of the moving mass; in general, I = 2dm. For a uniform material of mass density , the mass moment of inertia can be expressed as

1.7

The rotational inertia about any reference axis, G, can be obtained from the parallel axis theorem as

1.8

Figure 1.2a Circular disk

Figure 1.2b Rectangular rod

Figure 1.3 Transitional mass and mass moment of inertia (F. Naeim, The Seismic Design Handbook, 2nd ed. (Dordrecht, Netherlands: Springer, 2001), reproduced with kind permission from Springer Science+Business Media B.V.)

The difference between mass and weight is sometimes confusing, particularly to those taking a first course in structural dynamics. The mass, m, is a measure of the quantity of matter, whereas the weight, w, is a measure of the force necessary to impart a specified acceleration to a given mass. The acceleration of gravity, g, is the acceleration that the gravity of the earth would impart to a free-falling body at sea level, which is 32.17 ft/sec2 or 386.1 in/sec2. For engineering calculations, the acceleration of gravity is often rounded to 32.2 ft/sec2, which results in 386.4 in/sec2 when multiplied by 12 in/ft. Therefore, mass does not equal weight but is related by the expression w = mg. To keep this concept straight, it is helpful to carry units along with the mathematical operations.

The concepts of the work done by a force, and of the potential and kinetic energies, are important in many problems of dynamics. Multiply both sides of Equation (1.2) by dv/dt and integrate with respect to time:

1.9

Because and , Equation (1.9) can be written as

1.10

The integral on the left side of Equation (1.10) is the area under the...