1. INTRODUCTION

1.1 VIBRATION AND STABILITY OF STRUCTURES

All bodies that have mass and elasticity have the potential to vibrate. Vibration considerations

are essential in engineering design and applications. Static stability is another important topic

for engineers in many disciplines. The purpose of this text is to present the subject of vibration and static stability of structures to the readers in an interesting way and to introduce some

common and uncommon methods of analysis, step by step, through worked examples.

Interactive multimedia programs that supplement this book can be used to see the influence of

factors such as the location of masses, supports or restraints and the properties of the structural

members on the vibration behaviour. The relationship between the vibration behaviour and the

static stability of structures is an interesting subject. We can get a glimpse of this relationship by observing the behaviour of some simple structures. As we carry out these observations, we will

define some of the common terms in vibration analysis.

UNatural FrequencyU

All structures that are in a state of stable, static equilibrium have a definite configuration

(shape). If such a structure that is at rest were subject to a force or disturbance, that configuration would change. If the structure is then released by removing the force, it would

tend to return to its original equilibrium configuration. The following example illustrates this

point.

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A ball resting (let us say comfortably) at the bottom (A) of a concave surface (see Figure 1.1.1) is in a state of stable, static equilibrium. If it is displaced to a

point A' and released, it would move

towards A. It would approach A with a

velocity, and therefore it would pass

this point and move further on the other

side till it comes to a momentary halt at

point A". If there are no frictional forces or air resistance, A' and A" would be at the same level,

and the ball would again move in the opposite direction, passing A to reach point A' again. The

ball will always have an acceleration towards the equilibrium state A. In the absence of friction

or air resistance, this vibratory motion can go on indefinitely, the ball moving between A' and

A". The speed at which the ball passes A is proportional to the frequency of this oscillation. If there are no external dynamic (time dependent) forces (as in this case), the motion is said to be a free vibration. In this example, although a force is required initially to displace the ball, once

released it vibrates freely. In this system, the motion of the ball takes place along one path only.

This is an example of a single degree of freedom system. This free vibration can take place only at a particular value of frequency, which is its natural frequency. The return period of vibration is the time taken to complete a full cycle, and is equal to the inverse of the natural frequency in

cycles per second.

UWhy Not For Ever?U

In reality, nothing goes forever, except perhaps examinations! Any frictional force, which

behaves like an opposition in parliament, causes energy loss during vibration, and the motion

will eventually cease. Such forces that absorb energy are classified as damping forces, and the vibration in the presence of these forces is called a damped free vibration. As there will always be some energy loss, an undamped free vibration exists only in theory and all free vibrations in

real life have an end. Nevertheless, in many practical situations the damping forces are not

significantly large to alter the natural frequencies noticeably, and in such cases the frequencies

A

A'A"

Figure 1.1.1 Stable equilibrium

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may be calculated by using an undamped model. However, the amplitude of vibration is

sensitive to damping, and for this reason artificial dampers are sometimes used to control the

vibration of structures.

UPoint of No Return!U

It can be shown easily that the natural frequency of the ball in the bowl in Figure 1.1.1 decreases

with the radius of curvature of the bowl. As the surface becomes flatter, the frequency

diminishes. When it is perfectly flat, the natural frequency will be zero. This means the return

period is infinity! The ball would never return to its original equilibrium state (A) if displaced and released at any arbitrary point A' (see Figure 1.1.2). It would stay in the displaced position, in a state of equilibrium. The equilibrium configuration is therefore indefinite. Such a state of

equilibrium is referred to as a critical or neutral state of equilibrium.

A third type of equilibrium called unstable equilibrium is illustrated in Figure 1.1.3. Any

disturbance to this system will initiate a motion, away from the original equilibrium state. There

will be no oscillation, and the system does not have a natural frequency. Mathematically, the

frequency may take an imaginary value.

URestoration of Equilibrium

In the above example, the gravity force played a vital role. The displacement of the ball from the

stable equilibrium position was resisted by a component of the gravitational force. This was,

therefore, a restoring force.

A'

A"

A

Figure 1.1.3. Unstable Equilibrium

A A'

Figure 1.1.2. Critical Equilibrium

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Another form of resistance is provided by the elastic behaviour of materials. The force-

displacement relationship for many engineering materials is linear within a limit called

proportional limit. Furthermore any change caused is reversible within the elastic limit. That is,

the displacement caused by a force will vanish once the force is removed. This linear, elastic

behaviour is well known in Physics as Hooke's law.

The structures that obey this law are referred to as linear elastic structures. In such structures,

the force or moment required to cause a unit displacement is a constant and is called the stiffness

or the stiffness coefficient. If a structure can be displaced in several forms, or along several

directions, then there will be several stiffness coefficients, all of which will remain constant

(independent of force) for linear elastic structures.

Consider the spring mass system shown in

Figure 1.1.4. If the mass is displaced from its

equilibrium state (A) to a point A', the elastic compressive force induced in the spring

would push it back towards A. Once the

initial disturbing force is removed (i.e. the mass released at A'), the mass will accelerate towards A. As it approaches A, the

compressive force induced in the spring will

decrease and will become zero when the mass

is at A. However, the motion does not end here. Since the mass has a velocity (and therefore a kinetic energy, but no strain energy in the spring) at this instance, it continues to move to the right, stretching the spring. The spring is now in tension; this tensile force pulls the mass

towards A. This causes a deceleration of the mass, and when it reaches A" it will have no

velocity (zero kinetic energy, but maximum potential energy due to the spring force). Now under the tensile force the mass accelerates towards A... So the motion goes on. In this example,

the restoring action is provided by the elastic force in the spring. This force is proportional to the

stiffness of the spring. A stiffer spring will exert a higher force resulting in increased

acceleration towards the equilibrium state and hence higher frequency. Therefore the natural

A

A'

A"

UFigure 1.1.4 Spring-Mass SystemU

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frequency increases with the stiffness. By applying Newton's second law to the mass, it can be

shown that the natural frequency is proportional to the squareroot of the stiffness of the spring.

UDegrees of FreedomU

The system in Figure 1.1.4 can vibrate in one particular mode only, ant it has only one natural

frequency. It is said to have one degree of freedom. The systems shown in Figure 1.1.5 have two degrees of freedom. Two independent co-ordinates are required to describe their motion.

System (a) can vibrate in two perpendicular directions and has therefore two modes (shapes) of vibration, each having a natural frequency. The masses in system (b) can also vibrate freely in two modes at two different natural frequencies. In the first mode the two masses vibrate in phase

(at any given time, they move in the same direction) and in the second mode they move in opposite directions (out of phase) as shown in the figure. A structure with a distributed mass such as the simply supported beam in Figure 1.1.6 has infinite number of degrees of freedom and

infinite number of modes in which it can vibrate freely. The first two modes are shown in the

figure. Such systems are referred to as continuous systems, while systems that have a finite

number of modes corresponding to the motion of finite number of rigid masses are called

discrete systems.

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Figure 1.1.5 Examples of Two Degrees of Freedom System

Figure 1.1.6 Vibration Modes of a Simply Supported Beam

UMoods and Modes: Forced and Free Vibration

When a structure is subject to an excitation (dynamic force) it responds by translating or rotating which may be described as undergoing a change in its geometry. If the frequency of excitation is

equal to one of its natural frequencies, then the amplitude of vibration may be very large (in the absence of damping the amplitude is indefinite) and this state is called resonance. In general, discrete and continuous systems may vibrate in a combination of more than one pure natural

mode (harmonics). The actual displacement would depend on initial conditions as well as on the natural frequency and modes. In the case of forced vibration, the dynamic displacement of the

First mode

Second mode One mode Other mode

System (a) System (b)

First mode Second mode

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system would depend on the system properties, its initial condition, and the applied dynamic

force. It is interesting to compare this vibration behaviour with human behaviour. Just as we

may have a behaviour pattern (that could be described as a personality - whether acquired or cultivated is a different matter) with certain ways of reacting when we experience different feelings (for example the way we smile when we are happy), physical systems posses natural frequencies and modes. However, the way we react to a particular situation depends UalsoU on

whether the day started off well and the nature of the situation, while the way a physical system

responds to an excitation depends on intial conditions and the properties of the excitation force.

In a forced vibration analysis, the dynamic displacement (response) of a system is expressed as the sum of the transient response, which depends on the natural frequencies and modes as well

as initial conditions, and the steady-state response, which is a function of the dynamic force and

the system properties.

UMinding their own business the orthogonal way

The natural vibration modes possess an interesting property. It is possible to induce a vibration

in one mode only without involving other shapes. However depending on initial conditions, in

general a response may consist of more than one mode. Even then, the restoring and inertial

forces associated with one mode of vibration do not do any work against or for, the vibration in

another mode. Take the case of System (a) in Figure 1.1.5. If the system were given an initial horizontal displacement, it would vibrate horizontally. During this vibration, if a vertical force is

applied to the mass, it would not affect the motion of the mass in the horizontal direction,

although the motion will consist of both horizontal and vertical translation. The mode may be

regarded as minding their own business. A neat mathematical statement that describes this

behaviour is called the orthogonality of modes. This property can be used to simplify the response calculations and also helps to find the natural frequencies in some iterative techniques.

U

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Stiffness and Natural Frequencies

The stiffness of a structure is an indication of its resistance to deformation under loading. In the

case of a structure that has been displaced from its equilibrium state, the restoring actions

increase with stiffness. The natural frequencies of a structure therefore increase with its

stiffness. This is why tall, slender structures having low stiffness have long return periods and

low natural frequencies.

We can now ask, what would happen if the stiffness of a structure corresponding to deformation

in a given mode is extremely small or zero? That is, what would happen if there is little or no

restoring action? To answer this question, let us go back to the relationship between the natural

frequency and stability of the ball in the bowl. The explanation given for the models in Figures

1.1.1 - 1.1.3 holds for any elastic structure. If the natural frequency is zero the return period

would be infinity, which means if subject to a disturbance the structure would never return to its original equilibrium state. By definition this is a critical state of equilibrium. Therefore one can

conclude that if a natural frequency of a structure is zero, the equilibrium corresponds to a

critical state in that mode. Any static load that causes a natural frequency of a structure to vanish

is therefore a critical load. We will now focus on some other examples to appreciate the

significance of the above statements.

UTuning Up by Tension!U

It is well known that string instruments are tuned by adjusting the tension. The tensile force in the string causes an increase in the natural frequencies. Hence the 'pitch' increases with tension.

One can easily 'feel' the increase in the stiffness of string caused by tightening. This is true for

beams, frameworks and other structures too. Tensile stresses/forces in general increase the

natural frequencies of structures corresponding to transverse modes while compressive

stresses/forces causes a decrease of these frequencies.

Let us take the case of a simply supported beam/column subject to compressive axial loading. The beam, having a distributed mass, has infinite number of natural modes each associated with

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a natural frequency. The natural frequencies of lateral (flexural) vibration decrease with axial compressive load. At a certain load, the fundamental (lowest) natural frequency becomes zero. This is the point of no return! The beam reaches a critical state and 'buckles'. At this state, the

beam would not have any lateral stiffness (it would feel 'soggy'). In practice imperfections in geometry and the nature of loading can cause failure associated with large displacements well

before reaching the critical load.

USmart Structures & Non-Destructive TestingU

For an initially straight simply supported beam, the square of all natural frequencies (P2P) decrease linearly with compressive axial load (P) as shown in Figure 1.1.7. If the natural frequencies of this beam were measured at small values of axial load, it would be possible to

extrapolate the lines in the plot and predict the critical loads without knowing the properties of

the beam. This linear relationship is approximately true for beams subject to other boundary conditions too. However its applicability in practical situations may be very limited due to the

presence of initial geometrical imperfections.

A definite theoretical relationship between the

flexural natural frequencies and static axial

loads can be obtained for any skeletal structure

using routine computer programs. By

comparing these with the measured frequencies

of prototype structures, some defects may be

detected. This may be useful in underwater

structures where visual inspection is difficult.

Hence, in addition to being useful in the

prediction of critical loads, the frequency

measurements may also be useful in

monitoring of existing structures.

P2P

B2PB2

P

B1PB2

P

PBC1B

PBC2B

P

PBcnB = nPUtUh Pcritical load

Figure 1.1.7. Variation of Natural Frequency and Axial Load

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The fact that some natural frequencies and modes depend on the axial force in beams has been

used to control the flexural vibration of beams subject to harmonic dynamic loading. This is achieved by measuring the amplitude of vibration when subject to an excitation with varying frequency, and adjusting the axial force (when the amplitude is excessive) until the amplitude is decreased to an acceptable level. The amplitude of response increases whenever the excitation

frequency approaches a natural frequency. If this were to happen, the natural frequency can be

moved away, by changing the axial force. Experimental results indicate that this form of

passive control is achievable in beams. This could form the basis of the Design of Smart

Structures that are designed to self-tune to stay away from resonance.

Nothing is Perfect!

One complicating factor in the vibration analysis of beams and frameworks is the effect of

geometric imperfections. While simple theoretical models have been in existence for perfect,

straight beams and frames for a long time, the effect of initial curvature has been the subject of recent research. The case of a slightly curved beam may be used to illustrate the effect of

curvature on the vibration.

A slightly curved beam (or a shallow arch) that is pinned at both ends is shown in Figure 1.1.8. Any lateral load on this arch would be resisted by the action of bending as well as axial straining.

Therefore its resistance to any imposed lateral forces or displacements is higher than that of a

straight beam having the same properties and boundary conditions. This means the curvature

increases the lateral stiffness of a beam, and hence some natural frequencies associated with

lateral motion. This effect depends on the longitudinal boundary conditions. For example, a

shallow arch that has a sliding simple support will behave like a straight beam, as it cannot

sustain axial straining (see Figure 1.1.9).

Figure 1.1.8. Pinned-Pinned Arch Figure 1.1.9. Arch on a Roller

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The vibration behaviour of other structures such as plates is also sensitive to geometric

imperfections. Studies in this field may have practical applications in aircraft and shipbuilding

industry, since ships and aircraft contain curved panels and are often subject to dynamic forces.

Vibration can be a hazard, but it can also be useful as described in some of the examples here. It

is an important subject for engineers in many disciplines. It is not a subject that can be condensed in a single book, and such a task is certainly beyond the ability of the author. This

book mainly covers the vibrational behaviour of some common continuous systems, and is

intended to take the readers through simple and straightforward steps which the author has used

in his own learning process.