Applied Structural and Mechanical Vibrations 2009 Part 5 doc

Let us consider the general equation of motion for a damped SDOF system 5.85with initial conditions and The Laplace transformation of both sides gives 5.86where, as customary, we are usi

where G1(s) and G2(s) are the transforms of g1(t) and g2(t), respectively From

a table of transforms we get

so that the convolution integral is zero for t<t1, leaving only the first term in

eq (5.83) and

for t>t1 The inverse transformation of eq (5.83) finally yields

which, aside from the constant f0, are exactly eqs (5.10) and (5.43)

So far, we have not yet considered the possibility of obtaining directly thefinal solution satisfying given initial conditions This is one advantage ofsolving linear differential equations with constant coefficients by the Laplacetransform method By standard methods one finds a general solutioncontaining arbitrary constants, and further calculations for the values of theconstants are needed to solve a particular problem The Laplace transforms

of derivatives given in Chapter 2 (eqs (2.39) and (2.40)) will now be used toclarify this point

Let us consider the general equation of motion for a damped SDOF system

(5.85)with initial conditions and The Laplace transformation

of both sides gives

(5.86)where, as customary, we are using lower-case letters for functions in thetime domain and capital letters for functions in the transformed domain

Solving for X(s) and rearranging leads to

(5.87)

The first term on the right-hand side (product of two functions of s)

transforms back to the convolution integral

(5.88a)the second term transforms back to (see any list of Laplace transforms)

From the discussion and the examples of preceding sections it appears that

both h(t), the impulse response function (IRF), and the frequency response function (FRF) H( ω) (or the transfer function H(s)) completely define the

dynamic characteristics of a linear system This fact suggests that we should

be able to derive one from the other and vice versa The key connection

between the two domains is established by the convolution theorem and bythe Fourier (or Laplace) transform of the Dirac delta function In Chapter 2(eq (2.29a)) we determined that the Fourier transform of the convolution of

two functions g1(t) and g2(t)—provided that —is theproduct of the two transformed functions With our definition of the Fourier

transform, as a formula this statement reads

and results, as we have seen, from an application of Fubini’s theorem

Now, since we know from Section 5.2 that the time-domain response x(t)

of a linear system is given by the convolution (Duhamel’s integral) between

the forcing function f(t) and the system’s IRF h(t), i.e.

we can Fourier transform both sides of this equation to get the input-outputrelationship in the frequency domain

(5.89)Equation (5.89) justifies eq (5.76a) from a more rigorous mathematical point

of view In fact the two equations (5.76a) and (5.89) are the same if we define

(5.90a)and

(5.90b)

In this light, note that the functions F( ω) and X(ω) are the Fourier transforms

of the functions f(t) and x(t), respectively, but the FRF H( ω) (eq (5.90a)) differs slightly from the definition of the Fourier transform of the IRF h(t)

(there is no 1/(2π) multiplying factor) This is a consequence of our definition

of the Fourier transform (eqs (2.15) and (2.16)) and of the fact that thefundamental input-output relationship for linear systems is almost alwaysfound in the form given by eq (5.76a) However, this is only a minorinconvenience since (Chapter 2) the position of the factor 1/(2π) is optional

so long as it appears in either the Fourier transform equation or the inverseFourier transform equation Hence, the inverse transform equationcorresponding to eq (5.90a) is

(5.91)

which, conforming to our definition of the inverse transform can be written

No such inconvenience arises in the case of Laplacetransforms because in this case, by virtue of the convolution theorem (eq(2.43)), the Laplace transform of the Duhamel integral leads to

(5.92)

where we defined X(s) as the Laplace transform of the response x(t).

With the above general developments in mind, we may now recall that,

for a linear system, the function h(t) represents the system’s response to a delta input, i.e x(t)=h(t) when Consequently, since

(eq (2.74)), and eqs (5.89) and (5.92) give respectively

(5.93)

which tell us that, in the case of δ-excitation, the Fourier (Laplace) transform

of the system’s response is precisely its frequency response (transfer) function,

a circumstance which is also exploited in experimental practice

With the definitions above, it is now not difficult to verify eq (5.76a)(and hence eq (5.93)) for the case of, say, a viscously damped SDOF system

In fact, in this case we know from the second of eqs (5.7a) that

so that, with the aid of a table of integrals from which we get

we can calculate the term

by noting that, in our specific case and

It is then left to the reader to show that the actual calculation leads to

(5.94)

where, as usual, If we multiply this result by

we obtain explicitly the right-hand side of eq (5.76a) for our viscously dampedSDOF system Then, the left hand-side can be obtained by virtue of eq (5.5)and it can be determined immediately that, as expected,

In the light of these considerations we can write the response of a system

to an input f(t), whose Fourier and Laplace transforms are F( ω) and F(s), as

(5.95)

or, when more convenient, as the inverse Laplace transform of the product

H(s)F(s) Thus the following three equivalent definitions of the FRF H( ω)

can be given:

1 H( ω) is 2π times the Fourier transform of h(t).

2 For a sinusoidal input, i.e is the coefficient of theresulting sinusoidal response

3 Provided that in the frequency range of interest, H( ω) equals

the ratio where X( ω) is the Fourier transform of x(t).

Figure 5.8 is a frequently found schematic representation of the fact thatthe dynamic characteristics of a ‘single-input single-output’ system are fully

defined by either h(t) or H( ω)

A final note of interest concerns two systems connected in cascade as shown

in Fig 5.9 Denoting by h(t) and H( ω) the IRFs and FRFs of the combined

Fig 5.8 Symbolic representation of a linear system.

Fig 5.9 Systems in cascade.

system, by and H1( ω) and H2( ω) the relevant functions of the

two subsystems we have

(5.96a)and

(5.96b)

5.5 Distributed parameters: generalized SDOF systems

Up to the present we have always considered the simplest type of SDOFsystem, i.e a system where all the parameters of interest—mass, dampingand elasticity—are represented by discrete localized elements This is, ofcourse, an idealized view However, even when more complicated modelling

is required for the case under study, we have to remember that the key aspect

of SDOF systems is that only one generalized coordinate is sufficient to

describe their motion; if this characteristic is maintained it can be shownthat the equation of motion of our SDOF system, no matter how complex,can always be written in the form

(5.97)

where z(t) is the single generalized coordinate and the symbols with asterisks

(not to be confused with complex conjugation) represent generalized physicalproperties—the generalized parameters—of our system with respect to the

coordinate z(t) This latter statement means that a different choice of this

coordinate leads to different values of the generalized parameters The possibility

of writing an equation such as (5.97) allows us to extend all the considerationsthat we have made so far to a broader class of systems: assemblages of rigidbodies with localized spring elements, systems with distributed mass, dampingand elasticity (bars, plates, etc.) or combinations thereof can be analysed inthis way once the relevant generalized parameters have been determined Theirvalues can be obtained, in general, from energy principles such as Hamilton’s

or the principle of virtual displacements (Chapter 3) but general standardizedforms of these expressions can be given for practical use

It is important to note that the SDOF behaviour of the system underinvestigation may sometimes correspond very closely to the real situationbut, more often, is merely an assumption based on the consideration thatonly a single vibration pattern (or deflected shape in case of continuoussystems) is developed For example, a beam that deforms in flexure is, as amatter of fact, a system with an infinite number of degrees of freedom, but

in certain circumstances a SDOF analysis can be good and accurate enoughfor all practical purposes

For continuous systems in particular, the success of this procedure—which

is a particular case of the assumed modes method (Chapter 9)—depends onthe validity of the assumption above and on an appropriate choice of a

characteristics of the system and also on the type of loading Ideally, theselected shape function should satisfy all the boundary conditions of theproblem At a minimum, it should satisfy the essential boundary conditions

In this light, a few definitions are given here and then some examples willclarify the considerations above

• An essential (or geometric) boundary condition is a specified condition

placed on displacements or slopes on the boundary of a physical body(e.g at the clamped end of a cantilever bar both displacement and slopemust be zero)

• A natural (or force) boundary condition is a condition on bending

moment and shear (e.g at the free end of a cantilever bar the bendingmoment and the shear force must be zero)

• A comparison function is a function of the space coordinate(s) satisfying

all the boundary conditions—essential and natural—of the problem at

hand, plus appropriate conditions of continuity up to an appropriateorder

• An admissible function is a function that satisfies the essential boundary

conditions and is continuous with its derivatives up to an appropriateorder For a specific problem, the class of comparison functions is asubset of the class of admissible functions

• An assumed mode (or shape function) is a comparison or an admissible

admissible function used to approximate the deformation of a continuousbody

Example 5.8 Let us consider the rigid bar of length L metres and mass m

kilograms shown in Fig 5.10 The angle θ of rotation about the hinge, where

at static equilibrium, can be chosen as the generalized coordinate The

vertical displacement z(t) of the tip of the bar can be another choice; for

small oscillations as shown in the figure

Since the bar is considered rigid, the system has distributed mass (alongthe length of the bar), localized stiffness, damping (the spring and the dashpot)

and is subjected to a localized force f(t).

For small oscillations about the equilibrium, we assume the shape function

The virtual displacement is then given by andfrom the principle of virtual displacements it is not difficult to obtain

Our method assumes that only one mode is developed during the motion

and represents the deflected shape u(x, t) of the beam as the product

(5.100)

where (x) is the chosen admissible function and z(t) is the unknown

generalized coordinate The principle of virtual displacements considers allthe forces that do work and reads

(5.101)where from the definition of potential energy V It will be

shown in later chapters that the strain potential energy of a beam undergoing

a transverse deflection u(x, t) is given by

Fig 5.11 Schematized beam on elastic foundation.

Similarly, for damping forces we get

(5.105)and for the distributed spring and external forces

(5.106)

(5.107)Addition of the various terms leads to

where all the integrals are taken between 0 and L The virtual displacement δz

is arbitrary and therefore can be cancelled out, leaving the equation of motion

of our system in the form of eq (5.97) where, after rearranging, the generalizedparameters are given by

The most general case of the type shown above consists of a system which

is a combination of distributed and localized masses, springs, dampers andexternal forces Again, the displacement is assumed of the form

where z(t) is the unknown generalized coordinate and the

following standardized expressions for the generalized parameters can begiven:

c1 at a distance x1 from the origin of the axes is given by in thesummation of eq (5.110) With regard to the generalized mass, the secondsummation accounts for the rotation effects of localized rigid-body masses:

I 0j is the mass moment of inertia of the jth mass and the first derivative of

at the point x j represents the rotation at that point Referring back to theexample of Fig 5.10, it is not difficult to see that the generalized parameters

of eqs (5.99a–d) are particular cases of eqs (5.109)–(5.112) where the assumedmotion was and the connection to the general case is given by

and the generalized mass accounts for translational motion of the centre of

mass (at x=L/2) and the rotational motion around the centre of mass (mass

moment of inertia )

Care must be taken in the calculation of the generalized stiffness when

‘destabilizing’ forces are acting, since they add a further contribution to eq(5.111) Destabilizing forces may arise in different situations and may be ofvarious nature Gravity, for example, can be such a force in the case of aninverted pendulum, where a mass M is mounted on the tip of a light rigidbar as in Fig 5.12

The reader is invited to determine that the effective stiffness of the system

can be written as k–k G where k G depends on the weight Mg When

the system becomes unstable

In the case of a simple beam under the action of the axial compressive load

only—if P does not depend on x and can be taken out of the integral—we can

obtain the critical buckling load from the condition as

(5.116)

which is, obviously, relative to the assumed shape (x).

As the simple examples above show, and as the word itself implies,destabilizing forces lead to stability problems Stability is a broad subject inits own right and extends outside our scope Some of its basic aspects will beconsidered when and if appropriate in the course of the book For the moment,

it suffices to say that stability is in general connected to situations in which

the physical parameters (m, k or c or their generalized counterparts; one or

more of them) become negative The motion is not well-behaved in thesecases and may diverge, i.e increase without bounds, with or withoutoscillating An example of diverging motion, even if no destabilizing forcesare active, is the undamped oscillator excited at resonance

One final word to point out that the assumed mode procedure can beextended to more complicated elements For example, if the elementundergoing flexure is two dimensional—i.e a rectangular membrane or aplate—we can assume the displacement of the centre as the generalized

coordinate z(t) and write

where, again, (x, y) is a reasonable shape function consistent with the

boundary conditions However, a good choice of the shape function becomesmore and more difficult as the number of dimensions increases and, as aconsequence, the method may lead to unreliable results

5.5.1 Rayleigh (energy) method and the improved Rayleigh

method

Often, in practical situations, the quantity of main concern is the fundamentalfrequency of vibration of a given structural or mechanical system When thesystem is complex, the exact determination of such a quantity may not be

an easy task, long computation time and difficult calculations being involved

in the process The basis of a class of approximate methods to obtain the

needed result is the so-called Rayleigh’s method.

When an undamped elastic system vibrates at its fundamental frequency,each part of the system executes simple harmonic motion about its equilibriumposition The principle of conservation of energy applies for such a systemand during the motion two extreme situations occur:

• All the energy is in the form of potential strain energy at maximumdisplacement

• All the energy is in the form of kinetic energy when the system passesthrough its equilibrium position (maximum velocity)

Conservation of total energy requires that the potential energy at maximumdisplacement must equal the kinetic energy at maximum velocity, i.e

(5.117)

The Rayleigh method calculates these maximum values, equates them andsolves for frequency, since this quantity always appears in the kinetic energyterm as a consequence of the simple harmonic motion of the system.The undamped harmonic oscillator of Fig 4.2 is the simplest example:the motion of such a system can be written as (Chapter 4)

the potential and kinetic energies are then

Their maximum values are

Equation (5.117) follows because, individually, the two terms above mustequal the total energy Solving for ωn we get the well-known result

(5.118)

Another example can be the beam of Fig 5.10 in free vibration; we make

it conservative by considering c=0 and f(t)=0: no energy is removed from the

system and no energy is fed into it We assume as before and aharmonic motion of the generalized coordinate given by Themaximum values of the potential and kinetic energies are now

equating the two energies and solving for frequency gives

as the Rayleigh quotient—can be appreciated.

Further generalization will be given in later chapters when appropriate

As a last example of SDOF system we consider a simple cantilever beam(i.e a beam that is clamped at one end and free at the other) that undergoes

flexural vibrations without energy loss during its motion We assume the

x-axis in the horizontal direction

The assumption characterizes the SDOF behaviour ofthis system and, again, characterizes the harmonic timedependence of this motion The maximum potential and kinetic energies are

(5.120)

(5.121)

respectively, where EI is the flexural rigidity, is the mass per unit length.

Equating and solving for the frequency gives

(5.122)

At this point it is interesting to test the effect of different choices for (x)

on the calculated frequency Since the exact deformation shape can only beobtained by solving the equation of motion (but in this case the value of thefundamental frequency would be determined also) and therefore it is notknown, we will try three trial functions

curve of a cantilever beam under uniform load In all of the cases above wecan calculate the Rayleigh quotient and obtain an approximate value for thefundamental frequency of our system.

After some calculation that the reader is invited to try, we get the followingresults:

(5.129)

The first consideration is that all of the trial functions produce a resultthat overestimates the exact value; this is a fundamental characteristic of theRayleigh quotient and will be proven rigorously on a mathematical basis

On physical grounds, one can observe that additional constraints must beapplied to the system if it is forced to vibrate in a shape that is different fromits natural one; these constraints add stiffness to the system and hence anincrease in frequency Obviously, if the exact shape function (the lowest order

eigenfunction) is used for (x), the result is eq (5.129).

In addition, we note that the degree of approximation is rather crude(27% high) for the first function, but satisfactory for the other two.Qualitatively, we can say that the more the trial function resembles the truedeflection shape, the more accurate the result will be A closer examinationrequires the analysis of the boundary conditions For the cantilever beamthe following boundary conditions must be satisfied:

1 zero displacement and slope at the clamped end (x=0), i.e

(5.130)which we recognize as essential boundary conditions;

2 zero bending moment and shear force at the free end (x=L), i.e.

(5.131)

which we recognize as natural boundary conditions

All the trial functions satisfy the conditions of eqs (5.130) but only 3(x)

satisfies all four; 1(x) does not satisfy the first of eqs (5.131) and 2(x) does

not satisfy the second of eqs (5.131)

In general, the deflection produced by a static load is a good candidatefor Ψ(x) because it automatically satisfies all the necessary boundary

conditions and simplifies the calculation of the potential strain energy thatcan be obtained as the work done by the static load to produce the desired

deflection Only the function (x) appears in this calculation and not its

second derivative

A common assumption is to choose the deflection shape that results fromthe application of the gravity load due to the mass of the structure In thiscase, the direction of gravity must be chosen to match the probabledeformation shape: in the analysis of the free vibrations of a vertical cantileverfor example, the direction of gravity must be horizontal if we are interested

in lateral motions of the structure Obviously, this does not correspond toanything real, it is just a useful expedient

There are two are the reasons that justify the assumption above:

1 It is not necessary to spend much time in the choice of an assumedshape because any reasonable function compatible with the essentialboundary conditions leads to acceptable results It will be shown inChapter 9 that the error on the calculated frequency is of the order of

ε2, if ε is the error of the assumed shape with respect to the exact one

2 The displacements in free vibration result, as a matter of fact, from theapplication of inertia forces and these forces depend, in their turn, onthe mass distribution in our system

This latter consideration, together with the serious disadvantage that the

method does not allow us to estimate e if the exact (x) is not known, leads

to the improved Rayleigh method, whose line of reasoning is as follows.

Suppose that the true deflection (x) is the same as the deflection produced

by an external load f(x) Then, deflection is produced by a

load zf(x) and the potential strain energy is the work done by this force to give the displacement z , i.e.

If, as before, we get

Equating to the maximum kinetic energy of eq (5.121) gives

(5.132)

which states that the load of eq (5.132), where we recognize inertia forces,

produces the exact vibration shape Equation (5.132) is true if (x) is the

true shape Our assumed shape, which we call now 0(x), is probably

different from the true one and hence the load will produce ashape different from 0(x), let us call it This function cannot becalculated because of the unknown factor, but intuition suggests that it

is likely to be a better approximation than 0 for the true deflected shape.Nevertheless, the function (not to be confused with thefunction of eqs (5.123) which was a particular 0 for the cantilever problem)can be obtained from

(5.133)and we can write the maximum potential energy as

Equating to E p,max of eq (5.134) gives now

(5.137)

Further iteration—that is, the use of 1 to obtain an even betterapproximate function 2 and use the latter to calculate the frequency—isgenerally not worth it

We still do not have an estimate for the error ε, but indirectly we can

have an idea by looking at the difference between the frequencies obtainedfrom eq (5.122) and eq (5.135) or (5.137) If this difference is large, thefunction 0 is not a very good approximation for the true deflected shapeand ε is large as well; if it is small, ε is small as well and 0 is a good choice.Now, going back to the cantilever problem, we show an application ofthe improved Rayleigh method We start from the function —which produced the result of eq (5.126)—and use the inertia forces tocalculate a better deflected shape We know from beam theory that

where 1 is the shape that results from the application of the forces on theright-hand side By integrating four times and calculating the constants ofintegration from the boundary conditions of eqs (5.130) and (5.131) we get

These values are much better estimates of the exact frequency (given by

eq (5.129)) and the large difference between the frequencies obtained from

eq (5.126) and (5.139)—26.7% with respect to the lower value—indirectlysuggests that the assumed 0 was not a good approximation for the truedeflected shape and hence the relative error on the frequency must havebeen large as well

5.6 Summary and comments

Chapter 5 continues the discussion on SDOF systems When the excitation

is not a simple sinusoidal function, the response of the system can be obtained

by means of various techniques, which obviously apply to harmonic excitation

as well The main distinction is between time-domain and frequency-domaintechniques

If the functions involved are analysed in the time domain, a fundamental

concept is the impulse response function h(t), whose convolution with the

forcing exciting function provides the time response of our SDOF system

This particular form of convolution is known as Duhamel integral, which,

in turn, can be visualized as a sum of the input excitation ‘weighted’ by an

appropriately shifted form of the impulse response h(t) As far as dynamic aspects are concerned, the function h(t) is an inherent property of the system

and characterizes it completely

In this light, the response to the frequently encountered situation ofloadings of short duration that may release a considerable amount of energy

can be considered One generally speaks of transient or shock loading,

depending on a comparison between the time duration of the input load andthe system’s period of oscillation The ratio between these two latter

quantities is the natural abscissa axis for the representation of shock spectra,

where the maximum response of the system is plotted on the ordinate axiswithout regard to the entire time history of the event Shock spectra areobtained considering an undamped SDOF system as a standard referenceand are widely used for design and comparison purposes in order to assessthe potential disruptive effects of various forms of shock

Another class of loadings is given by periodic (i.e with a repetitive pattern

in time) functions Fourier’s theorem states that a general periodic (and behaved) signal is the superposition of an infinite number of simple sinusoidalfunctions with frequencies that are all integral multiples of a value ω0 Itfollows that its mathematical form is a convergent Fourier series of suchfunctions and, owing to the principle of superposition, the response of alinear system is a similar Fourier series as well Amplitudes and phases aremodified between input and output (excitation and response), but it is not sofor the frequency content and even if, rigorously, an infinite number of termsappear in the mathematical representation of the above series, a finite andlimited number of terms often suffices for all practical purposes

well-The generalization of Fourier series to nonperiodic signals leads to theFourier and Laplace transformation integrals, which constitute the basis ofthe frequency-domain approach Besides the fact that they often allow asimplification of the mathematics required to solve specific problems, their

importance cannot be overstated and the fundamental concepts of frequency

directly from their application

These latter functions play a crucial role in almost every aspect of linearvibration analysis They completely characterize a linear system in the

frequency domain and—as h(t) in the time domain—they are inherent

properties of the system under study Given these similarities, logic dictates

that it must be possible to obtain H(ω)—or H(s)—from h(t) and vice versa The connection is the Fourier (or Laplace) transform: h(t) and H(ω) are a Fourier transform pair and, likewise, h(t) and H(s) are a Laplace transform

pair

Unfortunately, both the time-domain and the frequency-domainapproaches often lead to integral expressions which cannot be evaluatedanalytically and, therefore, recourse must then be made to computercalculations on ‘sampled’ versions of the original signals This samplingprocess is not harmless and its effects will be considered in Part II of thebook which deals with electronic instrumentation It is, however, important

to point out right away that some care must be exercised in these cases if we

do not want to run into undesirable consequences

The last part of the chapter shows how, in some circumstances, an SDOFanalysis can be extended to a wide class of more complex systems whensome basic assumptions on the system’s behaviour can be made or when theneeded results can be accepted with a reasonable degree of approximation

The concept of generalized parameters is introduced in order to obtain a

SDOF equation of motion or to calculate an approximate value of the

fundamental frequency by means of the Rayleigh energy method The

assumption that only one vibration pattern (or ‘shape’) is developed duringthe motion is particularly useful in an approximate examination of continuoussystems, where the static deflection under an appropriate load—often theirown weight—is a good choice for the assumed vibration shape in most cases

The assumed shape then satisfies automatically the essential and natural

boundary conditions of the problem Nevertheless, simpler shapes satisfying

only the essential boundary conditions can be chosen if great accuracy is notneeded Needless to say, the exact value of frequency is obtained if one hasthe luck (or the physical insight) to choose the correct deformed shape

The improved Rayleigh method provides more accuracy in the calculation

of the fundamental frequency—which is always overestimated when theassumed shape is not correct—and allows an indirect qualitative evaluation

of the error with respect to the (unknown) exact value by looking at theimprovement of the first one or two iterations involved in the process Furtheriterations are, in general, not needed

1 Cooley, J.W and Tukey, J.W., An algorithm for the machine calculation of complex

Fourier series, Mathematics of Computation, 19, 297–301, 1965.

2 Harris, C.M (ed.), Shock and Vibration Handbook, 3rd edn, McGraw-Hill,

New York, 1988.

3 Jacobsen, L.S and Ayre, R.S., Engineering Vibrations, McGraw-Hill, New York,

1958.

4 Erdélyi, A., Magnus, W., Oberhettingher, F and Tricomi, F.G., Tables of Integral

Transforms, 2 vols, McGraw-Hill, New York, 1953.

5 Thomson, W.T., Laplace Transformation, 2nd edn, Prentice Hall, Englewood

Cliffs, NJ, 1960.

6 Graff, K.F., Wave Motion in Elastic Solids, Dover, New York, 1991.

7 Inman, D.J., Engineering Vibrations, Prentice Hall, Englewood Cliffs, NJ, 1994.

on the scope of the investigation In some circumstances the assumption iscorrect; in some other cases it may lead to a description of the system’sdynamic behaviour within an acceptable degree of accuracy but, in manyother cases, the assumption is just an extreme oversimplification leading toinaccurate results which have almost nothing to do with the real situation.

Since, a priori, the true behaviour of a real system is in general not known,

the assessment of the validity of the results obtained from a SDOF analysismay not be an easy task Therefore, in order to obtain a meaningfuldescription for a wide class of systems, more complex representations areneeded from the outset

When one coordinate is not sufficient to characterize the motion of a

given system, one speaks rightfully of two-, three-,…n- or, in general,

multiple-degree-of-freedom (MDOF) systems; where the number refers to theindependent coordinates necessary to describe completely the vibrationphenomenon

The SDOF model enables us to explain—without particular mathematicaldifficulties—many fundamental concepts such as free and forced vibrations,natural frequency and resonance Broadly speaking, all of these conceptscan be extended to MDOF models However, some important differenceswill appear; in anticipation we can say that the natural vibration of an MDOFsystem may occur at a number of different frequencies Each one of themcorresponds to a particular pattern (or ‘shape’ to give a pictorial view) of

the system’s motion and these different configurations, known as natural or

normal modes of vibration, play a crucial role in almost every aspect of

further analysis

As discussed in Chapter 3, a set of n simultaneous ordinary differential

equations of motion—one for each degree of freedom—must now be obtained

in order to mathematically describe our system and a proper choice of the