Vibrations Part One

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Chapter 5

Vibrations

5.1 Introduction to vibration

The subject of vibration - a subset of dynamics of rigid and deformable bodies - isdedicated to the study of oscillatory motions of bodies and forces associated withthem.

All physical bodies, and structures made of them, posses the mass and stiffnessand are thus capable of vibration. In certain engineering cases vibration plays aparamount role, in others it is negligible and is of no interest from a technical point

of view. To decide which type is the problem under consideration is a matter of vibration analysis and ne engineering judgement.In this chapter different ways are presented for ascertaining the character of

vibrations in terms of natural frequencies and modes of vibration. A number of examples of simplied cases of real world structures are treated by basic approachesof classical mechanics, numerical mathematics and Matlab programming.

The most difcult step in vibration analysis is the process of modelling - itconsists of the creation of a mathematical model of the system or structure to beconsidered. The mathematical models fall into ‘two basic categories, i.e. continu-ous or discrete parameter models. An example of a continuous model is a thin beamtreated according to Bernoulli-Euler hypothesis, an example of a discrete model isa lattice - a structure composed of mass particles and massless springs. A contin-uous model leads to partial differential equations while a discrete one to ordinarydifferential equations.

In order to understand the behaviour of real systems a series of simplied caseswill be shown and treated in this chapter. Often parts of systems can be treated rigidand the number of degrees of freedom can thus be reduced. Although all vibratingsystems are subject to damping - since energy is always dissipated - it is oftenadvantageous to treat systems without damping since in case of small damping thesystem behaviour is inuenced only slightly.

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CHAPTER 5. VIBRATIONS 302

We distinguish two general classes of vibration, namely free and forced.

Free vibration occurs when the system vibrates under action of internal forcesonly; mathematically the free vibration is fully described by a so-called homoge-neous integral of the corresponding equation of motion and numerically leads to thesolution of the generalized eigenproblem that produces eigenvalues and eigenvec-tors.

Eigenvalues (mathematical entities) directly correspond to eigenfrequencies orto natural frequencies (mechanical entities) while eigenvectors are natural modes of vibrations. It can be shown that a discrete system with n degrees of freedom hasn pairs of eigenfrequencies and eigenmodes. Continuous systems have unboundedspectrum - the number of their natural frequencies and modes of vibrations is in-

nite. Free vibration of a linear system can always be expressed as a superpositionof all eigenmodes whose amplitudes depend on initial conditions.Forced vibration takes place under the excitation of external forces. When the

excitation is oscillatory we talk about steady-state vibration, in other cases we talk about transient vibration.

Assembling the equations of motion is an important step in the process of deter-mination the vibration behaviour of the system. There are several methods for do-ing that, namely - the free body diagram, the principle of virtual work, Lagrangianmethod (equations) and Hamiltonian principle, etc. To nd more about these meth-ods see chapter the Dynamics.

In this chapter we will concentrate on methods leading to the determinationof oscillatory motion by analytical and numerical methods. We will devote ourattention to linear systems and to simple 1D cases believing that the understandingthe procedures described here will allow the reader to use them effectively in morecomplicated cases as well. A list of books and papers pertinent for the subject is inthe bibliography.

Matlab programs, accompanying the text, are used not only for the computa-tion itself but also for reading. They are not written in the most possible efcientway, the emphasis is given to the explanation of both mechanics and programmingprinciples.

The reader of the text is assumed to be familiar with basics of mechanics of solids, matrix algebra manipulations, standard calculus analysis and with rudimentsof Matlab programming skills.

5.2 Harmonic and periodic motions

One-dimensional harmonic motion of a particle can be described by a sine (or co-sine) function of time in the form

(5.1)


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where

is the displacement measured in [m],

is amplitude of motion [m], is angular frequency [rad/s], is time [s], is an initial phase, phase constant or epoch [rad].

The angular frequency is related to the cyclic frequency by

(5.2)

where has the dimension [1/s].When the harmonic motion is related to the rotary motion we can consider as

revolutions per second and dene

(5.3)

as R.P.M. or revolutions per minute.If we add to , then the value of must remain unaltered. This requires that

the argument of the sine has been increased by . From it follows that .The quantity is called period and is

(5.4)

The velocity and acceleration of a harmonic motion are given by derivatives of displacement with respect to time

(5.5)

where

and

are amplitudes of velocity and acceleration respectively.The motion is harmonic if it is described by a sine or cosine function in time.The motion is periodic if it has the same pattern after a period or after its

integer multiples. For the function describing it we can write , for

Combination (superposition) of two harmonic motions being expressed by

(5.6)

gives a simple harmonic (sinusoidal) wave if and only if their angular frequenciesare the same, i.e.

. In such a case

(5.7)


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where

(5.8)

(5.9)

The sum of two harmonic motions with different angular frequencies is not har-monic, i.e.

. It could, however, be periodic if theratio of frequencies is a rational number.

A special case of interest is when the frequencies are slightly different. Let usconsider the motion in the form

(5.10)

Simple trigonometric manipulation reveals that

(5.11)

The resulting motion could be considered as a sine wave with angular frequency and with varying amplitude

. Every time theamplitude reaches a maximum, there is said to be a beat. Beats are well pronouncedif the difference of frequencies is small, i.e. if .

Read carefully the following Matlab program VHMharpoh1a.m . Observe theresults shown in Fig. 5.1, Fig. 5.2 and Fig. 5.3. Observe what happens as the inputparameters are varied.

% VHMharpoh1a% addition of harmonic motionsclearu10=10; u20=20; % amplitudesom1=0.3; om2=1.05; % frequenciesfi1=0.1; fi2=0.4; % phase anglest=0:pi/64:12*pi; % time rangeu1=u10*sin(om1*t+fi1); % the first motion

u2=u20*sin(om2*t+fi2); % the second motionuv=u1+u2; % add

figure(1)subplot(3,1,1); plot(t,u1,’k’); axis(’off’)subplot(3,1,2); plot(t,u2,’k’); axis(’off’)subplot(3,1,3); plot(t,uv,’k’);title(’sum of two periodic motions’,’fontsize’,13);xlabel(’does not generally give a periodic motion’,’fontsize’,13);print VHMharpohf1.eps -deps


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Template T2 – The Jacobi transformation matrix.t = 0f o r i = 1 t o n

t(i,i) = 1endc = c o s ( ) ; s = s i n ( )t(p,p) = c; t(p,q) = s; t(q,p) = -s; t(q,q) = c;

The Jacobi method uses the transformation matrix in the form shown in tem-plate T2 with trigonometrical entries in -th and -th columns and rows only, withones on the diagonal and zeros everywhere else. In each iteration the procedure is

based on such a choice of rotations that leads to zeroing the largest out of out-of-diagonal element. For details see [25]. The generalized Jacobi method could beconceived in such a way that the transformation of generalized eigenvalue probleminto a standard form could be avoided. See [2].

A command help eig reveals how to proceed when solving standard and/orgeneralized eigenproblem in Matlab.

help eig

EIG Eigenvalues and eigenvectors.E = EIG(X) is a vector containing the eigenvalues of a squarematrix X.

[V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and afull matrix V whose columns are the corresponding eigenvectors sothat X*V = V*D.

[V,D] = EIG(X,’nobalance’) performs the computation with balancingdisabled, which sometimes gives more accurate results for certainproblems with unusual scaling.

E = EIG(A,B) is a vector containing the generalized eigenvaluesof square matrices A and B.

[V,D] = EIG(A,B) produces a diagonal matrix D of generalizedeigenvalues and a full matrix V whose columns are thecorresponding eigenvectors so that A*V = B*V*D.

See also CONDEIG, EIGS.

Solving a vibration problem dened by mass and stiffness matrices and in engineering leads to a generalized eigenvalue problem. Matlab eig command isthen as follows

[V,D] = EIG(K,M),


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where is an eigenmode matrix, storing eigenvectors columnwise and is a

diagonal matrix containing eigenvalues on corresponding diagonal entries. Eigen-frequencies are square roots of eigenvalues and are not stored according to theirmagnitudes.


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5.8.2 Numerical methods for transient problems

Transient problems of solid mechanics in their discretized form are usually de-scribed by a system of ordinary second order differential equations. The spatialdomain is usually discretized by nite elements, by which the partial differentialequations of continua in space and time variables are replaced by a system of or-dinary differential equations in time. This system is then discretized in time usingvarious time integration methods.

The aim of computational procedures used for the solution of transient problemsis to satisfy the equation of motion, not continually, but at discrete time intervalsonly. It is assumed that in the considered time span

all the discretizedquantities at times are known, while the quantities at times

are to be found. The quantity , being the time step, need notnecessarily be constant throughout the integration process.

Time integration methods can be broadly characterized as explicit or implicit.Explicit time integration algorithmsExplicit formulations follow from the equations of motion written at time .

Corresponding quantities are denoted with a left superscript as in the equation of motion written in the form . The residual force isthe difference of external and internal forces.

The vector of internal forces can alternatively be written as

where ,

and

are the mass, viscous damping and stiffness matrices,respectively and , and are nodal accelerations, velocities and displacements.In structural solid mechanics problems the mass matrix does not change withtime.

A classical representative of the explicit time integration algorithm is the centraldifference scheme.

If the transient problem is linear, then the stiffness and damping matrices areconstant as well. Substituting the central difference approximations for velocitiesand accelerations

(5.57)

into the equations of motion written at time we get a system of algebraic equationsfrom which we can solve for displacements at time namely

where the effective quantities are

(5.58)

The last term in the expression for the effective force indicates that the processis not self-starting. A possible Matlab implementation is as follows.


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function [disn,veln,accn] = ...

VTRcedif(dis,diss,vel,acc,xm,xmt,xk,xc,p,h)% central difference method

% dis,vel,acc ...... displacements, velocities, accelerations% at the begining of time step ... at time t% diss ............. displacements at time t - h% disn,veln,accn ... corresponding quantities at the end% of time step ... at time t + h% xm ............... mass matrix% xmt .............. effective mass matrix% xk ............... stiffness matrix% xc ............... damping matrix% p ................ loading vector at the end of time step% h ................ time step

% constantsa0=1/(h*h);a1=1/(2*h);a2=2*a0;a3=1/a2;

% effective loading vectorr=p-(xk-a2*xm)*dis-(a0*xm-a1*xc)*diss;% solve system of equations for displacementsdisn=xmt\r;

% new velocities and accelerationsaccn=a0*(diss - 2*dis + disn);veln=a1*(-diss + disn);% end of VTRcedif

The process is effective if the mass matrix is made diagonal by a suitable lump-ing process. The damping matrix needs to be diagonal as well. The inversion of

is then trivial and instead of a matrix equation we simply have the set of indi-vidual equations for each degree of freedom and no matrix solver is needed.

Stability analysis of explicit and implicit schemes has been studied for a long

time. Park [21] has investigated stability limits and stability regions for both linearand nonlinear systems. A comprehensive survey showing a variety of approaches ispresented by Hughes in [3]. See also [1].

The explicit methods are only conditionally stable; the stability limit is approx-imately equal to the time for elastic waves to transverse the smallest element. Thecritical time step securing the stability of the central difference method for a linearundamped system is (see [21])

being the maximum eigenfrequency,related to the maximum eigenvalue

of the generalized eigenvalue problem by . Practical calculations show that the result is also appli-

cable to nonlinear cases, since each time step in nonlinear response can roughly be


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considered as a linear increment of the whole solution.

Finding the maximum eigenvalue of the system is an expensive task, especiallyin nonlinear cases where eigenvalues of the system change with time and economyconsiderations require us to change the time step as the solution evolves. It hasbeen proved, however, that the maximum eigenvalue of the system is bounded bythe maximum eigenvalue of all the elements assembled in the considered system(see [9]) which simplies the problem considerably and permits to march in timeefciently, using a variable time step. This allows, for example, decreasing the valueof the time step if the system stiffens. The proper choice of time step for explicitmethods was until recently a matter of a judicious engineering judgment. Today,many nite element transient codes do not rely on the user and calculate the correct

time step automatically.Explicit time integration methods are employed mostly for the solution of non-linear problems, since implementing complex physical phenomena and constitutiveequations is then relatively easy. The stiffness matrix need not be assembled so thatno matrix solver is required, which saves computer storage and time. The main dis-advantage is the conditional stability, which clearly manifests itself in linear prob-lems, where the solution quickly blows up if the time step is larger than the criticalone. In nonlinear problems results calculated with a ’wrong’ step could contain asignicant error and may not show immediate instability.

Implicit time integration algorithmsThe implicit formulations stem from equations of motion written at time ;

unknown quantities are implicitly embedded in the formulation and the system of algebraic equations must be solved to ’free’ them. In structural dynamic problemsimplicit integration schemes give acceptable solutions with time steps usually oneor two orders of magnitude larger than the stability limit of explicit methods.

Perhaps the most frequently used implicit methods belong to the Newmark fa-mily. The Newmark integration scheme is based upon an extension of the linear ac-celeration method, in which it is assumed that the accelerations vary linearly withina time step.

The Newmark method consists of the following equations [17]

(5.59)

(5.60)

which are used for the determination of three unknowns , and .The parameters and determine the stability and accuracy of the algorithm andwere initially proposed by Newmark as and thus securing theunconditional stability of the method, which means that the solution, for any set of


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initial conditions, does not grow without bound regardless of the time step. Un-

conditional stability itself does not secure accurate and physically sound results,however. See [3], [12], [27].

With the values of and mentioned above, the method is sometimes referredto as the constant-average acceleration version of the Newmark method, and iswidely used for structural dynamic problems. In this case the method conservesenergy.

For linear problems the mass, damping and stiffness matrices are constant andthe method leads to the repeated solution of the system of linear algebraic equationsat each time step giving the displacements at time by solving the system,

where the effective quantities are

(5.61)

where

(5.62)

The last two parameters are used for calculating of the accelerations and velocitiesat time

(5.63)

An efcient implementation of the Newmark methods for linear problems requiresthat direct methods (e.g. Gauss elimination) be used for the solution of the system of algebraic equations. The effective stiffness matrix is positive denite, which allowsproceeding without a search for the maximum pivot. Furthermore the matrix isconstant and thus can be factorized only once, before the actual time marching, andat each step only factorization of the right hand side and backward substitution is

carried out. This makes Newmark method very efcient; the treatment of a problemwith a consistent mass matrix requires even less oating point operations than thatusing the central difference method.

A possible Matlab implementation is as follows.

function [disn,veln,accn] = ...VTRnewmd(beta,gama,dis,vel,acc,xm,xd,xk,p,h)

% Newmark integration method%% beta, gama coefficients% dis,vel,acc displacements, velocities, accelerations


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% at the begining of time step

% disn,veln,accn corresponding quantities at the end% of time step% xm,xd mass and damping matrices% xk effective rigidity matrix% p loading vector at the end of time step% h time step%% constantsa1=1/(beta*h*h);a2=1/(beta*h);a3=1/(2*beta)-1;a4=(1-gama)*h;a5=gama*h;a1d=gama/(beta*h);a2d=gama/beta-1;a3d=0.5*h*(gama/beta-2);% effective loading vectorr = p + . . .

xm*(a1*dis+a2*vel+a3*acc)+xd*(a1d*dis+a2d*vel+a3d*acc);% solve system of equations for displacements xkdisn=xk\r;% new velocities and accelerationsaccn=a1*(disn-dis)-a2*vel-a3*acc;veln=vel+a4*acc+a5*accn;% end of VTRnewmd

Examples of using central difference and the Newmark methods are in 5.10.2.If

and

the method is still unconditionally stable but apositive algorithmic damping is introduced into the process. With

a nega-tive damping is introduced, which eventually leads to an unbounded response. Withdifferent values of parameters , , the Newmark scheme describes a whole seriesof time integration methods, which are sometimes called the Newmark family. Forexample if

and

, it is a well known Fox-Goodwin formula, whichis implicit and conditionally stable, else if

and , then the Newmark’smethod becomes a central difference method, which is conditionally stable and ex-plicit.

The algorithmic damping introduced into the Newmark method, by setting theparameter

and calculating the other parameter as

is fre-quently used in practical computations, since it lters out the high-frequency com-ponents of the mechanical system’s response. This damping is generally viewed asdesirable, since the high-frequency components are very often mere artifacts of -nite element modelling, they have no physical meaning and are consequences of thediscrete nature of the nite element model and its dispersive properties. See [19].

It is known that algorithmic damping adversely inuences the lower modes inthe solution. To compensate for the negative inuence of algorithmic damping on


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the lower modes behaviour Hilber [11], Hilber and Hughes [10] modied Newmark

method with the intention of ensuring adequate dissipation in the higher modes andat the same time guaranteeing that the lower modes are not affected too strongly.The results of detailed numerical testing of Hilber and classical Newmark integra-tion methods appear in [18].

A comprehensive survey of the algorithmizations of Newmark methods and alsoof other implicit integration methods is presented in [27].

Matlab offers very efcient tools for the integration of ordinary differentialequations of the rst order. They are based on Runge-Kutta methods (see [13]) of different orders with an automatic choice of the time step. Use Matlab commandshelp ODEFILE and

ODE23, ODE45, ODE113, ODE15S, ODE23S, ODE23T, ODE23TBoptions handling: ODESET, ODEGEToutput functions: ODEPLOT, ODEPHAS2, ODEPHAS3, ODEPRINTodefile examples: ORBITODE, ORBT2ODE, RIGIDODE, VDPODE

for more details.

An example showing a possible use of Matlab build in procedure named ode45for the solution of the forced vibration of an one-degree-of-freedom system withprescribed initial conditions is provided by the program VTRtode.m and functionalprocedure vtronedofnum.m .

Using ode procedures one has to rewrite each differential equation of the second

order into a system of two differential equations of the rst order. In our case wecould proceed as follows

(5.64)

Dening , and

we have

(5.65)

Introducing a new variable we get two rst order differential equations in-stead of (5.65), namely

(5.66)

Renaming variables so that they form a ’vector’ by

(5.67)

we nally have

(5.68)


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% VTRtode

% test ode procedure for one dof ... see vtronedofnum.m% m*xddot + b*xdot + k*x = A*sin(omega*t)% xddot = -b*xdot/m - k*x/m + A*sin(omega*t)/m% xddot = -B*xdot -OMEGA2*x + A0*sin(omega*t)% let’s introduce% xdot = z; xddot = zdot;% then% zdot = -B*z -OMEGA*x + A0*sin(omega*t)% define new variables% y1 = x ... displacement% y2 = z . .. velocity% and finally we have two ordinary diff. equations% ydot1 = y2% ydot2 = -B*y2 - OMEGA*y1 + A0*sin(omega*t)%cleary0 = [1 0]; % initial conditionstspan = [0 2.5]; % time domainm = 400; % massk = 60000; % stiffnessb = 1000; % damping coeffB = b/m;OMEGA2 = k/m; % square of natural frequency% f = 2; % excitation frequency in Hz% omega = 2*pi*f; % corresponding circular frequency

omega = 10*sqrt(OMEGA2);A0 = 1000; % amplitude[t,y]=ode45(’vtronedofnum’,tspan,y0,[],B,OMEGA2,A0,omega);figure(1)subplot(2,1,1); plot(t,y(:,1),’k’,’linewidth’,2);title(’displacement’);gridsubplot(2,1,2); plot(t,y(:,2),’k’,’linewidth’,2);title(’velocity’);grid[omega sqrt(OMEGA2)]print VTRtode -deps; print VTRtode -dmeta% end of VTRtode

function dy = vtronedofnum(t,y,flag,B,OMEGA2,A0,omega);% define the ode equation of the second order% m*xddot + b*xdot + k*x = A*sin(omega*t)% xddot = -b*xdot/m - k*x/m + A*sin(omega*t)/m% xddot = -B*xdot -OMEGA2*x + A0*sin(omega*t)% let’s introduce% x d o t = z% then% zdot = -B*z -OMEGA2*x + A0*sin(omega*t)% define new variables% ydot1 = y2


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% ydot2 = -B*y2 - OMEGA2*y1 + A0*sin(omega*t)

% define new variables% y1 = x ... displacement% y2 = z . .. velocity% and finally we have two ordinary diff. equations% whose matlab interpretation is

dy = zeros(2,1);dy(1) = y(2);dy(2) = -B*y(2) - OMEGA2*y(1) + A0*sin(omega*t);

% end of vtronedofnum

The output of VTRtode

is in Fig. 5.15

0 0.5 1 1.5 2 2.5−1

−0.5

0

0.5

1

1.5displacement

0 0.5 1 1.5 2 2.5−30

−20

−10

0

10

20velocity

Figure 5.15: Particle displacement and velocity of the one-degree-of freedom sys-tem - numerical solution by ode45 procedure

Vibrations Part One

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