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7 Response of a One-degree-of-freedom System

7.1 INTRODUCTION

In this chapter we examine the dynamic behaviour of a one-degree-of-freedom system under forced deterministic and random loading.

Consider first a simple mass-spring system as shown in Figure 7.1. The behaviour of the system can be analysed in terms of the displacement u. More complex systems will be represented by many coordinates, but by applying a transformation it is possible to work in terms of generalised coordinates, the behaviour of each being governed by an uncoupled equation. In this way each coordinate can be studied effectively as a one-degree-of-freedom system, as shown in this chapter. We shall see how to carry out this transformation in Chapter 8.

The system of Figure 7.1 will be in equilibrium when the system is at

Figure 7.1 Spring-mass system

240

RESPONSE OF A ONE-DEGREE-OF-FREEDOM SYSTEM 241

rest and the force in the spring k is equal to the weight of the mass (weight = mg, where g is the acceleration due to gravity). Hence:

(7.1)

where us denotes the static displacement. If the mass is displaced a further distance u it will experience a restoring force Fa, such that:

(7.2)

Taking into account the static equilibrium, we have:

(7.3)

D'Alembert's law states that at each moment in time the restoring force will be:

(7.4)

where is the acceleration. Here we have, from (7.3) and (7.4), that:

(7.5)

The solution of this equation represents the free vibrations of the spring-mass system, which has a harmonic solution of the type:

(7.6)

where A and B are constants to be determined from the initial conditions. Substituting (7.6) into (7.5) we find:

(7.7)

Thus

(7.8)

or (7.9)

CDT is the natural frequency of the system. The displacement is governed by equation (7.6), and we can now

determine A and B from the initial conditions of the system (i.e. the conditions at time t = 0). If we define u0 as the displacement of the

242 RESPONSE OF A ONE-DEGREE-OF-FREEDOM SYSTEM

system at t = 0 and 0 as the velocity of the system at t = 0, we find that:

(7.10)

Hence the displacement can now be written:

(7.11)

This function can be represented by a single function:

(7.12)

where

Function (7.12) can be plotted as shown in Figure 7.2(a). C is the amplitude of the displacements and a is the phase angle. The time between two peaks is called the period T:

(7.13)

The frequency fis sometimes expressed in hertz or cycles per second:

(7.14)

We can also plot the velocity or derivative of (7.12), i.e.

(7.15)

The initial velocity is ii0; see Figure 7.2(b). Consider now that the spring-mass system is subjected to a

harmonic varying force (Figure 7.3). A simple force, for instance, is:

(7.16)

The governing equation for the system is then:

(7.17)

where co is the forced frequency and P is the amplitude of the force. The solution of (7.17) consists of a complementary solution for the

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Figure 7.2 Plots of (a) displacement, (b) velocity

Figure 7.3 Forced vibrations of a spring-mass system


homogeneous equation and a particular solution for the F loading. In what follows we always neglect the transient part or complementary solution and concentrate on the particular solution. A trivial parti-cular solution for (7.17) is:

(7.18)

Substituting (7.18) into (7.17) we obtain:

(7.19)

Hence:

(7.20)

or (7.21)

Figure 7.4 Response of spring-mass system

We could add (7.21) to (7.11) to have the general solution, but as we are not interested in the initial state let us assume that u0 = u0 = 0


and investigate the behaviour of equation (7.21), which can be written as:

(7.22)

where us is the static deflection. The term between brackets is called the 'dynamic amplification' of the system and can be plotted as a function of co, as shown in Figure 7.4.

Note that for co = T the amplitude of vibration tends to infinity. This value coT is called the resonance frequency for the system.

7.2 FORCED VIBRATIONS OF A DAMPED SYSTEM

In practice the amplitude is bounded, owing to the damping of the system. If we consider the case of viscous damping represented by the dashpot of Figure 7.5, where the motion is resisted by the viscosity of the fluid, a new force Fv can be added to equation (7.4), i.e.

(7.23)

Figure 7.5 Dashpot-spring-mass system


This force acts in the same direction as the spring force and is equal to c. Hence:

(7.24)

For the case of forced response we can write:

(7.25)

where P is the amplitude of the exciting force and a> the forced frequency. Hence:

(7.26)

For the particular integral we can try the following solutions:

(7.27)

which can also be expressed as:

(7.28)

which implies that A = U sin a, B = U cos a. Substitution of this solution into (7.26) gives:

(7.29)

The first term between brackets gives:

(7.30)

and the second term gives:

(7.31)

Thus the amplitude ratio is:

(7.32)

It is useful to represent the exciting force (7.25) and the solution (7.28) in vector form (Figure 7.6). The angle a represents the difference


Figure 7.6 Vector representation of forced damped vibrations

in phase between the applied force F and the response u. This is produced by the damping term.

The equilibrium equation (7.26) could now be written as:

We see that the damping force leads the displacement by 90. This is because it is in the opposite direction to the velocity. The inertia force instead is in phase with the displacement. The vector interpretation of equations (7.31) and (7.32) is now quite evident. We can investigate these expressions further by writing the second in the form of equation (7.22), i.e.

(7.33)

where a>T = k/m and y = c/2mT = damping factor. We can now plot (u/us), which is called the magnification factor and is a function only of damping and frequency (Figure 7.7). Note that tana can also be written as:

(7.34)

The phase angle a is also plotted in Figure 7.7. It is interesting to note that in the region /T

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Figure 7.7 Magnification factor and phase angle as function of frequency

Figure 7.8 Resonance vector diagram

Figure 7.9 Inertia-dominant behaviour


= cot the vector representation of the vibration can be seen in Figure 7.8, where now a = 90. The amplitude at resonance is:

When /r > 1 the angle a tends to 180 and the force P is used mainly to overcome the large inertia of the system (Figure 7.9).

7.3 COMPLEX RESPONSE METHOD

The use of complex algebra simplifies the forced vibration response of a damped system and it is important in order to find the random vibration response of the system. We can assume that the two functions, for the impressed form F and the displacement u, can be represented as:

(7.35)

(7.36)

Note that the U response lags behind the applied force by the a angle.

For the case of a force such as the one given in equation (7.25) we have:

(7.37) and

where Im{ } means the imaginary part of the complex number. If the impressed force F were a cosine we would take:

(7.38)

where Re { } means the real part of the complex number. In general we can work with equations (7.35) and (7.36), defining:

(7.39)

where U = C/*exp(-ia).


Let us apply this complex analysis to equation (7.26), written now as:

(7.40)

If the solution is of the form 7.39 we obtain:

(7.41)

The complex response U is then:

(7.42)

Since U = U* exp(- i'ot) we can write:

(7.43)

and the phase angle is, as before:

(7.44)

It is important to note that P and U do not need to be real, and more generally can be taken as complex. The function F then becomes similar to a term of the complex Fourier series seen in Chapter 2.

Equation (7.42) is usually written as:

(7.45)

where if (co) is the complex frequency response function. Note that the complex frequency response can be written as:

(7.46)

The result is the same as the one we obtain by applying the Fourier transform to equation (7.41), i.e.

(7.47)

or

Solving this system we obtain:

(7.48)


with H() = Z(oo we obtain the spectral density (see section 2.2), i.e.

(7.53)

Therefore (7.52) becomes:

(7.54)

or

(7.55)

This expression relates the spectral density of the forces to the spectral density of the response or displacement. We can now obtain the variance of the displacements:

(7.56)


7.5 APPROXIMATE SOLUTIONS

The analysis of structural systems usually requires the use of an approximate method, 'exact' solutions being limited to very simple structure configurations. Two approximate methods of analysis are the finite-element method and the boundary-element one, which in contrast to finite elements discretises only the external surface of the continuum. Both methods are based on weighted-residual principles. It is important to understand how these principles can be applied in practice. In what follows we illustrate them for a simple column such as the one shown in Figure 7.10. The system will be reduced to a one-degree-of-freedom system after a series of simplifications.

Figure 7.10 Simple column

The equilibrium equation for a beam element is the following fourth-order equation:

(7.57)

where p{x) are the applied forces along the beam, E is the modulus of elasticity and / the moment of inertia; u are the transverse displace-ments and A is the cross-sectional area. The boundary conditions are of two types: Essential or displacement conditions on the Sj part of the boundary, of the type:

(7.58)


where the bar indicates a known value of displacement u or rotation 0(0 = du/dz). Natural or force conditions on the S2 part of the boundary, which are moment M and shear Q:

(7.59)

The bar denotes the applied (known) forces. Those without it are the internal components. Note that S = S1 + S2.

Initial conditions are not needed as we are only considering the steady-state solution.

One way of finding an approximate solution for these equations is by weighting equation (7.57) and the boundary conditions (7.58) and (7.59) in the following way:

(7.60)

where W are weighting functions that are assumed to satisfy the essential boundary conditions, i.e. W and d Wjdz are identically zero on Sj. We assume that the shapes of the ^functions are the same as the u functions we take as approximate solutions; this leads to the following form of the principle of virtual displacements:

(7.61)

Integrating equation (7.61) by parts twice, we obtain the best-known expression for virtual displacements, i.e.

(7.62)


Consider now the beam shown in Figure 7.10, with boundary conditions:

(7.63)

and

(7.64)

The principle of virtual displacements, equation (7.62), then becomes:

(7.65)

with the function for u satisfying the essential boundary conditions (7.63) at z = 0, that is the part of the boundary S2.

If we now assume an approximate function for u, such that:

(7.66) then similarly

where u represents the horizontal displacement at the top of the column and g(z) is a 'shape' function, i.e. it represents the shape of the column. Substituting (7.66) into (7.65) we find:

v,.u7)

As u is arbitrary we can simply write: (7.68)

where

K, M and F are the equivalent stiffness, mass and force coefficients for the one-degree-of-freedom system.

We can similarly include the damping term into the equation. This will be illustrated in what follows. The more general equilibrium


equation will then be:

(7.70)

where

(7.71)

7.6 APPLICATION OF ONE-DEGREE-OF-FREEDOM SYSTEM ANALYSIS

Consider a concrete offshore structure, which can be idealised as shown in Figure 7.11. The complexity of a typical offshore structure is such that this idealisation is suspect. However, the response of these structures tends to occur predominantly in the first mode, which indicates that a one-degree-of-freedom model may be useful as a preliminary design tool. In addition the analysis gives us an insight into the way in which the more complex cases described in Chapter 8 can be solved.

Note that the column is of length / in a sea of depth d. The mass of

Figure 7.11 One-degree-of-freedom idealisation


the platform, mc, is assumed to be concentrated at z = /. The stiffness and area are constant.

For the idealisation shown in Figure 7.11 we have the following equivalent values:

EI equivalent stiffness of the column, N m2

Ac equivalent (concrete) area of the column, m2

In addition we define:

A volume of water displaced per unit length, i.e. the cross-sectional area, m2

mc mass of the platform, kg p density of the water, kg/m3

pc density of the concrete, kg/m3

We have the following drag and inertia coefficients:

where cd is the drag coefficient (1.0 for cylinders) and cm is the inertia coefficient (also 1.0 for cylinders). The equilibrium equation is (7.70):

(7.72)

where u is the displacement at the top of the column (x = /). The term C includes the structural and hydrodynamic damping. M is obtained by addition of the mass of the column, the mass of the platform and the hydrodynamic mass. Note that the CA term does not enter into M because CA only affects the water particle accelerations.

If the shape of the column is assumed to be g(z\ where z = z//, the M term in equation (7.72) becomes:

(7.73)

The inertia term for the column is:

(7.74)

The natural frequency of the system is:

(7.75)


In order to write the C term (where C is formed by the addition of the structural plus hydrodynamic damping: C = Cs + CH ) in its usual form C = 2Myo)T, let us consider the hydrodynamic damping term. The drag coefficient in Morison's formula was written as:

(7.76)

and is multiplied by (vx li), where the contribution can be passed to the left-hand side and combined with Cs.

The hydrodynamic damping contribution for the column is:

(7.77)

One needs first to compute the variance of the water particle velocity, av which is a function of the velocity spectrum, i.e.

(7.78)

Hence the deviation is:

(7.79)

Once the term CH has been computed we can write the percentage of critical damping as:

(7.80)

where ys is the structural damping contribution. The equation of motion can now be written as:

(7.81)

The F(t) term is computed from the contribution of the v, v terms in Morison's equation, i.e.

(7.82)

where Y\ = a cos cot and r\' = a sin cot. (Note that the column is now taken to be at x = 0.)


The generalised force for the column can now be written as:

or

(7.83)

The spectral density function for this force is:

(7.84)

Note that in this case the cross spectral density terms relating vx and vx disappear, owing to their different phases.

The transfer function for displacements is obtained by substituting uV exp(io)i), F = F Qxp(icot) into the equation of motion. This gives:

(7.85)

Hence:

with

The following relationship applies 'for the spectral densities:

(7.86)

where

(7.87)


The spectral density of the force is:

(7.88)

where | A()\2 is the force transfer function defined in equation (7.84). Once Sgg(co) is known its variance can be calculated:

(7.89)

As one is working with a Gaussian process with zero mean, it is possible to calculate the probability of 7 being within a certain value kav\ for instance, for k = 3 the probability is 99.7 per cent. Alternatively, knowing that the peaks of a narrow-band Gaussian process have a Rayleigh distribution, i.e.

(7.90)

one can compute the most probable maximum deflection (or stress) for a given storm.

The expected maximum value of the response can be approximated by:

(7.91)

where T is the duration of the storm Tm is the mean period, given by:

(7.92)

In addition to oc we can calculate the variance of any other quantity such as stresses or moments. Assume, for instance, that the bending moment at the base of the column is related to the displacement at the top by a function B such that:

(7.93)

The spectral density for this moment is now:

(7.94)


Example 7.1

Assume we have an offshore structure that can be approximated to the one-degree-of-freedom structure shown in Figure 7.11 with the following characteristics:

(a)

Ac = 29 m2 (cross-sectional area of concrete)

A = 78 m3/m (total volume of water displaced per unit length)

p = 103 kg/m3 (density of water)

pc = 2.5 x 103 kg/m3 (density of concrete)

D = 10 m (diameter of the column)

The drag and inertia coefficients for the equivalent column are:

(b)

The wave spectrum used is the one given by Pierson and Moskowitz for a wind velocity W 20 m/s. The deflected shape of the structure will be approximated by g(z) = z2, where z = z/L Hence the mass of the system can be written:

(c)

and the stiffness is:

(d)

One can now find the natural frequency of the system, tor :

(e)


Figure 7.12 Variation of water particle velocity with depth

In order to calculate the damping constant y one needs to compute the hydrodynamic damping constant CH, having first found the variance ov at different heights. The variation of av for the spectrum under consideration is shown in Figure 7.12 and was obtained by integrating numerically equation (7.78). Next one calculates CH using equation (7.77), which gives:

The damping constant can now be found, i.e.

For the structural damping the value ys = 0.05 was taken. Hence y = 0.06 for this case. The next step is to evaluate using numerical integration the force spectra given by equation (7.84). They are shown in Figure 7.13, where the drag and inertia contributions can be seen. The transfer function H(co) for a one-degree-of-freedom system can be computed using formulae (7.87), and the results are plotted in Figure 7.14. Next the values of the transfer function are multiplied by


Figure 7.13 Spectral density of (a) waves, {b) generalised force F

the spectral density SFF to obtain the response spectrum, i.e.

(h)

which is shown in Figure 7.14. Integrating this spectrum, the variance of the generalised displacement can be obtained, i.e.

(i)

Numerical integration of equation (i) gives:

The probability of the U value being within 3ag = 0.912 m is


Figure 7.14 (a) Transfer function, (b) spectral density of response

9 9 . 7 PCr CCnt. T h e Tfmrr\i*r\t ran alert fv r\Y%in\ru*A i tt

(j)

Alternatively we could have calculated the multiplier of og using equation (7.91), i.e.

(k)

Bibliography

Brebbia, C. A., et al, Vibrations of engineering structures, Southampton University Press (1974)

Brebbia, C. A., and Connor, J. J., Fundamentals of finite element techniques for structural engineers, Butterworths (1973)


Dym, C. L., and Shames, I. H., Solid mechanics: a variational approach, McGraw-Hill (1973)

Thomson, W. T., Vibration theory and applications, Allen and Un win (1966) Warburton, G. B., The dynamical behaviour of structures, Pergamon Press (1964)

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