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COMPLEX MODAL ANALYSIS OF A CONTINUOUS MODEL WITH RADIATION DAMPING
Journal of Sound and Vibration (1996) 192(1), 15–33
COMPLEX MODAL ANALYSIS OF A CONTINUOUS MODEL WITH RADIATION DAMPING G. O Istituto di Scienza delle Costruzioni, Universita` di Catania, Viale A. Doria 6, 95125 Catania, Italy
A. S Istituto di Ingegneria Civile ed Energetica, Universita` di Reggio Calabria, Via E. Cuzzocrea 48, 89100 Reggio Calabria, Italy (Received 5 May 1994, and in final form 25 November 1994) With reference to a simplified model for soil–structure interaction problems, this paper describes the use of the complex mode superposition method for the analysis of the dynamical response of continuous systems. Analytical expressions are derived for complex frequencies and modes of vibration. Two sets of orthogonality conditions are established for the model and used in decoupling the equations of motion. Numerical applications showing the convergence of the method and allowing for the comparison with other methods of analysis are presented. Finally a physical interpretation for the complex modes of vibration is provided and the source of radiation damping is clearly expounded. 7 1996 Academic Press Limited
1. INTRODUCTION
The dynamical response of structures is greatly affected by damping which may be from many sources and usually is difficult to evaluate [1]. For this reason the mathematical models used in the applications of engineering practice adopt either linear viscous or hysteretic damping. Classical damping [2,3], where the modes of vibration of the damped structure coincide with those of the undamped one, is generally used because the calculation of frequencies and modes of vibration can be performed by means of well-established methods of eigenvalue analysis [4,5]. However, there are some important engineering problems where damping is no longer of the classical type and/or cannot be treated as such [3]. Amongst these are structures with base isolation [6,7] and those with special damping devices [8]. Another large class is that of the soil–structure interacting systems [3,9–12]. Here, an important source of damping is energy radiation across the boundary at infinity. In all those cases the modes of vibration for the damped system no longer coincide with those of the undamped one and in fact are complex as are the corresponding frequencies. The calculation of complex frequencies and modes of vibration [13] has been considered in the literature for discrete structures and applications to vibration analysis have been presented [14]. The authors are not aware of the application of the same kind of analysis to continuous systems. The object of the present paper is to show how complex modal analysis can be used in the calculation of the dynamical response of continuous systems. A simplified model 15 0022–460X/96/160015 + 19 $18.00/0
7 1996 Academic Press Limited
16
. .
which allows a closed form calculation for the frequencies and modes of vibration is used as a basis for the demonstration of the mode superposition method in complex form. The physical meaning of the complex frequencies and modes of vibration is clearly stated and the proper orthogonality conditions are established for the model. Although the model has been devised as a simple tool for the demonstration of the use of complex modal analysis for continuous systems, it has nevertheless some physical characteristics which enable an understanding to be gained of the source and effects of radiation damping in soil–structure interacting systems. These aspects will also be described, but the limitations of the model will be clearly stated. The ideas described above will be illustrated by means of numerical applications showing the effect of radiation damping on frequencies and modes of vibration and on the dynamical response. Finally, the convergence of the mode superposition method is also shown through numerical examples.
2. THE PHYSICAL MODEL
The physical model has been selected in such a way as to be amenable to a closed form solution for both the natural frequencies and the modes of vibration. This permits an easy characterization of the model and a clear understanding of the physical aspects of the problem of radiation damping. The model, however, may not be quite realistic in the interpretation of the behaviour of soil–structure interacting systems for reasons which will be explained later. The actual model is composed of a shear beam of finite length L, mass per unit length m and shear rigidity GA attached to a semi-infinite shear beam of mass per unit length ms and shear rigidity Gs As ; see Figure 1. Due to the infinite length, the overall static stiffness of the semi-infinite beam, i.e., the force applied to the near end required to produce a unit displacement, is zero. This is contrary to our physical insight and, in fact, a better model for soil–structure interaction problems could be provided by the shear cone introduced by Meek and Veletsos [15]. Nevertheless, the model presented here is preferable in a qualitative study, intended essentially to demonstrate the aspects of the complex modal superposition method in the calculation of the dynamical response of continuous structures. The shear cone model would only require a numerical procedure for the calculation of the frequencies of vibration, but then the analysis would follow along similar lines. The present model could be used also to make a good approximation in the
Figure 1. The physical model: a shear beam of finite length represents the structure; a semi-infinite shear beam represents the soil.
17
description of the behaviour of a soil layer of finite depth resting on a different layer of infinite depth.
3. THE GOVERNING EQUATIONS
It is assumed in what follows that the free field motion at the soil surface is known. In the present model this amounts to the knowledge of the motion at the near end of the semi-infinite beam when the beam of finite length simulating the structure or the soil layer is absent. The equation of motion for the structure or the soil layer takes the form [3] GAu0(z,t) − mu¨ (z,t) = mu¨g (t), where u(z,t) is the displacement of the structure relative to the ground or base layer and u¨g (t) is the prescribed free field ground acceleration. The equation of motion for the soil or base layer takes the form Gs As u0s (z,t) − ms u¨s (z,t) = 0, where us (z,t) is the displacement in the ground, additional to the free field motion. The two equations of motion need to be supplemented by proper boundary conditions which may be written as follows: GAu'(L,t) = 0,
u(0,t) = us (0,t),
GAu'(0,t) = Gs As u's (0,t).
These ensure that the shear force vanishes at all times at the free end of the structure and that displacements and shear forces are continuous at the interface between the finite and the semi-infinite beams. A fourth condition should ensure the vanishing of displacements and shear forces at the far end of the semi-infinite beam and will be discussed later.
4. FREQUENCIES OF VIBRATION
In free vibrations the free field ground motion is non-existent, i.e., ug (t) = 0, and the time dependence of the structure and ground motions is harmonic: u(z,t) = f(z)exp{ivt},
us (z,t) = fs (z)exp{ivt}.
Therefore the following equations must be satisfied: GAf0(z) + mv 2f(z) = 0,
Gs As f0(z) + ms v 2fs (z) = 0,
or by setting b 2 = mv 2/GA and bs2 = ms v 2/Gs As , f0(z) + b 2f(z) = 0,
f0s (z) + bs2 fs (z) = 0.
The general solution of the above equations may be written as f(z) = C1 exp{ibz} + C2 exp{−ibz},
fs (z) = C3 exp{ibs z} + C4 exp{−ibs z}.
By assuming that the frequency parameter a = bL = vzmL 2/GA is complex with a negative imaginary part, the vanishing of displacements and forces at the far end of the semi-infinite beam requires the vanishing of the integration constant C4 . The additional boundary conditions GAf'(L) = 0,
f(0) = fs (0),
GAf'(0) = Gs As f's (0),
. .
18
lead to the algebraic eigenvalue problem
&
'& ' & '
a exp{ia} −a exp{−ia} 0 1 1 −1 m −m −1
C1 0 C2 = 0 , C3 0
(1)
where m = zmGA/ms Gs As is a dimensionless structural parameter. The frequency equation is therefore given by a(cosa + im sina) = 0, from which it may be seen that the problem admits zero frequency. The remaining frequencies are obtained as solutions of the equation cosa + im sina = 0. By setting a = p + iq this complex equation gives rise, after some algebraic manipulations, to the two real equations cosp(coshq − m sinhq) = 0,
sinp(sinhq − m coshq) = 0,
solutions of which may be written as
$ %
(2a,b)
$ %
(3a,b)
pn = 2(n − 1)p,
n = 1,2,3,...,
1 m+1 qn = ln 2 m−1
p pn = 2(2n − 1) , 2
n = 1,2,3,...,
1 1+m qn = ln 1−m 2
for m q 1 and
for m Q 1. It is important to realize that the imaginary part of the frequency parameter is independent of n. The actual frequencies are therefore complex even if the model is elastic, pointing to the existence of a source of energy dissipation, the nature of which will be discussed later. The frequency parameter an may be related to the dimensionless natural frequency p˜n and damping ratio jn of a viscously damped one-degree-of-freedom system as follows [14]: an = p˜n ( 2 z1 − jn2 + ijn ),
p˜n = zpn2 + qn2 ,
jn = qn /p˜n .
5. MODES OF VIBRATION
Once the frequencies of vibration have been evaluated, the corresponding modes of vibration may be calculated by replacing the frequency parameter an for a in equations (1) and solving for C1 , C2 and C3 . The nth vibration mode therefore takes the form fn (z) = cos[an (1 − z)],
fsn (z) = cosan exp{idan z},
where d = m(ms /m) and z = z/L. It should be noticed that the modes of vibration are complex and occur in conjugate pairs. One set of modes of vibration corresponds to the positive values of pn , while the conjugate set corresponds to the negative values of pn . Two exceptions to the general trend are worth mentioning. For pn = 0 the frequency is purely imaginary and the mode takes the form fn (z) = cosh[qn (1 − z)],
fsn (z) = coshqn exp{−dqn z},
19
and is real. For a = 0 the mode is real and unity over all the length of the model: i.e., f(z) = 1, fs (z) = 1. In both cases the conjugate pairs are in fact coincident, while in all other cases they are distinct.
6. PHYSICAL INTERPRETATION OF COMPLEX MODES OF VIBRATION
The complex modes fn associated to the complex frequency vn do not in themselves have any physical relevance. In fact the motion of the structural system would take the form unc (z,t) = Cn exp{sn t}fn (z),
(4)
where sn = ivn and Cn is a constant which is related to the initial conditions. It is easy to see that the conjugate of expression (4) also satisfies the governing equations and therefore defines a mathematically admissible motion u¯nc (z,t) = C n exp{s¯ n t}f n (z).
(5)
The sum of expressions (4) and (5) is obviously real and therefore provides a physically admissible motion: i.e., un (z,t) = 2Re[unc (z,t)]. In terms of real algebra, this may be written more expressively as un (z,t) = exp{−jn =vn =t}=fn (z)={An cos[vDn t + un (z)] + Bn sin[vDn t + un (z)]},
(6)
after having set Cn = 12 (An − iBn ),
tanun (z) =
Im[fn (z)] , Re[fn (z)]
vDn = =vn =z1 − jn2 ,
where =·= denotes the modulus of a complex number. Unlike classical modes, where un (z) = 0 or un (z) = p, here the phase angle un (z) may take any value depending on the ratio of the imaginary to the real part of the complex function fn (z). Any harmonic motion with frequency vDn takes the above form where the real constants An and Bn may be found from the initial conditions. The modal shape is time dependent because the displacement does not reach its maximum value simultaneously at all points. However, as may be seen from equation (6), a modal shape repeats itself after each period. It should be noticed that the motion described by equation (6) is damped even though the system is elastic.
7. ORTHOGONALITY CONDITIONS
When the mode superposition method is used in dynamical problems, two sets of orthogonality conditions are usually required in order to uncouple the equations of motions in principal co-ordinates [3]. In the following, the appropriate orthogonality conditions for the problem at hand will be derived. Because the interest is restricted to the beam of finite length, the orthogonality conditions will be established with reference to this subsystem only. The equations of motion may be written for mode n and m as GAf0n + vn2 mfn = 0,
GAf0m + vm2 mfm = 0.
. .
20
By multiplying the first equation by fm and the second by fn and integrating over the length L it follows that
g
L
GAf0n fm dz + vn2
0
g
g
L
mfnfm dz = 0,
0
L
GAf0m fn dz + vm2
0
g
L
mfnfm dz = 0.
0
Integrating by parts and taking account of the boundary conditions, the previous equations may be written as vn2
g
L
g
L
g
mfn fm dz − ivn zms Gs As fn (0)fm (0) −
0
vm2
L
GAf'n f'm dz = 0,
(7)
0
mfn fm dz − ivm zms Gs As fn (0)fm (0) −
0
g
L
GAf'n f'm dz = 0.
(8)
0
Subtracting equation (8) from equation (7), one finds (vn2 − vm2 )
g
L
mfn fm dz − i(vn − vm )zms Gs Asfn (0)fm (0) = 0,
0
which, for vn $ vm , leads to the first orthogonality condition: (vn + vm )
g
L
mfn fm dz − izms Gs As fn (0)fm (0) = 0.
(9)
0
Subtraction of equation (8) multiplied by vn from equation (7) multiplied by vm provides (vn − vm )vn vm
g
L
mfn fm dz + (vn − vm )
0
g
L
GAf'n f'm dz = 0,
0
which, for vn $ vm , gives the second orthogonality condition: vn vm
g
L
mfn fm dz +
0
g
L
GAf'n f'm dz = 0.
(10)
0
8. RADIATION DAMPING
Upon setting sn = ivn the orthogonality conditions become sn sm
g
L
0
(sn + sm )
g
mfn fm dz −
g
L
GAf'n f'm dz = 0,
0
L
mfn fm dz + zms Gs As fn (0)fm (0) = 0.
0
To each eigenvalue sn there corresponds a conjugate one s¯ n as may be realized from
21
Figure 2. Dynamic stiffness functions for soil models: (a) present model; (b) cone model. f = (GA/L)(k + ica)u.
expressions (2) and (3) for the natural frequency parameters. By taking sm = s¯ n and by noticing that sn + s¯ n = −2jn =vn = and sn s¯ n = =vn =2, it follows that =vn =2 =
f0L GA=f'n =2 dz kn = , mn f0L m=fn =2 dz
2jn =vn = =
zms Gs As =fn (0)=2 f0L m=fn =2 dz
=
cn . mn
(11,12)
From expression (12) it follows that the soil in this model is represented by a series of modal linear viscous dashpots with damping constant cn = zms Gs As =fn (0)=2. Furthermore, expressions (11) and (12) could also be written as =vn =2 =
L f−a GA=f'n =2 dz , L f−a m=fn =2 dz
2jn =vn = =
zms Gs As =fn (−a)=2 L f−a m=fn =2 dz
,
as may easily be realized, if the equations of motion are integrated over the full length of the model. It is therefore clear that the loss of energy which generates damping occurs at the far end of the semi-infinite beam. It is also worth noticing that if modes and frequencies were real, expression (11) would take a quite familiar form.
22
. . 9. THE MODE SUPERPOSITION METHOD
In what follows, the use of the mode superposition method [3,14] for the evaluation of the dynamical response of the system to a free field motion in the sense specified in section 3 will be outlined. First the modal impulsive response functions will be derived. Then these will be used to evaluate the response to a general free field motion specified in terms of the acceleration u¨g (t). Finally the frequency response function is evaluated in closed form in order to check the convergence of the complex mode superposition method. 9.1. The equation of motion for impulsive free field accelerations may be written as GAu0(z,t) − mu¨ (z,t) = mId(t)
(13)
for the structural model or soil layer, with I = f u¨g (t)dt denoting the acceleration impulse and with d(t) being Dirac’s delta function. In the standard mode superposition method, the unknown displacement function u(z,t) admits the following expansion in terms of eigenfunctions: 0+ 0−
a
u(z,t) = s yn (t)fn (z). n=1
Here, contrary to the classical method, both vibration modes fn (z) and principal co-ordinates yn (t) are complex functions of the spatial co-ordinate z and time t. Due to the impulsive nature of the loading, the principal co-ordinates are expected to take on the expression yn (t) = Cn exp{ivn t}, which implies y˙n (t) = ivn yn (t),
y¨n (t) = −vn2 yn (t).
Figure 3. Imaginary part of the frequency parameter a = p + iq as a function of the material property ratio m = zmGA/ms Gs As .
23
Figure 4. Modes of vibration of the superstructure: (a) structure on a fixed base; (b) and (c) structure on soft soils. – – – –, Reference line; —— ——, real part; ——, imaginary part. For (a) m=0, q = 0; for (b) m = 0·20, q = 1·199; for (c) m = 1·50, q = 0·805.
By using the eigenfunction expansion in the equation of motion (13) this becomes a
s [GAyn (t)f0n (z) − my¨n (t)fn (z)] = mId(t). n=1
Upon multiplication by fm (z) and integration over the length L it follows that a
$ g
L
s yn (t) n=1
0
GAf0n (z)fm (z)dz − y¨n (t)
g
L
0
%
mfn (z)fm (z)dz = Id(t)
g
L
0
mfm (z)dz.
. .
24
Integration by parts and application of boundary conditions leads to the equation a
6 $
g
s yn (t) −ivn zms Gs As fn (0)fm (0) − n=1
L
GAf'n (z)f'm (z)dz
0
−y¨n (t)
g
7
L
mfn (z)fm (z)dz = Id(t)
0
g
%
L
mfm (z)dz.
(14)
0
By using the orthogonality condition (10), this may be written as a
6 $
s yn (t) −ivn zms Gs As fn (0)fm (0) + vn vm n=1
−y¨n (t)
g
L
mfn (z)fm (z)dz
0
g
7
L
mfn (z)fm (z)dz = Id(t)
0
g
%
L
mfm (z)dz.
0
After some algebraic manipulations, and making use of the relationships between the principal co-ordinates and their derivatives, the following equation is found: a
s n=1
6$
izms Gs As fn (0)fm (0) − (vn + vm )
g
L
0
mfn (z)fm (z)dz
% 7
y¨n (t) = Id(t) vn
g
L
mfm (z)dz. (15)
0
It may be seen that the term in square brackets represents the orthogonality condition (9) and therefore vanishes for vn $ vm . Therefore, the summation of equation (14) and equation (15) reduces to only one term, thus decoupling the equation of motion. The uncoupled equations of motion may be written as 2M n y¨n (t) + C n y˙n (t) = −L n Id(t),
Figure 5. Modal impulse response functions for the displacement at the top of the superstructure: —— ——, impulse response function from the superposition of the first 10 modal responses. m = 0·20.
25
Figure 6. Modal impulse response functions for the displacement at the top of the superstructure from the first 10 modal contributions: —— ——, structure interacting with the soil; ——, structure on a fixed base.
where L n =
g
L
mfn (z)dz,
M n =
0
g
L
mfn2 (z)dz,
C n = zms Gs As fn2 (0).
0
Integration of the modal equations over the small interval of time [0−,0+] provides the result 2M n y˙n (0+) + C n yn (0+) = −L n I, from which the constant Cn may be derived: Cn = −
I f0L mfn (z)dz 2ivn f0L mfn2 (z)dz + zms Gs As fn2 (0)
.
The modal impulse response function therefore takes on the expression hnc (z,t) = Cn fn (z)exp{ivn t} where the constant Cn is evaluated for I = 1. It should be noted that this function is complex, and that for the calculation of the dynamical response a formulation in terms of real algebra may be more convenient. This can easily be achieved by combining each modal contribution with that of its conjugate: i.e., hnr (z,t) = 2Re[Cn fn (z)exp{ivn t}]. In this way each modal impulse response function is real and includes the contribution of both the mode and its conjugate. Although this function is formally real, it still requires complex algebra for its calculation. However, by defining the real functions bn (z) = 2Re[Cn fn (z)],
gn (z) = 2Im[Cn fn (z)],
an (z) = jn bn (z) − z1 − jn2 gn (z),
the impulse response function may be expressed entirely in terms of real algebra as hnr (z,t) = an (z)=vn =hn (t) + bn (z)h n (t),
. .
26 where
hn (t) = (1/vDn )exp{−jn =vn =t}sinvDn t represents the impulse response function of a one-degree-of-freedom linear viscous system with natural frequency =vn =, damping ratio jn and damped frequency vDn = =vn =z1 − jn2 . 9.2. The response to a general free field ground motion may be obtained by using the mode superposition method and the previously defined impulse response function as follows: a
u(z,t) = s
g
t
u¨g (t)hnr (z,t − t)dt.
n=1 0
In this expression the unit impulse has been replaced by I = u¨g (t)dt and the load history has been divided into a succession of impulses in the standard way of dynamical analysis. For computational purposes, it is convenient to express the dynamical response in terms of response integrals as a
u(z,t) = s [an (z)Vn (t) + bn (z)D n (t)],
(16)
n=1
where Vn (t) = =vn =Dn (t) = =vn =
g
t
u¨g (t)hn (t − t)dt,
D n (t) =
0
g
t
u¨g (t)h n (t − t)dt.
0
9.3. When the free field ground motion is harmonic the response integrals defined in the previous section may be evaluated in closed form: Dn (t) =
exp{iv¯ t} − [cosvDn t + (jn /z1 − jn2 + iv¯ )sinvDn t]exp{−jn =vn =t} u¨g0 , =vn =2 − v¯ 2 + i2jn =vn =v¯
D n (t) = {iv¯ exp{iv¯ t} + [vDn sinvDn t + jn =vn =cosvDn t − (jn /z1 − jn2 + iv¯ ) ×(vDn cosvDn t − jn =vn =sinvDn t)]exp{−jn =vn =t}}
u¨g0 . =vn =2 − v¯ 2 + i2jn =vn =v¯
The dynamical response can still be evaluated by using expression (16). It may be worth noticing that the stationary response may be obtained by removing from the response integrals the transient part, to obtain Dn (v¯ ) =
u¨g0 , =vn =2 − v¯ 2 + i2jn =vn =v¯
D n (v¯ ) =
iv¯ u¨g0 . =vn =2 − v¯ 2 + i2jn =vn =v¯
The stationary response to harmonic free field ground motions may also be obtained directly in closed form without making use of the mode superposition method. Actually this solution may be used to demonstrate the convergence of the method. The derivation of the stationary harmonic response may be obtained directly from the equations of motion by assuming free field harmonic displacements of the type us (z,t) = ug0 cosbs z exp{iv¯ t}.
27
By applying the proper boundary conditions, the following expressions are obtained for the displacements and the accelerations in the top layer or in the structure: u0 (z) = u¨0 (z) = −a 2
cosa(1 − z/L) − cosa − im sina ug0 , cosa + im sina cosa(1 − z/L) − cosa − im sina GA u . cosa + im sina mL 2 g0
The expression for the shear force is also useful for application purposes: T0 (z) =
GA a sina(1 − z/L) u . L cosa + im sina g0
10. NUMERICAL APPLICATIONS
Some numerical applications are presented in order to illustrate the concepts and methods of analysis reported in the previous sections. First of all, the equivalence of the soil model to a simple one-degree-of-freedom system is shown. Then a quantitative and qualitative representation of the modal radiation damping and its effect on the frequencies of vibration is presented. The graphical representation of a few modal shapes is provided. The modal impulse response functions are presented for a few modes of vibration, together with the corresponding frequency response functions. Finally the mode superposition method is applied to the case of a stationary harmonic free field ground motion showing the convergence to the exact solution. 10.1. It has been shown [16] that the semi-infinite beam representing the soil may be replaced by a one-degree-of-freedom system equivalent to a linear viscous dashpot with constant C = GA/Lm and restoring force f=i
GA a u. L m
Figure 7. Modal impulse response functions for the shear force at the base of the superstructure: —— ——, impulse response function from the superposition of the first 10 modal responses. m = 0·20.
28
. .
Figure 8. Modal impulse response functions for the shear force at the base of the superstructure from the first 10 modal contributions: —— ——, structure interacting with the soil; ——, structure on a fixed base.
This is quite different from other soil models [15,17,18] where the restoring force is expressed as f = (GA/L)(k + ica)u. As should have been expected, due to the semi-infinite length, the stiffness coefficient vanishes in the present model. Although this may not be realistic for soil–structure interaction problems, it nevertheless has the advantage of being amenable to simple analytical solutions which are suitable for illustrating the concept of the complex mode superposition method for continuous systems. Furthermore, the model reproduces exactly the behaviour of a soil layer of finite depth resting on a different layer of infinite depth. It should also be noticed that the restoring force provided by the soil layer of infinite depth is linearly dependent on the frequency, just as is the viscous part of the cone model; see Figure 2. 10.2. The frequencies of vibration of the structural model on a fixed base are provided by the parameter pn = (2n − 1)p/2 for negligible damping. For the soil–structure interacting model, the frequencies of vibration take the form an = pn + iqn , where
b b
1 m+1 qn = ln 2 m−1
depends only on the relative properties of the structure and the soil. For the generic frequency, the real part pn is given by either expression (2a) or expression (3a) according to whether m q 1 or m Q 1. In the latter case, pn coincides with the corresponding value for the structure on a fixed base, while in the former case there is a −p/2 shift with respect to this value. It is worth noticing that the real part of the complex frequency is in fact coincident with the damped frequency, while the imaginary part is responsible for the amplitude decay of the oscillations. Therefore for m Q 1 the frequencies of the damped motion coincide with those of the structure on a fixed base, while for m q 1 there is a −p/2 shift. The
29
imaginary part is independent of the order of the mode and depends on the ratio m as shown in Figure 3. The general pattern described here is violated by the zero frequency which, besides having a zero real part, has also a zero imaginary part and the corresponding mode is not damped. 10.3. The first three modes of vibration are shown in Figure 4 for two values of the material property ratio m against the corresponding modal shapes of the structure on a fixed base. It may be seen that for m = 0·20 the real parts of the frequency parameter coincide with those of the structure on a fixed base (m = 0), while the imaginary part is constant for all modes. It is worth noticing that for this case the real parts of the modes quite closely resemble the modal shapes of the structure on a fixed base. However, the presence of the imaginary parts ensures that each section vibrates with a different phase lag. The case of m = 1·50 (m q 1) shows the −p/2 shift on the real part of the frequency parameter, while the imaginary part is constant for all modes. It is interesting that in this case the imaginary part of the first mode vanishes. Furthermore, each mode presents large displacements at
Figure 9. Frequency response functions for the displacement at the top of the superstructure: (a) amplitude; (b) phase. m = 0·20.
30
. .
Figure 10. Frequency response functions for the shear force at the base of the superstructure: (a) amplitude; (b) phase; m = 0·20.
the base of the structure and the imaginary parts are much more pronounced than the corresponding ones for m = 0·20. It should also be observed that the very first mode has been omitted in the representation, being a constant translation of the superstructure associated to a zero frequency, i.e., zero real and imaginary part. 10.4. A few modal impulse functions have been drawn for the material parameter ratio m = 0·20 for the displacement at the top and for the shear force at the base of the superstructure. The excitation is provided by a free field acceleration impulse I at the interface between the superstructure and the soil. In Figure 5 the modal impulse response functions for the displacement at the top of the structure are shown for n = 0,1,2. It can be seen that the first contribution, which corresponds to the constant translational real mode (n = 0), gives rise to a constant displacement. This is consistent with the absence of damping in this mode. The subsequent two modes (n = 1,2) are clearly damped, as may be seen from the amplitude decay of the impulse response functions. In the same figure, the impulse response function obtained by the superposition of the first 10 modes is also shown. This clearly shows a damped oscillation about an average value provided by the constant contribution of the undamped first mode.
31
Figure 11. Amplitude of the frequency response function for the acceleration of the relative motion at the surface of the top layer for different values of the material property ratio m.
The comparison with the results for the structure on a fixed base is shown in Figure 6. While the latter shows an undamped oscillation of zero mean, because no damping is considered in the superstructure, the former shows the behaviour which has just been described because of radiation damping towards the far end of the infinite beam. The modal impulse response functions for the shear force at the base of the superstructure are shown in Figure 7 for n = 1,2,3. For n = 0 the impulse response function is equal to zero because the spatial derivative f'0 (z) = 0, for the corresponding mode is a rigid translation. The amplitude decay of the modal response functions due to radiation damping may be observed from the figure. The superposition of the first 10 modal response functions provides a clear picture of the shape of the overall impulse response function for the shear force at the base. This resembles a decaying square wave with period close to that of the first mode of vibration. The comparison on the basis of the first 10 modal contributions of this response function with that of the structure on a fixed base is shown in Figure 8. The only difference appears
Figure 12. Amplitude of the frequency response function for the acceleration of the absolute motion at the surface of the top layer for different values of the material property ratio m.
32
. .
to be the amplitude decay of the response function for the structure interacting with the soil which is due to radiation damping, while the structure on a fixed base is undamped. 10.5. The frequency response functions for the displacement at the top of the superstructure calculated from a number of modal contributions (n = 3,5) are shown in Figure 9 against the exact frequency response function derived at the end of section 10.3. Figure 9(a) shows the amplitude, while Figure 9(b) shows the phase angle. It may be noticed how just a few modal contributions provide a very good approximation of the exact response. The frequency response functions for the shear force at the base of the superstructure are shown in Figure 10. Figure 10(a) shows the amplitude, while Figure 10(b) shows the phase angle. Once more it may be seen how only a few modal contributions are needed in order to provide an adequate approximation to the exact response. 10.6. As has already been mentioned, the model presented here may be used to represent the behaviour of a soil layer of finite depth resting on a firmer layer of infinite depth. An interesting problem in seismic engineering is the evaluation of the amplitude of the motion on the surface of the top layer being known the amplitude of the motion at the interface. The solution to this problem obtained by means of wave propagation methods has been reported by Wakabayashi [19] (see section 2.3.3, pp. 62–67). The amplitude of the frequency response function for the acceleration at the surface of the top layer is shown in Figure 11. This, however, is the acceleration of the relative motion. When the acceleration at the base of the layer is added to this, the total surface acceleration is obtained, which is shown in Figure 12, being identical to the result reported by Wakabayashi [19]. 11. CONCLUSIONS
In this paper, the complex mode superposition method has been used for the evaluation of the dynamical response of a simple continuous model with radiation damping. This model may be used in order to obtain qualitative results for the behaviour of structures interacting with the soil and for the description of the dynamical behaviour of a soil layer of finite depth resting on another one of infinite depth. For the application of the mode superposition method, analytical expressions have been obtained for the complex frequencies and modes of vibration. Two set of orthogonality conditions have been derived and used for decoupling the equations of motion. A novel technique, which is more in line with the classical mode superposition method, has been used in order to derive the principal co-ordinates. The excitation is provided by a free field motion in the bottom layer specified by the surface acceleration. Although the applications have been carried out for impulsive and stationary harmonic excitations, any type of ground motion may be considered. The model may appear to be rather unrealistic for soil–structure interaction problems inasmuch as its dynamic stiffness possesses only the imaginary part. However, it was used for the rather pleasant feature of providing analytical expressions for frequencies and modes of vibration in order to demonstrate the complex mode superposition method for continuous systems. More realistic models, such as the cone model, are amenable to the same type of analysis although the frequencies may than have to be derived numerically. The numerical applications have been used to illustrate the main aspects of the analysis including the effects of radiation damping on frequencies and modes of vibration and on the dynamical response of the model. Finally some results have been
33
compared with those available in the literature. In conclusion, this paper shows that when complex frequencies and modes of vibration may be derived, the complex modes superposition method can be successfully applied for the calculation of the dynamical response of continuous structures. ACKNOWLEDGMENTS
This work has been financially supported by the National Group for Defence against Earthquakes of the Italian Research Council (GNDT/CNR) and by the Italian Ministry for University and Scientific and Technological Research (MURST). REFERENCES 1. S. H. C 1970 Journal of Sound and Vibration 11(1), 3–18. The role of damping in vibration theory. 2. T. H. C and M. E. O’K 1971 ASME Journal of Applied Mechanics 32, 583–588. Classical normal modes in damped linear dynamic systems. 3. R. W. C and J. P 1993 Dynamics of Structures, second edition. New York: McGraw-Hill. 4. J. H. W 1965 The Algebraic Eigenvalue Problem. Oxford: Clarendon Press. 5. J. H. W and C. R 1971 Linear Algebra. New York: Springer-Verlag. 6. J. M. K 1993 Earthquake-resistant Design with Rubber. New York: Springer-Verlag. 7. J. M. K 1994 Proceedings of the 10th European Conference on Earthquake Engineering, Vienna, Austria. The development of isolators and isolation components for earthquake-resistant design. 8. J. A. I, F. L-A, D. F, H. K and J. M. K 1994 Proceedings of the 10th European Conference on Earthquake Engineering, Vienna, Austria. Experimental verification of seismic effectiveness of TMDs with viscoelastic materials. 9. J. E. L 1982 in ASME Earthquake Ground Motion and its Effects on Structures, AMD 53 (S. K. Datta, editor), 41–57. New York. Linear soil–structure interaction: a review. 10. A. S. V 1977 in Structural and Geotechnical Mechanics, A Volume Honoring N. M. Newmark (W. J. Hall, editor), 333–361; Englewood Cliffs, NJ: Prentice-Hall. Dynamics of structure–foundation systems. 11. G. B. W 1978 Earthquake Engineering and Structural Dynamics 6, 535–556. Soil–structure interaction for tower structures. 12. J. B 1976 ASCE Journal of Engineering Mechanics 102(EM 5), 771–786. Modal analysis for building–soil interaction. 13. W. C. H and M. F. R 1964 Dynamics of Structure. Clifton, NJ: Prentice-Hall. See pp. 313–337. 14. A. S. V and C. E. V 1986 Earthquake Engineering and Structural Dynamics 14, 217–243. Modal analysis of non-classically damped linear systems. 15. J. W. M and A. S. V 1974 Proceedings of the Fifth World Conference on Earthquake Engineering, IAEE Rome, Italy, 2610–2613. Simple models for foundations in lateral and rocking motion. 16. G. O and A. S 1994 Proceedings of the Fifth International Conference on Recent Advances in Structural Dynamics, Southampton, U.K., 775–784. Complex modal analysis of a soil–structure interacting model. 17. J. W. M and J. P. W 1992 ASCE Journal of Geotechnical Engineering 118(5), 667–685. Cone models for homogeneous soil. 18. J. W. M and J. P. W 1992 ASCE Journal of Geotechnical Engineering 118(5), 686–703. Cone models for soil layer on rigid soil. 19. M. W 1986 Design of Earthquake-resistant Structures. New York: McGraw-Hill. See pp. 62–67.
COMPLEX MODAL ANALYSIS OF A CONTINUOUS MODEL WITH RADIATION DAMPING G. O Istituto di Scienza delle Costruzioni, Universita` di Catania, Viale A. Doria 6, 95125 Catania, Italy
A. S Istituto di Ingegneria Civile ed Energetica, Universita` di Reggio Calabria, Via E. Cuzzocrea 48, 89100 Reggio Calabria, Italy (Received 5 May 1994, and in final form 25 November 1994) With reference to a simplified model for soil–structure interaction problems, this paper describes the use of the complex mode superposition method for the analysis of the dynamical response of continuous systems. Analytical expressions are derived for complex frequencies and modes of vibration. Two sets of orthogonality conditions are established for the model and used in decoupling the equations of motion. Numerical applications showing the convergence of the method and allowing for the comparison with other methods of analysis are presented. Finally a physical interpretation for the complex modes of vibration is provided and the source of radiation damping is clearly expounded. 7 1996 Academic Press Limited
1. INTRODUCTION
The dynamical response of structures is greatly affected by damping which may be from many sources and usually is difficult to evaluate [1]. For this reason the mathematical models used in the applications of engineering practice adopt either linear viscous or hysteretic damping. Classical damping [2,3], where the modes of vibration of the damped structure coincide with those of the undamped one, is generally used because the calculation of frequencies and modes of vibration can be performed by means of well-established methods of eigenvalue analysis [4,5]. However, there are some important engineering problems where damping is no longer of the classical type and/or cannot be treated as such [3]. Amongst these are structures with base isolation [6,7] and those with special damping devices [8]. Another large class is that of the soil–structure interacting systems [3,9–12]. Here, an important source of damping is energy radiation across the boundary at infinity. In all those cases the modes of vibration for the damped system no longer coincide with those of the undamped one and in fact are complex as are the corresponding frequencies. The calculation of complex frequencies and modes of vibration [13] has been considered in the literature for discrete structures and applications to vibration analysis have been presented [14]. The authors are not aware of the application of the same kind of analysis to continuous systems. The object of the present paper is to show how complex modal analysis can be used in the calculation of the dynamical response of continuous systems. A simplified model 15 0022–460X/96/160015 + 19 $18.00/0
7 1996 Academic Press Limited
16
. .
which allows a closed form calculation for the frequencies and modes of vibration is used as a basis for the demonstration of the mode superposition method in complex form. The physical meaning of the complex frequencies and modes of vibration is clearly stated and the proper orthogonality conditions are established for the model. Although the model has been devised as a simple tool for the demonstration of the use of complex modal analysis for continuous systems, it has nevertheless some physical characteristics which enable an understanding to be gained of the source and effects of radiation damping in soil–structure interacting systems. These aspects will also be described, but the limitations of the model will be clearly stated. The ideas described above will be illustrated by means of numerical applications showing the effect of radiation damping on frequencies and modes of vibration and on the dynamical response. Finally, the convergence of the mode superposition method is also shown through numerical examples.
2. THE PHYSICAL MODEL
The physical model has been selected in such a way as to be amenable to a closed form solution for both the natural frequencies and the modes of vibration. This permits an easy characterization of the model and a clear understanding of the physical aspects of the problem of radiation damping. The model, however, may not be quite realistic in the interpretation of the behaviour of soil–structure interacting systems for reasons which will be explained later. The actual model is composed of a shear beam of finite length L, mass per unit length m and shear rigidity GA attached to a semi-infinite shear beam of mass per unit length ms and shear rigidity Gs As ; see Figure 1. Due to the infinite length, the overall static stiffness of the semi-infinite beam, i.e., the force applied to the near end required to produce a unit displacement, is zero. This is contrary to our physical insight and, in fact, a better model for soil–structure interaction problems could be provided by the shear cone introduced by Meek and Veletsos [15]. Nevertheless, the model presented here is preferable in a qualitative study, intended essentially to demonstrate the aspects of the complex modal superposition method in the calculation of the dynamical response of continuous structures. The shear cone model would only require a numerical procedure for the calculation of the frequencies of vibration, but then the analysis would follow along similar lines. The present model could be used also to make a good approximation in the
Figure 1. The physical model: a shear beam of finite length represents the structure; a semi-infinite shear beam represents the soil.
17
description of the behaviour of a soil layer of finite depth resting on a different layer of infinite depth.
3. THE GOVERNING EQUATIONS
It is assumed in what follows that the free field motion at the soil surface is known. In the present model this amounts to the knowledge of the motion at the near end of the semi-infinite beam when the beam of finite length simulating the structure or the soil layer is absent. The equation of motion for the structure or the soil layer takes the form [3] GAu0(z,t) − mu¨ (z,t) = mu¨g (t), where u(z,t) is the displacement of the structure relative to the ground or base layer and u¨g (t) is the prescribed free field ground acceleration. The equation of motion for the soil or base layer takes the form Gs As u0s (z,t) − ms u¨s (z,t) = 0, where us (z,t) is the displacement in the ground, additional to the free field motion. The two equations of motion need to be supplemented by proper boundary conditions which may be written as follows: GAu'(L,t) = 0,
u(0,t) = us (0,t),
GAu'(0,t) = Gs As u's (0,t).
These ensure that the shear force vanishes at all times at the free end of the structure and that displacements and shear forces are continuous at the interface between the finite and the semi-infinite beams. A fourth condition should ensure the vanishing of displacements and shear forces at the far end of the semi-infinite beam and will be discussed later.
4. FREQUENCIES OF VIBRATION
In free vibrations the free field ground motion is non-existent, i.e., ug (t) = 0, and the time dependence of the structure and ground motions is harmonic: u(z,t) = f(z)exp{ivt},
us (z,t) = fs (z)exp{ivt}.
Therefore the following equations must be satisfied: GAf0(z) + mv 2f(z) = 0,
Gs As f0(z) + ms v 2fs (z) = 0,
or by setting b 2 = mv 2/GA and bs2 = ms v 2/Gs As , f0(z) + b 2f(z) = 0,
f0s (z) + bs2 fs (z) = 0.
The general solution of the above equations may be written as f(z) = C1 exp{ibz} + C2 exp{−ibz},
fs (z) = C3 exp{ibs z} + C4 exp{−ibs z}.
By assuming that the frequency parameter a = bL = vzmL 2/GA is complex with a negative imaginary part, the vanishing of displacements and forces at the far end of the semi-infinite beam requires the vanishing of the integration constant C4 . The additional boundary conditions GAf'(L) = 0,
f(0) = fs (0),
GAf'(0) = Gs As f's (0),
. .
18
lead to the algebraic eigenvalue problem
&
'& ' & '
a exp{ia} −a exp{−ia} 0 1 1 −1 m −m −1
C1 0 C2 = 0 , C3 0
(1)
where m = zmGA/ms Gs As is a dimensionless structural parameter. The frequency equation is therefore given by a(cosa + im sina) = 0, from which it may be seen that the problem admits zero frequency. The remaining frequencies are obtained as solutions of the equation cosa + im sina = 0. By setting a = p + iq this complex equation gives rise, after some algebraic manipulations, to the two real equations cosp(coshq − m sinhq) = 0,
sinp(sinhq − m coshq) = 0,
solutions of which may be written as
$ %
(2a,b)
$ %
(3a,b)
pn = 2(n − 1)p,
n = 1,2,3,...,
1 m+1 qn = ln 2 m−1
p pn = 2(2n − 1) , 2
n = 1,2,3,...,
1 1+m qn = ln 1−m 2
for m q 1 and
for m Q 1. It is important to realize that the imaginary part of the frequency parameter is independent of n. The actual frequencies are therefore complex even if the model is elastic, pointing to the existence of a source of energy dissipation, the nature of which will be discussed later. The frequency parameter an may be related to the dimensionless natural frequency p˜n and damping ratio jn of a viscously damped one-degree-of-freedom system as follows [14]: an = p˜n ( 2 z1 − jn2 + ijn ),
p˜n = zpn2 + qn2 ,
jn = qn /p˜n .
5. MODES OF VIBRATION
Once the frequencies of vibration have been evaluated, the corresponding modes of vibration may be calculated by replacing the frequency parameter an for a in equations (1) and solving for C1 , C2 and C3 . The nth vibration mode therefore takes the form fn (z) = cos[an (1 − z)],
fsn (z) = cosan exp{idan z},
where d = m(ms /m) and z = z/L. It should be noticed that the modes of vibration are complex and occur in conjugate pairs. One set of modes of vibration corresponds to the positive values of pn , while the conjugate set corresponds to the negative values of pn . Two exceptions to the general trend are worth mentioning. For pn = 0 the frequency is purely imaginary and the mode takes the form fn (z) = cosh[qn (1 − z)],
fsn (z) = coshqn exp{−dqn z},
19
and is real. For a = 0 the mode is real and unity over all the length of the model: i.e., f(z) = 1, fs (z) = 1. In both cases the conjugate pairs are in fact coincident, while in all other cases they are distinct.
6. PHYSICAL INTERPRETATION OF COMPLEX MODES OF VIBRATION
The complex modes fn associated to the complex frequency vn do not in themselves have any physical relevance. In fact the motion of the structural system would take the form unc (z,t) = Cn exp{sn t}fn (z),
(4)
where sn = ivn and Cn is a constant which is related to the initial conditions. It is easy to see that the conjugate of expression (4) also satisfies the governing equations and therefore defines a mathematically admissible motion u¯nc (z,t) = C n exp{s¯ n t}f n (z).
(5)
The sum of expressions (4) and (5) is obviously real and therefore provides a physically admissible motion: i.e., un (z,t) = 2Re[unc (z,t)]. In terms of real algebra, this may be written more expressively as un (z,t) = exp{−jn =vn =t}=fn (z)={An cos[vDn t + un (z)] + Bn sin[vDn t + un (z)]},
(6)
after having set Cn = 12 (An − iBn ),
tanun (z) =
Im[fn (z)] , Re[fn (z)]
vDn = =vn =z1 − jn2 ,
where =·= denotes the modulus of a complex number. Unlike classical modes, where un (z) = 0 or un (z) = p, here the phase angle un (z) may take any value depending on the ratio of the imaginary to the real part of the complex function fn (z). Any harmonic motion with frequency vDn takes the above form where the real constants An and Bn may be found from the initial conditions. The modal shape is time dependent because the displacement does not reach its maximum value simultaneously at all points. However, as may be seen from equation (6), a modal shape repeats itself after each period. It should be noticed that the motion described by equation (6) is damped even though the system is elastic.
7. ORTHOGONALITY CONDITIONS
When the mode superposition method is used in dynamical problems, two sets of orthogonality conditions are usually required in order to uncouple the equations of motions in principal co-ordinates [3]. In the following, the appropriate orthogonality conditions for the problem at hand will be derived. Because the interest is restricted to the beam of finite length, the orthogonality conditions will be established with reference to this subsystem only. The equations of motion may be written for mode n and m as GAf0n + vn2 mfn = 0,
GAf0m + vm2 mfm = 0.
. .
20
By multiplying the first equation by fm and the second by fn and integrating over the length L it follows that
g
L
GAf0n fm dz + vn2
0
g
g
L
mfnfm dz = 0,
0
L
GAf0m fn dz + vm2
0
g
L
mfnfm dz = 0.
0
Integrating by parts and taking account of the boundary conditions, the previous equations may be written as vn2
g
L
g
L
g
mfn fm dz − ivn zms Gs As fn (0)fm (0) −
0
vm2
L
GAf'n f'm dz = 0,
(7)
0
mfn fm dz − ivm zms Gs As fn (0)fm (0) −
0
g
L
GAf'n f'm dz = 0.
(8)
0
Subtracting equation (8) from equation (7), one finds (vn2 − vm2 )
g
L
mfn fm dz − i(vn − vm )zms Gs Asfn (0)fm (0) = 0,
0
which, for vn $ vm , leads to the first orthogonality condition: (vn + vm )
g
L
mfn fm dz − izms Gs As fn (0)fm (0) = 0.
(9)
0
Subtraction of equation (8) multiplied by vn from equation (7) multiplied by vm provides (vn − vm )vn vm
g
L
mfn fm dz + (vn − vm )
0
g
L
GAf'n f'm dz = 0,
0
which, for vn $ vm , gives the second orthogonality condition: vn vm
g
L
mfn fm dz +
0
g
L
GAf'n f'm dz = 0.
(10)
0
8. RADIATION DAMPING
Upon setting sn = ivn the orthogonality conditions become sn sm
g
L
0
(sn + sm )
g
mfn fm dz −
g
L
GAf'n f'm dz = 0,
0
L
mfn fm dz + zms Gs As fn (0)fm (0) = 0.
0
To each eigenvalue sn there corresponds a conjugate one s¯ n as may be realized from
21
Figure 2. Dynamic stiffness functions for soil models: (a) present model; (b) cone model. f = (GA/L)(k + ica)u.
expressions (2) and (3) for the natural frequency parameters. By taking sm = s¯ n and by noticing that sn + s¯ n = −2jn =vn = and sn s¯ n = =vn =2, it follows that =vn =2 =
f0L GA=f'n =2 dz kn = , mn f0L m=fn =2 dz
2jn =vn = =
zms Gs As =fn (0)=2 f0L m=fn =2 dz
=
cn . mn
(11,12)
From expression (12) it follows that the soil in this model is represented by a series of modal linear viscous dashpots with damping constant cn = zms Gs As =fn (0)=2. Furthermore, expressions (11) and (12) could also be written as =vn =2 =
L f−a GA=f'n =2 dz , L f−a m=fn =2 dz
2jn =vn = =
zms Gs As =fn (−a)=2 L f−a m=fn =2 dz
,
as may easily be realized, if the equations of motion are integrated over the full length of the model. It is therefore clear that the loss of energy which generates damping occurs at the far end of the semi-infinite beam. It is also worth noticing that if modes and frequencies were real, expression (11) would take a quite familiar form.
22
. . 9. THE MODE SUPERPOSITION METHOD
In what follows, the use of the mode superposition method [3,14] for the evaluation of the dynamical response of the system to a free field motion in the sense specified in section 3 will be outlined. First the modal impulsive response functions will be derived. Then these will be used to evaluate the response to a general free field motion specified in terms of the acceleration u¨g (t). Finally the frequency response function is evaluated in closed form in order to check the convergence of the complex mode superposition method. 9.1. The equation of motion for impulsive free field accelerations may be written as GAu0(z,t) − mu¨ (z,t) = mId(t)
(13)
for the structural model or soil layer, with I = f u¨g (t)dt denoting the acceleration impulse and with d(t) being Dirac’s delta function. In the standard mode superposition method, the unknown displacement function u(z,t) admits the following expansion in terms of eigenfunctions: 0+ 0−
a
u(z,t) = s yn (t)fn (z). n=1
Here, contrary to the classical method, both vibration modes fn (z) and principal co-ordinates yn (t) are complex functions of the spatial co-ordinate z and time t. Due to the impulsive nature of the loading, the principal co-ordinates are expected to take on the expression yn (t) = Cn exp{ivn t}, which implies y˙n (t) = ivn yn (t),
y¨n (t) = −vn2 yn (t).
Figure 3. Imaginary part of the frequency parameter a = p + iq as a function of the material property ratio m = zmGA/ms Gs As .
23
Figure 4. Modes of vibration of the superstructure: (a) structure on a fixed base; (b) and (c) structure on soft soils. – – – –, Reference line; —— ——, real part; ——, imaginary part. For (a) m=0, q = 0; for (b) m = 0·20, q = 1·199; for (c) m = 1·50, q = 0·805.
By using the eigenfunction expansion in the equation of motion (13) this becomes a
s [GAyn (t)f0n (z) − my¨n (t)fn (z)] = mId(t). n=1
Upon multiplication by fm (z) and integration over the length L it follows that a
$ g
L
s yn (t) n=1
0
GAf0n (z)fm (z)dz − y¨n (t)
g
L
0
%
mfn (z)fm (z)dz = Id(t)
g
L
0
mfm (z)dz.
. .
24
Integration by parts and application of boundary conditions leads to the equation a
6 $
g
s yn (t) −ivn zms Gs As fn (0)fm (0) − n=1
L
GAf'n (z)f'm (z)dz
0
−y¨n (t)
g
7
L
mfn (z)fm (z)dz = Id(t)
0
g
%
L
mfm (z)dz.
(14)
0
By using the orthogonality condition (10), this may be written as a
6 $
s yn (t) −ivn zms Gs As fn (0)fm (0) + vn vm n=1
−y¨n (t)
g
L
mfn (z)fm (z)dz
0
g
7
L
mfn (z)fm (z)dz = Id(t)
0
g
%
L
mfm (z)dz.
0
After some algebraic manipulations, and making use of the relationships between the principal co-ordinates and their derivatives, the following equation is found: a
s n=1
6$
izms Gs As fn (0)fm (0) − (vn + vm )
g
L
0
mfn (z)fm (z)dz
% 7
y¨n (t) = Id(t) vn
g
L
mfm (z)dz. (15)
0
It may be seen that the term in square brackets represents the orthogonality condition (9) and therefore vanishes for vn $ vm . Therefore, the summation of equation (14) and equation (15) reduces to only one term, thus decoupling the equation of motion. The uncoupled equations of motion may be written as 2M n y¨n (t) + C n y˙n (t) = −L n Id(t),
Figure 5. Modal impulse response functions for the displacement at the top of the superstructure: —— ——, impulse response function from the superposition of the first 10 modal responses. m = 0·20.
25
Figure 6. Modal impulse response functions for the displacement at the top of the superstructure from the first 10 modal contributions: —— ——, structure interacting with the soil; ——, structure on a fixed base.
where L n =
g
L
mfn (z)dz,
M n =
0
g
L
mfn2 (z)dz,
C n = zms Gs As fn2 (0).
0
Integration of the modal equations over the small interval of time [0−,0+] provides the result 2M n y˙n (0+) + C n yn (0+) = −L n I, from which the constant Cn may be derived: Cn = −
I f0L mfn (z)dz 2ivn f0L mfn2 (z)dz + zms Gs As fn2 (0)
.
The modal impulse response function therefore takes on the expression hnc (z,t) = Cn fn (z)exp{ivn t} where the constant Cn is evaluated for I = 1. It should be noted that this function is complex, and that for the calculation of the dynamical response a formulation in terms of real algebra may be more convenient. This can easily be achieved by combining each modal contribution with that of its conjugate: i.e., hnr (z,t) = 2Re[Cn fn (z)exp{ivn t}]. In this way each modal impulse response function is real and includes the contribution of both the mode and its conjugate. Although this function is formally real, it still requires complex algebra for its calculation. However, by defining the real functions bn (z) = 2Re[Cn fn (z)],
gn (z) = 2Im[Cn fn (z)],
an (z) = jn bn (z) − z1 − jn2 gn (z),
the impulse response function may be expressed entirely in terms of real algebra as hnr (z,t) = an (z)=vn =hn (t) + bn (z)h n (t),
. .
26 where
hn (t) = (1/vDn )exp{−jn =vn =t}sinvDn t represents the impulse response function of a one-degree-of-freedom linear viscous system with natural frequency =vn =, damping ratio jn and damped frequency vDn = =vn =z1 − jn2 . 9.2. The response to a general free field ground motion may be obtained by using the mode superposition method and the previously defined impulse response function as follows: a
u(z,t) = s
g
t
u¨g (t)hnr (z,t − t)dt.
n=1 0
In this expression the unit impulse has been replaced by I = u¨g (t)dt and the load history has been divided into a succession of impulses in the standard way of dynamical analysis. For computational purposes, it is convenient to express the dynamical response in terms of response integrals as a
u(z,t) = s [an (z)Vn (t) + bn (z)D n (t)],
(16)
n=1
where Vn (t) = =vn =Dn (t) = =vn =
g
t
u¨g (t)hn (t − t)dt,
D n (t) =
0
g
t
u¨g (t)h n (t − t)dt.
0
9.3. When the free field ground motion is harmonic the response integrals defined in the previous section may be evaluated in closed form: Dn (t) =
exp{iv¯ t} − [cosvDn t + (jn /z1 − jn2 + iv¯ )sinvDn t]exp{−jn =vn =t} u¨g0 , =vn =2 − v¯ 2 + i2jn =vn =v¯
D n (t) = {iv¯ exp{iv¯ t} + [vDn sinvDn t + jn =vn =cosvDn t − (jn /z1 − jn2 + iv¯ ) ×(vDn cosvDn t − jn =vn =sinvDn t)]exp{−jn =vn =t}}
u¨g0 . =vn =2 − v¯ 2 + i2jn =vn =v¯
The dynamical response can still be evaluated by using expression (16). It may be worth noticing that the stationary response may be obtained by removing from the response integrals the transient part, to obtain Dn (v¯ ) =
u¨g0 , =vn =2 − v¯ 2 + i2jn =vn =v¯
D n (v¯ ) =
iv¯ u¨g0 . =vn =2 − v¯ 2 + i2jn =vn =v¯
The stationary response to harmonic free field ground motions may also be obtained directly in closed form without making use of the mode superposition method. Actually this solution may be used to demonstrate the convergence of the method. The derivation of the stationary harmonic response may be obtained directly from the equations of motion by assuming free field harmonic displacements of the type us (z,t) = ug0 cosbs z exp{iv¯ t}.
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By applying the proper boundary conditions, the following expressions are obtained for the displacements and the accelerations in the top layer or in the structure: u0 (z) = u¨0 (z) = −a 2
cosa(1 − z/L) − cosa − im sina ug0 , cosa + im sina cosa(1 − z/L) − cosa − im sina GA u . cosa + im sina mL 2 g0
The expression for the shear force is also useful for application purposes: T0 (z) =
GA a sina(1 − z/L) u . L cosa + im sina g0
10. NUMERICAL APPLICATIONS
Some numerical applications are presented in order to illustrate the concepts and methods of analysis reported in the previous sections. First of all, the equivalence of the soil model to a simple one-degree-of-freedom system is shown. Then a quantitative and qualitative representation of the modal radiation damping and its effect on the frequencies of vibration is presented. The graphical representation of a few modal shapes is provided. The modal impulse response functions are presented for a few modes of vibration, together with the corresponding frequency response functions. Finally the mode superposition method is applied to the case of a stationary harmonic free field ground motion showing the convergence to the exact solution. 10.1. It has been shown [16] that the semi-infinite beam representing the soil may be replaced by a one-degree-of-freedom system equivalent to a linear viscous dashpot with constant C = GA/Lm and restoring force f=i
GA a u. L m
Figure 7. Modal impulse response functions for the shear force at the base of the superstructure: —— ——, impulse response function from the superposition of the first 10 modal responses. m = 0·20.
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. .
Figure 8. Modal impulse response functions for the shear force at the base of the superstructure from the first 10 modal contributions: —— ——, structure interacting with the soil; ——, structure on a fixed base.
This is quite different from other soil models [15,17,18] where the restoring force is expressed as f = (GA/L)(k + ica)u. As should have been expected, due to the semi-infinite length, the stiffness coefficient vanishes in the present model. Although this may not be realistic for soil–structure interaction problems, it nevertheless has the advantage of being amenable to simple analytical solutions which are suitable for illustrating the concept of the complex mode superposition method for continuous systems. Furthermore, the model reproduces exactly the behaviour of a soil layer of finite depth resting on a different layer of infinite depth. It should also be noticed that the restoring force provided by the soil layer of infinite depth is linearly dependent on the frequency, just as is the viscous part of the cone model; see Figure 2. 10.2. The frequencies of vibration of the structural model on a fixed base are provided by the parameter pn = (2n − 1)p/2 for negligible damping. For the soil–structure interacting model, the frequencies of vibration take the form an = pn + iqn , where
b b
1 m+1 qn = ln 2 m−1
depends only on the relative properties of the structure and the soil. For the generic frequency, the real part pn is given by either expression (2a) or expression (3a) according to whether m q 1 or m Q 1. In the latter case, pn coincides with the corresponding value for the structure on a fixed base, while in the former case there is a −p/2 shift with respect to this value. It is worth noticing that the real part of the complex frequency is in fact coincident with the damped frequency, while the imaginary part is responsible for the amplitude decay of the oscillations. Therefore for m Q 1 the frequencies of the damped motion coincide with those of the structure on a fixed base, while for m q 1 there is a −p/2 shift. The
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imaginary part is independent of the order of the mode and depends on the ratio m as shown in Figure 3. The general pattern described here is violated by the zero frequency which, besides having a zero real part, has also a zero imaginary part and the corresponding mode is not damped. 10.3. The first three modes of vibration are shown in Figure 4 for two values of the material property ratio m against the corresponding modal shapes of the structure on a fixed base. It may be seen that for m = 0·20 the real parts of the frequency parameter coincide with those of the structure on a fixed base (m = 0), while the imaginary part is constant for all modes. It is worth noticing that for this case the real parts of the modes quite closely resemble the modal shapes of the structure on a fixed base. However, the presence of the imaginary parts ensures that each section vibrates with a different phase lag. The case of m = 1·50 (m q 1) shows the −p/2 shift on the real part of the frequency parameter, while the imaginary part is constant for all modes. It is interesting that in this case the imaginary part of the first mode vanishes. Furthermore, each mode presents large displacements at
Figure 9. Frequency response functions for the displacement at the top of the superstructure: (a) amplitude; (b) phase. m = 0·20.
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. .
Figure 10. Frequency response functions for the shear force at the base of the superstructure: (a) amplitude; (b) phase; m = 0·20.
the base of the structure and the imaginary parts are much more pronounced than the corresponding ones for m = 0·20. It should also be observed that the very first mode has been omitted in the representation, being a constant translation of the superstructure associated to a zero frequency, i.e., zero real and imaginary part. 10.4. A few modal impulse functions have been drawn for the material parameter ratio m = 0·20 for the displacement at the top and for the shear force at the base of the superstructure. The excitation is provided by a free field acceleration impulse I at the interface between the superstructure and the soil. In Figure 5 the modal impulse response functions for the displacement at the top of the structure are shown for n = 0,1,2. It can be seen that the first contribution, which corresponds to the constant translational real mode (n = 0), gives rise to a constant displacement. This is consistent with the absence of damping in this mode. The subsequent two modes (n = 1,2) are clearly damped, as may be seen from the amplitude decay of the impulse response functions. In the same figure, the impulse response function obtained by the superposition of the first 10 modes is also shown. This clearly shows a damped oscillation about an average value provided by the constant contribution of the undamped first mode.
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Figure 11. Amplitude of the frequency response function for the acceleration of the relative motion at the surface of the top layer for different values of the material property ratio m.
The comparison with the results for the structure on a fixed base is shown in Figure 6. While the latter shows an undamped oscillation of zero mean, because no damping is considered in the superstructure, the former shows the behaviour which has just been described because of radiation damping towards the far end of the infinite beam. The modal impulse response functions for the shear force at the base of the superstructure are shown in Figure 7 for n = 1,2,3. For n = 0 the impulse response function is equal to zero because the spatial derivative f'0 (z) = 0, for the corresponding mode is a rigid translation. The amplitude decay of the modal response functions due to radiation damping may be observed from the figure. The superposition of the first 10 modal response functions provides a clear picture of the shape of the overall impulse response function for the shear force at the base. This resembles a decaying square wave with period close to that of the first mode of vibration. The comparison on the basis of the first 10 modal contributions of this response function with that of the structure on a fixed base is shown in Figure 8. The only difference appears
Figure 12. Amplitude of the frequency response function for the acceleration of the absolute motion at the surface of the top layer for different values of the material property ratio m.
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. .
to be the amplitude decay of the response function for the structure interacting with the soil which is due to radiation damping, while the structure on a fixed base is undamped. 10.5. The frequency response functions for the displacement at the top of the superstructure calculated from a number of modal contributions (n = 3,5) are shown in Figure 9 against the exact frequency response function derived at the end of section 10.3. Figure 9(a) shows the amplitude, while Figure 9(b) shows the phase angle. It may be noticed how just a few modal contributions provide a very good approximation of the exact response. The frequency response functions for the shear force at the base of the superstructure are shown in Figure 10. Figure 10(a) shows the amplitude, while Figure 10(b) shows the phase angle. Once more it may be seen how only a few modal contributions are needed in order to provide an adequate approximation to the exact response. 10.6. As has already been mentioned, the model presented here may be used to represent the behaviour of a soil layer of finite depth resting on a firmer layer of infinite depth. An interesting problem in seismic engineering is the evaluation of the amplitude of the motion on the surface of the top layer being known the amplitude of the motion at the interface. The solution to this problem obtained by means of wave propagation methods has been reported by Wakabayashi [19] (see section 2.3.3, pp. 62–67). The amplitude of the frequency response function for the acceleration at the surface of the top layer is shown in Figure 11. This, however, is the acceleration of the relative motion. When the acceleration at the base of the layer is added to this, the total surface acceleration is obtained, which is shown in Figure 12, being identical to the result reported by Wakabayashi [19]. 11. CONCLUSIONS
In this paper, the complex mode superposition method has been used for the evaluation of the dynamical response of a simple continuous model with radiation damping. This model may be used in order to obtain qualitative results for the behaviour of structures interacting with the soil and for the description of the dynamical behaviour of a soil layer of finite depth resting on another one of infinite depth. For the application of the mode superposition method, analytical expressions have been obtained for the complex frequencies and modes of vibration. Two set of orthogonality conditions have been derived and used for decoupling the equations of motion. A novel technique, which is more in line with the classical mode superposition method, has been used in order to derive the principal co-ordinates. The excitation is provided by a free field motion in the bottom layer specified by the surface acceleration. Although the applications have been carried out for impulsive and stationary harmonic excitations, any type of ground motion may be considered. The model may appear to be rather unrealistic for soil–structure interaction problems inasmuch as its dynamic stiffness possesses only the imaginary part. However, it was used for the rather pleasant feature of providing analytical expressions for frequencies and modes of vibration in order to demonstrate the complex mode superposition method for continuous systems. More realistic models, such as the cone model, are amenable to the same type of analysis although the frequencies may than have to be derived numerically. The numerical applications have been used to illustrate the main aspects of the analysis including the effects of radiation damping on frequencies and modes of vibration and on the dynamical response of the model. Finally some results have been
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compared with those available in the literature. In conclusion, this paper shows that when complex frequencies and modes of vibration may be derived, the complex modes superposition method can be successfully applied for the calculation of the dynamical response of continuous structures. ACKNOWLEDGMENTS
This work has been financially supported by the National Group for Defence against Earthquakes of the Italian Research Council (GNDT/CNR) and by the Italian Ministry for University and Scientific and Technological Research (MURST). REFERENCES 1. S. H. C 1970 Journal of Sound and Vibration 11(1), 3–18. The role of damping in vibration theory. 2. T. H. C and M. E. O’K 1971 ASME Journal of Applied Mechanics 32, 583–588. Classical normal modes in damped linear dynamic systems. 3. R. W. C and J. P 1993 Dynamics of Structures, second edition. New York: McGraw-Hill. 4. J. H. W 1965 The Algebraic Eigenvalue Problem. Oxford: Clarendon Press. 5. J. H. W and C. R 1971 Linear Algebra. New York: Springer-Verlag. 6. J. M. K 1993 Earthquake-resistant Design with Rubber. New York: Springer-Verlag. 7. J. M. K 1994 Proceedings of the 10th European Conference on Earthquake Engineering, Vienna, Austria. The development of isolators and isolation components for earthquake-resistant design. 8. J. A. I, F. L-A, D. F, H. K and J. M. K 1994 Proceedings of the 10th European Conference on Earthquake Engineering, Vienna, Austria. Experimental verification of seismic effectiveness of TMDs with viscoelastic materials. 9. J. E. L 1982 in ASME Earthquake Ground Motion and its Effects on Structures, AMD 53 (S. K. Datta, editor), 41–57. New York. Linear soil–structure interaction: a review. 10. A. S. V 1977 in Structural and Geotechnical Mechanics, A Volume Honoring N. M. Newmark (W. J. Hall, editor), 333–361; Englewood Cliffs, NJ: Prentice-Hall. Dynamics of structure–foundation systems. 11. G. B. W 1978 Earthquake Engineering and Structural Dynamics 6, 535–556. Soil–structure interaction for tower structures. 12. J. B 1976 ASCE Journal of Engineering Mechanics 102(EM 5), 771–786. Modal analysis for building–soil interaction. 13. W. C. H and M. F. R 1964 Dynamics of Structure. Clifton, NJ: Prentice-Hall. See pp. 313–337. 14. A. S. V and C. E. V 1986 Earthquake Engineering and Structural Dynamics 14, 217–243. Modal analysis of non-classically damped linear systems. 15. J. W. M and A. S. V 1974 Proceedings of the Fifth World Conference on Earthquake Engineering, IAEE Rome, Italy, 2610–2613. Simple models for foundations in lateral and rocking motion. 16. G. O and A. S 1994 Proceedings of the Fifth International Conference on Recent Advances in Structural Dynamics, Southampton, U.K., 775–784. Complex modal analysis of a soil–structure interacting model. 17. J. W. M and J. P. W 1992 ASCE Journal of Geotechnical Engineering 118(5), 667–685. Cone models for homogeneous soil. 18. J. W. M and J. P. W 1992 ASCE Journal of Geotechnical Engineering 118(5), 686–703. Cone models for soil layer on rigid soil. 19. M. W 1986 Design of Earthquake-resistant Structures. New York: McGraw-Hill. See pp. 62–67.