The action of external time-varying loads on a continuous system leads to a nonhomogeneous partial differential equation. In general, in the light of the preceding sections, it is not difficult to obtain the governing differential equation also because—once the mass and stiffness operators have been introduced—its formal structure is similar to the matrix equation governing the forced vibrations of MDOF systems. However, now the boundary conditions come into play and we must pay due attention to them.
Two general methods of solutions can be identified:
1. integral transform methods, which are particularly well suited to systems with infinite or semi-infinite extension in space (note, in fact, that in these cases the concept of normal mode loses its meaning) and for problems with time-dependent boundary conditions;
2. the mode superposition method.
Here, we concentrate our attention on method 2, which we can call ‘the modal approach’. The relevant equation of motion is written as
(8.172)
where and, for brevity of notation, we indicate with x the set of space variables. Note that now f is a forcing function representing an external action (it is the counterpart of vector f of eq (7.1)) and must not be confused with the f functions of the preceding sections in this chapter, where this symbol was often adopted to identify a general function to be used for the specific needs of that part only. Equation (8.172) must be supplemented by the set of p boundary conditions
(8.173) Now, by virtue of the results given in Section 8.7.1, the general response w can be expressed in terms of a superposition of eigenfunctions φi multiplied by a set of time-dependent generalized coordinates yi(t), i.e.
(8.174)
which is the infinite-dimensional counterpart of eq (7.2). If we substitute the solution (8.174) into eq (8.172) and then take the inner product of the resulting equation by we get
which, owing to the orthogonality relationships (8.128), reduces to the infinite set of 1-DOF uncoupled equations
(8.175) whose solution is given by (Chapter 5 and eq (7.7))
(8.176)
where yj(0) and j(0) are the initial jth modal displacement and velocity, respectively. Moreover, it is not difficult to show that we can obtain these quantities from the inner products
(8.177)
where and are the initial conditions in physical coordinates. The analogy with eq (7.5b) is evident.
A few remarks can be made at this point.
1. First, it is apparent that the first step of the whole approach is the solution of the differential (symmetrical) eigenvalue problem satisfying the appropriate boundary conditions. The resulting set of eigenvalues and orthonormal eigenfunctions make it possible to express the general solution to the free vibrations problem (i.e. eq (8.172) with f=0) as in eq (6.50), which reads
(8.178)
For example, suppose that we want to consider the longitudinal motion of a uniform clamped-free rod of length L and à mass per unit length. Let the
initial conditions be such that and Since the mass orthonormal functions are given by
we can calculate the rod free motion by means of eq (8.178). To this end we must first evaluate the inner product
(the calculation is not difficult and is left to the reader) and then obtain the free motion as
where we know from eq (8.49) that
2. When expressed in normal coordinates, the kinetic and potential energy of free vibration assume the particularly simple forms
(8.179)
so that the Lagrangian has no coupling terms between the coordinates and is simply the Lagrangian function of an infinite number of independent harmonic oscillators.
3. The final remark has to do with the apparent discrepancy according to which our system may be equally well described by a continuous system of coordinates w(x, t) or by a discrete one, i.e. yj(t). The general feeling is that this second set of coordinates cannot describe the same number of degrees
of freedom as the first one, although this number is infinite in both cases.
However, it must be noted that, broadly speaking, the definition of normal coordinates itself somehow incorporates the boundary conditions from the outset, and this is not the case for w(x, t). In fact, the coordinates w(x, t) could be a priori chosen as completely independent of each other and this would allow us—in principle—to describe also discontinuous motions of our system. By contrast, the expansion in normal modes requires that the functions representing the point-by-point displacement of our system be reasonably well-behaved.
So, although the normal mode description is, as a matter of fact, mathematically more restrictive, it really does not put any significant restriction on the physical problem because the boundary conditions and the good behaviour of the displacement functions are dictated by the physical nature of our system and, ultimately, by the physics of the phenomena we are trying to describe. Also, note that for MDOF systems there was no need to make a similar remark because, whatever the coordinate system we choose to adopt, the elements of the stiffness matrix automatically take care of the boundary conditions.
We can now go back to the main discussion of this section and note that in Chapter 7—where we dealt with the response of MDOF systems—we gave special emphasis to the modal impulse response functions and to the modal frequency response functions, also showing their relationships with impulse response functions and frequency response functions expressed in physical coordinates. We want now to extend those concepts to the case of distributed parameter systems.
Let us consider an undamped system: from the preceding chapters we know that the jth modal impulse response function hj(t) is the solution of the equation
(8.180) where δ is the Dirac’s delta function. Moreover, we also know that (eqs (5.7a) and (5.7b))
(8.181) where the term representing the jth generalized mass does not appear in the denominator of the right-hand side of eq (8.181) because we are considering mass-orthonormal eigenfunctions.
Now, if for simplicity we assume eq (8.176) reduces to (8.182)
If we further assume that the excitation is an impulse applied at position x=xk at time we can t=0 write
so that
(8.183)
which must be substituted in eq (8.182) to obtain yj(t).
Then, from eq (8.174) we can obtain the response in physical coordinates as
(8.184) More specifically, we can consider the response at the point x=xm, i.e.
(8.185a) and note that this is just the displacement response of point xm to a unit impulse applied at point xk at the instant t=0. In other words, eq (8.185a) represents the impulse response function so that we can write
(8.185b) which is the continuous systems counterpart of eq (7.37b) where hjk denoted the physical coordinates (displacement) impulse response function at the jth DOF due to an unit force impulse applied at the kth DOF at t=0.
Alternatively, we can consider the frequency domain and note that the jth modal frequency response function (receptance in this case) can be obtained by assuming a forcing function in sinusoidal form. This means that
so that eqs (8.175) become
(8.186) hence, assuming a response which is also in sinusoidal form, we have
(8.187a)
so that the jth receptance FRF is
(8.187b) If we now consider a harmonic forcing function of unit amplitude applied at the point x=xk, i.e.
we have the modal response is then
(8.188) and the (steady-state) solution in physical coordinates is given by
(8.189a) so that the response at point x=xm is
(8.189b) Finally, if we observe that the physical coordinate response at point xm due to a harmonic excitation at xk can be expressed in the general form
where is, by definition, the physical coordinates receptance function corresponding to points xm and xk, we get from eq (8.189b)
(8.190) which is the (undamped) continuous systems counterpart of eq (7.28).
Note that from eqs (8.185b) and (8.190) we get
(8.191)
which show that the reciprocity theorem holds. Moreover, the generalization
of the second of eqs (8.191) to a general FRF function H(ω) other than receptance is straightforward and reads
(8.192) Note that, for reasons outlined in Section 7.4.1, we only consider FRF functions in the forms of receptances, mobilities and accelerances; hence, for our purposes and unless otherwise stated, the symbol H will always be tacitly assumed to mean a FRF functions in one of these forms.
For its importance in modal testing, we refer to eqs (8.189b) and (8.190) to point out that a concentrated excitation force applied at a point xk where for some index j results in no excitation of all these modes. In other words, for example, if we excite a beam by means of a concentrated load at mid-span, all even natural frequencies and modes will not contribute to the measured response. By the same token, if we pluck at mid-length a guitar string tuned to give the note A at 440 Hz, the second (A at 880 Hz), fourth (A at 1760 Hz) etc. harmonics will be missing and the sound we hear will be the superposition of the fundamental note A (440 Hz) plus the third (E at 1320 Hz), the fifth (C# at 2200 Hz) etc.
harmonics. On the other hand, if we pluck the same string at one-third of its length we will still hear the same fundamental note A at 440 Hz, but the harmonic content of the sound will be different.
8.9.1 Forced response of continuous systems: some examples Example 8.1 Consider a vertical clamped-free beam of length L, mass per unit length à, and flexural rigidity EI subjected to an excitation in the form of a lateral base displacement It is easy to realize that this situation, for example, can be used as a first approximation to model the response of a tall, slender building to an earthquake excitation. For our purposes, we ignore the fact that the definition of an appropriate g(t) is very difficult in this case and it is one of the most uncertain steps of the analysis.
To solve this problem, we need to consider eq (8.55a); we write the beam displacement y(x, t) as
(8.193) and substitute it into eq (8.55a). Note that u(x, t) represents the displacement of the beam relative to the rigid-body translation of the ground. We get
(8.194) where it is evident that the inertia forces depend on the total motion, whereas
the stiffness (and damping, if it were included in the analysis) forces depend only on the relative motion. The r.h.s. of eq (8.194) is the effective earthquake force and it is usually indicated with the symbol feff. In the light of the discussion of preceding sections, it follows that—assuming the system initially at rest—we have (eq (8.182))
and hence
where ωj are given by eq (8.66b) and the φj are the eigenfunctions (8.67), in which the constant C1 has been chosen so as to make them mutually mass- orthonormal.
The relative response in physical coordinates is then obtained as the superposition
(8.195) where, in general, for seismic excitation the minus sign in (8.195) is irrelevant and—owing to the complicated expressions of the eigenfunctions and of the ground acceleration —the integrations must be performed numerically.
In general, only a few terms of the series must be considered in order to obtain a satisfactory representation of the actual response so that, in the end, we are brought back to the case of an n-DOF system, where only a finite number (n) of modal coordinates is needed to describe the response.
The total response y(x, t) is finally obtained according to eq (8.193).
A simpler case arises if we consider the longitudinal motion of our system;
the excitation g(t) is now a vertical displacement and the relevant equation of motion is eq (8.42). The general form of the relative displacement is still given by eq (8.195) but the eigenvalues ωj are given by the first of eq (8.49) and the eigenfunctions are given by eq (8.50), where In this circumstance, it is not difficult to show that the space integration in (8.195) results in
(8.196)
and only the time integration needs to be performed numerically.
Example 8.2. Let us now consider a uniform clamped-free rod of length L and mass per unit length à excited by a tip load at the free end, i.e.
If the rod is at rest before the excitation occurs the first two terms on the r.h.s. of eq (8.176) are zero and
(8.197)
so that eq (8.176) reduces to
(8.198)
because If we further assume that the explicit
form of p(t) is a unit step (Heaviside) function θ(t), i.e.
we can substitute θ(t) into eq (8.198), perform the integration by noting that
and obtain
(8.199) Finally the displacement in physical coordinates is obtained as the super- position
(8.200) where in the second expression we take into account the explicit form of Also, it is worth pointing out that eq (8.200) is dimensionally correct because, since we have assumed a unit force, the dimensions of w(x, t) are displacement per unit force (i.e. m/N).
At this point, it is interesting to note that the system above can be analysed either as:
1. an excitation-free system with a time-dependent boundary condition at x=L, or
2. a forced vibration problem with homogeneous boundary conditions.
As stated at the beginning of the preceding section, free-vibration problems with nonhomogeneous boundary conditions are often tackled by an integral transform (Laplace or Fourier) approach; however, the modal approach can also be adopted in consideration of the fact that a boundary value problem of type (1) can usually be transformed into a boundary value problem of type (2) (e.g. Courant and Hilbert [20] or Mathews and Walker [21]).
In general, it appears that in these cases a disadvantage of the modal approach—which is essentially a ‘standing waves solution’—is that the resultant series converges quite slowly and many terms must be included in order to achieve a reasonable accuracy. By contrast, depending on how the inverse transformation is carried out, the Laplace transform method allows us the possibility to obtain a solution either in terms of standing waves or in terms of travelling waves (waves being reflected back and forth within the rod). This latter possibility—the travelling wave approach—leads to a solution in the form of a rapidly converging series, thus making this strategy more attractive. However, on physical grounds, we may argue that the time scale in which we are interested suggests the type of solution to adopt; in fact, the travelling wave solution converges rapidly when we consider the short-term response of our system whereas, if the long-term response is desired, more and more terms are needed. The situation is reversed for the modal solution:
as time progresses, the terms corresponding to higher modes die out because of damping and we are left with a series in which, say, only the first two or three terms have a significant contribution.
We will not consider an integral transform strategy of solution here (the interested reader is referred to Meirovitch [22]) but, using the rod example above, we will show how a problem of type (1) can be transformed in a problem of type (2).
Our rod problem can be formulated as a type (1) problem in the following form:
(8.201a)
(8.201b)
where eq (8.201a) is the homogeneous equation of motion and eq (8.201b) are the boundary conditions. Since one boundary condition (the second) is nonhomogeneous we assume the solution of our problem in the form
(8.202) where the term —which we can define in compact notation as
—is a so-called ‘pseudostatic’ displacement brought about by the boundary motion and v(x, t) is the displacement relative to the support displacement. Mathematically, the function ust is chosen in such a way as to make the boundary conditions for v(x, t) homogeneous. On physical grounds, the usual assumption made for the choice of ust is that no inertia forces (i.e., no accelerations) are produced by the application of the support motion;
hence the name ‘pseudostatic’. For our case, this assumption implies that ust obeys the equation
(8.203) from which follows (provided that
(8.204) Moreover, given the expression (8.202), the boundary conditions (8.201b) become
from which—if we want homogeneous boundary conditions for v(x, t)—it follows
(8.205)
Enforcing the boundary conditions (8.205) on the solution (8.204) leads to (8.206)
The transformation of the problem (8.201) into a type (2) problem is complete when we determine the nonhomogeneous equation of motion for the relative displacement v(x, t): this is simply accomplished by substituting eq (8.202) into eq (8.201a) and results in
(8.207a)
where the r.h.s. of eq (8.207) has clearly the dimensions of N/m and, for short, can be indicated with the symbol feff (effective force).
Equations (8.207a), (8.206) plus the homogeneous boundary conditions
(8.207b)
constitute our type (2) boundary value problem which fits into the scheme of problems that can be more effectively tackled by the modal approach. In this light, we expand v(x, t) in a series of eigenfunctions and calculate the normal coordinates as prescribed in eq (8.182) (note that, from eq (8.206),
we get i.e.
(8.208)
Upon substituting the explicit expressions of φj and in eq (8.208), the space integral within braces gives
so that eq (8.208) becomes
(8.209)
Now, in the problem we are considering, we assumed that p(t) is the Heaviside function θ(t); since (eq (2.67a) or (2.84)) the time integral of eq (8.209) becomes
where we take into account the properties of the Dirac’s delta function (eq (2.69)) and the explicit expression of hj. The final steps consist of substituting this result in eq (8.209), writing explicitly the series expansion of v(x, t), i.e.
and putting it all back together into the solution (8.202) which becomes
(8.210)
This result must be compared with eq (8.200) and it is not difficult to show that they are equal. This is due to the fact that the function (π2x/8L) can be expanded in a Fourier series as (the proof is left to the reader)
so that—after performing the product in eq (8.200)—the first term is exactly (x/EA), i.e. the function (x) of the pseudostatic displacement. The advantage of including explicitly the pseudostatic displacement from the outset lies in the more rapid convergence of the series (8.210) as compared to the series (8.200), the pseudostatic displacement representing the average position about which the vibration takes place.
Example 8.3. In modal testing, we are often concerned with the response of a given system to an impulse loading. So, consider the rod of Example 8.2 subjected to a unit impulse applied at x=L at t=0. The response in physical coordinates at x=L is given by eqs (8.185) and reads
(8.211)
This result should be hardly surprising because we know from Chapter 5 (eq (5.42)) that the impulse response function is the time derivative of the Heaviside response function. So, in this circumstance we could have ignored eq (8.185) by simply noting that the result (8.211) can be obtained by calculating the time derivative of eq (8.200) and by substituting x=L in it.
On the other hand, the receptance FRF can be obtained from eq (8.190):
at x=L this is
(8.212) In the light of preceding chapters, we expect that h(L, L, t) and H(L, L, ω) form a Fourier transform pair. However, the Fourier transform of eq (8.211) does not exist, but we may note that the Laplace transform of eq (8.211) does exist and is given by
where s is the (complex) Laplace operator and can be expressed as
Hence, leaving aside mathematical rigour for a moment, we see that we can arrive at eq (8.212) by first taking the Laplace transform of eq (8.211) and then letting This mathematical trick is just for purposes of illustration and it would not be needed if the system had some amount of positive damping;
as a matter of fact, this is always the case for real systems whose time response and FRFs (eq (8.211) and (8.212)) do not go to infinity when
Example 8.4. Consider now the case of a constant force P moving at a constant velocity V along an Euler-Bernoulli beam simply supported at both ends. The engineering importance of this case is evident because this example can be used to model a number of common situations, the simplest one being a heavy vehicle travelling across a bridge deck. We also make the reasonable assumption that the mass of the vehicle is small in comparison with the beam mass (the bridge deck) and it does not alter appreciably its eigenvalues and eigenfunctions.
Mathematically, the moving load can be represented as
(8.213) and, with reference to eq (8.182), we obtain