Applied Structural and Mechanical Vibrations 2009 Part 7 pot

premultiply both sides by P T and postmultiply by P to get which we can write as where we define for brevity of notation From the above it follows that which, after pre- and postmultiplication of both sides by P and P T , respectively, leads to (7.26) so that the solution (7.24) can be written as (7.27) and the (jk)th element of the receptance matrix can be explicitly written as (7.28) Now, the term in brackets in eq (7.28) looks, indeed, familiar and a slightly different approach to the problem will clarify this point. For a proportionally damped system, the equations of motion (7.22) can be uncoupled with the aid of the modal matrix and written in normal coordinates as (7.29) Each equation of (7.29) is a forced SDOF equation with sinusoidal excitation. We assume a solution in the form where j is the complex amplitude response. Following Chapter 4, we arrive at the steady-state solution (the counterpart of eq (4.42)), (7.30) where Copyright © 2003 Taylor & Francis Group LLC By definition, the frequency response function (FRF) is the coefficient H( ω ) of the response of a linear, physically realizable system to the input ; with this in mind we recognize that (7.31) is the jth modal (because it refers to normal, or modal, coordinates) FRF. If we define the n×1 vector of response amplitudes we can put together the n equations (7.29) in the matrix expression (7.32) and the passage to physical coordinates is accomplished by the transformation (7.2), which, for sinusoidal solutions, translates into the relationship between amplitudes Hence (7.33) which must be compared to eq (7.27) to conclude that (7.34a) Equation (7.34a) establishes the relationship between the FRF matrix (R) of receptances in physical coordinates and the FRF matrix of receptances in modal coordinates. This latter matrix is diagonal because in normal (or modal) coordinates the equations of motion are uncoupled. This is not true for the equations in physical coordinates, and consequently R is not diagonal. Moreover, appropriate partitioning of the matrices on the right-hand side of eq (7.34a) leads to the alternative expression for the receptance matrix (7.34b) where the term is an (n×1) by (1×n) matrix product and hence results in an n×n matrix. From eq (7.34a) or (7.34b) it is not difficult to determine that (7.35) i.e. R is symmetrical; this conclusion can also be reached by inspection of eq (7.28) where it is evident that This result is hardly surprising. In fact, owing to the meaning of the term R jk (i.e. eq (7.25)), it is just a different statement of the reciprocity theorem considered in Section 7.2. Copyright © 2003 Taylor & Francis Group LLC 7.4 Time-domain and frequency-domain response In Section 7.2, eq (7.7b) represents, in the time domain, the normal coordinate response of a proportionally damped system to a general set of applied forces. Since we pass to physical coordinates by means of the transformation u=Py, we have (7.36) so that the n×n matrix of impulse response functions in physical coordinates is given by (7.37a) Explicitly, the (jk)th element of matrix (7.37a) is written (7.37b) and it is evident that or equivalently, On the other hand, if we take the Fourier transform of both sides of eqs (7.6), we get (7.38) where we have called Y j ( ω ) and Φ j ( ω ) the Fourier transforms of the functions y j (t) and respectively. If we form the column vectors and where is the (element by element) Fourier transform of f, we obtain from eq (7.38) (7.39) Copyright © 2003 Taylor & Francis Group LLC Now, since u(t)=Py(t) it follows that and eq (7.39) leads to (7.40) which is the frequency-domain counterpart of the time-domain equation (7.36). Summarizing the results above and referring to the discussion of Chapter 5 about impulse-response functions and frequency-response functions, we can say that—as for the SDOF case—the modal coordinates functions h j (t) and are a Fourier transform pair and fully define the dynamic characteristics of our n-DOF proportionally damped system. In physical coordinates, the dynamic response of the same system is characterized by the matrices h(t) and R( ω ) whose elements are given, respectively, by eqs (7.37b) and (7.28). These matrices are also a Fourier transform pair (Section 5.4), i.e. (7.41) which is not unexpected if we consider that the Fourier transform is a linear transformation. Also, from the discussion of Chapter 5, it is evident that the considerations of this section apply equally well if ω is replaced by the Laplace operator s and the FRFs are replaced by transfer functions in the Laplace domain. Which transform to use is largely dictated by a matter of convenience. A note about the mathematical notation In general an FRF function is indicated by the symbol H( ω ) and, consequently, a matrix of FRF functions can be written as H( ω ). However, as shown in Table 4.3, H( ω ) can be a receptance, a mobility or an accelerance (or inertance) function; in the preceding sections we wrote R( ω ) because, specifically, we have considered only receptance functions, so that R( ω ) is just a particular form of H( ω ). Whenever needed we will consider also the other particular forms of H( ω ), i.e. the mobility and accelerance matrices and we will indicate them, respectively, with the symbols V( ω ) and A( ω ) which explicitly show that the relevant output is velocity in the first case and acceleration in the second case. Obviously, the general FRF symbol H( ω ) can be used interchangeably for any one of the matrices R( ω ), V( ω ) or A( ω ). By the same token, H(s) is a general transfer function and R(s), V(s) or A(s) are the receptance, mobility and accelerance transfer functions. Finally, it is worth noting that some authors write FRFs as H(i ω ) in order to remind the reader that, in general, FRFs are complex functions with a real Copyright © 2003 Taylor & Francis Group LLC and imaginary part or, equivalently, that they contain both amplitude and phase information. We do not follow this symbolism and write simply H( ω ). 7.4.1 A few comments on FRFs In many circumstances, one may want to consider an FRF matrix other than R( ω ). The different forms and definitions are listed in Table 4.3 and it is not difficult to show that, for a given system, the receptance, mobility and accelerance matrices satisfy the following relationships: (7.42) which can be obtained by assuming a solution of the form (7.23) and noting that (7.43) where we have defined the (complex) velocity and acceleration amplitudes v and a. However, the definitions of Table 4.3 include also other FRFs, namely the dynamic stiffness, the mechanical impedance and the apparent mass which, for the SDOF case are obtained, respectively, as the inverse of receptance, mobility and accelerance. This is not so for an MDOF system. Even if in this text we will generally use only R( ω ), V( ω ) or A( ω ), the reader is warned against, say, trying to obtain impedance information by calculating the reciprocals of mobility functions. In fact, the definition of a mobility function V jk , in analogy with eq (7.25), implies that the velocity at point j is measured when a prescribed force input is applied at point k, with all other possible inputs being zero. The case of mechanical impedance is different because the definition implies that a prescribed velocity input is applied at point j and the force is measured at point k, with all other input points having zero velocity. In other words, all points must be fixed (grounded) except for the point to which the input velocity is applied. Despite the fact that this latter condition is also very difficult (if not impossible) to obtain in practical situations, the general conclusion is that (7.44) where we used for mechanical impedance the frequently adopted symbol Z. Similar relations hold between receptance and dynamic stiffness and between Copyright © 2003 Taylor & Francis Group LLC accelerance and apparent mass. So, in general [1], the FRF formats of dynamic stiffness, mechanical impedance and apparent mass are discouraged because they may lead to errors and misinterpretations in the case of MDOF systems. Two other observations can be made regarding the FRF which are of interest to us: • The first observation has to do with the reciprocity theorem. Following the line of reasoning of the preceding section where we determined (eq (7.35)) that the receptance matrix is symmetrical, it is almost straightforward to show that the same applies to the mobility and accelerance matrices. • The second observation is to point out that only n out of the n 2 elements of the receptance matrix R( ω ) are needed to determine the natural frequencies, the damping factors and the mode shapes. We will return to this aspect in later chapters but, in order to have an idea, suppose for the moment that we are dealing with a 3-DOF system with distinct eigenvalues and widely spaced modes. In the vicinity of a natural frequency, the summation (7.28) will be dominated by the term corresponding to that frequency so that the magnitude can be approximated by (eqs (7.28) and (7.34b)) (7.45) where j, k=1, 2, 3. Let us suppose further that we obtained an entire column of the receptance matrix, say the first column, i.e. the functions R 11 , R 21 and R 31 ; a plot of the magnitude of these functions will, in general, show three peaks at the natural frequencies ω 1 , ω 2 and ω 3 and any one function can be used to extract these frequencies plus the damping factors ζ 1 , ζ 2 and ζ 3 . Now, consider the first frequency ω 1 : from eq (7.45) we get the expressions (7.46) where the terms on the right-hand side are known. If we write explicitly eqs (7.46), we obtain three equations in three unknowns which can be solved to obtain p 11 , p 21 and p 31 , i.e. the components of the eigenvector p 1 . Then, the phase information on the three receptance functions can be used to assign a plus or minus sign to each component (phase at ω 1 is either +90° or –90°) and determine completely the first eigenvector. The same procedure for ω 2 Copyright © 2003 Taylor & Francis Group LLC and ω 3 leads, respectively, to p 2 and p 3 and, since the choice of the first column of the receptance matrix has been completely arbitrary, it is evident that any one column or row of an FRF matrix (receptance, mobility or accelerance) is sufficient to extract all the modal parameters. This is fundamental in the field of experimental modal analysis (Chapter 10) in which the engineer performs an appropriate series of measurements in order to arrive at a modal model of the structure under investigation. Kramers-Kronig relations Let us now consider a general FRF function. If we become a little more involved in the mathematical aspects of the discussion, we may note that FRFs, regardless of their origin and format, have some properties in common. Consider for example, an SDOF equation in the form (4.1) (this simplifying assumption implies no loss of generality and it is only for our present convenience). It is not difficult to see that a necessary and sufficient condition for a function f(t) to be real is that its Fourier transform F( ω ) have the symmetry property which, in turn, implies that Re[F( ω )] is an even function of ω , while Im[F( ω )] is an odd function of ω . Since H( ω ) is the Fourier transform of the real function h(t), the same symmetry property applies to H( ω ) and hence (7.47) where, for brevity, we write H Re and H Im for the real and imaginary part of H, respectively. In addition, we can express h(t) as (7.48) divide the real and imaginary parts of H( ω ) and, since h(t) must be real, arrive at the expression (7.49) where the change of the limits of integration is permitted by the fact that, owing to eqs (7.47), the integrands in both terms on the r.h.s. are even functions of ω . Copyright © 2003 Taylor & Francis Group LLC If we now introduce the principle of causality—which requires that the effect must be zero prior to the onset of the cause—and consider the cause to be an impulse at t=0, it follows that h(t) must be identically zero for negative values of time. The two terms of eq (7.49) are even and odd functions of time and so, if h(t) is to vanish for all t<0, we have (7.50) for all positive values of t. In other words, the two terms of eq (7.49) are each equal to h(t)/2 when t is positive but cancel out when t is negative. Equation (7.50) constitutes another restriction on the mathematical properties of the real and imaginary parts of an FRF and means that if we know H Re ( ω ), we can compute H Im ( ω ) and vice versa. The explicit relations between H Re and H Im can be found by writing the relation where the lower limit of integration can be set to zero because we assumed h(t)=0 for t<0. Next, by separating the real and imaginary parts of H( ω ) we obtain (7.51) In addition, from eq (7.49) we have which (introducing the dummy variable of integration) can be substituted in the second of eqs (7.51) to give Copyright © 2003 Taylor & Francis Group LLC and hence, since it can be shown that we can perform the time integration to obtain the result (7.52) where the symbol P indicates that it is necessary to take the Cauchy principal value of the integral because the integrand possesses a singularity. By following a similar procedure and noting that from eq (7.50) we can also write we can introduce this expression into the first of eqs (7.51) to obtain (7.53) Equations (7.52) and (7.53) are known as Kramers-Kronig relations. Note that they are not independent but they are two alternative forms of the same restriction on H( ω ) imposed by the principle of causality. The conclusion is that for any given ‘reasonable’ choice of H Re on the real axis there exists one and only one ‘well-behaved’ form of H Im . The terms ‘reasonable’ and ‘well-behaved’ are deliberately vague because a detailed discussion involves considerations in the complex plane and would be out of place here: however, the reader can intuitively imagine that, for example, by ‘reasonable’ we mean continuous and differentiable and such as to allow the Kramers-Kronig integrals to converge. We will not pursue this subject further because, in the field of our interest, the Kramers-Kronig relations are unfortunately of little practical utility. In fact, even with numerical integration, the integrals are very slowly convergent and experimental errors on, say, H Re may produce anomalies in H Im which can be easily misinterpreted and vice versa. Nevertheless, the significance of the Kramers-Kronig relations is mainly due to the fact that they exist and that their very existence reflects the fundamental relation between cause and effect, a concept of paramount importance in our quest for an increasingly refined and complete description of the physical world. 7.5 Systems with rigid-body modes Consider now an undamped system with m rigid-body modes. From the equations of motion Copyright © 2003 Taylor & Francis Group LLC [...]... real parts) dies away as and arrive at the solution (7. 75a) or, alternatively (7. 75b) Next, once again by virtue of eq (7. 70), we can partition the matrices S and ST as in eqs (7. 71a and b) and obtain from which it follows that (7. 76a) or, equivalently (7. 76b) From the definition of receptance matrix and from eqs (7. 76a) and (7. 76b) we get the n×n matrix (7. 77) whose (jk)th element is obtained as (7. 78a)... in the form (7. 70) By virtue of eq (7. 70), the 2n×2n matrices S and ST can be partitioned into (7. 71a) and (7. 71b) where the orders of Z, ZT and diag( j) are n×2n, 2n×n and 2n×2n, respectively With this in mind, noting that we can recover the displacement solution from eq (7. 69) as (7. 72) which represents the response of our system to an arbitrary excitation 7. 6.1 Harmonic excitation and receptance... Francis Group LLC and its (jk)th element is (7. 59b) Note that the expansion (7. 56) on the basis of modes which are not mass orthonormal results in a term Mii in the denominator of the first sum on the right-hand side of eqs (7. 59a) and (7. 59b) and in a term Mii in the denominator of the second sum Equations (7. 59a) and (7. 59b) are, respectively, the counterpart of eqs (7. 34b) and (7. 28) for an undamped... model of eqs (7. 96a, b) and (7. 97) There is, however, a subtle conceptual problem Our FRFs must be Copyright © 2003 Taylor & Francis Group LLC the Fourier transform of real impulse response functions, and this implies (Section 7. 4) that the conditions (7. 47) apply This is not the case for the FRFs of eqs (7. 96b) and (7. 97) ; furthermore, if these latter are modified to agree with eqs (7. 47) it follows... as it is not difficult to (7. 95) and recognize the receptance matrix (7. 96a) whose (jk)th element is given by (7. 96b) and the last expression has been written by taking eq (7. 91) into account The reader is invited to consider the more general case of proportional hysteretic damping expressed by eqs (7. 93a) and (7. 93b) and show that we arrive at (7. 97) where now we have (7. 98) Despite the discussion... partitioned into two n×2n matrices as (7. 83a) and, by the same token, it is also convenient to partition S–1 into two 2n×n matrices as (7. 83b) Furthermore (7. 84) and hence (7. 85) which is the displacement response of our system to an arbitrary excitation 7 Again, the case of harmonic excitation can be obtained as a particular case of eq (7. 85) The jth participation factor is now (7. 86) Copyright © 2003 Taylor... be determined) reads (7. 56) where we assume all modes to be mass orthonormal Equation (7. 56) can be substituted in eq (7. 54) to obtain a somewhat lengthy expression which, in turn, can be premultiplied by to give (7. 57a) and premultiplied by to give (7. 57b) so that eq (7. 56) becomes (7. 58) which can be compared to eq (7. 55) to conclude that the receptance matrix is written as (7. 59a) Copyright © 2003... writing the result as Hence (7. 64) It may now be useful to show how, with a more compact notation, we can arrive at the result (7. 64) in matrix form Let us write eq (7. 61) as (7. 65) where S is the 2n×2n matrix of eigenvectors and substitute (7. 65) in eq (7. 60b) and premultiply by ST to obtain now, (7. 66) Without loss of generality, we can assume equation and arrive at the matrix (7. 67) which is the matrix... transform method can be directly applied to eq (7. 1) to obtain (7. 99) where U=U(s) and F(s) are, respectively, the Laplace transforms of u(t) and f(t), s is the Laplace operator and are the vectors of initial displacements and velocities For zero initial conditions eq (7. 99) can be rewritten as (7. 100) and consequently (7. 101) where the last expression on the right-hand side for G–1 can be found in... as a particular case of eq (7. 64) The jth participation factor is now (7. 73) where Without loss of generality we can assume zero initial conditions and the normalization condition then, eq (7. 64) becomes (7. 74) Copyright © 2003 Taylor & Francis Group LLC Since we are mainly interested in the steady-state solution, we can drop the second term on the right-hand side which (if the system is stable and . matrices S and S T as in eqs (7. 71a and b) and obtain from which it follows that (7. 76a) or, equivalently (7. 76b) From the definition of receptance matrix and from eqs (7. 76a) and (7. 76b) we get. is in the form (7. 70) By virtue of eq (7. 70), the 2n×2n matrices S and S T can be partitioned into (7. 71a) and (7. 71b) where the orders of Z, Z T and diag( j ) are n×2n, 2n×n and 2n×2n, respectively. With. system is stable and all eigenvalues have negative real parts) dies away as and arrive at the solution (7. 75a) or, alternatively (7. 75b) Next, once again by virtue of eq (7. 70), we can partition the