Theorem A.5. If has eigenvalues the following statements are equivalent: 1. A is normal. 2. A is unitarily diagonalizable. 3. 4. There is a orthonormal set of n eigenvectors of A. The equivalence of 1 and 2 in Theorem A.5 is often called the spectral theorem for normal matrices. For our present purposes we recall that a Hermitian matrix is just a special case of normal matrix and we stress that—as expected—the statement of the theorem says nothing about A having distinct eigenvalues (and in fact, two or more eigenvalues could be equal). Then, summarizing the results of the preceding discussion we can say that a complex Hermitian matrix (or a real symmetrical matrix) A: 1. has real eigenvalues; 2. is always nondefective (which means that—regardless of the existence of multiple eigenvalues—there always exists a set of n linearly independent eigenvectors, which, in addition are mutually orthogonal); 3. is unitarily (orthogonally) similar to the diagonal matrix of eigenvalues diag( j ). Moreover, the unitary (orthogonal) similarity matrix is the matrix X of eigenvectors in which the jth column is the jth eigenvector. We close this section by briefly considering special classes of Hermitian matrices. A n×n Hermitian matrix A is said to be positive definite if (A.32a) for all nonzero vectors If the strict equality in eq (A.32a) is weakened to (A.32b) then A is said to be positive semidefinite. Moreover, by simply reversing the inequalities in eqs (A.32a) and (A.32b), we can define the concept of negative definite and negative semidefinite matrices. Note that, if A is Hermitian, the definitions above tacitly imply that the term x H Ax—which is called the Hermitian form generated by A—is always a real number and so we can also speak of positive definite Hermitian form (eq (A.32a)) or positive semidefinite Hermitian form (eq (A.32b)). The real counterparts of Hermitian forms are called quadratic forms and are expressions of the type x T Ax, where A is a real symmetrical matrix. Quadratic forms arise naturally in many branches of physics and engineering, and—as we also saw throughout many chapters of this book—the subject of Copyright © 2003 Taylor & Francis Group LLC engineering vibrations is no exception. Clearly, the appropriate definition of a positive definite matrix reads in this case (A.33) for all nonzero vectors Similarly, the relation for all nonzero vectors defines a positive semidefinite matrix. For our purposes, the following result will suffice and we refer the interested reader to specialized literature for more details. Theorem A.6. A Hermitian matrix is positive semidefinite if and only if all its eigenvalues are nonnegative. It is positive definite if and only if all its eigenvalues are positive (clearly, this same theorem applies for real symmetrical matrices). Finally, it is left to the reader to show that the trace and the determinant of a positive definite matrix are also positive. A.4 Matrices and linear operators Some aspects of the strict relationship between linear operators on a vector space and matrices have been somehow anticipated in Section A.1. Given a basis in an n-dimensional vector space V on a scalar field , the statement that the mapping (i.e. the mapping that associates the vector to its components relative to the chosen basis ) is an isomorphism constitutes a fundamental result which allows us to manipulate vectors by simply operating on their components. In fact, according to these developments, we saw in Section A.1 how the components of a vector change when we choose a different basis in the same vector space (in mathematical terminology, the sentence ‘ is an isomorphism but it is not a canonical isomorphism’ translates this fact that is indeed injective and surjective, but the coordinates of a given vector change under a change of basis and therefore depend on the choice of the basis). In a similar way, when we have to deal with linear operators on a vector space, it can be shown that—after a basis has been chosen in the space V— any given linear operator is represented by a n×n matrix and it can also be shown that—given a basis in V—the mapping that associates a given linear operator with its representative matrix relative to the chosen basis is an isomorphism between the vector space of linear operators from V to V and the vector space of square matrices Simple examples of such mapping are the null operator—i.e. the operator for which Zx=0 for all —which is represented by the null matrix and the identity operator—i.e. the operator for which Ix=x for all —which is represented by the unit matrix. In general, however, when a different basis is chosen in V, the same linear operator is represented by a different matrix. So, the question arises: since Copyright © 2003 Taylor & Francis Group LLC different matrices may represent the same linear operator, what is the relationship between any two of them? The answer to this question is that any two matrices which represent the same linear operator are similar. Let us examine these points in more detail. First of all we must determine what we mean by a matrix representation of a given linear operator. To this end, let V be a n-dimensional vector space and let be a linear transformation on V. If we choose a basis in the vector space, the action of T on any vector is determined once one knows the vectors because any x has a unique representation and linearity implies Now, since every vector Tu i , in turn, can be written as a linear combination (A.34) the n 2 coefficients t ki can be arranged in a square matrix T, which is called the matrix representation of the operator T relative to the basis The entries of the matrix clearly depend on the chosen basis and this fact can be emphasized by indicating this matrix by [T] u so that, by choosing a different basis in V, we will obtain the matrix representation [T] v of T. At this point, before examining the relationship between two different representations of T we need a preliminary result: we will show that—in a given n-dimensional vector space in which two basis and have been chosen—the ‘change-of-basis’ matrix is always nonsingular. In fact, since we can write (A.35) where i=1, 2,…, n in the first equation (and the n 2 coefficients c ji can be arranged in a square matrix which is the change-of-basis matrix from the basis to and j=1, 2,…,n in the second equation (and the n 2 coefficients ij can be arranged in a square matrix which is the change-of-basis matrix from the basis to Then from eqs (A.35) we get and since any vector can be expressed uniquely as a linear combination of the vectors the term within brackets must satisfy Copyright © 2003 Taylor & Francis Group LLC (A.36a) By the same token, it can also be shown immediately (A.36b) Equations (A.36a) and (A.36b) in matrix form read, respectively (A.37) meaning that (or, equivalently, ). Therefore, a change-of- basis matrix C is always nonsingular. Also, with a slight change of notation, we can re-express the result of Section A.1 by noting that, since any vector can be written as (A.38) we can substitute the first of eqs (A.35) into the first of eqs (A.38) to obtain which is equivalent to the matrix equation (A.39) where the notation [x] v means that we are considering the components of x relative to the basis Similarly, [x] u indicates the components of x relative to the basis and indicates the change-of-basis matrix from to The rather cumbersome (but self-explanatory) notation of eq (A.39) will now serve our purposes in order to obtain the relation between two matrix representations of the same linear operator. In fact, in terms of components the action of a linear operator T on a vector x can be written Copyright © 2003 Taylor & Francis Group LLC (A.40) where we defined Now, substituting eq (A.39) and its counterpart for the vector y into the second of eqs (A.40) yields so that premultiplying both sides by the matrix we get which implies (compare with the first of eqs (A.40)) (A.41) that is, the matrices [T] u and [T] v are similar, the similarity matrix being the change-of-basis matrix C. Example A.5. As a simple example in 2 , let us consider the two bases and Explicitly, the first of eqs (A.35) now reads and we can immediately obtain so that we can form the change-of-basis matrix Copyright © 2003 Taylor & Francis Group LLC Similarly, from the second of eqs (A.35) we obtain the change-of-basis matrix so that, as expected (eqs (A.35)) or, according to the more cumbersome notation above, Now, consider the linear transformation which acts on a vector as follows: (the proof of linearity is left to the reader). The representative matrix of T relative to the basis is obtained from the equations from which it follows that and finally we get from eq (A.39) which is exactly, as can be directly verified from the equations the representative matrix of T relative to the basis If, in addition, the two bases that we consider in the complex (real) linear space V are orthonormal bases—this obviously implies that an inner product has been defined in V—the similarity matrix is unitary (orthogonal). Copyright © 2003 Taylor & Francis Group LLC In fact, let for example V be a real n-dimensional linear space and let and be two orthonormal basis in V. Then, from the first of eqs (A.35) and from the orthogonality condition we get so that the equality reads in matrix form (A.42) which implies and shows that, in a real linear space, we pass from one orthonormal basis to another orthonormal basis by means of an orthogonal change-of-basis matrix. In terms of linear operators, this means that two different matrix representations A and B of the same linear operator are orthogonally similar and B=C T AC, where C is the change-of- basis matrix. Clearly, if V is a complex linear space, we get C H C=I (i.e. C is unitary; recall that the inner product in a complex space is not homogeneous in one of the slots) and the matrices A and B are unitarily similar, that is B=C H AC. We will not go into further details here, but a final observation is in order: specific properties of linear operators are reflected by specific characteristics of the matrices which may represent such operators; these characteristics, in turn, are generally invariant under a similarity transformation. As an illustrative example of this situation, it can be shown that if a square matrix A is Hermitian, then S H AS is Hermitian for all this is because a Hermitian matrix represents a Hermitian operator and another matrix representing the same operator must necessarily retain this characteristic (the definition of Hermitian operator is beyond our scopes and the interested reader is referred to specific literature). Also recall the corollary to Theorem A.4 stating that eigenvalues are invariant under a similarity transformation: this circumstance reflects the fact that eigenvalues are intrinsic characteristics of a given linear operator and do not change when different matrices are used for its representation. In the light of these considerations, we may recall the discussion on n- DOF systems (see Chapters 6 and 7, and also Chapter 9 for some important results on the characterization of eigenvalues) and note that the stiffness and mass of a given vibrating system can be envisioned as (symmetrical) linear operators on the system’s n-dimensional configuration space. Then, the essence of the modal approach consists of finding the orthogonal basis—the basis of eigenvectors—in which such operators have a diagonal representation. Copyright © 2003 Taylor & Francis Group LLC Solving the eigenvalue problem is the process by which we determine the basis of eigenvectors. The inconvenience of dealing with a generalized eigenvalue problem rather than with a standard eigenvalue problem translates into the fact that we have to diagonalize simultaneously two matrices instead of diagonalizing a single matrix. As stated before, however, this is only a minor inconvenience which does not significantly modify the essence of the mathematical treatment. Reference 1. Wilkinson, J.H., The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1988. Further reading Bickley, W.G. and Thomson, R.S.H.G., Matrices—Their Meaning and Manipulation, The English Universities Press, 1964. Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge University Press, 1985. Pettofrezzo, A.J., Matrices and Transformations, Dover, New York, 1966. Quarteroni, A., Sacco, A. and Saleri, F., Matematica Numerica, Springer-Verlag, Italy, 1998. Shephard, G.C., Spazi Vettoriali di Dimensioni Finite, Cremonese, Rome, 1969 (In Italian). (Translated from the original English edition Vector Spaces of Finite Dimension, University Mathematical Texts, Oliver & Boyd.) Voïevodine, V., Algèbre Linéaire, Mir, Moscow, 1976 (in French). Copyright © 2003 Taylor & Francis Group LLC B Some considerations on the assessment of vibration intensity B.1 Introduction In a number of circumstances one of the main tasks of vibration analysis is to ‘assess the vibration intensity’. This phrase, which is rather vague, can be interpreted as assigning to a specific vibration phenomenon a ‘figure of merit’ which can be used to predict the potential damaging effects, if any, of such a phenomenon. In these cases, one also speaks of ‘assessment of vibration severity’. Given the very large number of possible practical situations, it is obvious that the primary factors to be considered are, broadly speaking, the type, nature and duration of the excitation and the physical system which is affected by the vibration. Accordingly, there exist a number of specialized fields of investigation which study different aspects of the problem and consider, for example, the effect of shocks and vibrations on humans, buildings, various types of structures, electronic components etc. In this appendix, also in the light of the fact that it can be extremely difficult to categorize a complex phenomenon with a single number (as a matter of fact, there seems to exist no internationally accepted standard), we will obviously limit ourselves to some general considerations. B.2 Definitions In order to ‘assess vibration intensity’, the first definition we consider is the so-called Zeller’s power (or strength) of vibration, which takes into account the acceleration amplitude a, in cm/s 2 , and the frequency v and is defined by the relation (B.1) Zeller’s power is in units of cm 2 /s 3 and in the second expression on the r.h.s. of eq (B.1) we call x (in cm) the displacement amplitude. Copyright © 2003 Taylor & Francis Group LLC [...]... threshold of about 80 Hz, the sensations and effects are extremely dependent on local conditions at the point of application (position, local damping due to clothing or footwear, etc.) The International Standard ISO 2631 [4] applies to vibrations in both vertical and horizontal directions and deals with random and shock vibration as well as harmonic vibration In this standard, three levels of human discomfort... Harris, C.M., Shock and Vibration Handbook, 3rd edn, McGraw-Hill, New York, 1988 Holmberg, R et al., Vibrations Generated by Traffic and Building Construction Activities, Swedish Council for Building Research, Stockholm, 1984 ISO/DIS 5349.2 (1984) Guidelines for the Measurement and the Assessment of Human Exposure to Hand-Transmitted Vibration Studer, J and Suesstrunk, A., Swiss Standard for Vibration... engineering and medicine is currently being investigated in even greater detail Copyright © 2003 Taylor & Francis Group LLC References 1 Zeller, W., Proposal for a measure of the strength of vibration, VDI, Zeitschrift, 77, 323 2 DIN 4150 (1986) Part 3, Structural Vibration in Buildings: Effects on Structures (Part 1 (Principles, Predetermination and Measurement of the Amplitude of Oscillations) and Part. .. pal is ‘annoying’ and 40 pal is ‘unpleasant’ B.2.1 Effects of vibrations on buildings As far the effects of vibrations on buildings are concerned, engineers are generally interested in the possibility of structural damage and the vibrar scale has been used by some researchers in order to give some general guidelines So, a vibration up to 30 vibrar covers the ‘light’ and ‘medium’ ranges and no damaging... proficiency boundary’ and the ‘exposure limit’ A completely different situation arises in the study of hand-arm vibrations induced by the use of working tools in heavy industry such as hammer drills, chainsaws, etc High vibration levels and long exposure periods may lead, in the long run, to serious effects and also to permanent damage In the hope of preventing such harmful effects, this important and interesting... The current German standard DIN 4150, Part 2 [2] deals with the effects of vibrations on people in residential buildings and considers the range of frequency from 1 to 80 Hz In this standard, the measured value of principal harmonic vibration is used to calculate a factor of intensity perception KB by means of the formula (B.5) where d is the displacement amplitude in millimetres and v is the principal... effects on people and buildings For example, a vibration with a Zeller power of 2 cm2/s3 is rated as ‘very light’, Z=50 is ‘measurable’ and produces small plaster cracks, Z=250 is ‘fairly strong’, Z=1000 is ‘strong’ and defines the beginning of the danger zone, Z=5000 is ‘very strong’ and produces serious cracking, Z=2×104 is ‘destructive’, Z=105 is ‘devastating’ and so on On the other hand, according... basis of human perceptions and on the effects on humans and structures Historically, many of such scales have been proposed through the centuries: the Gastaldi scale (1564), the Pignataro scale (178 3) and the Rossi-Forel scale (1883) Nowadays, a modified version of the Mercalli-Cancani-Sieberg (MCS) scale is widely used in Europe: it is called the modified Mercalli (MM) scale and consists of 12 levels... epicentre, where superficial waves are predominant and different relations are needed B.2.2 Effects of vibrations on humans Although some vibration phenomena may not cause any structural damage, they can be annoying to the occupants of residential buildings, offices, etc In this regard, it should be noted that the human body is extremely sensitive to vibrations and amplitudes as low as 0.1 µm may be detected... the factors which influence ‘human sensitivity’ to vibrations are: • • • • • position (standing, sitting, lying down); direction of incidence with respect to the spine; age and sex; personal activity (resting, working, walking, running, etc.); frequency of occurrence and time of day The ‘intensity of perception’ depends on: • • • displacement, velocity and acceleration amplitudes; duration of exposure; . Standard ISO 2631 [4] applies to vibrations in both vertical and horizontal directions and deals with random and shock vibration as well as harmonic vibration. In this standard, three levels of human discomfort. DIN 4150 (1986) Part 3, Structural Vibration in Buildings: Effects on Structures. (Part 1 (Principles, Predetermination and Measurement of the Amplitude of Oscillations) and Part 2 (Influence. current German standard DIN 4150, Part 2 [2] deals with the effects of vibrations on people in residential buildings and considers the range of frequency from 1 to 80 Hz. In this standard, the measured