where G 1 (s) and G 2 (s) are the transforms of g 1 (t) and g 2 (t), respectively. From a table of transforms we get so that the convolution integral is zero for t<t 1 , leaving only the first term in eq (5.83) and for t>t 1 . The inverse transformation of eq (5.83) finally yields which, aside from the constant f 0 , are exactly eqs (5.10) and (5.43). So far, we have not yet considered the possibility of obtaining directly the final solution satisfying given initial conditions. This is one advantage of solving linear differential equations with constant coefficients by the Laplace transform method. By standard methods one finds a general solution containing arbitrary constants, and further calculations for the values of the constants are needed to solve a particular problem. The Laplace transforms of derivatives given in Chapter 2 (eqs (2.39) and (2.40)) will now be used to clarify this point. Let us consider the general equation of motion for a damped SDOF system (5.85) with initial conditions and The Laplace transformation of both sides gives (5.86) where, as customary, we are using lower-case letters for functions in the time domain and capital letters for functions in the transformed domain. Solving for X(s) and rearranging leads to (5.87) Copyright © 2003 Taylor & Francis Group LLC The first term on the right-hand side (product of two functions of s) transforms back to the convolution integral (5.88a) the second term transforms back to (see any list of Laplace transforms) (5.88b) and we have already considered the third term whose inverse transform is (5.88c) The sum of the three expressions (5.88a, b and c) finally gives the general response which is, as expected, eq (5.19) and shows explicitly how this method takes directly into account the initial conditions in the calculation of the response to external excitation. 5.4 Relationship between time-domain response and frequency-domain response From the discussion and the examples of preceding sections it appears that both h(t), the impulse response function (IRF), and the frequency response function (FRF) H( ω ) (or the transfer function H(s)) completely define the dynamic characteristics of a linear system. This fact suggests that we should be able to derive one from the other and vice versa. The key connection between the two domains is established by the convolution theorem and by the Fourier (or Laplace) transform of the Dirac delta function. In Chapter 2 (eq (2.29a)) we determined that the Fourier transform of the convolution of two functions g 1 (t) and g 2 (t)—provided that —is the product of the two transformed functions. With our definition of the Fourier Copyright © 2003 Taylor & Francis Group LLC transform, as a formula this statement reads and results, as we have seen, from an application of Fubini’s theorem. Now, since we know from Section 5.2 that the time-domain response x(t) of a linear system is given by the convolution (Duhamel’s integral) between the forcing function f(t) and the system’s IRF h(t), i.e. we can Fourier transform both sides of this equation to get the input-output relationship in the frequency domain (5.89) Equation (5.89) justifies eq (5.76a) from a more rigorous mathematical point of view. In fact the two equations (5.76a) and (5.89) are the same if we define (5.90a) and (5.90b) In this light, note that the functions F( ω ) and X( ω ) are the Fourier transforms of the functions f(t) and x(t), respectively, but the FRF H( ω ) (eq (5.90a)) differs slightly from the definition of the Fourier transform of the IRF h(t) (there is no 1/(2π) multiplying factor). This is a consequence of our definition of the Fourier transform (eqs (2.15) and (2.16)) and of the fact that the fundamental input-output relationship for linear systems is almost always found in the form given by eq (5.76a). However, this is only a minor inconvenience since (Chapter 2) the position of the factor 1/(2π) is optional so long as it appears in either the Fourier transform equation or the inverse Fourier transform equation. Hence, the inverse transform equation corresponding to eq (5.90a) is (5.91) Copyright © 2003 Taylor & Francis Group LLC which, conforming to our definition of the inverse transform can be written No such inconvenience arises in the case of Laplace transforms because in this case, by virtue of the convolution theorem (eq (2.43)), the Laplace transform of the Duhamel integral leads to (5.92) where we defined X(s) as the Laplace transform of the response x(t). With the above general developments in mind, we may now recall that, for a linear system, the function h(t) represents the system’s response to a delta input, i.e. x(t)=h(t) when Consequently, since (eq (2.74)), and eqs (5.89) and (5.92) give respectively (5.93) which tell us that, in the case of δ-excitation, the Fourier (Laplace) transform of the system’s response is precisely its frequency response (transfer) function, a circumstance which is also exploited in experimental practice. With the definitions above, it is now not difficult to verify eq (5.76a) (and hence eq (5.93)) for the case of, say, a viscously damped SDOF system. In fact, in this case we know from the second of eqs (5.7a) that so that, with the aid of a table of integrals from which we get we can calculate the term by noting that, in our specific case and It is then left to the reader to show that the actual calculation leads to (5.94) Copyright © 2003 Taylor & Francis Group LLC where, as usual, If we multiply this result by we obtain explicitly the right-hand side of eq (5.76a) for our viscously damped SDOF system. Then, the left hand-side can be obtained by virtue of eq (5.5) and it can be determined immediately that, as expected, In the light of these considerations we can write the response of a system to an input f(t), whose Fourier and Laplace transforms are F( ω ) and F(s), as (5.95) or, when more convenient, as the inverse Laplace transform of the product H(s)F(s). Thus the following three equivalent definitions of the FRF H( ω ) can be given: 1. H( ω ) is 2π times the Fourier transform of h(t). 2. For a sinusoidal input, i.e. is the coefficient of the resulting sinusoidal response 3. Provided that in the frequency range of interest, H( ω ) equals the ratio where X( ω ) is the Fourier transform of x(t). Figure 5.8. is a frequently found schematic representation of the fact that the dynamic characteristics of a ‘single-input single-output’ system are fully defined by either h(t) or H( ω ) A final note of interest concerns two systems connected in cascade as shown in Fig. 5.9. Denoting by h(t) and H( ω ) the IRFs and FRFs of the combined Fig. 5.8 Symbolic representation of a linear system. Fig. 5.9 Systems in cascade. Copyright © 2003 Taylor & Francis Group LLC system, by and H 1 ( ω ) and H 2 ( ω ) the relevant functions of the two subsystems we have (5.96a) and (5.96b) 5.5 Distributed parameters: generalized SDOF systems Up to the present we have always considered the simplest type of SDOF system, i.e. a system where all the parameters of interest—mass, damping and elasticity—are represented by discrete localized elements. This is, of course, an idealized view. However, even when more complicated modelling is required for the case under study, we have to remember that the key aspect of SDOF systems is that only one generalized coordinate is sufficient to describe their motion; if this characteristic is maintained it can be shown that the equation of motion of our SDOF system, no matter how complex, can always be written in the form (5.97) where z(t) is the single generalized coordinate and the symbols with asterisks (not to be confused with complex conjugation) represent generalized physical properties—the generalized parameters—of our system with respect to the coordinate z(t). This latter statement means that a different choice of this coordinate leads to different values of the generalized parameters. The possibility of writing an equation such as (5.97) allows us to extend all the considerations that we have made so far to a broader class of systems: assemblages of rigid bodies with localized spring elements, systems with distributed mass, damping and elasticity (bars, plates, etc.) or combinations thereof can be analysed in this way once the relevant generalized parameters have been determined. Their values can be obtained, in general, from energy principles such as Hamilton’s or the principle of virtual displacements (Chapter 3) but general standardized forms of these expressions can be given for practical use. It is important to note that the SDOF behaviour of the system under investigation may sometimes correspond very closely to the real situation but, more often, is merely an assumption based on the consideration that only a single vibration pattern (or deflected shape in case of continuous systems) is developed. For example, a beam that deforms in flexure is, as a matter of fact, a system with an infinite number of degrees of freedom, but in certain circumstances a SDOF analysis can be good and accurate enough for all practical purposes. Copyright © 2003 Taylor & Francis Group LLC For continuous systems in particular, the success of this procedure—which is a particular case of the assumed modes method (Chapter 9)—depends on the validity of the assumption above and on an appropriate choice of a shape (or trial) function (x) which, in turn, depends on the physical characteristics of the system and also on the type of loading. Ideally, the selected shape function should satisfy all the boundary conditions of the problem. At a minimum, it should satisfy the essential boundary conditions. In this light, a few definitions are given here and then some examples will clarify the considerations above. •An essential (or geometric) boundary condition is a specified condition placed on displacements or slopes on the boundary of a physical body (e.g. at the clamped end of a cantilever bar both displacement and slope must be zero). •A natural (or force) boundary condition is a condition on bending moment and shear (e.g. at the free end of a cantilever bar the bending moment and the shear force must be zero). •A comparison function is a function of the space coordinate(s) satisfying all the boundary conditions—essential and natural—of the problem at hand, plus appropriate conditions of continuity up to an appropriate order. •An admissible function is a function that satisfies the essential boundary conditions and is continuous with its derivatives up to an appropriate order. For a specific problem, the class of comparison functions is a subset of the class of admissible functions. •An assumed mode (or shape function) is a comparison or an admissible admissible function used to approximate the deformation of a continuous body. Example 5.8. Let us consider the rigid bar of length L metres and mass m kilograms shown in Fig. 5.10. The angle θ of rotation about the hinge, where at static equilibrium, can be chosen as the generalized coordinate. The vertical displacement z(t) of the tip of the bar can be another choice; for small oscillations as shown in the figure. Since the bar is considered rigid, the system has distributed mass (along the length of the bar), localized stiffness, damping (the spring and the dashpot) and is subjected to a localized force f(t). For small oscillations about the equilibrium, we assume the shape function The virtual displacement is then given by and from the principle of virtual displacements it is not difficult to obtain Copyright © 2003 Taylor & Francis Group LLC Our method assumes that only one mode is developed during the motion and represents the deflected shape u(x, t) of the beam as the product (5.100) where (x) is the chosen admissible function and z(t) is the unknown generalized coordinate. The principle of virtual displacements considers all the forces that do work and reads (5.101) where from the definition of potential energy V. It will be shown in later chapters that the strain potential energy of a beam undergoing a transverse deflection u(x, t) is given by (5.102) where we indicated for simplicity of notation It follows from eq (5.102) (5.103) since For the inertia forces we get (5.104) since and Fig. 5.11 Schematized beam on elastic foundation. Copyright © 2003 Taylor & Francis Group LLC Similarly, for damping forces we get (5.105) and for the distributed spring and external forces (5.106) (5.107) Addition of the various terms leads to where all the integrals are taken between 0 and L. The virtual displacement δ z is arbitrary and therefore can be cancelled out, leaving the equation of motion of our system in the form of eq (5.97) where, after rearranging, the generalized parameters are given by (5.108a) (5.108b) (5.108c) (5.108d) It is now evident how the particular choice of the shape function affects the generalized parameters of the system. Moreover, it is also clear that the application of Hamilton’s principle leads the same expressions of eqs (5.108a–d). The most general case of the type shown above consists of a system which is a combination of distributed and localized masses, springs, dampers and external forces. Again, the displacement is assumed of the form Copyright © 2003 Taylor & Francis Group LLC [...]... and calculating the constants of integration from the boundary conditions of eqs (5. 130) and (5. 131) we get (5. 138) which can be used in eq (5. 1 35) to obtain (5. 139) or in eq (5. 137) to get (5. 140) Copyright © 2003 Taylor & Francis Group LLC These values are much better estimates of the exact frequency (given by eq (5. 129)) and the large difference between the frequencies obtained from eq (5. 126) and. .. the cantilever problem) can be obtained from (5. 133) and we can write the maximum potential energy as (5. 134) Equating to the maximum kinetic energy gives (5. 1 35) At this point it seems reasonable to proceed further and use kinetic energy, so that 1 also for the (5. 136) Copyright © 2003 Taylor & Francis Group LLC Equating to Ep,max of eq (5. 134) gives now (5. 137) Further iteration—that is, the use of... series, Mathematics of Computation, 19, 297–301, 19 65 2 Harris, C.M (ed.), Shock and Vibration Handbook, 3rd edn, McGraw-Hill, New York, 1988 3 Jacobsen, L.S and Ayre, R.S., Engineering Vibrations, McGraw-Hill, New York, 1 958 4 Erdélyi, A., Magnus, W., Oberhettingher, F and Tricomi, F.G., Tables of Integral Transforms, 2 vols, McGraw-Hill, New York, 1 953 5 Thomson, W.T., Laplace Transformation, 2nd edn,... by The maximum values of the potential and kinetic energies are now equating the two energies and solving for frequency gives (5. 119) where the generalized stiffness and mass are the same as in eqs (5. 99a) and (5. 99c) It is clear at this point that Rayleigh’s method is strictly related to the assumed modes method (which is used to obtain the equation of motion), and the generality of the symbolic relation... Copyright © 2003 Taylor & Francis Group LLC we obtain (5. 113) The virtual work done by P(x) is finally obtained as (5. 114) The term of eq (5. 114) must now be added to eq (5. 101) to give a generalized stiffness in the form where k* is as before and (5. 1 15) In the case of a simple beam under the action of the axial compressive load only—if P does not depend on x and can be taken out of the integral—we can obtain... approximate function 2 and use the latter to calculate the frequency—is generally not worth it We still do not have an estimate for the error ε , but indirectly we can have an idea by looking at the difference between the frequencies obtained from eq (5. 122) and eq (5. 1 35) or (5. 137) If this difference is large, the function 0 is not a very good approximation for the true deflected shape and ε is large as... end and free at the other) that undergoes flexural vibrations without energy loss during its motion We assume the xaxis in the horizontal direction The assumption characterizes the SDOF behaviour of this system and, again, characterizes the harmonic time dependence of this motion The maximum potential and kinetic energies are (5. 120) (5. 121) respectively, where EI is the flexural rigidity, Equating and. ..where z(t) is the unknown generalized coordinate and the following standardized expressions for the generalized parameters can be given: (5. 109) (5. 110) (5. 111) (5. 112) where the integrals take into account the distributed parameters of the system under investigation (the symbols with a caret (^) are intended per unit length) and the summations take into account the discrete elements For... respect to the stiffness matrix: by virtue of eq (6.34), eq (6.32) gives (6. 35) which is precisely the stiffness-orthogonality condition of the two eigenvectors When i=j, the left-hand side of eqs (6.34) and (6. 35) is different from zero and we can write (6.36) where the scalars Mii and Kii are called the modal or generalized mass and the modal or generalized stiffness of the ith mode, respectively These... per unit length (5. 122) At this point it is interesting to test the effect of different choices for (x) on the calculated frequency Since the exact deformation shape can only be obtained by solving the equation of motion (but in this case the value of the fundamental frequency would be determined also) and therefore it is not known, we will try three trial functions (5. 123) (5. 124) (5. 1 25) All of these . times and calculating the constants of integration from the boundary conditions of eqs (5. 130) and (5. 131) we get (5. 138) which can be used in eq. (5. 1 35) to obtain (5. 139) or in eq (5. 137). Taylor & Francis Group LLC system, by and H 1 ( ω ) and H 2 ( ω ) the relevant functions of the two subsystems we have (5. 96a) and (5. 96b) 5. 5 Distributed parameters: generalized SDOF. forces we get (5. 1 05) and for the distributed spring and external forces (5. 106) (5. 107) Addition of the various terms leads to where all the integrals are taken between 0 and L. The virtual