Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 1, pp. 195-212, Warsaw 2004 VIBRATION ENERGY FLOW IN RECTANGULAR PLATES Jacek Cieślik Department of Robotics and Machine Dynamics, University of Science and Technology, Kraków e-mail: ghciesli@cyf-kr.edu.pl The paper presents obtained from literature review formulas on structural intensity calculations. The formulas involve loads (forces andmoments) and strains (linear and angular) which enabled the estimation of structural sur- face intensity for beams, and plates considered here as simple structural elements. The numerical method of intensity evaluation was based on com- plexmodal analysiswith the use of the finite elementmethod.Thepresented calculation results lead to the assessment of distribution of structural inten- sity vectors on the surface of a steel rectangular plate. Themodels included the source of vibrations (force excitation) and sink of energy (damper) with known position of application. The changes of the finite elements grid den- sity enabled detailed investigation of the total vibration energy flow in the analysed plates. Key words: vibration energy flow, structural intensity, intensity vector field 1. Introduction The quantity of structural intensity was introduced with the purpose of extension of the description of vibration phenomena in structures with the application of vector fields. It has found exceptional application to investi- gations of vibration energy flow in deformable elastic bodies. The structural intensity represents the averaged in time mechanical energy flow through the unit area perpendicular to the direction of the flow (Gavric et al., 1997). Be- cause of themeasurement possibilities, the experimental analysis is conducted in a thin region close to the surface, and is mainly dedicated to thin-walled structures. The analysis in deeper parts of constructional elements is out of practical meaning mainly due to their thin-wall nature. The analysis of the spatial distribution of structural intensity vector fields enables determination and location of transmission paths, sources and sinks of energy of vibrations in 196 J.Cieślik application to mechanical systems. It gives the particular information on the streams of energy flow, which ismuchmore advantageous than othermethods used earlier to such kind of analysis. Thebasic formulations and relationships allowing one to calculate the com- ponents of structural intensity vectors for simple constructional elements such as beams, plates, shells and pipes were given in Gavric and Pavic (1993), Gavric et al. (1997). Development of the intensity methods in application to analysis of the rigid bodies started in themiddle of 70s of the last century.The papers by Noiseux (1970), Pavic (1976) and Verheij (1976) cover mainly the development ofmeasuringmethods.As a result, therewere twomainmethods of measurements elaborated: simplified – using the definitional dependencies (Pavic, 1976; Verheij, 1976) and exact based on the wave theory of vibrations (Gavric et al., 1997; Pavic, 1992). The used measurement methods still have disadvantages in their application to practical cases. They are a compromise between the measurement accuracy and simplicity of the measuring system, meaning the number of measuring points. The structural intensity can not be measured directly. Methods for the measurement use deflections (strains) instead of stress components. Significant development of the structural intensity applications was noted in the second half of 80s and the beginning of 90s, when works showing the possibility of numerical calculation of structural intensity (Gavric, 1991; Ga- vric andPavic, 1993) appeared. Fundamental dependencies were based on the assumption of one or two dimensional structures such as beams or plates. The numerical approach was based on themodal model. The possibility of computational analysis of the structural intensity is very promising due to the prospects of use of already elaborated for other purposes finite element models of mechanical structures. The practical examination on the computational way of the vibration energy flow in complex structures consisting of beams, plates and shells have not been solved satisfactory till now. Main problems were found in the necessity of consideration of complex boundary conditions, complexity of the analysed real structures and applied models of damping. Themethod of numerical analysis of structural intensity fields can be used for themodalmodels obtained from the experimental modal analysis (Gavric, 1997; Gavric et al., 1997). As shown the experiments presented below, the modalmodel should cover at least first sixty eigenfrequencies. This conclusion is the result of low convergence of the method used, and indicates important contribution of high frequencymodes into the vibration energy transfer in the structures. Vibration energy flow in rectangular plates 197 2. Structural intensity in thin-walled structural elements For a steady state of vibration the surface structural intensity can be eva- luated as a complex quantity (Gavric and Pavic, 1990) S̃σklvl(ω)= Ik(ω)+ jJk(ω) (2.1) where ω = 2πf is the angular frequency, f is the frequency of vibrations, S̃σklvl(ω) is the cross spectrum function of complex components of the stress and particle velocity. In practical cases, the real part of the structural intensity, called the active intensity, is only analysed.Only the real part of the intensity is responsible for the energy transfer. The imaginary part represents the energy conservation in the system, and is connected with standing waves. The damping in the structure is necessary for the existence of vibration energy flow. In the case of low damping in the analysed structure, the examination of vibration energy is limited. A curl of the structural intensity is observed instead of the energy flow. An instantaneous value of the real part of the structural intensity ik(t) is a time dependent vector equal to the change of energy density in an infinitively small volume (Gavric, 1991). Its kth component is given by the equation ik(t)=−σkl(t)vl(t) l=1,2,3 (2.2) where vl(t) is the lth component of the velocity vector, and σkl(t) is the klth component of the stress tensor. The averaged in time value of (2.2) represents the net energy flow in the mechanical structure (Gavric, 1991) Ik = 〈ik(t)〉 (2.3) in the direction of the kth coordinate of a rectangular frame of reference corresponding to the analysed structural element. The formulations for measurements methods are based on easily measu- red quantities such as deflections, strains and velocities. The values of partial derivatives are approximated with the finite difference method (Carniel and Pascal, 1985). This causes an instantaneous parallel measurement to be done in thirteen points. The method used for the measurement is inapplicable to calculations. The components of the structural intensity vector can be calculated for beams, plates and shells as functions of the following variables: bending and twistingmoments, shear forces, linear and angular displacements. This appro- ach significantly differs from the measurement methods. 198 J.Cieślik 2.1. Structural intensity in rods and beams A simplest structure, which could provide the energy transportation, is an unbounded thin rod excited to axial vibrations. The structural intensity is related to the axial line of the rod and equal to I = √ Eρ v2 (2.4) where E is Young’s modulus, ρ – mass density, ν – rod axial velocity. The only stress occurring in the rod is the axial stress which is in phase with the velocity of vibrations. In the case of a prismatic beam exposed to a single excitation frequency the vibration velocities are found from the complex representation of the di- splacements. The stress in the beam is in general a result of bending, shear and twisting. The component along themain axis of the beam is (Gavric and Pavic, 1993) Ix =− ω 2 Im [ Ñũ∗0+ Q̃yṽ ∗ 0 + Q̃zw̃ ∗ 0 + T̃θ ∗ x+M̃yθ ∗ y +M̃zθ ∗ z ] (2.5) where Ñ are tension forces, Q̃y and Q̃z – shear forces, T̃ – torque, M̃y, M̃z – bending moments, θ̃x, θ̃y, θ̃z – angular displacements, ũ0, ṽ0, w̃0 – linear displacements of the beam neutral line according to the x, y, z axis, respectively. 2.2. Structural intensity in plates In the vibration of thin plates the bending and longitudinal waves are the dominant ones. For the longitudinal waves, the displacements do not vary with depth of the plate. In the case of bending vibrations, the displacements causedby shear forces, bendingandtwistingmoments areaddedup.Structural intensity vector of a plate subjected to all mentioned kinds of loads is the sum of three components. For bendingmotion of the plate, the displacements caused by the bending are related to the in-plane displacements as follows ux =− h 2 ∂uz ∂x uy =− h 2 ∂uz ∂y (2.6) In numerical calculations, realised by means of the finite element method, the structural intensity is related to the neutral undeformed middle plane of the plate. In a general case, the plate is subjected to bending, tension and twisting motions. On this assumption, the following relationships on the Vibration energy flow in rectangular plates 199 structural intensity components are derived for thin plates (Gavric andPavic, 1993; Cieślik, 2002a) Ix =− ω 2 Im [ Ñxũ ∗ 0+ Ñxyṽ ∗ 0 + Q̃xw̃ ∗ 0 +M̃xθ ∗ y −M̃xyθ ∗ x ] (2.7) Iy =− ω 2 Im [ Ñyṽ ∗ 0 + Ñyxũ ∗ 0+ Q̃yw̃ ∗ 0 −M̃yθ ∗ x+M̃yxθ ∗ y ] where: Ñx, Ñy – in-plane normal forces, Ñxy = Ñyx – in-plane tangential forces, Q̃x, Q̃y – shear forces, M̃xy = M̃yx – torques, M̃x, M̃y – bending moments. There are assumed small deformations which enabled the superposition of independent displacements for flat finite element of plate or shell type. 3. Calculation of structural intensity components 3.1. Modal approach to structural intensity calculations Complex displacements and stresses needed for structural intensity evalu- ation can be obtained by themodal approach. The software used in computa- tion with the finite element method uses in most cases the real stiffness and mass matrices. As a result of calculations the real displacements and stresses are obtained. The procedure of structural intensity calculation is based on the complex response of the structurewith amodal representation of the structure without dissipation (Gavric, 1990). The damping is considered in two forms. The structural internal damping of the structure is taken into consideration asmodal damping. The additional damping (energy absorber, sink), placed in the known location, is treated as a external loading R̃=−S̃X̃ (3.1) where S̃ is the additional stiffness matrix, X̃ – displacement vector. An excitation by a sinusoidal force in the form F̃ exp(jωt) was assumed. The equation of motion of the structure including two models of damping is (Gavric and Pavic, 1993) −ω2MX̃+KX̃ = F̃ + R̃ (3.2) where M and K are the mass and stiffness matrices. 200 J.Cieślik The complex response of the structure is obtained from the calculations of eigenvalues. The internal proportional damping is considered by use of com- plex eigenfrequencies. The real values of the eigenfrequencies matrix ω20 are replaced with the values including themodal loss factor ηi for the ith mode. The complex eigenfrequencies are then equal to ω̃20i =(1+ jηi)ω 2 0i. The complex response of the structure is obtained from the calculations done with a post-processor program operating on the values obtained from the classical FEM model in order to compute the complex modal response in modal coordinates. For the purpose of structural intensity calculations an own program written in the MatLab environment was elaborated (Cieślik, 2001, 2002b). Program realises calculations as well as presents the results in a graphical form. 3.2. Calculations of structural intensity based on the finite element method The structural intensity, defined by (2.2), depends on particle velocity and stress. As a result of the calculations, one can get displacements in the nodes and stresses in a point inside a finite element. In the process of calculations these values are need to be found in the same points. Because the accuracy of stress calculations is less than that for displacements, the values of the structure response are taken in points of highest accuracy for the stress. The displacements in these points are foundwith the help of a shape functions for the finite element from relative displacements of the element. The shape functionmatrix is, in a general case, related to the local system of coordinates for the finite element. The vector of modal displacements have to be defined in the same coordinate system. The displacements inside the finite element are also related to the local system of coordinates. Application of the procedure allows one to find the vector containing proper displacements of element centroides for a given eigenvalue. Complex displacements and stres- ses, both related to the element centroide and necessary for evaluation of the structural intensity, are calculated by modal superposition from definitions (2.5) and (2.7). The obtained structural intensity is complex and expressed in the local coordinate system connected with the finite element. It should be transformed into the general system of coordinates connected with the system of finite elements, the analysed structure. Vibration energy flow in rectangular plates 201 4. Case study – rectangular plate For the purpose of verification of the structural intensity applicability to the identification of discontinuities, a numerical experiment was performed. The model chosen for the analysis was a three-dimensional structure of a rectangular plate convenient for the numerical modelling of the structural intensity. Two general cases were discussed: the change of width and change of thickness in the middle part of the plate. The results were compared with the results obtained for a rectangular homogenous plate. In the calculations, the same own program for calculations of complex modalmodel allowing the consideration of additional localised damping in the system, was used. 4.1. Analysed model Homogeneous rectangular plate Themodel chosen for the calculations was a homogeneous rectangular flat plate.Theplate hadmechanical properties of a structural steel anddimensions of 1.5m in width, 2.5m in length and thickness of 10−2m. The four finite element method models were arranged using the NASTRAN software. The models differed in number of finite elements. The plate was divided into 60, 240, 960 and 3840 elements. In each case, the model consisted of the same square shell elements of the QUAD4 type. The excitation and damping force were introduced to the model. The harmonic excitation force was applied to the plate at the place indicated in figures by the star. The damping force proportional to the velocity of vibration was applied to the plate at the place indicated in figures by the triangle. The magnitude of the damping force was set to 103N. The direction of its action was chosen perpendicularly to the plane of the plate. The locations of the damping and excitation force were the same for all models. Themodel of a simply supported plate was chosen. The only feasible mo- tion was rotation around the edges of the plate. Translatorymotions of edges in any directions were neglected. The model was most characteristic for the most technical cases of the plate mounting in practice. Rectangular plate with various thickness and width The next models chosen for the calculations were rectangular flat plates with various thickness in themiddle. The plates hadmechanical properties of a structural steel and dimensions of 1.5m in width and 2.5m in length with 202 J.Cieślik thickness of 10−2m. Four different models made up with the finite element method were prepared using the NASTRAN software. Three models differed in the thickness of the middle part of the plate. The fourth model had two symmetrical rectangular cut-outs in the middle part. In the square region (1× 1m) in the middle of the plate the thickness was changed to 0.5 ·0−2m and 2 ·10−2m, respectively. The model consisted of 450 square shell elements of the QUAD4 type. The plate with the cut-out had a constant thickness for all its area. The model consisted of 362 square shell elements of the sameQUAD4 type. The dimensions and shape of square elements were the same for all four cases. Similarly, as in the analysis of the homogenous plate, the excitation and damping forces were introduced to the model. The harmonic excitation force was applied to the plate at the place indicated in figuresby the star.Thedamping force proportional to the velocity of vibration was applied to the plate at the place indicated in figures by the triangle. Themagnitude of the damping force was set to 103N.The direction of its action was chosen perpendicularly to the plane of the plate. Themodel of a simply supported platewas chosen. The only feasiblemotionwas rotation around the edges of the plate. 4.2. Results of calculations Homogeneous rectangular plate The main target of the analysis was testing the convergence of numerical solution and determination of the number ofmode shapes necessary to obtain the solutionwith a specified accuracy. Calculationswere carried out for chosen numbers of the first mode shapes, accordingly: 10, 13, 16, 20, 25, 32, 40, 50, 63, 80 and 100. In cases of low numbers ofmode shapes, significant changes in the distribution of vectors for the same density of the mesh of elements were observed. The positions of the excitation and damping forces were not clearly shownby the structural intensity vectors distribution.As the number ofmode shapes increased, the more and more regular and ordered vector fields of the structural intensity were observed, Fig.1-Fig.4. For the purpose of the assessment of the necessary number ofmode shapes required for the achievement of the correct and convergent numerical solution a relativemeasure of convergencewas chosen.Themeasurewas the ratio of the difference of the vector magnitude in a chosen observation point, calculated for the two consecutive cases of calculations for the nth and (n−1)th number of the mode shape, to its value calculated for the (n− 1)th number of the mode shape. In each of four cases of the finite element mesh, the measure of convergence was less than 0.01 for the number of 63 mode shapes taken into Vibration energy flow in rectangular plates 203 Fig. 1. Distribution of the structural intensity vectors for a simply supported rectangular plate. Model consists of 60 elements. 100 mode shapes calculated Fig. 2. Distribution of the structural intensity vectors for a simply supported rectangular plate. Model consists of 240 elements. 100mode shapes calculated account. Consequently, it was assumed that the number greater than 60mode shapes corresponds to the exact numerical model of the plate, and properly represents the vibration energy flow in the system. It wasmentioned above that themagnitude of structural intensity vectors decrease with the increase in the number of finite elements. This allows one to draw the conclusion that in the analysis of the energy flow it is not sufficient to observe only the intensity vectors.Thebettermeasure of the energyflowseems 204 J.Cieślik Fig. 3. Distribution of the structural intensity vectors for a simply supported rectangular plate. Model consists of 960 elements. 100mode shapes calculated Fig. 4. Distribution of the structural intensity vectors for a simply supported rectangular plate. Model consists of 3840 elements. 100mode shapes calculated tobe the total energyflowthroughaclosedareaaroundthepointsof excitation and damping or through the whole width of the plate. For that reason the methodof summationof the structural intensityvectormagnitude in theclosed areawas applied. In the general case, the total energy flow shouldbe examined through the cross section perpendicular to the known or assumed direction of energy transfer indicated by an ordered distribution of the structural intensity vectors. Vibration energy flow in rectangular plates 205 Fig. 5. Regions of summation of the structure intensity vectors: (a) in the whole width of the plate in the middle part, (b) around the point of damping force application Fig. 6. Results of summation of the structure intensity vectors in chosen cross-sections. The plate model consisted of 960 elements 206 J.Cieślik Examples of the summationare shown inFig.5.Figure 6 shows the changes of the energy flow calculated from the vector field in consecutive cross sections of the plate and along the curves surrounding the excitation and sink of the vibration energy. Significant changes are observed for the area around the excitation and sink. This is due to the changes of stress values in the region of force application point. For other cross sections along the whole width of the plate only slight changes in the total energy are observed. The process of structural intensity values summation was done for the horizontal x and vertical y directions parallel to the sides of the plate. Rectangular plate with various thickness and width Themain target of the analysis was the testing of the effect of plate thick- ness and shapeon thedistributionof structural intensity vectors.Theobtained results of calculation are shown in Fig.7-Fig.10. Particularly in the region of abrupt change in the plate thickness, the distribution of structural intensity vectors changes with respect to the rest of the plate. The positions of the excitation and damping forces are clearly shown by the structural intensity vectors distribution for the case of a homogenous and thin plate. Fig. 7. Distribution of the structural intensity vectors for a simply supported rectangular plate with a cut-out in the middle part In all cases however, the distribution of intensity vectors in themiddle part has a similar shape. The distribution of vectors reveals a distinct direction of the energy flow from the excitation to the sink point. For a plate thicker in the middle part, a vorticity field in the region of the thicker part is observed. The changes of thickness are marked by the abrupt change in the direction andmagnitude of the intensity vectors. Comparing the obtained results of calculations for rectangular plates pre- sented here as the distribution of structural intensity vectors over the area of Vibration energy flow in rectangular plates 207 Fig. 8. Distribution of the structural intensity vectors for a simply supported rectangular plate of constant thickness. Vectors normalised to the unit length Fig. 9. Distribution of the structural intensity vectors for a simply supported rectangular plate being thinner in the middle. Vectors normalised to the unit length Fig. 10. Distribution of the structural intensity vectors for a simply supported rectangular plate being thicker in the middle. Vectors normalised to the unit length 208 J.Cieślik the plate, one can notice that there is no similarity with the distribution of displacements for eachmode shape. This conclusion results from the fact that themode shapes are connected to the standingwaves formed in the plate. In a systemwith small internal damping the energy flow is not observed or at least it is very low. The effect of the energy flow between the structure elements is observed only for systems with high internal damping caused e.g. by abrupt local changes of properties or a localised damping force. 4.3. Conclusions The modal analysis based on dense grid of finite elements enabled deta- iled analysis of vibration energy transportation. The results obtained from calculations are presented in a graphical form in Fig.1-Fig.10. The structural intensity vector distributions over the surface of elements of theplate structure are shown. Very specific disturbances of the distribution in points of applica- tion of the damping element and application of the exciting force are clearly seen. In the region far from connections as well as points of application of the external damping or excitation, the directions of vectors are ordered parallel and steady regarding the vector magnitude. It means a steady regular flow of the vibration energy in one direction defined by the sense of the vectors. In some places, especially those close to the point of change in the thickness, one can observe the circulation of structural intensity vectors. This effect of vortices, represented by rotational distribution of the intensity vectors, means circulation and conservation of the vibration energy in the system. The distribution of vectors reveals rotational character of the vector field due to wave reflections and probable places of dissipation of mechanical ener- gy. This effect should be verified by experimental measurement. A particular analysis leads to the conclusion that the intensity vector distribution, by na- ture, enables localisation of the energy sink in large surface elements. This is a significant advantage as compared to other identification methods. The distribution of the structural intensity vectors gives a qualitative cha- racteristic of vibration energy transportation in mechanical systems. Intro- duction of an additional measure in the form of an integral of the structural intensity vector component perpendicular to a certain closed surface, e.g. ele- ment cross-section, enables quantitative assessment of energy transfer paths and energy balance in the structure. The presentedmethod of calculation of the structural intensity vector ena- bles its evaluation for a chosen frequency range and mode shapes (Gavric, 1991). The frequency range of the analysis can not be exactly defined in real numbers. It should be rather related to the frequency range covering appro- Vibration energy flow in rectangular plates 209 ximately the first hundredmode shapes of the analysed structural element or entire structure. This relates the frequency range with mechanical properties and dimensions of the structure. The derived relationships connect the structural intensity with linear and angular displacements, forces and moments applied to the structures like be- ams, plates and shells. Themethod of structural intensity estimation is based on the calculation of displacements in nodes and stresses in inner points, i.e. Gauss points of finite elements. The calculations are carried out with the ap- plication of complex modal parameters counted by the use of the numerical modal analysis based on the finite element method. The damping is considered in two ways: as internal damping represented by modal damping and localised damping by the force proportional to the displacement. Due to complex notation used in the analysis, different models of damping are applicable. The main disadvantage of the method is its poor convergence. The num- ber of modes which are used in the calculation process should be properly chosen and relatively big. Usually, for calculation of the displacements and their derivatives, the number of modes is taken in such a way so the highest eigenfrequency applied in the calculations is several times higher than the fre- quency of excitation. In the presented case, the number ofmode shapes should be grater than 60. Some authors, however, claim the important contribution of high frequencymodes into vibration energy transfer in structures. The structural intensity is the product of stress and velocity for the given point of the structure. Abrupt changes of stress, which occur in points close to the excitation or discontinuities, cannotbewell describedbya limited number of lower modes. This creates the main problem of accuracy of the calculation method. The method of analysis of the structural intensity distribution has been applied as a new diagnostic method for the assessment of failure which could occur in structural elements under working loads. It enables the investigation of regions of high concentration of the vibration energy flow, particularly those exposed to the risk of damage or propagating vibration and sound waves to the environment. It can be also considered as the identification of regions in which additional damping could be applied for lowering of the vibration level and resulting noise radiation. Examples of the examination of the structural intensity distribution in elements of a rectangular plate have confirmed the method usability to the investigation of vibration energy transportation paths and places of its con- centration.Themethod enables a quantitative analysis basedon the structural 210 J.Cieślik intensity vector distribution. A quantitative analysis based on calculated va- lues of the power of vibrations in particular elements of a structure is possible as well. The appliedmodal approachwith some limitations enables themaking use of the results of experimental modal analysis for the evaluation of structural intensity vectors in real structures. With respect to the efficiency of analysis of complex structures and limita- tion of large FEMmodels, the structure intensity method seems to be a very promising tool for the energy vibration flow investigation in the middle fre- quency range. Themiddle frequency range is defined here as the range where dimensions of components of a certain structure are small or comparable to thewave length. In themiddle frequency range, the deterministicmethods, as e.g. FEM, need a dense grid of elements for the investigation of local changes in the structure. On the other hand, such structures do not fulfil assumptions for statistical methods used in the high frequency range. Such a heuristic defi- nition of themiddle frequency range does not specify its real boundary values. Regarding the results of the investigations shown above, it can be stated that themiddle frequency range for the purpose of the structural intensitymethod approximately begins at the frequency corresponding to the 20thmode shape and ends at the 100th. However, the convergence of the structural intensity calculation method is still poor, andmode shapes of a higher order should be taken into account. Acknowledgements The presented research was partly sponsored by the State Committee for Scien- tific Research (KBN) under Grant No. 5T07C01023 ”Theoretical and experimental elaboration of optimal quality assessment method of production and diagnostics of machines”. References 1. AhmidaK.M., PereiraA.K.A.,Arruda J.R.F., 1998,Predicting the total structural intensity inbeamstructuresusing the spectral elementmethod,Proc. of the Internoise, 2, 675-678 2. Bendat J.S., Piersol A.G., 1986,Random data, 2nd ed., John Wiley, New York Vibration energy flow in rectangular plates 211 3. 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PavicG., 1992,Vibroacoustical energyflowthrough straightpipes,J. of Sound and Vibration, 154, 411-429 20. Romano A.J., Williams E.G., Russo K.L., Schuette L.C., 1992, On the instantaneous structural intensity in a thin cylindrical shell, Proc. of the Internoise, p.521 21. Williams E.G., 1991, Structural intensity in thin cylindrical shells, J. of the Acoustical Society of America, 86, 1, 1615-1622 22. Verheij J.W., 1976, Cross-spectral density methods for measuring structure borne power flow on beams and pipes, J. of Sound and Vibration, 70, 133-138 Przepływ energii drganiowej w płytach prostokątnych Streszczenie W pracy przedstawiono uzyskane na podstawie danych literaturowych zależności określające natężenie dźwięków strukturalnych. Zależności te uwzględniają odkształ- cenia (liniowe i kątowe) oraz obciążenia (siły i momenty) pozwalające wyznaczyć wartość natężenia dźwięków strukturalnych dla belek i płyt rozpatrywanych jako ele- menty konstrukcyjne. Omówiono metodę wyznaczania wartości natężenia dźwięków strukturalnychnapodstawie zespolonychparametrówmodalnychuzyskiwanychdrogą analizymodalnej z użyciemmetody elementówskończonych.Wpracypodano również przykład obliczeniowy, którego wyniki pozwoliły na ocenę rozkładu wartości wekto- ra natężenia dźwięków strukturalnych na powierzchni prostokątnej płyty stalowej. Wmodelu przyjęto zadane położenie źródła drgań (siła wymuszająca) i rozpraszania energii (siła tłumiąca). Zmiany gęstości sieci elementów skończonych pozwoliły na szczegółową analizę całkowitego przepływu energii drganiowej w badanej płycie. Manuscript received September 12, 2003; accepted for print November 4, 2003