Stockbridge-Type Damper Effectiveness Evaluation: Part II The Influence of the Impedance Matrix Terms on the Energy Dissipated

Transcription

1 1470 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 4, OCTOBER 2003 Stockbridge-Type Damper Effectiveness Evaluation: Part II The Influence of the Impedance Matrix Terms on the Energy Dissipated Giorgio Diana, Alessandra Manenti, Claudia Pirotta, and Andrea Zuin Abstract This paper deals with a methodology to evaluate the 2 2 mechanical impedance matrix of a nonsymmetric Stockbridge-type damper, based on damper translational tests on a shaker and on a 6 d.o.f model of the damper itself. A series of comparisons with data measured on a laboratory span is reported, in order to investigate the effect of the single matrix terms on the global energy dissipation. The results of a software simulating the aeolian vibration behavior of a cable+damper system, accounting for the most important terms of the damper impedance matrix, are also presented. Index Terms Aeolian vibration, damper 2 2 complex impedance matrix, Stockbridge-type damper.,,, NOMENCLATURE Force transmitted by the damper to the cable. Torque transmitted by the damper to the cable. Damper s clamp translation. Damper s clamp rotation. Damper s clamp translation (sine wave). Damper s clamp rotation (sine wave). Mechanical impedance matrix elements. Mechanical impedance matrix. Damper masses displacement. Damper masses rotation.,, Damper mass, damping, and stiffness matrices. Damper messenger cable stiffness matrices. Damper messenger cable hysteretic damping constant. Messenger cables length.,, Damper masses (1,2) and clamp(c) mass.,, Damper masses (1,2) and clamp(c) moment of inertia., Frequency (rad/s). I. INTRODUCTION AS introduced in [1], which reports the results of the first part of this research, the efficiency of the Stockbridge-type damper is one of the leading aspects in designing new lines: Manuscript received April 3, The authors are with the Department of Mechanics Politecnico di Milano, Milano 20158, Italy ( giorgio.diana@polimi.it). Digital Object Identifier /TPWRD according to the actual standards [2] [4], the efficiency of a damper for a certain conductor and span can be valued according to two different philosophies. In one case, basic system, the energy dissipated by the system cable+damper is measured on a laboratory span [vibrated at the cable allowable strain (i.e., 150 microstrains peak to peak)] and then compared to the energy input from the wind: the damper is efficient if the energy dissipated is greater or equal to the energy input from the wind in all of the frequency range affected by aeolian vibrations. This method does not allow for the evaluation of the conductor amplitude of vibration on the real span. In the other case, direct method, the damper mechanical impedance is measured through translational tests on an electro-dynamic exciter and the aeolian vibration amplitudes (and strains) on the real span are computed through software, generally based on the energy balance principle [(EBP): the steady state amplitudes are those for which the energy input from the wind and the energy dissipated by the cable+damper balance]: the damper is efficient if the computed strains are lower than the allowable ones in all of the aeolian vibrations frequency range. As already observed in [1], the first method cannot be seen as a damping system design tool, due to the fact that at any change of the system (conductor, tension, damper type and/or position, number of dampers.) a new set of tests should be performed and this is surely time consuming and could be economically nonaffordable. The second method contains approximations whose entity must be carefully examined and identified: in fact, with respect to the actual standards [3], [4] the damper nonlinear response is not taken into account because the damper translational test is performed on a shaker at only one constant vibration velocity; the effect of the damper clamp rotation (terms and of the damper impedance matrix 1 ) is not considered, nor the torque transmitted by the damper due to translation of its clamp (term ). So only one term is measured (the force per unit displacement ) and then introduced in the software. These two facts may cause errors in the definition of the conductor+damper vibration mode shape, in the evaluation of the 1 The damper impedance matrix is defined by F M = f f u f f ' where F and M are the complex amplitudes of the force and torque transmitted by the damper to the cable, due to an harmonic displacement of the damper clamp with amplitudes u (translation) and ' (rotation) /03$ IEEE

2 DIANA et al.: STOCKBRIDGE-TYPE DAMPER EFFECTIVENESS EVALUATION: PART II 1471 energy dissipated by the damper and, finally, in the computation of the conductor amplitudes of vibration and strains. The first part of the research [1] has been devoted to an extensive measurement campaign with the aim of quantifying the difference between the damper dissipated energy as measured on the laboratory span and as valued through shaker translational tests: the damper clamp has been made a dynamometer one and so it has been possible to measure on the laboratory span the force and torque transmitted by the damper as a function of the damper clamp translation and rotation. The contribution of these two terms to the global energy losses and the comparison with the force translational term measured on the shaker allowed for a deeper understanding of the problem. However, the single contribution of the various terms of the damper mechanical impedance matrix still remained to be investigated, because only the global force and the global torque transmitted by the damper have been measured. The second part of the research, here exposed, consisted in the development of a methodology to define the damper mechanical impedance matrix, with a double purpose. quantifying the contribution of the single four terms of the damper impedance matrix to the dissipated energy: this allows for the evaluation of the actual direct method limitations and also for understanding which are the matrix most significant terms; improving the direct method : a proposal is made for the modification of the EBP-based software, introducing the damper impedance matrix both for the computation of the cable deflection shape and of the dissipated energy. The proposed methodology does not require to complicate the experimental part of the direct method, in fact, as it will be shown in the paper, it has the advantage of using the same translational test actually specified by the standard. The paper is organized in the following parts: comparison between the basic and the direct methods; damper mechanical impedance matrix evaluation; evaluation of the matrix terms contribution to the energy losses (through comparison with the laboratory span measurements); modified EBP-based software: comparison with the laboratory span measurements; conclusions and future work. II. COMPARISON BETWEEN THE BASIC AND THE DIRECT METHODS The following Figs. 1 4 report the comparison between the basic and the direct methods, in terms of energy dissipated by the damper. The tested cable is an ACSR Rail conductor strung on a 46-m laboratory span at N [1]. The damper has been clamped at 1.5 m far from the hinge, for a series of tests and at 2.5 m far from the hinge for another series of tests and different frequencies have been excited in order to explore different damper working conditions. Each of them is characterized by the ratio between damper clamp translation ( ) and damper clamp rotation (radians) [the mean among the values relevant to the different vibration amplitudes tested is considered]. Fig. 1. Energies dissipated at 10 Hz. Wavelength: 12.8 m. Damper at 1.5 m from the hinge. u =' =7; u =u =0:92; ' =' =0:17. Fig. 2. Energies dissipated at 23 Hz. Wavelength: 5.52 m. Damper at 1.5 m from the hinge. u =' =0:7; u =u =0:38; ' =' =0:14. Fig. 3. Energies dissipated at 22 Hz. Wavelength: 5.8 m. Damper at 2.5 m from the hinge. u =' =0:45; u =u =0:59; ' =' =0:91. In each figure, the following information is reported to clearly characterize the cable/damper interaction: frequency, mode wavelength, and damper position. Important information is also given: the mode distortion due to the damper presence in terms of the ratio between the damper clamp measured amplitude of vibration and the cable-without-damper-amplitude ( ) and also of the ratio between the measured damper clamp rotation and the cable-without-damper-rotation ( ). In each figure, several curves and points are reported: those interesting for this paragraph are: the curve span, which represents the energy dissipated by the damper as measured on

3 1472 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 4, OCTOBER 2003 Fig. 4. Energies dissipated at 28 Hz. Wavelength: 4.56 m. Damper at 2.5 m from the hinge. u =' =0:1; u =u =0:9; ' =' =1. the laboratory span (i.e., basic method), the curve shaker, which represents the energy dissipated by the damper evaluated through the term measured on the shaker and the displacement measured on the laboratory span and, finally, the points software, which represent the energy evaluated through the term measured on the shaker and the displacement evaluated through the EBP-based software developed by the Politecnico di Milano Mechanical Department researchers 2 (i.e., direct method). Going through the Figs. 1 4, it appears that if the damper vibrates far from a node (Figs. 1 and 2), the compared dissipated energies differ for a small amount. The nearer the damper is to a vibration node (the lower becomes), the more relevant the gap is. This is due to the effect of the rotation of the damper clamp, which produces both a torque ( ) and a force variation ( ). These terms together with the one are neglected in the curve shaker and this makes the difference with respect to the curve span which, obviously includes all of the terms. This result seems in contrast with Hagedorn findings [5]. In fact, his research showed that the moments introduced by the damper into the cable are of little or no importance with regard to the energy absorbed, while local bending strains on the cable could be strongly affected. According to [5], it is to be put in evidence that these terms cause also an error in the definition of the cable+damper deflection shape and this further error contribution is reflected by the software computed energies (the points software in Figs. 1 4). It can be observed that, in the case reported in Fig. 4, the error in the deflection shape computation and the error due to the missing impedance matrix terms compensate each other, thus giving an occasionally very good software result. Obviously, the nearer the damper is to a vibration node, the more relevant the contribution due to rotation will be. However, these results do not clarify which of the impedance matrix neglected terms are more important in the different situations, described by the parameter. 2 The software allows for using damper mechanical impedances f measured at a number of vibration velocities, in order to consider the damper nonlinear behavior, and the cable+damper vibration mode is computed accounting for the cable flexural stiffness. It is also useful to observe how the cable deflection shape distortion, due to the damper, changes in the different situations considered. In particular, it seems to be small when the damper is close to a node (Figs. 3 and 4), while it increases as the damper distance from the node increases (Figs. 1 and 2). This finding depends, obviously, also on the cable/damper system considered. As reported in [1], the Stockbridge-type damper used for these tests has been modified and the clamp has been made a dynamometer one, by means of four miniaturized quartz load cells (they can be seen in Fig. 5); moreover, in the span tests, the vertical translation and rotation of the clamp have been measured, by means of very light ICP accelerometers. Using this technique, it was possible to separate the contribution due to force (this contribution is reported as curve span-f in Figs. 1 4) and to torque out of the damper absorbed energy (this contribution, obviously, is the difference between the span and span-f curves). Some result is given in [1] and can also be drawn from Figs. 1 4, showing how the contribution due to torque is very low (4 to 8% of the force contribution) for high Figs. 1 and 2), while it becomes 15% of the force contribution for around 0.5. Approaching the node, the torque contribution increases and becomes even greater than the force contribution. It is however to be observed that only the relative contributions of the total force and of the total torque have been identified: As a matter of fact, the torque term is made of a torque due to translation ( ) and a torque due to rotation ( ) which cannot be separated and the same holds for the force term. Comparing the span, span-f, and shaker curves in Figs. 1 4, it is possible to draw some important conclusion: the force-due-to-rotation term, which can be evaluated through the difference between the span-f and the shaker curves seems to become not negligible as soon as decreases. The torque term, whose two components and cannot be separated, also becomes important for very small. It is then confirmed that a correct simulation of the dynamic behavior of a cable+damper system should account for the complete damper stiffness matrix, or, at least, for its more relevant terms. As already observed in the introduction, it is not easy to set up a damper vibration test allowing for translation and rotation contemporaneously, also because of the strong nonlinearity of the damper. However, simply using two load cells (Fig. 5: Force 1 and Force 2), it is very easy to get force and torque per unit of displacement ( and ) from a damper translational test. The force is given by the mean of the two force signals, while the torque is given by the difference between the two signals, multiplied by the distance between the two force transducers. The test can be repeated for different translation velocities, accounting for the damper nonlinear response (nonlinearity with respect to translation). In this respect, the damper can be seen as a linearized system for each tested translation velocity and then its stiffness matrix can be considered symmetrical (i.e., ). With this position, only the term is missing to complete the damper impedance matrix. This last term can be computed through a damper mathematical model, as described in the following.

4 DIANA et al.: STOCKBRIDGE-TYPE DAMPER EFFECTIVENESS EVALUATION: PART II 1473 Fig. 6. The Stockbridge-type damper analytical model. where Fig. 5. The Stockbridge-type damper on the shaker. This procedure is different from the one proposed by Hardy and Leblond [6], whose theoretical assessment of the complete 2 2 stiffness matrix of a symmetric damper was based on the measured dynamic translational stiffness only. Next steps will be to evaluate the complete mechanical impedance matrix of a damper and, through comparison with the span data, to identify their relative importance. III. ANALYTICAL MODEL OF THE DAMPER This section deals with the development of an analytical model in order to evaluate the damper impedance matrix. It has already been observed that the dynamical behavior of a Stockbridge-type damper shows a relevant nonlinear response as a function of the amplitude of vibration. Moreover, the energy dissipated by the messenger cables is due to the relative slide of the strands, which cannot be represented by a simple viscous model. Therefore, a rheological model should be defined, in order to obtain an unique analytical model working in all of the needed range of amplitudes and frequencies. Finally, the identification procedure of the related parameters would be very hard and could be a possible source of errors. For these reasons, a much more robust and easy to identify linear model has been chosen, representing the damper linearized dynamic behavior for each damper clamp translation velocity considered. The damper is assumed rigidly clamped on the cable and its motion in a plane. Therefore, as shown in Fig. 6, there are six degrees of freedom:,,, related to the damper masses motion, related to the clamp motion. Defining the vector, the following equation can be written: where and are the 4 4 stiffness matrices of the left and right messenger cables, assumed as beam elements, and, are force and torque applied to the clamp. Clamp displacement and rotation, are assumed to be harmonic Reordering the vectors and as where and reordering rows and columns of the matrices accordingly, we get [See equation (2) at the bottom of the page] the solution is (3) Due to the harmonic motion imposed, the messenger cables hysteretic damping can be defined by a matrix proportional to the stiffness matrix (1) where is the hysteretic damping constant. (2)

5 1474 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 4, OCTOBER 2003 The first (2) allows to define the vector for a given motion applied to the clamp and described by the vector ; the vector of the forces can be finally computed through the second equation. This procedure can be also written in a matrix form, substituting the expressions (3) into (2) and defining the matrices,,, and as follows: Substituting matrices,,, and in (2), we get (4) and finally (5) Fig. 7. f : numerical versus experimental (amplitude and phase). Test velocity: 50 mm/s. where the matrix is the searched damper impedance matrix, which is a complex function of the circular frequency, already represented through its four terms (see note 3 ): The impedance matrix depends on the parameters,,, and, which are the flexural stiffness ( ) and hysteretic damping constant ( ) of the left and right messenger cables, indicated with pedix 1 and 2, respectively. Comparing the and analytical to the experimental curves, by means of a minimization procedure, a set of parameters,,, and for each tested velocity has been identified, which represent the best fitting between analytical and experimental curves. In Figs. 7 12, the and analytical and experimental curves are reported for the tested velocities: Figs. 7 and 8 refer to 50 mm/s, Figs. 9 and 10 to 100 mm/s, and Figs. 11 and 12 to 150 mm/s. As already stated, the curves and are identical as a consequence of the symmetry of the impedance matrix, while the term can be analytically evaluated (Fig. 13). The parameters,,, and have been identified for a given translation velocity of the damper clamp, and the problem nonlinearity does not guarantee that the residuals in the rotation terms are of the same entity as the residuals in the translation terms. Fig. 8. f : numerical versus experimental (amplitude and phase). Test velocity: 50 mm/s. IV. EVALUATION OF THE IMPEDANCE MATRIX TERMS CONTRIBUTION TO THE ENERGY LOSSES This section summarizes the results obtained by comparing the laboratory span measurements to the computed damper dissipated energy. The dissipated energy has been computed using the four damper transfer functions (, experimental 3 The damper impedance matrix is defined by F M = f f u f f ' where F and M are the complex amplitudes of the force and torque transmitted by the damper to the cable, due to a harmonic displacement of the damper clamp with amplitudes u (translation) and ' (rotation). Fig. 9. f : numerical versus experimental (amplitude and phase). Test velocity: 100 mm/s. curves,, analytical curve) and the damper clamp displacement and rotation measured on the laboratory span.

6 DIANA et al.: STOCKBRIDGE-TYPE DAMPER EFFECTIVENESS EVALUATION: PART II 1475 Fig. 10. f : numerical versus experimental (amplitude and phase). Test velocity: 100 mm/s. Fig. 13. Analytical transfer functions: f : Amplitude and phase. Velocity: 100 mm/s. Fig. 11. f : numerical versus experimental (amplitude and phase). Test velocity: 150 mm/s. dissipated by the damper, for the four cases (frequencies) considered in the third paragraph. The reported percentages are obtained as the mean of the values relevant to the different amplitudes tested at each frequency. The total force and total torque energies computed through the damper impedance matrix are also reported to have a direct comparison with the correspondent measured values, reported in the last two columns of the table. This table aims at evidencing the contribution of the four impedance matrix terms. As can be observed, if the damper rotation is small, the only significant term is the one, while a different conclusion can be outlined when the clamp rotation increases: it is clearly shown in Table I that, as the ratio decreases, the contribution of the other impedance matrix terms becomes more important: it can be of the order of, reaching 60% when the damper is very close to a node. It must be pointed out that the values of displacement and rotation, used to compute the damper dissipated energy, are those measured on the laboratory span and, considering the contribution only, there will be an error also in the cable+damper deflection shape evaluation, which will contribute an additional error in the energy computation, as will be shown in the next paragraph. V. MODIFIED EBP-BASED SOFTWARE: COMPARISON WITH THE LABORATORY SPAN MEASUREMENTS Fig. 12. f : numerical versus experimental (amplitude and phase). Test velocity: 150 mm/s. The computed and measured terms are resumed in Table I, in terms of percentage energies with respect to the total energy The EBP-based software developed by the Politecnico di Milano Mechanical Department researchers schematizes the cable by stranded beam elements and then it is easy to account for both the force and the torque transmitted by the damper. The complex cable+damper deflection shape is computed imposing the end conditions due to the span extremities and to the damper. The software modification has been made considering that the three terms,, and are obtained through direct measurement on the shaker, while the term must be evaluated through the damper mathematical model described in the fourth paragraph. As shown in Table I, the contribution to

7 1476 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 18, NO. 4, OCTOBER 2003 TABLE I IMPEDANCE MATRIX TERMS CONTRIBUTION TO THE ENERGY LOSSES TABLE II PERCENTAGE ERRORS OF THE STANDARD AND THE MODIFIED SOFTWARE The table shows how the modified software allows a significant error reduction. 4 the energy dissipation is very small, except for the 28-Hz condition (i.e., when the damper is, practically, in a node of the cable defection shape). So it has been decided to introduce in the new software only the first three terms,, and (which are the most significant), such avoiding the use of the damper mathematical model or of more complex measurement procedures. In Figs. 1 4 (third paragraph), together with the already discussed curves, the results of the standard software (based on the term only) software points and of the modified software soft-new points are shown for some amplitude of vibration. The analytical (modified software) and experimental deflection shapes have also been compared and it has been observed that the cable+damper deflection shape can be fairly reproduced by the modified software if a suitable cable flexural stiffness value is chosen. In particular, the cable flexural stiffness generally assumed in the simulation programs is around, where the value is the stiffness related to the wire no-slippage condition. This value is kept constant along the cable length and allows to fit analytical to experimental cable natural frequencies. In order to analytically reproduce the cable+damper measured deflection shape, different values have to be used: in particular, the adopted stiffness value is of the order of about one half the stiffness value when the damper is far from a node and it causes a big distortion of the cable deflection shape (i.e., and/or sensibly different from unity see Figs. 1 and 2). In case the damper is close to a node, it causes a small distortion of the cable deflection shape (i.e., and close to unity see Figs. 3 and 4) and results to be close to the value. Table II reports in numerical terms what is shown in Figs. 1 4, that is the percentage error of the damper energies evaluated through the two methods with respect to the measured one. VI. CONCLUSIONS AND FUTURE WORK The following main results can be put in evidence. 1) The present research allowed to emphasize how the damper clamp rocking effect cannot generally be neglected neither for the cable deflection shape definition nor for the damper energy dissipation evaluation. Its contribution to the total damper energy is generally of the order of, reaching even when the damper is very close to a node. 2) The developed damper mathematical model, without aiming at faithfully reproducing the damper behavior, allowed for the evaluation of the contribution to the damper dissipated energy due to the rotation and translation terms of the damper mechanical impedance matrix. 3) The modified EBP software here proposed accounts, at least in a simplified way, for the rotation terms and represents an improvement of the standard EBP-based software. 4) For particular cases, such as strait or river crossings, in which more than one damper per span extremity is required, the correct computation of the cable deflection shape and of the damper energy also when the damper is close to a node becomes of paramount importance. This aspect of the problem will be the object of future research development. REFERENCES [1] G. Diana, A. Cigada, M. Belloli, and M. Vanali, Stockbridge-type damper effectiveness evaluation: Part I Comparison between tests on span and on the shaker, IEEE Trans. Power Delivery, vol. 18, pp , Oct., [2] Guide on Conductor Self-Damping Measurements, IEEE Std , May 26, [3] Guide on the Measurement of the Performance of Aeolian Vibration Dampers for Single Conductors, IEEE Std , [4] Requirements and Tests for Stockbridge-type Aeolian Vibration Dampers, IEC , [5] P. Hagedorn, On the computation of damped wind-excited vibrations of overhead transmission lines, J. Sound Vib., vol. 83, no. 2, pp , [6] A. Leblond and C. Hardy, On the estimation of a complex stiffness matrix of symmetric Stockbridge type dampers, in Proc. 3rd Int. Symp. Cable Dynamics, Trondheim, Norway, Aug The result relevant to the 28-Hz case has already been commented on in the third paragraph.

8 DIANA et al.: STOCKBRIDGE-TYPE DAMPER EFFECTIVENESS EVALUATION: PART II 1477 Giorgio Diana was born in He received the mechanical engineering degree in 1961 from Politecnico di Milano, Milano, Italy. Currently, he is Professor of Applied Mechanics at Politecnico di Milano since He is also the Director of the Mechanical Department in the same university. His research interests are in the field of fluido-elasticity, rotor-dynamics, vibration problems in mechanical engineering, railway vehicle dynamics, interaction between pantograph, and catenary. He is the author of many papers at national and international conferences and reviews. Dr. Diana is a member of Cigre (WG11-SC22-Chairman of TF1) and a member of the technical committee on rotor-dynamics of IFToMM. He is also a member of the Senato Accademico of Politecnico di Milano. Claudia Pirotta was born in She received the mechanical engineering degree in 2000 from Politecnico di Milano. She is currently pursuing the Ph.D. degree at the same university. Her research interests include the field of experimental and analytical behavior of overhead transmission line conductors and fluido-elasticity. Alessandra Manenti was born in She received the mechanical engineering degree and the Ph.D. degree in applied mechanics from Politecnico di Milano, Milano, Italy, in 1982 and 1987, respectively. Currently, she is a Professor in Mechanical Measurements at Politecnico di Milano. Her research interests include the field of experimental and analytical behavior of overhead transmission line conductors, rotor-dynamics, and statistical data analysis. Andrea Zuin was born in He received the mechanical engineering degree from Politecnico di Milano, Milano, Italy, in Currently, he is a Researcher in Applied Mechanics at Politecnico di Milano since His research interests include the experimental and analytical behavior of overhead transmission line conductors and the interaction between pantograph and overhead line.