Vibro-acoustic analysis of noise generation from a full ... - CiteSeerX
Vibro-acoustic analysis of noise generation from a full scale model of modular bridge expansion joint Jhabindra P. Ghimirea), Yasunao Matsumotob) and Hiroki Yamaguchic) (Received: 28 February 2008; Revised: 28 August 2008; Accepted: 1 September 2008)
Noises generated from modular bridge expansion joints during vehicle pass-bys have been causing local environmental problems recently. Previous experimental studies showed that possible causes of dominant noise components generated from the bottom side of the joint might be different from those from the top side. The objective of this study was to obtain theoretical insights into the mechanism of noise generation from the bottom side of the joint for which a main noise source might be structural vibration of the joint. Vibro-acoustic analysis was conducted based on the information from a full-scale model of modular expansion joint obtained in previous experimental studies. The dynamic behavior of the joint model was investigated by using the finite element method (FEM) and the sound field inside the cavity located beneath the joint model was analyzed by using the boundary element method (BEM). Indirect BEM was used to calculate the sound pressure inside the cavity with the velocity response obtained by the FE analysis as a boundary condition. The frequency range considered in the analysis was 20– 400 Hz where dominant frequency components were observed in the noise measured in the cavity beneath the joint in the previous experiment. It was intended to interpret numerical results obtained by a model developed with available mechanical properties of the joint components to seek a general understanding of the noise generation mechanism of the modular expansion joint. It was observed that the peaks in the spectrum of noise inside the cavity were due to resonances of structural vibration modes of the joint and/or resonances of acoustic modes of the cavity. There was evidence that showed possible interaction between structural modes of the joint and the acoustic modes of the cavity. © 2008 Institute of Noise Control Engineering. Primary subject classification: 21.2.2; Secondary subject classification: 75
1
INTRODUCTION
Bridges constructed recently tend to have longer and continuous-type spans owing to advanced technology in bridge construction, such as cable-supported type bridges and base-isolated viaducts. Those bridges require greater movement capability in multidimension for bridge expansion joints. Modular bridge expansion joint allows greater movements in translation and rotation required by long and continuous span a)
b)
c)
Civil and Environmental Engineering, Saitama University, 255 Shimo-Ohkuo, Sakura, Saitama, 338-8570, JAPAN Civil and Environmental Engineering, Saitama University, 255 Shimo-Ohkuo, Sakura, Saitama, 338-8570, JAPAN, email: [email protected] Civil and Environmental Engineering, Saitama University, 255 Shimo-Ohkuo, Sakura, Saitama, 338-8570, JAPAN, email: [email protected]
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bridges. The main components of the joint consists of a set of several parallel steel I-beams (referred to as center beams in this paper) aligned perpendicular to the bridge longitudinal axis, which is supported by a set of several parallel steel H-beams (referred to as support beams) in parallel with the bridge axis. This structure divides the total movement required for the joint into small movements required to accommodate each gap between the center beams. Because of this advantage, the application of the modular expansion joint has been increased these days. However, it has been recognized that the modular bridge expansion joint causes noise as vehicles pass over it1 and that noise has induced regional environmental problems recently in Japan and elsewhere2. Although possible techniques to reduce noise generated from the modular bridge expansion joint have been reported, e.g., Fobo3, the mechanism of noise generation from the modular bridge expansion joint has
not been understood fully. A series of studies have investigated possible noise generation mechanism of the modular expansion joint by conducting noise and vibration measurements with a full-scale joint model during car pass-by along with impact testing of the joint model4,5. Finite element analysis of the joint model was also conducted to understand the dynamic characteristics of the joint model5. It was concluded from these studies that the noise generated from the top side of the joint may be related to resonances of air column within the gap formed by a rubber sealing between two adjacent center beams and a car tire. The noise generated from the bottom side of the joint may be attributed to sound radiation from vibrations of the joint structure, such as the center beams. Those findings imply that, in a practical noise control problem, sources of joint noise may be dependent on the location of receiver. For example, in low-rise residential buildings along viaducts that are quite common in congested cities, residents may be affected mainly by the noise generated from the bottom side of the joint. If the noise generation from the bottom side of the joint is caused by vibration of the joint structure, understanding of the dynamic characteristics of the joint structure can be useful to develop effective measures for noise reduction. The dynamic response of the modular expansion joint subjected to traffic loading was investigated analytically by Steenbergen6 and experimentally by Ancich et al.7, although the objective of these studies was to investigate the durability of the joint structure exposed to repeated dynamic loadings due to vehicle pass-bys. In relation to the mechanism of the noise generation from the bottom side of modular expansion joint, Ancich and Brown2 measured noise and vibration of modular expansion joints installed in bridges and discussed the possibility of the interaction between structural vibration modes of the joint and acoustic modes of the cavity located beneath the joint. They reported the effectiveness of a Helmholtz absorber installation into the cavity beneath the joint in noise reduction, which implies possible contribution of acoustic modes of the cavity to noise generation. The objective of this study was to obtain theoretical insights into the contribution of the vibro-acoustic characteristics of a system consisting of the modular expansion joint and the cavity beneath the joint to the noise generation from the bottom side of the joint. Numerical vibro-acoustic analysis of the full-scale joint model used in the previous studies4,5 was conducted by using finite element method (FEM) / boundary element method (BEM) approach. FEM was used to calculate the dynamic response of the expanNoise Control Eng. J. 56 (6), Nov-Dec 2008
Fig. 1—Plan and cross section of full-scale test joint model. sion joint and indirect BEM was used for acoustic analysis of the cavity beneath the joint. A numerical model was developed based on available mechanical properties of the joint components. Assumptions were made for model properties that were not available in literature and other sources. From the numerical results obtained in such a manner, it was intended to obtain understanding of the noise generation mechanism that can be applied to modular expansion joint in general. In particular, possible mechanisms of noise generation from the bottom side of the joint were discussed based on structural vibration modes of the joint and acoustic modes of the cavity.
2
FULL SCALE JOINT MODEL AND EXPERIMENT
Figure 1 shows the full scale joint model used in the previous studies4,5. The joint model was set up in the compound of the Kawaguchi Metal Industries Co., Ltd. in Saitama, Japan. The joint model consisted of three center beams that were supported by four support beams. Control mechanisms consisting of a steel plank (referred to as control beam) and rubber springs installed between the top surface of a control beam and the bottom flange of a center beam were designed to prevent excessive displacement between the two center beams. There were steel frames placed vertically to accommodate rubber springs between the frame and the bottom flange of a support beam and a polyamide bearing between the bottom flange of a center beam and the top flange of a support beam. There were polyamide bearings beneath the ends of the support beams and rubber springs placed on top of the ends of support beams. Additionally, the modular bridge expansion joint has a drainage capability with the installation of a rubber sealing into each gap between two adjacent center beams. The joint model was mounted on top of a test cavity constructed under the ground level to represent the cavity beneath the joint between adjacent bridge girders in real bridges (Fig. 2). There was an 443
Center beam Spring
Support beam
Fig. 2—Cross-sectional views of the cavity beneath the joint.
Frame
opening at each end of the cavity. The wall of the cavity was covered by the Styrofoam that provided sound absorption within the cavity. In order to identify the dynamic characteristics of the joint, impact testing was carried out in the previous study5. The vertical and lateral acceleration of the center beams were measured at different locations and, additionally, the sound pressure was measured inside the cavity beneath the joint at 1.0 m below the top surface of the center of the joint. Several measurements were carried out by impacting the center beam from the top at different locations. The details of the experimental procedure and the results of experimental modal analysis were presented in Ravshanovich et al.5 A transfer function was calculated by using the impact force measured by an impact hammer as input and the acceleration of the beam or the sound pressure inside the cavity as output.
3 3.1
Bearing
VIBRO-ACOUSTIC ANALYSIS OF FULL SCALE MODEL Analysis Procedure
In the vibro-acoustic analysis conducted in this study, firstly, a finite element model of the full-scale joint model was developed so as to identify the modal
(a)
(b)
Fig. 3—FE model of the expansion joint: (a) 3-D view; (b) Cross-section view. parameters of the joint. The velocity responses of the joint to different point loadings were then calculated by modal superposition to compare the numerical results with the experimental results from the impact testing. Finally, the velocity response obtained was used as the boundary condition in the boundary element analysis of acoustic field in the cavity beneath the joint.
3.2
Finite Element Analysis of Expansion Joint 3.2.1 Model description Figure 3 shows the FE model of the joint developed in this study. The center beams, the support beams, the control beams and the frames were modeled with shell elements so that the vibration response of the joint could be transferred directly to the boundary element model as velocity boundary condition. Bearings and rubber springs were simplified and modeled with spring elements as in the FE model developed in the previous study5. Geometric and material properties of
Table 1—Cross-section, size and materials for steel members. Cross-section (mm) Young’s Modulus 共N / m2兲 Poisson ratio
Center beam I-shape, 140⫻ 80⫻ 21⫻ 12.5 2.06⫻ 1011 0.3
Support beam H-shape, 120⫻ 90⫻ 20⫻ 12 2.06⫻ 1011 0.3
Control beam Rectangular, 80⫻ 25 2.06⫻ 1011 0.3
Table 2—Spring constants of bearings and springs. Center beams Vertical (N/m) Lateral (N/m) Torsion (Nm) Rotation (Nm)
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Polyamide bearing 5.67⫻ 108 2.02⫻ 108 3.94⫻ 106 2.15⫻ 105
Rubber spring 1.50⫻ 106 5.07⫻ 105 9.76⫻ 103 5.56⫻ 102
Noise Control Eng. J. 56 (6), Nov-Dec 2008
Support beams Polyamide bearing 5.97⫻ 108 0 0 0
Rubber spring 5.83⫻ 106 1.97⫻ 106 3.89⫻ 104 2.27⫻ 103
Control beams Control spring 3.52⫻ 105 1.19⫻ 105 2.32⫻ 103 1.34⫻ 102
Openings Point Loads Joint-cavity boundary 1m
Measurement point Y
Third center beam
Y X
Z
Fig. 4—Position and direction of point loads for velocity response evaluation. different components in the FE joint model are shown in Tables 1 and 2. The size of a typical FE mesh for shell elements in the analysis was 40 mm to represent the cross section of the structural components of the joint, such as the center beams (Fig. 3). The ANSYS 10.0 was used in the analysis described in this section.
3.2.2
Velocity response
Velocity response of the expansion joint was calculated with harmonic point loads of unit magnitude in the frequency range from 20 to 400 Hz with an interval of 1 Hz. The point loads were applied in the lateral and vertical directions simultaneously at the center of the top flange of the second center beam, as shown in Fig. 4, to represent the loading on the joint during the impact testing experiment. In Fig. 4, the circle mark at the center of the third center beam represents one of the measurement positions during the impact testing, and the data at this location was used for comparison between numerical and experimental results in this paper, as an example. Modal-based forced response analysis was carried out by using the structural modes identified as described in the preceding section. All 211 structural vibration modes identified from the FE analysis in the frequency range between 0 – 800 Hz were considered in the analysis, since structural modes up to a frequency approximately double the maximum frequency of interest should be considered in the response calculation8. A modal damping ratio of 1% was assumed for all structural modes considered. The velocity response was calculated at all nodes of the FE model. The LMS Virtual Lab Rev 6A was used in the analysis described in this section.
3.3
Boundary Element Analysis of Acoustic Field
The velocity response obtained in the FE analysis as described above was used as a boundary condition in Noise Control Eng. J. 56 (6), Nov-Dec 2008
(a)
X
Z
(b)
Fig. 5—Boundary element model of the cavity for acoustic analysis: (a) Full 3-D model; (b) Sectional view showing measurement point. BE analysis of the acoustic field in the cavity beneath the joint. The BE model of the cavity developed in this study is shown in Fig. 5. The boundary of the cavity was discretized into quadrilateral and triangular elements. The maximum size of acoustic boundary element was set as 100 mm so that the maximum frequency that could be analyzed was 566 Hz according to a general recommendation that an accepted mesh size would be less than one-sixth of shortest wavelength of interest. The top boundary of the BE model was exactly the same as the FE model of the joint so that the velocity response of the joint could be transferred directly to the BE model. In this analysis, it was assumed that vibration response calculated from the in vacuo structural modes of the joint calculated separately can be used to analyze the acoustic field in the cavity, i.e., a weak coupling was assumed between the expansion joint and cavity beneath it (the dynamic behavior of the expansion joint was not influenced by the fluid in the cavity)9. According to the information received from the manufacturer of the Styrofoam, the sound absorption capacity of the Styrofoam was low in the frequency below 500 Hz. Therefore, the effect of the Styrofoam attached to the walls of the cavity beneath the joint was not considered in the analysis. The bare walls and floor of the cavity were considered rigid in the model. This assumption reduced computational cost in the numerical analysis. The sound speed and the density of the air used in the BE analysis were 340 m / s and 1.225 kg/ m3, respectively. Modal acoustic transfer vector (MATV) technique in indirect BEM (IBEM) was used to calculate the sound pressure inside the cavity. In this method, acoustic transfer vectors (ATVs), which are the transfer functions relating the normal velocities at the boundary to the sound pressure at a measurement point (i.e., a field point), were calculated. Modal parameters and 445
Openings
Y X
Z
Fig. 6—FE model of the cavity for acoustic modal analysis. modal participation factors of structural modes were then used to calculate the sound pressure at a field point. The details of ATV and MATV can be found in literature8,10. The field point used in this analysis was at the center of the cavity and 1.0 meter below the top surface of the joint: this point was equivalent to the measurement location for sound pressure in the impact testing experiment. The frequency range considered for the acoustic analysis was 20– 400 Hz. The LMS Virtual Lab Rev 6A was used for the acoustic analysis.
3.4
Finite Element Modal Analysis of the Cavity Beneath the Joint
Finite element analysis of the cavity beneath the joint shown in Fig. 2 was conducted to identify the acoustic modal parameters of the cavity. The FE model of the cavity developed is shown in Fig. 6. The acoustic space inside the cavity was modeled by cubical fluid element of mesh size 130 mm so that acoustic modes having natural frequencies up to 435 Hz could be calculated based on the relation between the mesh size and the highest frequency as described above. The openings on both sides of the cavity were modeled by applying impedance boundary condition. A characteristic acoustic impedance of 416 kg/ m2 s (i.e., the product of the sound speed and the air density used above) was applied to model these openings11.
4 4.1
respectively. The peaks in the velocity responses of the joint calculated in the numerical analysis were attributed to vibration modes of the joint structure. For example, the peaks in the numerical results at 112 Hz and 161 Hz in the lateral response were interpreted as the excitation of lateral bending modes of the center beam at 111.97 Hz and 161.49 Hz, respectively. These modes had one of the anti-nodes at the loading point, i.e., the center of the second center beam. For the vertical response, the peaks in the numerical results at 160 Hz, 189 Hz, and 241 Hz were due to the excitation of structural modes at 160.35 Hz, 189.35 Hz and 241.59 Hz, respectively. These modes had dominant vertical bending of the center beams with one of the anti-nodes at the loading point. Figures 7 and 8 also show the corresponding experimental data5. Two sets of experimental data are presented to show the repeatability of the measurement: the measurement appeared to be highly repeatable at frequencies below 300 Hz. Although similar trend could be observed in the experimental and
RESULTS Velocity Response of the Joint
Figures 7 and 8 show the mobility frequency response functions calculated for the velocity response at the center of the third center beam (i.e., the location shown in Fig. 4) in the lateral and vertical directions, 446
Fig. 7—Velocity response at the measurement point in lateral direction. Key:¯¯¯¯, model;——, experiment-1;——, experiment-2.
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Fig. 8—Velocity response at the measurement point in vertical direction. Key:¯¯¯¯, model;——, experiment-1;——, experiment-2.
Fig. 9—Sound pressure inside the cavity and natural frequencies of structural and acoustic modes. Key:¯¯¯¯, model;——, experiment-1;——, experiment-2;⫻, structural mode;䉭, acoustic mode. numerical results of the velocity responses, there were discrepancies in the magnitude and the peak frequencies of the response. Possible reasons include errors in the modeling of the expansion joint, for example, errors in the spring constants used for rubber bearings and springs, and errors in the damping ratio chosen in the response calculation. Also, the loadings exerted by the impact hammer might not have been applied to the center beam at 45 degrees from the ground surface as in the numerical analysis.
4.2
Sound Pressure Response Inside the Cavity
The numerical frequency response function between the sound pressure in the cavity and the input force is shown in Fig. 9. Additionally, the natural frequencies of the vibration modes of the joint and the acoustic modes of the cavity are indicated in Fig. 9 for comparison with the peak frequencies in the frequency response function. The peak frequencies in the noise appeared to correspond to the natural frequencies of the vibration modes of the joint, the natural frequencies of acoustic modes of the cavity, or both. This comparison is discussed in a later part of this paper. In the figure, the numerical result is compared with the experimental data measured in the impact testing5. Two sets of experimental data in Fig. 9 show that the measurement appeared to be repeatable at frequencies higher than about 120 Hz. Although a similar trend could be observed in the experimental and numerical results, there were discrepancies in the magnitude and the peak frequencies of the frequency response functions for several possible reasons. The discrepancies in the peak frequencies of the frequency response function calculated for the sound pressure, compared to Noise Control Eng. J. 56 (6), Nov-Dec 2008
the corresponding experimental data, may be partly due to differences in the natural frequencies of the vibration modes of the joint between the experimental and numerical results as discussed in the previous section. Additionally, there might be minor effects of the Styrofoam in the cavity beneath the joint in the experimental data, which were neglected in the numerical analysis. Spatial variation of the sound pressure around the measurement point appeared to be significant at higher frequencies, according to parametric investigation conducted separately, so that some discrepancies could be expected at higher frequencies if the field point for the numerical calculation of sound pressure was not identical to the measurement point in the experiment.
5 5.1
DISCUSSION Sound Generation Mechanism
Sound generation from the interaction of the vibrating structure with the enclosed acoustic cavity discussed in previous studies9,12 may be similar to sound generation from the joint model and the cavity beneath it investigated in this study. According to those previous studies, the peaks in the frequency response function calculated for the sound pressure in the cavity, as observed in Fig. 9, may be attributed to three different mechanisms: (1) major contribution of structural mode(s) of the joint with possible minor contribution of acoustic mode(s) of the cavity, (2) significant contribution from both structural mode(s) of the joint and acoustic mode(s) of the cavity, and (3) major contribution of acoustic mode(s) of the cavity with possible minor contribution of structural mode(s). For example, the peak in the frequency response function for the sound pressure at around 160 Hz in Fig. 9 may be attributed to the resonance of the structural mode at 160.35 Hz (vertical bending mode) and the acoustic modes at 161.10 Hz and 161.71 Hz. These structural and acoustic modes are shown in Fig. 10. The excitation of this structural mode was observed in the velocity response in the vertical direction shown in Fig. 8. The acoustic mode at 161.10 Hz shown in Fig. 10 had low amplitude at the boundary with the joint so that this acoustic mode may have had little interaction with the structural mode. Although the acoustic mode at 161.71 Hz shown in Fig. 10 may have had higher possibility of interaction with the structural mode, the magnitude around the measurement point of this acoustic mode was low. Therefore, the peak in the frequency response function for the sound pressure at around 160 Hz may be caused mainly by the excitation of the structural mode with some minor effects from the acoustic modes. Similar discussion could be made on the peaks of the frequency response function for the 447
Fig. 10—Structural and acoustic modes generating sound around 160 Hz: (a) Structural mode at 160.35 Hz; (b) Acoustic mode at 161.10 Hz and (c) Acoustic mode at 161.70 Hz. sound pressure at around 112 Hz, 188 Hz, and 230 Hz where structural modes may have contributed more than the acoustic modes. The sound pressure peak at 268 Hz may be related to the structural mode of the joint at 267.6 Hz (lateral bending mode) and the acoustic modes at 267.91 Hz and 269.40 Hz. This structural mode and the acoustic modes are shown in Fig. 11. Although the velocity response was small at around 266 Hz in Fig. 7 because of the small vibration of the third center beam in the structural mode shown in Fig. 11, this vibration mode may be excited. The acoustic modes at 267.91 Hz and 269.40 Hz may be excited by structural vibration of the joint because of high amplitude at the boundary with the joint. Therefore, the peak in the frequency response
for the sound pressure at around 268 Hz may have been caused by the excitation of both the structural and acoustic modes. For some peaks in the frequency response function for the sound pressure, the contribution may be mainly from the resonance of acoustic modes. For example, Fig. 9 shows that there was no structural mode in the frequency range between 367 Hz and 397 Hz, while there were many acoustic modes in this frequency range. The sound pressure peak at around 380 Hz could be due to the resonance of the acoustic modes at 380.38 Hz, 381.86 Hz, and 383.58 Hz, along with many other acoustic modes around this frequency. Some of the acoustic modes at around 380 Hz are shown in Fig. 12.
Fig. 11—Structural and acoustic modes generating sound around 268 Hz: (a) Structural mode at 267.6 Hz; (b) Acoustic mode at 267.91 Hz and (c) Acoustic mode at 269.40 Hz. 448
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Fig. 12—Acoustic modes generating sound around 380 Hz: (a) Acoustic mode at 380.38 Hz; (b) Acoustic mode at 381.86 Hz and (c) Acoustic mode at 383.58 Hz. 5.2
Reliability of Discussion Based on Numerical Results
There were differences in the frequency response observed between the numerical results obtained from the model developed in this study and the experimental results from the impact testing in the previous study5, as shown in Figs 7–9. The preceding sections show examples of those differences and discuss possible causes. Model updating was not attempted in this study because the experimental data available were limited while there were several sets of model properties that can be optimized. Although this was the limitation of this study, it was assumed that reasonable theoretical insights into the noise generation mechanism could be obtained from the model based on the reliable numerical technique with available mechanical properties and assumptions for model properties. The properties of structural vibration modes obtained from the numerical model were compared with the modal parameters identified in the impact testing5, although the details are not presented in this paper. There was a reasonable agreement between the numerical and experimental modal properties in vibration modes dominated by vertical bending of the beams, such as the mode shown in Fig. 10. This may partially support the reliability of the discussion based on the numerical results obtained in this study. However, there were some discrepancies in vibration modes dominated by lateral motions of the joint, including rigid body mode and lateral bending. A similar conclusion was reported in the previous study5 that compared modal properties identified in the experimental modal analysis with those in the analytical modal analysis with a 3-D frame model. Noise Control Eng. J. 56 (6), Nov-Dec 2008
Figure 13 shows the vertical velocity responses of the joint measured at the location shown in Fig. 4 during pass-bys of two types of vehicle (i.e., an ordinary car and a wagon) at two different speeds (i.e., 40 and 50 km/ h)4,5. The right wheels of those vehicles passed over the measurement and loading point shown in Fig. 4. Dominant frequency components in the vertical velocity responses were observed at frequencies below 220 Hz for all the experimental conditions, although there were variations caused by different conditions, as expected. Comparison of Fig. 13 with Fig. 8 shows similar characteristics in the results between the vehicle pass-by experiment and the impact testing, such as peaks at about 155 and 185 Hz and decreases at higher frequencies, although direct
Fig. 13—Velocity responses in vertical direction at the measurement point during vehicle pass-bys over control beams. Key:——, ordinary car at 40 km/ h;——, ordinary car at 50 km/ h;¯¯¯¯, wagon at 40 km/ h;-·-··, wagon at 50 km/ h. 449
comparison cannot be made because Fig. 13 includes the effect of loading spectra while Fig. 8 shows the frequency responses. Those similarities observed in Figs 8 and 13 imply that the theoretical insights into the dynamic characteristics of the joint and their interaction with the acoustic characteristics of the cavity obtained from impact testing data could be useful in noise control in real situation.
6
CONCLUSIONS
Vibro-acoustic analysis of the full-scale model of the modular bridge expansion joint was carried out so as to obtain theoretical insights into the mechanism of noise generation from the bottom side of the joint. In the numerical analysis, sound field beneath the joint was investigated by FEM-BEM approach in the frequency range 20– 400 Hz. Modal analysis was also conducted so as to understand the dynamic characteristics of the joint structure and the acoustic properties of the cavity beneath the joint. It was observed that dominant frequency components in the sound pressure generated inside the cavity was due to vibration modes of the joint structure and/or acoustic modes of the cavity. Some dominant frequency components in the sound pressure inside the cavity beneath the joint may be caused mainly by the vibration modes of the joint structure and others may be caused by the excitation of acoustic modes by the vibration modes of the structure. At higher frequencies where the modal density of the acoustic modes was high, there were dominant frequency components in the sound pressure that may be caused mainly by the excitation of acoustic modes of the cavity with minor contribution from vibration modes of the joint structure.
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7
ACKNOWLEDGMENT
The support from the personnel of the Kawaguchi Metal Industries Co., Ltd. is gratefully acknowledged.
8
REFERENCES
1. G. Ramberger, “Structural bearings and expansion joints for bridges”, Structural Engineering Documents 6. Zurich Switzerland: IABSE, (2002). 2. E. J. Ancich and S. C. Brown, “Engineering methods of noise control for modular bridge expansion joints”, Acoust. Aust., 32(3), 101–107, (2004). 3. W. Fobo, “Noisy neighbors”, Bridge Design & Engineering, 35, 75, (2004). 4. Y. Matsumoto, H. Yamaguchi, N. Tomida, T. Kato, Y. Uno, Y. Hiromoto and K. A. Ravshanovich, “Experimental investigation of noise generated by modular expansion joint and its control”, Doboku Gakkai Ronbunshuu A, 63(1), 75–92, (2007) [In Japanese]. 5. K. A. Ravshanovich, H. Yamaguchi, Y. Matsumoto, N. Tomida and S. Uno, “Mechanism of noise generation from a modular expansion joint under vehicle passage”, Eng. Struct., 29, 2206– 2218, (2007). 6. M. J. Steenbergen, “Dynamic response of expansion joints to traffic loading”, Eng. Struct., 11(5), 541–554, (2004). 7. E. C. Ancich, G. J. Chirgwin and S. C. Brown, “Dynamic anomalies in a modular bridge expansion joint”, J. Bridge Eng., 6(12), 1677–1690, (2006). 8. R. Citarella, L. Federico and A. Cicatiello, “Modal acoustic transfer vector approach in FEM-BEM vibro-acoustic analysis”, Eng. Anal. Boundary Elem., 31(3), 248–258, (2007). 9. S. M. Kim and M. J. Brennan, “A compact matrix formulation using the impedance and mobility approach for the analysis of structural-acoustic systems”, J. Sound Vib., 223(1), 97–113, (1999). 10. O. von Estorff and O. Zaleski, “Efficient acoustic calculations by the BEM and frequency interpolated transfer functions”, Eng. Anal. Boundary Elem., 27, 683–694, (2003). 11. LMS Virtual Lab Rev 6A Manual, LMS International, Leuven, Belgium, (2006) (referring to Munjal, “Acoustics of Ducts and Mufflers”, John Wiley and Sons, 1987). 12. E. H. Dowell, G. F. Gorman, III and D. A. Smith, “Acoustoelasticity: General theory, acoustic natural modes and forced response to sinusoidal excitation, including comparison with experiment”, J. Sound Vib., 52(4), 519–542, (1977).
Noises generated from modular bridge expansion joints during vehicle pass-bys have been causing local environmental problems recently. Previous experimental studies showed that possible causes of dominant noise components generated from the bottom side of the joint might be different from those from the top side. The objective of this study was to obtain theoretical insights into the mechanism of noise generation from the bottom side of the joint for which a main noise source might be structural vibration of the joint. Vibro-acoustic analysis was conducted based on the information from a full-scale model of modular expansion joint obtained in previous experimental studies. The dynamic behavior of the joint model was investigated by using the finite element method (FEM) and the sound field inside the cavity located beneath the joint model was analyzed by using the boundary element method (BEM). Indirect BEM was used to calculate the sound pressure inside the cavity with the velocity response obtained by the FE analysis as a boundary condition. The frequency range considered in the analysis was 20– 400 Hz where dominant frequency components were observed in the noise measured in the cavity beneath the joint in the previous experiment. It was intended to interpret numerical results obtained by a model developed with available mechanical properties of the joint components to seek a general understanding of the noise generation mechanism of the modular expansion joint. It was observed that the peaks in the spectrum of noise inside the cavity were due to resonances of structural vibration modes of the joint and/or resonances of acoustic modes of the cavity. There was evidence that showed possible interaction between structural modes of the joint and the acoustic modes of the cavity. © 2008 Institute of Noise Control Engineering. Primary subject classification: 21.2.2; Secondary subject classification: 75
1
INTRODUCTION
Bridges constructed recently tend to have longer and continuous-type spans owing to advanced technology in bridge construction, such as cable-supported type bridges and base-isolated viaducts. Those bridges require greater movement capability in multidimension for bridge expansion joints. Modular bridge expansion joint allows greater movements in translation and rotation required by long and continuous span a)
b)
c)
Civil and Environmental Engineering, Saitama University, 255 Shimo-Ohkuo, Sakura, Saitama, 338-8570, JAPAN Civil and Environmental Engineering, Saitama University, 255 Shimo-Ohkuo, Sakura, Saitama, 338-8570, JAPAN, email: [email protected] Civil and Environmental Engineering, Saitama University, 255 Shimo-Ohkuo, Sakura, Saitama, 338-8570, JAPAN, email: [email protected]
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bridges. The main components of the joint consists of a set of several parallel steel I-beams (referred to as center beams in this paper) aligned perpendicular to the bridge longitudinal axis, which is supported by a set of several parallel steel H-beams (referred to as support beams) in parallel with the bridge axis. This structure divides the total movement required for the joint into small movements required to accommodate each gap between the center beams. Because of this advantage, the application of the modular expansion joint has been increased these days. However, it has been recognized that the modular bridge expansion joint causes noise as vehicles pass over it1 and that noise has induced regional environmental problems recently in Japan and elsewhere2. Although possible techniques to reduce noise generated from the modular bridge expansion joint have been reported, e.g., Fobo3, the mechanism of noise generation from the modular bridge expansion joint has
not been understood fully. A series of studies have investigated possible noise generation mechanism of the modular expansion joint by conducting noise and vibration measurements with a full-scale joint model during car pass-by along with impact testing of the joint model4,5. Finite element analysis of the joint model was also conducted to understand the dynamic characteristics of the joint model5. It was concluded from these studies that the noise generated from the top side of the joint may be related to resonances of air column within the gap formed by a rubber sealing between two adjacent center beams and a car tire. The noise generated from the bottom side of the joint may be attributed to sound radiation from vibrations of the joint structure, such as the center beams. Those findings imply that, in a practical noise control problem, sources of joint noise may be dependent on the location of receiver. For example, in low-rise residential buildings along viaducts that are quite common in congested cities, residents may be affected mainly by the noise generated from the bottom side of the joint. If the noise generation from the bottom side of the joint is caused by vibration of the joint structure, understanding of the dynamic characteristics of the joint structure can be useful to develop effective measures for noise reduction. The dynamic response of the modular expansion joint subjected to traffic loading was investigated analytically by Steenbergen6 and experimentally by Ancich et al.7, although the objective of these studies was to investigate the durability of the joint structure exposed to repeated dynamic loadings due to vehicle pass-bys. In relation to the mechanism of the noise generation from the bottom side of modular expansion joint, Ancich and Brown2 measured noise and vibration of modular expansion joints installed in bridges and discussed the possibility of the interaction between structural vibration modes of the joint and acoustic modes of the cavity located beneath the joint. They reported the effectiveness of a Helmholtz absorber installation into the cavity beneath the joint in noise reduction, which implies possible contribution of acoustic modes of the cavity to noise generation. The objective of this study was to obtain theoretical insights into the contribution of the vibro-acoustic characteristics of a system consisting of the modular expansion joint and the cavity beneath the joint to the noise generation from the bottom side of the joint. Numerical vibro-acoustic analysis of the full-scale joint model used in the previous studies4,5 was conducted by using finite element method (FEM) / boundary element method (BEM) approach. FEM was used to calculate the dynamic response of the expanNoise Control Eng. J. 56 (6), Nov-Dec 2008
Fig. 1—Plan and cross section of full-scale test joint model. sion joint and indirect BEM was used for acoustic analysis of the cavity beneath the joint. A numerical model was developed based on available mechanical properties of the joint components. Assumptions were made for model properties that were not available in literature and other sources. From the numerical results obtained in such a manner, it was intended to obtain understanding of the noise generation mechanism that can be applied to modular expansion joint in general. In particular, possible mechanisms of noise generation from the bottom side of the joint were discussed based on structural vibration modes of the joint and acoustic modes of the cavity.
2
FULL SCALE JOINT MODEL AND EXPERIMENT
Figure 1 shows the full scale joint model used in the previous studies4,5. The joint model was set up in the compound of the Kawaguchi Metal Industries Co., Ltd. in Saitama, Japan. The joint model consisted of three center beams that were supported by four support beams. Control mechanisms consisting of a steel plank (referred to as control beam) and rubber springs installed between the top surface of a control beam and the bottom flange of a center beam were designed to prevent excessive displacement between the two center beams. There were steel frames placed vertically to accommodate rubber springs between the frame and the bottom flange of a support beam and a polyamide bearing between the bottom flange of a center beam and the top flange of a support beam. There were polyamide bearings beneath the ends of the support beams and rubber springs placed on top of the ends of support beams. Additionally, the modular bridge expansion joint has a drainage capability with the installation of a rubber sealing into each gap between two adjacent center beams. The joint model was mounted on top of a test cavity constructed under the ground level to represent the cavity beneath the joint between adjacent bridge girders in real bridges (Fig. 2). There was an 443
Center beam Spring
Support beam
Fig. 2—Cross-sectional views of the cavity beneath the joint.
Frame
opening at each end of the cavity. The wall of the cavity was covered by the Styrofoam that provided sound absorption within the cavity. In order to identify the dynamic characteristics of the joint, impact testing was carried out in the previous study5. The vertical and lateral acceleration of the center beams were measured at different locations and, additionally, the sound pressure was measured inside the cavity beneath the joint at 1.0 m below the top surface of the center of the joint. Several measurements were carried out by impacting the center beam from the top at different locations. The details of the experimental procedure and the results of experimental modal analysis were presented in Ravshanovich et al.5 A transfer function was calculated by using the impact force measured by an impact hammer as input and the acceleration of the beam or the sound pressure inside the cavity as output.
3 3.1
Bearing
VIBRO-ACOUSTIC ANALYSIS OF FULL SCALE MODEL Analysis Procedure
In the vibro-acoustic analysis conducted in this study, firstly, a finite element model of the full-scale joint model was developed so as to identify the modal
(a)
(b)
Fig. 3—FE model of the expansion joint: (a) 3-D view; (b) Cross-section view. parameters of the joint. The velocity responses of the joint to different point loadings were then calculated by modal superposition to compare the numerical results with the experimental results from the impact testing. Finally, the velocity response obtained was used as the boundary condition in the boundary element analysis of acoustic field in the cavity beneath the joint.
3.2
Finite Element Analysis of Expansion Joint 3.2.1 Model description Figure 3 shows the FE model of the joint developed in this study. The center beams, the support beams, the control beams and the frames were modeled with shell elements so that the vibration response of the joint could be transferred directly to the boundary element model as velocity boundary condition. Bearings and rubber springs were simplified and modeled with spring elements as in the FE model developed in the previous study5. Geometric and material properties of
Table 1—Cross-section, size and materials for steel members. Cross-section (mm) Young’s Modulus 共N / m2兲 Poisson ratio
Center beam I-shape, 140⫻ 80⫻ 21⫻ 12.5 2.06⫻ 1011 0.3
Support beam H-shape, 120⫻ 90⫻ 20⫻ 12 2.06⫻ 1011 0.3
Control beam Rectangular, 80⫻ 25 2.06⫻ 1011 0.3
Table 2—Spring constants of bearings and springs. Center beams Vertical (N/m) Lateral (N/m) Torsion (Nm) Rotation (Nm)
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Polyamide bearing 5.67⫻ 108 2.02⫻ 108 3.94⫻ 106 2.15⫻ 105
Rubber spring 1.50⫻ 106 5.07⫻ 105 9.76⫻ 103 5.56⫻ 102
Noise Control Eng. J. 56 (6), Nov-Dec 2008
Support beams Polyamide bearing 5.97⫻ 108 0 0 0
Rubber spring 5.83⫻ 106 1.97⫻ 106 3.89⫻ 104 2.27⫻ 103
Control beams Control spring 3.52⫻ 105 1.19⫻ 105 2.32⫻ 103 1.34⫻ 102
Openings Point Loads Joint-cavity boundary 1m
Measurement point Y
Third center beam
Y X
Z
Fig. 4—Position and direction of point loads for velocity response evaluation. different components in the FE joint model are shown in Tables 1 and 2. The size of a typical FE mesh for shell elements in the analysis was 40 mm to represent the cross section of the structural components of the joint, such as the center beams (Fig. 3). The ANSYS 10.0 was used in the analysis described in this section.
3.2.2
Velocity response
Velocity response of the expansion joint was calculated with harmonic point loads of unit magnitude in the frequency range from 20 to 400 Hz with an interval of 1 Hz. The point loads were applied in the lateral and vertical directions simultaneously at the center of the top flange of the second center beam, as shown in Fig. 4, to represent the loading on the joint during the impact testing experiment. In Fig. 4, the circle mark at the center of the third center beam represents one of the measurement positions during the impact testing, and the data at this location was used for comparison between numerical and experimental results in this paper, as an example. Modal-based forced response analysis was carried out by using the structural modes identified as described in the preceding section. All 211 structural vibration modes identified from the FE analysis in the frequency range between 0 – 800 Hz were considered in the analysis, since structural modes up to a frequency approximately double the maximum frequency of interest should be considered in the response calculation8. A modal damping ratio of 1% was assumed for all structural modes considered. The velocity response was calculated at all nodes of the FE model. The LMS Virtual Lab Rev 6A was used in the analysis described in this section.
3.3
Boundary Element Analysis of Acoustic Field
The velocity response obtained in the FE analysis as described above was used as a boundary condition in Noise Control Eng. J. 56 (6), Nov-Dec 2008
(a)
X
Z
(b)
Fig. 5—Boundary element model of the cavity for acoustic analysis: (a) Full 3-D model; (b) Sectional view showing measurement point. BE analysis of the acoustic field in the cavity beneath the joint. The BE model of the cavity developed in this study is shown in Fig. 5. The boundary of the cavity was discretized into quadrilateral and triangular elements. The maximum size of acoustic boundary element was set as 100 mm so that the maximum frequency that could be analyzed was 566 Hz according to a general recommendation that an accepted mesh size would be less than one-sixth of shortest wavelength of interest. The top boundary of the BE model was exactly the same as the FE model of the joint so that the velocity response of the joint could be transferred directly to the BE model. In this analysis, it was assumed that vibration response calculated from the in vacuo structural modes of the joint calculated separately can be used to analyze the acoustic field in the cavity, i.e., a weak coupling was assumed between the expansion joint and cavity beneath it (the dynamic behavior of the expansion joint was not influenced by the fluid in the cavity)9. According to the information received from the manufacturer of the Styrofoam, the sound absorption capacity of the Styrofoam was low in the frequency below 500 Hz. Therefore, the effect of the Styrofoam attached to the walls of the cavity beneath the joint was not considered in the analysis. The bare walls and floor of the cavity were considered rigid in the model. This assumption reduced computational cost in the numerical analysis. The sound speed and the density of the air used in the BE analysis were 340 m / s and 1.225 kg/ m3, respectively. Modal acoustic transfer vector (MATV) technique in indirect BEM (IBEM) was used to calculate the sound pressure inside the cavity. In this method, acoustic transfer vectors (ATVs), which are the transfer functions relating the normal velocities at the boundary to the sound pressure at a measurement point (i.e., a field point), were calculated. Modal parameters and 445
Openings
Y X
Z
Fig. 6—FE model of the cavity for acoustic modal analysis. modal participation factors of structural modes were then used to calculate the sound pressure at a field point. The details of ATV and MATV can be found in literature8,10. The field point used in this analysis was at the center of the cavity and 1.0 meter below the top surface of the joint: this point was equivalent to the measurement location for sound pressure in the impact testing experiment. The frequency range considered for the acoustic analysis was 20– 400 Hz. The LMS Virtual Lab Rev 6A was used for the acoustic analysis.
3.4
Finite Element Modal Analysis of the Cavity Beneath the Joint
Finite element analysis of the cavity beneath the joint shown in Fig. 2 was conducted to identify the acoustic modal parameters of the cavity. The FE model of the cavity developed is shown in Fig. 6. The acoustic space inside the cavity was modeled by cubical fluid element of mesh size 130 mm so that acoustic modes having natural frequencies up to 435 Hz could be calculated based on the relation between the mesh size and the highest frequency as described above. The openings on both sides of the cavity were modeled by applying impedance boundary condition. A characteristic acoustic impedance of 416 kg/ m2 s (i.e., the product of the sound speed and the air density used above) was applied to model these openings11.
4 4.1
respectively. The peaks in the velocity responses of the joint calculated in the numerical analysis were attributed to vibration modes of the joint structure. For example, the peaks in the numerical results at 112 Hz and 161 Hz in the lateral response were interpreted as the excitation of lateral bending modes of the center beam at 111.97 Hz and 161.49 Hz, respectively. These modes had one of the anti-nodes at the loading point, i.e., the center of the second center beam. For the vertical response, the peaks in the numerical results at 160 Hz, 189 Hz, and 241 Hz were due to the excitation of structural modes at 160.35 Hz, 189.35 Hz and 241.59 Hz, respectively. These modes had dominant vertical bending of the center beams with one of the anti-nodes at the loading point. Figures 7 and 8 also show the corresponding experimental data5. Two sets of experimental data are presented to show the repeatability of the measurement: the measurement appeared to be highly repeatable at frequencies below 300 Hz. Although similar trend could be observed in the experimental and
RESULTS Velocity Response of the Joint
Figures 7 and 8 show the mobility frequency response functions calculated for the velocity response at the center of the third center beam (i.e., the location shown in Fig. 4) in the lateral and vertical directions, 446
Fig. 7—Velocity response at the measurement point in lateral direction. Key:¯¯¯¯, model;——, experiment-1;——, experiment-2.
Noise Control Eng. J. 56 (6), Nov-Dec 2008
Fig. 8—Velocity response at the measurement point in vertical direction. Key:¯¯¯¯, model;——, experiment-1;——, experiment-2.
Fig. 9—Sound pressure inside the cavity and natural frequencies of structural and acoustic modes. Key:¯¯¯¯, model;——, experiment-1;——, experiment-2;⫻, structural mode;䉭, acoustic mode. numerical results of the velocity responses, there were discrepancies in the magnitude and the peak frequencies of the response. Possible reasons include errors in the modeling of the expansion joint, for example, errors in the spring constants used for rubber bearings and springs, and errors in the damping ratio chosen in the response calculation. Also, the loadings exerted by the impact hammer might not have been applied to the center beam at 45 degrees from the ground surface as in the numerical analysis.
4.2
Sound Pressure Response Inside the Cavity
The numerical frequency response function between the sound pressure in the cavity and the input force is shown in Fig. 9. Additionally, the natural frequencies of the vibration modes of the joint and the acoustic modes of the cavity are indicated in Fig. 9 for comparison with the peak frequencies in the frequency response function. The peak frequencies in the noise appeared to correspond to the natural frequencies of the vibration modes of the joint, the natural frequencies of acoustic modes of the cavity, or both. This comparison is discussed in a later part of this paper. In the figure, the numerical result is compared with the experimental data measured in the impact testing5. Two sets of experimental data in Fig. 9 show that the measurement appeared to be repeatable at frequencies higher than about 120 Hz. Although a similar trend could be observed in the experimental and numerical results, there were discrepancies in the magnitude and the peak frequencies of the frequency response functions for several possible reasons. The discrepancies in the peak frequencies of the frequency response function calculated for the sound pressure, compared to Noise Control Eng. J. 56 (6), Nov-Dec 2008
the corresponding experimental data, may be partly due to differences in the natural frequencies of the vibration modes of the joint between the experimental and numerical results as discussed in the previous section. Additionally, there might be minor effects of the Styrofoam in the cavity beneath the joint in the experimental data, which were neglected in the numerical analysis. Spatial variation of the sound pressure around the measurement point appeared to be significant at higher frequencies, according to parametric investigation conducted separately, so that some discrepancies could be expected at higher frequencies if the field point for the numerical calculation of sound pressure was not identical to the measurement point in the experiment.
5 5.1
DISCUSSION Sound Generation Mechanism
Sound generation from the interaction of the vibrating structure with the enclosed acoustic cavity discussed in previous studies9,12 may be similar to sound generation from the joint model and the cavity beneath it investigated in this study. According to those previous studies, the peaks in the frequency response function calculated for the sound pressure in the cavity, as observed in Fig. 9, may be attributed to three different mechanisms: (1) major contribution of structural mode(s) of the joint with possible minor contribution of acoustic mode(s) of the cavity, (2) significant contribution from both structural mode(s) of the joint and acoustic mode(s) of the cavity, and (3) major contribution of acoustic mode(s) of the cavity with possible minor contribution of structural mode(s). For example, the peak in the frequency response function for the sound pressure at around 160 Hz in Fig. 9 may be attributed to the resonance of the structural mode at 160.35 Hz (vertical bending mode) and the acoustic modes at 161.10 Hz and 161.71 Hz. These structural and acoustic modes are shown in Fig. 10. The excitation of this structural mode was observed in the velocity response in the vertical direction shown in Fig. 8. The acoustic mode at 161.10 Hz shown in Fig. 10 had low amplitude at the boundary with the joint so that this acoustic mode may have had little interaction with the structural mode. Although the acoustic mode at 161.71 Hz shown in Fig. 10 may have had higher possibility of interaction with the structural mode, the magnitude around the measurement point of this acoustic mode was low. Therefore, the peak in the frequency response function for the sound pressure at around 160 Hz may be caused mainly by the excitation of the structural mode with some minor effects from the acoustic modes. Similar discussion could be made on the peaks of the frequency response function for the 447
Fig. 10—Structural and acoustic modes generating sound around 160 Hz: (a) Structural mode at 160.35 Hz; (b) Acoustic mode at 161.10 Hz and (c) Acoustic mode at 161.70 Hz. sound pressure at around 112 Hz, 188 Hz, and 230 Hz where structural modes may have contributed more than the acoustic modes. The sound pressure peak at 268 Hz may be related to the structural mode of the joint at 267.6 Hz (lateral bending mode) and the acoustic modes at 267.91 Hz and 269.40 Hz. This structural mode and the acoustic modes are shown in Fig. 11. Although the velocity response was small at around 266 Hz in Fig. 7 because of the small vibration of the third center beam in the structural mode shown in Fig. 11, this vibration mode may be excited. The acoustic modes at 267.91 Hz and 269.40 Hz may be excited by structural vibration of the joint because of high amplitude at the boundary with the joint. Therefore, the peak in the frequency response
for the sound pressure at around 268 Hz may have been caused by the excitation of both the structural and acoustic modes. For some peaks in the frequency response function for the sound pressure, the contribution may be mainly from the resonance of acoustic modes. For example, Fig. 9 shows that there was no structural mode in the frequency range between 367 Hz and 397 Hz, while there were many acoustic modes in this frequency range. The sound pressure peak at around 380 Hz could be due to the resonance of the acoustic modes at 380.38 Hz, 381.86 Hz, and 383.58 Hz, along with many other acoustic modes around this frequency. Some of the acoustic modes at around 380 Hz are shown in Fig. 12.
Fig. 11—Structural and acoustic modes generating sound around 268 Hz: (a) Structural mode at 267.6 Hz; (b) Acoustic mode at 267.91 Hz and (c) Acoustic mode at 269.40 Hz. 448
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Fig. 12—Acoustic modes generating sound around 380 Hz: (a) Acoustic mode at 380.38 Hz; (b) Acoustic mode at 381.86 Hz and (c) Acoustic mode at 383.58 Hz. 5.2
Reliability of Discussion Based on Numerical Results
There were differences in the frequency response observed between the numerical results obtained from the model developed in this study and the experimental results from the impact testing in the previous study5, as shown in Figs 7–9. The preceding sections show examples of those differences and discuss possible causes. Model updating was not attempted in this study because the experimental data available were limited while there were several sets of model properties that can be optimized. Although this was the limitation of this study, it was assumed that reasonable theoretical insights into the noise generation mechanism could be obtained from the model based on the reliable numerical technique with available mechanical properties and assumptions for model properties. The properties of structural vibration modes obtained from the numerical model were compared with the modal parameters identified in the impact testing5, although the details are not presented in this paper. There was a reasonable agreement between the numerical and experimental modal properties in vibration modes dominated by vertical bending of the beams, such as the mode shown in Fig. 10. This may partially support the reliability of the discussion based on the numerical results obtained in this study. However, there were some discrepancies in vibration modes dominated by lateral motions of the joint, including rigid body mode and lateral bending. A similar conclusion was reported in the previous study5 that compared modal properties identified in the experimental modal analysis with those in the analytical modal analysis with a 3-D frame model. Noise Control Eng. J. 56 (6), Nov-Dec 2008
Figure 13 shows the vertical velocity responses of the joint measured at the location shown in Fig. 4 during pass-bys of two types of vehicle (i.e., an ordinary car and a wagon) at two different speeds (i.e., 40 and 50 km/ h)4,5. The right wheels of those vehicles passed over the measurement and loading point shown in Fig. 4. Dominant frequency components in the vertical velocity responses were observed at frequencies below 220 Hz for all the experimental conditions, although there were variations caused by different conditions, as expected. Comparison of Fig. 13 with Fig. 8 shows similar characteristics in the results between the vehicle pass-by experiment and the impact testing, such as peaks at about 155 and 185 Hz and decreases at higher frequencies, although direct
Fig. 13—Velocity responses in vertical direction at the measurement point during vehicle pass-bys over control beams. Key:——, ordinary car at 40 km/ h;——, ordinary car at 50 km/ h;¯¯¯¯, wagon at 40 km/ h;-·-··, wagon at 50 km/ h. 449
comparison cannot be made because Fig. 13 includes the effect of loading spectra while Fig. 8 shows the frequency responses. Those similarities observed in Figs 8 and 13 imply that the theoretical insights into the dynamic characteristics of the joint and their interaction with the acoustic characteristics of the cavity obtained from impact testing data could be useful in noise control in real situation.
6
CONCLUSIONS
Vibro-acoustic analysis of the full-scale model of the modular bridge expansion joint was carried out so as to obtain theoretical insights into the mechanism of noise generation from the bottom side of the joint. In the numerical analysis, sound field beneath the joint was investigated by FEM-BEM approach in the frequency range 20– 400 Hz. Modal analysis was also conducted so as to understand the dynamic characteristics of the joint structure and the acoustic properties of the cavity beneath the joint. It was observed that dominant frequency components in the sound pressure generated inside the cavity was due to vibration modes of the joint structure and/or acoustic modes of the cavity. Some dominant frequency components in the sound pressure inside the cavity beneath the joint may be caused mainly by the vibration modes of the joint structure and others may be caused by the excitation of acoustic modes by the vibration modes of the structure. At higher frequencies where the modal density of the acoustic modes was high, there were dominant frequency components in the sound pressure that may be caused mainly by the excitation of acoustic modes of the cavity with minor contribution from vibration modes of the joint structure.
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7
ACKNOWLEDGMENT
The support from the personnel of the Kawaguchi Metal Industries Co., Ltd. is gratefully acknowledged.
8
REFERENCES
1. G. Ramberger, “Structural bearings and expansion joints for bridges”, Structural Engineering Documents 6. Zurich Switzerland: IABSE, (2002). 2. E. J. Ancich and S. C. Brown, “Engineering methods of noise control for modular bridge expansion joints”, Acoust. Aust., 32(3), 101–107, (2004). 3. W. Fobo, “Noisy neighbors”, Bridge Design & Engineering, 35, 75, (2004). 4. Y. Matsumoto, H. Yamaguchi, N. Tomida, T. Kato, Y. Uno, Y. Hiromoto and K. A. Ravshanovich, “Experimental investigation of noise generated by modular expansion joint and its control”, Doboku Gakkai Ronbunshuu A, 63(1), 75–92, (2007) [In Japanese]. 5. K. A. Ravshanovich, H. Yamaguchi, Y. Matsumoto, N. Tomida and S. Uno, “Mechanism of noise generation from a modular expansion joint under vehicle passage”, Eng. Struct., 29, 2206– 2218, (2007). 6. M. J. Steenbergen, “Dynamic response of expansion joints to traffic loading”, Eng. Struct., 11(5), 541–554, (2004). 7. E. C. Ancich, G. J. Chirgwin and S. C. Brown, “Dynamic anomalies in a modular bridge expansion joint”, J. Bridge Eng., 6(12), 1677–1690, (2006). 8. R. Citarella, L. Federico and A. Cicatiello, “Modal acoustic transfer vector approach in FEM-BEM vibro-acoustic analysis”, Eng. Anal. Boundary Elem., 31(3), 248–258, (2007). 9. S. M. Kim and M. J. Brennan, “A compact matrix formulation using the impedance and mobility approach for the analysis of structural-acoustic systems”, J. Sound Vib., 223(1), 97–113, (1999). 10. O. von Estorff and O. Zaleski, “Efficient acoustic calculations by the BEM and frequency interpolated transfer functions”, Eng. Anal. Boundary Elem., 27, 683–694, (2003). 11. LMS Virtual Lab Rev 6A Manual, LMS International, Leuven, Belgium, (2006) (referring to Munjal, “Acoustics of Ducts and Mufflers”, John Wiley and Sons, 1987). 12. E. H. Dowell, G. F. Gorman, III and D. A. Smith, “Acoustoelasticity: General theory, acoustic natural modes and forced response to sinusoidal excitation, including comparison with experiment”, J. Sound Vib., 52(4), 519–542, (1977).
