LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

Transcription

1 Seismic Analysis of a Low-Rise Base-Isolated Structural System by Halit Kaplan and Ahmet H. Aydilek reprinted from Journal of LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL VOLUME 25 NUMBER MULTI-SCIENCE PUBLISHING COMPANY LTD.

2 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL Pages Seismic Analysis of a Low-Rise Base-Isolated Structural System Halit Kaplan 1 and Ahmet H. Aydilek 2 1 The Sciemtitic and Technological Research Council of Turkey (TUBITAK) 06100, Kavakliders, Ankara, Turkey. 2 Department of Civil and Environmental Engineering 1163 Glenn Martin Hall University of Maryland College Park, MD 20742, U.S.A. aydilek@eng.umd.edu received 30th March 2006 Key Words: Earthquake, base isolation, mechanical springs, low-rise structure. ABSTRACT: A base-isolation system was developed for earthquake protection of low-rise structures. The system incorporates spherical supports for the base, a specially designed spring-cam system to keep the base supported under normal conditions, and moves for the duration of the earthquake under the constraint of a spring with optimized stiffness characteristics. The dynamic behaviour of a three-story concrete structure with and without the base isolation subjected to the Taft and El Centro earthquake loads was investigated. The results indicated that the absolute peak acceleration and displacement as well as shear forces decreased significantly with the application of a base isolation system, and it is possible to achieve 87 to 94% reduction in the maximum accelerations and transmitted forces. The movement of the base relative to the ground was less than 0.15 m in the optimized system, and the springs were not fully compressed at any time during application of the earthquake loads. The maximum induced vertical forces as a result of the spherical base support were found to be less than 1.5 % of the weight of the structure. Since the system performance is highly dependent on the rapid unlocking of the cams in the event of a seismic disturbance, careful consideration should be given to the optimal design of the spring-cam system. 1. INTRODUCTION In seismically critical regions, the use of base isolation systems is considered as a means of minimizing the earthquake effects. Several active and passive isolation systems are currently being used for this purpose, including rubber-steel composite isolators, frictional pendulum systems, active tendon mechanisms, rolling and sliding systems, tuned mass, liquid absorbers, and suspension mechanisms (Wang and Reinhjorn 1989, Kuroda and Saruta 1989, Mayes et al. 1990, Kareem 1994, Koike and Murata 1994, Mostaghel and Davis 1997, Jahilal and Utku 1998, Almazan et al. 1998, Zhou and Lu 1998, Kaplan and Seireg 2001). The base isolation systems have been traditionally applied to various structures under earthquake loads (Fujita et al. 1994, Wang and Liu 1994, Youssef et al. 1994, Kaplan and Seireg 2000, Kaplan and Seireg 2002). Research has also been extended to investigate the applicability of these systems in designing low-rise structures (Shing et al. 1996, Matheu et al. 1998), and wood structures (Symans 2002). Marano and Greco (2003) isolated the base of a low-rise building by using high damping rubber bearings. A stochastic approach was used to model the seismic acceleration acting at the base. The displacements and dissipation of energy decreased as a result Vol. 25 No

3 Seismic Analysis of a Low-Rise Base-Isolated Structural System of base isolators. Shake table tests on a five-storey benchmark model with base isolation were conducted by Samali et al. (2003). Both translational and torsional responses of the structure were reduced by isolating the base with either laminated rubber or lead-core bearings. Isolation system damping for which acceleration of the structure attains a minimum value was used by Jangid and Kelly (2001). Hybrid base-isolation systems were also used for protection of an eight-storey structure under earthquake loads (Tzan and Panadalidas 1994), and a power rate reaching law hybrid control method developed by Zhao et al. (2000) provided comparable results with a model traditionally being used. Yang and Huang (1998) showed that the response of a structure to earthquake excitations can be effectively reduced using a base-isolation system formed of elastic bearings. In their study, each storey was modeled with translation and torsion, and the optimal point for mounting the construction equipment was shown to be the one where the equipment remains undisplaced during vibration. Existing studies indicated that the base-isolation systems can significantly decrease the peak acceleration, displacement, and shear loads in a fixed-base structure (Lin and Hone 1993, Lin et al. 1995). Use of the base-isolation systems can be quite helpful in decreasing the displacement and shear loads in residential buildings typically constructed in seismically sensitive regions. The objective of this study was to investigate the performance of a low-rise base-isolated residential building under different earthquake excitations using a newly developed baseisolation system. To meet this objective, a fixed-base three-storey concrete structure was subjected to the 1940 El Centro and the 1952 Tafts earthquake loads and the response to these loads was analyzed. Comparisons were made between the observed performance of the rigidly supported (fixed-base) and base-isolated structures. 2. SYSTEM MODEL FOR THE CONVENTIONALLY DESIGNED FIXED- BASE STRUCTURE The design analysis undertaken in this study suggests that a relatively simple, robust active control can be implemented to protect a structure. A system similar to the one developed for a 40-storey structure (Fujita et al. 1994, Kaplan and Seireg 2001) and specifically the one developed for low-rise buildings (Kurota and Saruta 1989) was used herein for a three-storey concrete residential building. The fixed-base system was modeled with three lumped masses and three stiffness coefficients (Figure 1). Structural damping was neglected for safety. The mass distributions for the first, second, and third floors were 79,000 kg, 58,000 kg, and 60,000 kg, respectively. The coefficients of stiffness were selected as 27, 900 ton/m, 16, 100 ton/m and 15, 100 ton/m, respectively, based on a trapezoidal stiffness distribution for the floors. The building is 10 m-wide and 10 m-high. A depth of 6.5 m was considered for the structure. The first, second, and third natural frequencies of the structure were calculated as wn 1 = 1.318, wn 2 = 3.334, and wn 3 = Hz, respectively. Figure 1: Lumped mass model of fixed-base three floor concrete structure 94 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

4 Halit Kaplan and Ahmet H. Aydilek Dynamic equations of the system model can be written as follows (Kaplan and Seireg 2000): (1) (2) (3) where (t) is the ground displacement, M 1, M 2, and M 3 are mass, C 1, C 2, and C 3 are damping, K 1, K 2, and K 3 are stiffness, and x 1, x 2, and x 3 displacement values for each floor, respectively. 3. SEISMIC RESPONSE OF THE FIXED-BASE STRUCTURE The fixed-base structure was separately subjected to the 1940 El Centro (M=6.7) and the 1952 Tafts (M=7.7) earthquake loads and its response to these loads was analyzed using the fourth-order Runge-Kutta numerical integration technique with variable step size (Matlab 2001). The numerical integration was performed for the duration of the simulated earthquake, such as 53 seconds in this case. Figure 2 provides the data for the two earthquakes. (a) (b) Figure 2 Temporal characteristics of acceleration and displacement for the (a) El Centro earthquake, and the (b) Taft earthquake Figure 3 summarizes the results of simulations for the El Centro earthquake loads. All acceleration and displacement values are absolute values and they will simply be called acceleration and displacement in the rest of the paper. Comparison of Figures 2a and 3 suggests that peak acceleration at the top of the building (the third floor) is approximately 3.5 times higher than the peak ground acceleration. The peak displacement on the third floor is 1.8 times of the ground displacement. As expected, the peak acceleration and displacement generally increased for the higher floors, and the maximum shear force occurred at the lower portion of the building. Vol. 25 No

5 Seismic Analysis of a Low-Rise Base-Isolated Structural System Figure 3 Seismic response of the fixed-base structure subjected to the El Centro earthquake loads Figure 3 (cond d) Seismic response of the fixed-base structure subjected to the E1 Centro earthquake loads 96 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

6 Halit Kaplan and Ahmet H. Aydilek 4. SYSTEM MODIFICATION USING A CONCAVE-BALL SUPPORT BASE A physical model comprising a spring-cam system previously developed by Kaplan and Seireg (2000) was used herein to isolate the base of the structure. The system improves the performance of the building against earthquake loads by controlling the displacement of the base and transfer of shear forces between the floors. The cams compress fully the springs under normal conditions and are released when an earthquake is sensed. A side view of the system with steel balls and concave base supports is shown in Figure 4. A diagrammatic representation of the spring-cam system along with the force-displacement curve of the spring is shown in Figure 5. Both the steel balls and the base support were assumed to be polymer-coated. Preliminary analysis by Kaplan (2002) indicated that a choice of twenty-five for the number of balls during the design was sufficient for the sensitivity of the analysis. In this case, each ball was placed between concave surfaces that tend to automatically restore balls and the base to their original position even if the system undergoes some rotational response. Another advantage of the use of concave surfaces is a reduction of the contact stresses as a result of their curvatures. Detailed description of the design can be found in Kaplan and Seireg (2000). Figure 4 (a) Side and (b) top view of the three-storey concrete base isolated structure with steel balls and concave base supports Figure 5 (a) The force displacement characteristics of the spring, and (b) diagrammatic representation of the spring cam system Vol. 25 No

7 Seismic Analysis of a Low-Rise Base-Isolated Structural System The spring function K b used as part of the base isolation system herein represents the equivalent restoring stiffness on the base as it contacts the springs which are placed radially (Figure 4). The spring function K b is not sensitive to the travel direction of the shock wave in this type of spring arrangement. A restoring function with an eight-spring arrangement for different directions of wave propagation indicated that K b can be defined by the following formula: (4) where K is the stiffness of each individual spring, and l is the directional cosine for each spring relative to the direction of displacement. Accordingly, K b = 2K is valid for the considered system. A detailed diagrammatic representation of the spring cam system is given in Figure 5b. 5. DYNAMIC EQUATIONS OF THE ISOLATED SYSTEM The dynamic equations of the base-isolated system can be written for two separate cases. The first case is the one in which the absolute relative displacement of the base (x b ) with respect to displacement of the ground ( (t)) is greater than the distance of the free movement between the spring and the base (b) that is, max x b - (t) b. The corresponding equation is: (5) where M b is the mass of the base, M s is the mass of the structure, and g is the gravitational acceleration. Note that Equation-5 is valid when the spring is being compressed. The vertical acceleration at the base, ÿ b, and the total horizontal force acting on the base. H, are defined with the following equations (see Appendix for details): (6) (7) where R is the radius of the concave support, and r is the radius of the hollow spherical balls. The effective coefficient of friction at the base, µ eff, was defined by Kaplan (2002) as follows: (8) where µ 0 is the coefficient of rolling friction. The second case is in which max x b - (t) < b and the differential equation of this system becomes: 98 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

8 Halit Kaplan and Ahmet H. Aydilek (9) (10) (11) (12) The damping effects are neglected (i.e., C 1 = C 2 = C 3 = 0 ) for simplicity, and the mass of isolated structure is defined by: (13) and M b is set to a practical value (14) And finally, the total vertical force acting on the base, V, can be defined as follows: (15) 6. DESIGN OPTIMIZATION The main goal of the design problem was to minimize the peak shear force in the isolated structure by constraining the relative displacement of the structure to the ground. An optimization scheme was developed for this purpose and is presented below. Since, well-stablished codes and standards are available for designing the structural component of the low-rise buildings in seismic sensitive regions, the optimization model was applied only to the base isolation system. The decision parameters of the system are the parts of the base isolated mechanism where concave seats are supported by the hollow spherical balls and the system is controlled by a spring-cam system. Therefore, the parameters considered during designing the base isolation system were R, r, K b, b, M b, M eff, and d. All the parameters were defined above, except the parameter d is the distance between the base and the fully compressed spring. Figure 6a provides a schematic of the base system. (a) Vol. 25 No

9 Seismic Analysis of a Low-Rise Base-Isolated Structural System Figure 6 (a) Illustration of the motion in the base system and the parameters considered during optimization scheme, and (b) change in shear forces for different relative base displacement values The only design variables considered in the optimization scheme were R, K b, and b, since the parameters M b, r and µ 0 were set to their practical values considering the geometry and material properties (Mostaghel and Davis 1997, Seireg 1998). A positive value of d denotes that the springs are not fully compressed at any time during the action of the disturbance. A reasonable value of 0.3 m was assumed for d to limit the movement of the base during the search for the solution. The objective function was selected to minimize the peak shear force at the critical storey in the isolated structure normalized by the maximum shear force in the fixed-base case and is given by: (16) or (17) where F i and F r are the peak shear forces transmitted to each floor for the isolated and the fixed-base case, respectively. The following constraints were considered for the optimization scheme: 1. The maximum displacement of the base relative to the ground motion should be set to a maximum allowable value at which the peak shear force (i.e., transmitted force) in the structure is minimized. The shear forces were determined for different peak values of relative displacements and are plotted in Figure 6b. As seen from the figure, the numerical minimum stress can be achieved by allowing the base to be displaced 0.15 m, at which an acceptably 100 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

10 Halit Kaplan and Ahmet H. Aydilek low transmitted force can be obtained. Therefore, a maximum value of b = 0.15 m was chosen for the optimization scheme: (18) The main purpose herein was to control the maximum displacement of the structure. Considering the fluctuations in the curve in Figure 6b, the spring was assumed to be fully compressed when max x b - (t) 0.30 m, and the boundary conditions became x b = (t) and ẋ b = (t) 2. Upper and lower limits were assigned to the radius of concave curvature (R), the spring stiffness (K b ), and the free movement distance between the spring and the base (b) considering the practicality of the application: (19) (20) (21) Based on the given design parameters, differential equations of motion were solved using the fourth-order Runge-Kutta numerical integration technique with a variable step-size. Similar to the fixed-base case, the numerical integration was performed for the duration of the simulated earthquake. Then, the relevant quantities in the objective function (maximum shear forces) were obtained indirectly utilizing the numerical simulation data. The Quasi-Newton method provided in the Matlab programming language was used to optimize the design parameters. The following optimum design parameters were used for simulating the response of the baseisolated system to the earthquake loads: R= 4 m, K b = (0.005) K 1 N/m, and b= 0.05 m. Figure 7 Acceleration versus time response of the base-isolated structure with subjected to the El Centro earthquake loads Vol. 25 No

11 Seismic Analysis of a Low-Rise Base-Isolated Structural System Figure 8 Displacement versus time response of the base isolated structure with base isolation subjected to the El Centro earthquake loads 7. SEISMIC RESPONSE OF THE BASE-ISOLATED SYSTEM In order to define the change in structural performance due to base isolation, acceleration and displacement of the base and each floor were plotted versus time in Figures 7 and 8, respectively. The El Centro earthquake data were used in calculations. As expected, the peak acceleration and displacement generally increased for the higher floors. However, comparisons with the fixed-base structure in Figure 3 indicate that the base isolation decreased the peak acceleration of the structure about 14 times. Similarly, the peak displacement of the base isolated structure was 0.13 m as compared to a peak value of 0.18 m determined for the fixed-base structure, which indicates a decrease of about 28%. Additionally, the acceleration or displacement value at any given time was lower for the isolated case. Figure 9 provides the relative displacement and shear forces in each storey. Since the earthquake performance of a storey is affected from the floors above and below it, relative displacements were necessary for defining the performance of the storeys. Additionally, the shear force transmitted from one floor to another was plotted versus time in Figure 10. The transmitted shear force was lower for the upper floors, similar to the behaviour observed for the fixed-base structure. A comparison of Figures 3 and 10 indicates that shear forces decreased significantly after modifying the system with a base isolation. For instance, examination of the plots revealed that the peak shear force transmitted into storey #l decreased from l500kn to about l30kn. In order to provide a direct comparison between the fixed-base and isolated case the peak values of acceleration, displacement, and shear forces were plotted for the El Centro and Taft earthquakes in Figures 11 and 12, respectively. The base isolation system caused a significant decrease in the acceleration, and displacement values when the structure was subjected to the El Centro earthquake or the Taft earthquake loads. The decreases in maximum acceleration, displacement, and shear force were 93%, 47%, and 94%, respectively under the Taft earthquake loads. The results indicated that the base isolation system increased the earthquake resistance of the structure significantly. In addition, the effective coefficient of 102 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

12 Halit Kaplan and Ahmet H. Aydilek friction of the base was small (Figure 13). Even though not presented herein, the observed movement of the base relative to the ground did not exceed 0.15 m when subjected to either of the earthquakes. Moreover, maximum induced vertical and horizontal forces as a result of the base support were determined to be less than 1.5% and 6%, respectively, of the weight of the structure. Figure 9 Relative displacement versus time response of each storey subjected to the El Centro earthquake loads Figure 10 Transmitted force (shear force) versus time response in each storey subjected to the El Centro earthquake loads Vol. 25 No

13 Seismic Analysis of a Low-Rise Base-Isolated Structural System Figure 11 Seismic responses of fixed-base and the base-isolated structure subjected to the El Centro earthquake loads Figure 12 Seismic responses of fixed-base and the base-isolated structure subjected to the Taft earthquake loads. 104 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

14 Halit Kaplan and Ahmet H. Aydilek Figure 13 Response of effective coefficient of rolling friction to the El Centro earthquake loads CONCLUSIONS A base isolation system that incorporates a spring-cam system with a spherical support was developed for earthquake protection of a low-rise building. The dynamic behaviour of a three-storey concrete structure subjected to the Taft and El Centro earthquakes was investigated. The results indicated that the measured maximum acceleration, displacement and shear forces decreased significantly with the application of a base isolation system. The decreases were 93%, 24%, and 87%, respectively, for the El Centro earthquake and 93%, 43%, and 94%, respectively, for the Taft earthquake. The movement of the base relative to the ground was less than 0.15 m in the optimized system, and the springs were not fully compressed at any time during disturbance. The maximum induced vertical forces as a result of the proposed spherical base support were found to be less than 1.5 % of the weight of the structure and, consequently, their effect on the structure can be neglected in comparison with the shear forces. Since the system performance is highly dependent on the rapid unlocking of the cams in the event of a seismic disturbance, careful consideration should be given to the design of a reliable cam release control. This can be achieved by spring loading each cam such that it would be normally unlocked. A hydraulic actuator may be used to force it rotate to the locking position under fluid pressure that would be constantly maintained at the design level during normal conditions. The actuator can be equipped with a quick response release valve for rapidly releasing the pressure and consequently unlocking the cam as soon as an earthquake is detected. REFERENCES Almazan, J. L., De la Llera, J. C., and Inaudi, J.C., Modeling Aspects of Structures Isolated With the Frictional Pendulum System, Earthquake Engineering and Structural Dynamics, Vol. 27, No. 8, pp Fujita S., Furuya, O., and Fujita, T., Dynamic Tests on High Damping Rubber Damper for Vibration Control of Tall Buildings, FA2-3; First World Conference on Structural Control, Los Angeles, California, USA. Vol. 25 No

15 Seismic Analysis of a Low-Rise Base-Isolated Structural System Jahilal, P. and Utku, S., Active Control In Passively Base Isolated Buildings Subjected to Lower Power Excitations, Computers and Structures, Vol. 66, No. 2-3, pp Jamgid, R.S. and Kelly, I.M., Base Isolation for Near-Fault Motions, Earthquake Engineering and Structural Dynamics Vol. 30, No. 5, pp Kaplan, H. and Seireg, A., A Computer Controlled System for Earthquake Protection of Structures, International Journal of Computer Applications in Technology, Vol.13,No.1-2,pp Kaplan, H. and Seireg, A., Optimal Design of a Base Isolated System for a High-Rise Steel Structure, Earthquake Engineering and Structural Dynamics, Vol. 30, No. 2, pp Kaplan, H., A Computer Controlled System for Earthquake Protection of Structures, Ph.D. Dissertation, University of Wisconsin-Madison, Madison, Wisconsin, USA, 519 p. Kaplam, H. and Seireg, A., A Base Isolation system for Bridges Subjected to Seismic Disturbances, Earthquake Engineering and Structural Dynamics, Vol. 31, No. 5,pp Kareem, A., The Next Generation of Tuned Liquid Dampers, First World Conference on Structural Control, Los Angeles, California, USA. Koike, Y. and Murata T., Development of V-Shaped Hybrid Mass Damper and its Applications to High-Rise Buildings First World Conference on Structural Control, Los Angeles, California, USA. Kuroda, T. and Saruta, M., Verification Studies on Base Isolation Systems by Full Scale Buildings, In Seismic, Shock and Vibration Isolation, Chung, H. and Fujita, T. (eds), Vol. 181, ASME, New York pp. 1-8 Lin, T.W., and Hone, C.C., Base Isolation by Free Rolling Rods Under Basement, Earthquake Engineering and Structural Dynamics Vol. 22, No. 3, pp Lin, T.W., Chern, C.C., and Hone, C.C., Experimental Study of Base Isolation by Free Rolling Rods, Earthquake Engineering and Structural Dynamics Vol. 24, No. 12, pp Matlab Version 5 User s Guide, Prentice Hall, Englewood Cliffs, New Jersey. Marano, G.C. and Greco, R., Efficiency of Base Isolation Systems in Structural Seismic Protection and Energetic Assessment, Earthquake Engineering and Structural Dynamics Vol. 32, No. 10, pp Matheu, E. E., Singh, M.P., and Beathe, C., Output-Feedback Sliding-Mode Control with Generalized Sliding Surface for Civil Structures under Earthquake Excitation, Earthquake Engineering and Structural Dynamics, Vol. 27, No. 3, pp Mayes, R.L., Jones, R., Bukle, G., and Eeri, M., Impediments to the Implementation of Seismic Isolation, Earthquake Spectra, Earthquake Engineering Research Institute, Vol. 6, No.2, pp JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

16 Halit Kaplan and Ahmet H. Aydilek Mostaghel, N. and Davis, T., Representations of Coulomb Friction for Dynamic Analysis, Earthquake Engineering and Structural Dynamics, Vol. 26, No. 5, pp Samali, B., Wu, Y.M., and Li, J., Shake Table Tests on Mass Eccentric Model with Base Isolation, Earthquake Engineering and Structural Dynamics Vol. 32, No. 9, pp Seireg, A., Friction and Lubrication m Mechanical Design, Mercel Dekker, New York, NY. Shing, P. B., Dixon, M. E., Kermiche, N., Su, R., and Frangopol M., Control of Building Vibrations with Active/Passive Devices, Earthquake Engineering and Structural Dynamics, Vol. 25, No. 10, pp Symans, M.D., Base Isolation and Supplemental Damping Systems for Seismic Protection of Wood Structures: Literature Review, Earthquake Spectra, Earthquake Engineering Research Institute, Vol. 18, No. 3, pp Tzan, S. R. and Pandelides. C.P., Hybrid Structural Control Using Viscoelastic Dampers and Active Control Systems, Earthquake Engineering and Structural/ Dynamics Vol. 23, No. 12, pp Wang, Y. P and Liu, C. J., Active Control of Sliding Structures under Strong Earthquakes, FP1-23; First World Conference on Structural Control, Los Angeles, California USA. Wang, Y. P. and Reinhjorn, A. M., Motion Control of Sliding Isolated Structure, In Seismic, Shock and Vibration Isolation, Chung, H. and Fujita, T. (eds), Vol. 181, ASME, New York. Yang, Y.B. and Huang, W. H., Equipment-Structure Interaction Considering the Effect of Torsion and Base Isolation, Earthquake Engineering and Structural Dynamics Vol. 27, No. 2, pp Youssef, N., Nuttall, B, Rahman, A., and Hata, O., Passive Control of the Los Angeles City Hall, FP2-54; First World Conference on Structural Control, Los Angeles, California, USA. Zhao, B., Lu, X., Wu, M., and Mei, Z., Sliding Mode Control of Buildings with Base-Isolation Hybrid Protective System, Earthquake Engineering and Structural Dynamics Vol. 29, No. 3, pp Zhou, Q. and Lu, X., Dynamic Analysis on Structures Base Isolated by A Ball System with Restoring Property, Earthquake Engineering and Structural Dynamics, Vol. 27, No. 8, pp Vol. 25 No

17 Seismic Analysis of a Low-Rise Base-Isolated Structural System APPENDIX Motion of the base and geometric relationships Figure A-l- Motion of the base and geometric relationships Figure A-2- Motion of the base on the spherical balls with the concave support (A-1) (A-2) (A-3) 108 JOURNAL OF LOW FREQUENCY NOISE, VIBRATION AND ACTIVE CONTROL

18 Halit Kaplan and Ahmet H. Aydilek (A-4) The vertical acceleration: (A-5) Total horizontal force at the base (A-6) Total vertical force at the base (A-7) where n is the number of the spherical balls. (A-8) (A-9) By substituting Equation (A-9) into Equation (A-6), (A-10) and H can also be approximated as: (A-11) Vol. 25 No