Dynamic Magnification Factor Research Articles

Simple design procedure of a friction damper for reducing seismic responses of a single-story structure

This study proposed a simple design procedure for determining the required damping force of a friction damper installed in a single-story structure. The analysis model was transformed into an equivalent mass-spring-dashpot system by approximating a nonlinear Coulomb damping force with an equivalent viscous damping force. A closed form solution for the dynamic magnification factor (DMF) for a steady-state response was derived using the energy balance equation. The equivalent viscous damping ratio was defined using the DMF at the natural frequency. The transfer function between input harmonic excitation and output structural response was obtained from the DMF, and the response reduction factor of the root mean square (RMS) of displacements with and without friction dampers was analytically determined. Using the proposed procedure the friction force required for satisfying a given target response reduction factor was obtained. The response reduction factors were obtained for the structures with different natural frequencies subjected to ten earthquake records. Based on the dynamic analysis results, it was concluded that the mean response reduction factors matched well with the target values.

Dynamic Analysis of a Composite Cable-Stayed Bridge: Escaleritas Viaduct

This work describes some of the most important results of the experimental and numerical analyses of Escaleritas Viaduct, Spain. Before the inauguration of this composite cable-stayed bridge in 2006, the bridge authority required a dynamic load test identifying, for instance, the natural vibration modes, the dynamic magnification factor, and the maximum vertical acceleration. The dynamic test was accompanied by numerical simulation performed in two different three-dimensional finite-element models, one of them composed of 145,000 shell elements. The correlation of test and analysis data is good and allows several interesting general conclusions to be drawn. It is shown that Escaleritas Viaduct complies with the requirements on the dynamic structural behavior defined in the standards.

단자유도 건물의 지진응답제어를 위한 마찰감쇠기 설계

Approximate analysis for a building installed with a friction damper is performed to get insight of its dynamic behavior. Energy balance equation is used to have a closed analytical form solution of dynamic magnification factor(DMF). It is found out that DMF is dependent on friction force ratio and resonance frequency. Approximation of DMF and equivalent damping ratio of a friction damper is proposed with such assumption that the building with a friction damper shows harmonic steady-state response and narrow banded response behavior near resonance frequency. Linear transfer function from input external force to output building displacement is suggested from the simplified DMF equation. Root mean square of a building displacement is derived under earthquake-like random excitation. Finally, design procedure of a friction damper is proposed by finding friction force corresponding to target control ratio. Numerical analysis is carried out to verify the proposed design procedure.

The Use of the Dynamic Magnification Factor in the Dynamic Analysis of Framed Structures

This paper examined the use of the dynamic magnification factor in the analysis of framed structures. It is a method of practice in dynamic analysis of structures to magnify static response by a dynamic magnification factor in order to obtain the equivalent dynamic response. This method has been applied to the dynamic analysis of many structures including bridges and some country’s codes of practice made specifications in respect of the dynamic magnification factor for the analysis and design of various types of structures subjected to dynamic excitation. The suitability of this method to the dynamic analysis of frames was investigated in this paper by carrying out static and dynamic analysis of four frames using the flexible frame model and the stiffness formulation. Dynamic responses were first obtained by direct analysis as solutions to the set of equations governing the motions of the frames and secondly by the magnification of the static responses using the dynamic magnification factors. By comparing the results obtained in both methods it was inferred that the practice of magnifying static responses to obtain their dynamic equivalents in frame analysis gives correct results only in the case of deflections and not in stresses. Finally, this practice should de discouraged or limited only to the case of deflections in the dynamic analysis of framed structures.

Nonlinear dynamics of an inclined beam subjected to a moving load

In this paper, the nonlinear dynamic response of an inclined pinned-pinned beam with a constant cross section, finite length subjected to a concentrated vertical force traveling with a constant velocity is investigated. The study is focused on the mode summation method and also on frequency analysis of the governing PDEs equations of motion. Furthermore, the steady-state response is studied by applying the multiple scales method. The nonlinear response of the beam is obtained by solving two coupled nonlinear PDEs governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-pint of the beam are obtained for various load velocity ratios and the outcome results have been illustrated and compared to the results with those obtained from traditional linear solution. The appropriate parametric study considering the effects of the linear viscous damping, the velocity of the traveling load, beam inclination angle under zero or nonzero axial load are carried out to capture the influence of the effect of large deflections caused by stretching effects due to the beam’s immovable ends. It was seen that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also in the case where the object leaves the inclined beam, its planar motion path is derived and the targeting accuracy is investigated and compared with those from the rigid solution assumption. Moreover, the stability analysis of steady-state response for the modes equations having quadratic nonlinearity was carried out and it was observed from the frequency response curves that for the considered parameters in the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon can be predicted for the longitudinal excitation.

A parametric study on the dynamics of urban transit maglev vehicle running on flexible guideway bridges

The purpose of this study is to develop a numerical model for a dynamic interaction analysis of an actively controlled maglev vehicle and flexible guideway structure. In addition, an investigation is performed of the effect of vehicle and guideway characteristics on dynamic responses of low and medium speed maglev systems. Dynamic governing equations are derived by combining the 5-dof maglev vehicle model, the modal properties of guideway structures and a LQG controller for electromagnetic suspension. An investigation is then carried out on the effect of vehicle models, vehicle speed, irregularity, guideway deflection ratio, span length, span continuity and damping ratio on dynamic responses of the relatively low speed maglev vehicle and guideway structures. From the numerical simulation, it is found that the air gap of the vehicle is strongly affected by vehicle speed, tract roughness and guideway deflection ratio. In particular, the guideway deflection ratio is the most influential parameter which governs the air gap. Continuous span girders are found to be effective in reducing the air gap because of its smooth curvature and small deflection slope near the support. However, the span length and damping ratio of the guideway structure do not affect the air gap. The overall dynamic magnification factor of the guideway girder is not severe compared to the traditional wheel type vehicle.

Nonlinear vibrations of an inclined beam subjected to a moving load

In this paper, the nonlinear dynamic responses of an inclined pinned-pinned Euler-Bernoulli beam with a constant cross section and finite length subjected to a concentrated vertical force traveling with constant velocity is investigated by using the mode summation method. Frequency analysis of the PDE's governing equations of motion for steady-state response is studied by applying multiple scales method. The nonlinear dynamic deflections of the beam are obtained by solving two coupled nonlinear PDE's governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-point of the beam are obtained for various load velocity ratios and the numerical results are compared with those obtained from traditional linear solution. It is found that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also stability analysis of the steady-state response for the modes equations having quadratic nonlinearity is carried out and it is observed from the amplitude response curves that for the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon are predicted for the longitudinal excitation.

Active multiple-tuned mass dampers for asymmetric structures considering soil-structure interaction

By resorting to Fourier transform, the equations of motion for the soil-asymmetric structure-active multiple-tuned mass dampers (AMTMD) interaction system are developed in the frequency domain under the ground acceleration. The criterion for searching the optimum parameters of the AMTMD is selected as the minimization of the minimum values of the maximum displacement dynamic magnification factors (DMF) of the asymmetric structure with the AMTMD. The estimation criterion of the effectiveness of the AMTMD is chosen as the ratio of the minimization of the minimum values of the maximum displacement DMF of the asymmetric structure with the AMTMD to the maximum displacement DMF of the asymmetric structure without the AMTMD. Employing these two criteria, the parametric studies for the influences of the normalized eccentricity ratio (NER), torsional to translational frequency ratio (TTFR), stiffness ratio of the soil relative to the structure, and height-to-base ratio of the soil-asymmetric structure interaction system are then carried out on both the effectiveness and robustness of the AMTMD. Simultaneously, the effectiveness of a single active-tuned mass damper (ATMD) with the optimum position is also presented and compared with that of the AMTMD. Extensive numerical simulations show that both the AMTMD and ATMD can effectively attenuate the translational and torsional responses of asymmetric structures built on soft soil foundation. Copyright © 2009 John Wiley & Sons, Ltd.

NONLINEAR TRANSIENT ANALYSIS OF A RAILWAY CONCRETE SLEEPER IN A TRACK SYSTEM

Railway sleepers in a track system are usually subjected to a wide range of loading conditions. A critical type of loading condition that causes cracking in the railway concrete sleepers is the dynamic transient wheel force. The transient wheel forces are often due to wheel or rail abnormalities. This paper presents a dynamic finite element model of a railway concrete sleeper in a track system, aimed at raising the consideration of dynamic effects in sleeper design. The railway concrete sleeper is modeled using the beam-on-elastic-foundation theory. Since in the actual tracks the ballast underneath does not provide any tensile resistance, the finite beam elements employed in this investigation take into account the bending and shear deformations, together with the tensionless nature of the elastic support. This paper places emphasis on the effect of the transient periods on the flexural responses of railway sleepers in track systems. Using the robust finite element software STRAND7, the finite element model of the railway concrete sleeper was previously established and validated against experimental data by the authors. The numerical analyses present the ratio between the dynamic and the static bending moment resultants, the dynamic magnification factor, of the railway concrete sleeper under different sinusoidal pulse durations.

Estimation of active multiple tuned mass dampers for asymmetric structures

This paper proposes the application of active multiple tuned mass dampers (AMTMD) for translational and torsional response control of a simplified two-degree-of-freedom (2DOF) structure, able to represent the dynamic characteristics of general asymmetric structures, under the ground acceleration. This 2DOF structure is a generalized 2DOF system of an asymmetric structure with predominant translational and torsional responses under earthquake excitations using the mode reduced-order method. Depending on the ratio of the torsional to the translational eigenfrequency, i.e. the torsional to translational frequency ratio (TTFR), of asymmetric structures, the following three cases can be distinguished: (1) torsionally flexible structures (TTFR < 1.0), (2) torsionally intermediate stiff structures (TTFR = 1.0), and (3) torsionally stiff structures (TTFR > 1.0). The even distribution of the AMTMD within the whole width and half width of the asymmetric structure, thus leading to three cases of installing the AMTMD (referred to as the AMTMD of case 1, AMTMD of case 2, AMTMD of case 3, respectively), is taken into account. In the present study, the criterion for searching the optimum parameters of the AMTMD is defined as the minimization of the minimum values of the maximum translational and torsional displacement dynamic magnification factors (DMF) of an asymmetric structure with the AMTMD. The criterion used for assessing the effectiveness of the AMTMD is selected as the ratio of the minimization of the minimum values of the maximum translational and torsional displacement DMF of the asymmetric structure with the AMTMD to the maximum translational and torsional displacement DMF of the asymmetric structure without the AMTMD. By resorting to these two criteria, a careful examination of the effects of the normalized eccentricity ratio (NER) on the effectiveness and robustness of the AMTMD are carried out in the mitigation of both the translational and torsional responses of the asymmetric structure. Likewise, the effectiveness of a single ATMD with the optimum positions is presented and compared with that of the AMTMD.

A blind prediction test of nonlinear analysis procedures for reinforced concrete shear walls

A full scale slice of a 7 story reinforced concrete building was tested on the shake table at the UCSD Engelkirk Structural Research Centre in 2006. As part of the research project, a blind prediction contest was sponsored to assess the capability of currently available analysis procedures to predict the seismic response of cantilever reinforced concrete shear wall structures. This paper describes an entry based on a nonlinear finite element model, using macro elements to represent both the shear and the flexural modes of behaviour. A comparison of the predicted response with the test results showed that the analysis procedure produced reasonable predictions of deformations for the lowest and highest of the four earthquakes but under-estimated response for the two moderate earthquakes by approximately 30%. For all earthquakes, the analysis base moment was much lower than the test value. Modifications to the procedure to improve the correlation were identified and implemented but did not remedy the deficit in base moments. Detailed results of the test program revealed that the causes for this discrepancy were the contribution to overturning results of gravity columns and the flange wall, neither of which had been included in the model. When these were incorporated the average error between test and analysis results was less than 10% for all earthquakes, well within acceptable limits for a design office type of model. The correlation of tests and analysis also provided useful information on design aspects for shear walls, such as the influence of secondary components and dynamic magnification factors.

Characteristics of linearly distributed parameter-based multiple-tuned mass dampers

Five multiple-tuned mass damper (MTMD) models with their natural frequencies being uniformly distributed around their mean natural frequency and eight MTMD models with the system parameters being uniformly distributed around their average values, respectively, have been recently presented by the second author. In order to further seek the MTMD models with high performance, the present paper will extend the above-mentioned work in terms of the simultaneously uniform distributions of three- or two-system parameters. Consequently, highlighted in this paper are four other MTMD models, referred to as general linearly distributed parameter-based multiple-tuned mass dampers (general LDP-MTMD). The structure is represented by the mode-generalized system corresponding to the specific vibration mode that needs to be controlled. The main objective of the present-study, thus is to estimate the performance with general LDP-MTMD with respect to the MTMD with identical damping coefficient and damping ratio but unequal stiffness and uniform distribution of masses (UM-MTMD3) by resorting to minimization of the minimum values of the maximum dynamic magnification factors (i.e. Min.Min.Max.DMF) of structures with these four LDP-MTMD models. It is again manifested in terms of the simulation results presented that it is preferable to select the optimum UM-MTMD3 or the optimum MTMD with identical stiffness and damping coefficient but unequal mass and uniform distribution of natural frequencies. Notwithstanding this, in comparison with the UM-MTMD3 when the total mass ratio is smaller than or equal to 0.02, due to significantly smaller strokes as well as almost the same effectiveness and robustness levels, the LDP-MTMD3 is also preferable with larger total number of dampers. Copyright © 2007 John Wiley & Sons, Ltd.

Control strategy of the lever-type active multiple tuned mass dampers for structures

The lever-type active multiple tuned mass dampers (LT-AMTMD), consisting of several lever-type active tuned mass dampers (LT-ATMD), is proposed in this paper to attenuate the vibrations of long-span bridges under the excitation directly acting on the structure, rather than through the base. With resorting to the derived analytical-expressions for the dynamic magnification factors of the LT-AMTMD structure system, the performance assessment then is conducted on the LT-AMTMD with the identical stiffness and damping coefficient but unequal mass. Numerical results indicate that the LT-AMTMD with the actuator set at the mass block can provide better effectiveness in reducing the vibrations of long-span bridges compared to the LT-AMTMD with the actuator set at other locations. An appealing feature of the LT-AMTMD with the actuator set at the mass block is that the static stretching of the spring may be freely adjusted in accordance with the practical requirements through changing the location of the support within the viable range while maintaining the same performance (including the same stroke displacement). Likewise, it is shown that the LT-AMTMD with the actuator set at the mass block can further ameliorate the performance of the lever-type multiple tuned mass dampers (LT-MTMD) and has higher effectiveness than a single lever-type active tuned mass damper (LT-ATMD). Therefore, the LT-AMTMD with the actuator set at the mass block may be a better means of suppressing the vibrations of long-span bridges with the consequence of not requiring the large static stretching of the spring and possessing a desirable robustness.

Performance and parametric study of infinite-multiple TMDs for structures under ground acceleration by H∞ optimization

A new design method, i.e. the Infinite-Multiple tuned mass dampers (IMTMD) method, is proposed for the optimum configuration of classical MTMD. Firstly, the transfer function (TF) of IMTMD is obtained by integration, and then the dynamic magnification factor (DMF) of the single-degree-of-freedom (SDOF) main structure with IMTMD under base acceleration excitation is investigated as well as MTMD. Using IMTMD the best and critical performance of MTMD composed of many TMDs can be obtained together with optimum design parameters, which can only be conjectured based on a lot of data observation in previous studies. Moreover, the IMTMD method has excellent efficiency of iteration and it can give correct optimum parameters numerically for MTMD composed of more than 20 TMDs. For convenience, two types of MTMD hypotheses proposed in the previous literature are selected as the examples for demonstration and discussion.

Investigation of response of systems with active multiple tuned mass dampers

The active multiple tuned mass dampers (AMTMD), consisting of several active tuned mass dampers (ATMD), with the uniform distribution of natural frequencies have been proposed by Li et al. (J. Struct. Eng. (ASCE) 2003; 129(7):972–977) to attenuate undesirable oscillations of structures under ground acceleration. Here, the open–closed-loop AMTMDs, referred to as the OCL-AMTMD, have been further investigated including both the open-loop and closed-loop AMTMD, referred, respectively, to as the OL-AMTMD and CL-AMTMD. Based on the modegeneralized system in the specific vibration mode being controlled (referred to as the main system), the expression for the dynamic magnification factor (DMF) of the structure with the OCL-AMTMD is then formulated. The criterion for the optimum searching can thus be defined as the minimization of the minimum values of the maximum DMF (i.e. min.min.max.DMF). With resorting to this criterion, the research is carried out on the parameters measuring both the effectiveness and robustness of the OCL-AMTMD. The parameters include the frequency spacing, average damping ratio, tuning frequency ratio, total number, total mass ratio, and structural and seismic normalized acceleration feedback gain factor (NAFGF). Via estimating the OL-AMTMD, CL-AMTMD, and OCL-AMTMD, it is found that the CL-AMTMD and OCL-AMTMD may further enhance the performance of the multiple tuned mass dampers (MTMD) and a single open–closed-loop active tuned mass damper (referred to as the OCL-ATMD). Likewise, the near-zero optimum average damping ratio found by Li and Liu (J. Struct. Eng. (ASCE) 2002; 128(10):1362–1365) will disappear in the CL-AMTMD and OCL-AMTMD. For the OL-AMTMD, however, there yet exists the near-zero optimum average damping ratio. Copyright © 2007 John Wiley & Sons, Ltd.

Seismic response of controlled structures with active multiple tuned mass dampers

Active multiple tuned mass dampers (referred to as AMTMD), which consist of several active tuned mass dampers (ATMDs) with identical stiffness and damping coefficients but varying mass and control force, have recently been proposed to suppress undesirable oscillations of structures under ground acceleration. It has been shown that the AMTMD can remarkably improve the performance of multiple tuned mass dampers (MTMDs) and is also more effective in reducing structure oscillation than single ATMDs. Notwithstanding this, good performance of AMTMD (including a single ATMD illustrated from frequency-domain analysis) may not necessarily translate into a good seismic reduction behavior in the time-domain. To investigate these phenomena, a three-story steel structure model controlled by AMTMD with three ATMDs was implemented in SIMULINK and subjected to several historical earthquakes. Likewise, the structure under consideration was assumed to have uncertainty of stiffness, such as ±15% of its initial stiffness, in the numerical simulations. The optimum design parameters of the AMTMD were obtained in the frequency-domain by implementing the minimization of the minimum values of the maximum dynamic magnification factors (DMF) of general structures with AMTMD. For comparison purposes, response analysis of the same structure with a single ATMD was also performed. The numerical analysis and comparison show that the AMTMD generally renders better effectiveness when compared with a single ATMD for structures subjected to historical earthquakes. In particular, the AMTMD can improve the effectiveness of a single ATMD for a structure with an uncertainty of stiffness of ±15% of its initial stiffness.

PERFORMANCE OF DUAL-LAYER MULTIPLE TUNED MASS DAMPERS FOR STRUCTURES UNDER GROUND EXCITATIONS

The dual-layer multiple tuned mass dampers (DL-MTMD) with a uniform distribution of natural frequencies are proposed, which consist of one large tuned mass damper (L-TMD) and an arbitrary number of small tuned mass dampers (S-TMD). The structure is represented by a generalized system corresponding to the specific vibration mode to be controlled. The criterion for assessing the optimum parameters and effectiveness of the DL-MTMD is based on the minimization of the minimum values of the maximum dynamic magnification factors (DMF) of the structure installed with the DL-MTMD. Also considered is the stroke of the DL-MTMD. The proposed DL-MTMD system is demonstrated to show higher effectiveness and robustness to the change in frequency tuning, in comparison to the multiple tuned mass dampers (MTMD) with equal total mass ratios. It is also demonstrated to be more effective than the dual tuned mass dampers (DTMD) with one large and one small tuned mass damper, but they maintain the same level of robustness to the change in frequency tuning. The DL-MTMD system can be easily manufactured as the optimum value for the linking dashpots between the structure and L-TMD is shown to be zero.

Estimating double tuned mass dampers for structures under ground acceleration using a novel optimum criterion

The double tuned mass dampers (DTMD), consisting of one larger mass block (i.e. one larger tuned mass damper (TMD)) and one smaller mass block (i.e. one smaller TMD), have been proposed to seek for the mass dampers with high effectiveness and robustness for the reduction of the undesirable vibrations of structures under the ground acceleration. The structure is represented by the mode-generalized system corresponding to the specific vibration mode that needs to be controlled. In light of the developed dynamic magnification factors (DMF) of the DTMD structure system, the criterion used for assessing the optimum parameters and effectiveness of the DTMD is selected as the minimization of the minimum values of the maximum DMF of the structure with the DTMD. With resorting to the maximum DMF of both the larger and smaller TMDs in the DTMD, the stroke of the DTMD is simultaneously investigated too. It is highlighted that a novel optimum objective function has been proposed in order to acquire high robust control system. Consequently, the two types of optimum goal functions (including the optimum goal function commonly used) have been applied for the optimum searching of the DTMD. The numerical results indicate that the DTMD designed in terms of the second type of optimum objective functions (i.e. the novel optimum objective function) practically provides the same effectiveness and robustness to the changes in the drift frequency ratio (DFR) as the multiple tuned mass dampers (MTMD) with the distributed natural frequencies with the total number of the TMD units equal to five and with equal total mass ratio. Likewise, the DTMD designed with resort to the second type of optimum objective functions can practically attain the same effectiveness as the TMD with equal total mass ratio. More importantly, in the robustness to the changes in the DFR, the DTMD is significantly better than the TMD, whereas in the robustness to the natural frequency tuning (NFT), measured by the frequency band width coefficient (FBWC), the DTMD is significantly better than the MTMD, thus manifesting that the DTMD is an advanced control device.

Study on the inertia effect of helical spring of the absorber on suppressing the dynamic responses of a beam subjected to a moving load

In the dynamic analysis of the classical absorber composed of a spring, a damper and a lumped mass, the mass of its helical spring (or spring mass) is usually neglected. The purpose of this paper is to present a modified absorber with inertia effect of the spring mass considered. In order to evaluate the vibration–reduction efficiency of the presented modified absorber, the forced vibration analyses of a single degree-of-freedom (dof) and a multiple dof structural systems are performed. The first analysis is to determine the dynamic magnification factor of a single dof spring–mass system, respectively, attached by a classical absorber and a modified absorber and subjected to a harmonic excitation. The second analysis is to determine the maximum dynamic responses of a multiple dof pinned–pinned beam, respectively, attached by the last two kinds of absorbers and subjected to a moving concentrated load. Because the second analysis is conducted using the conventional finite element method (FEM), in additional to the mass matrix, damping matrix and stiffness matrix of the classical and modified absorbers, the expressions for calculating the optimum parameters of the last two absorbers associated with any order of vibration mode of the pinned–pinned beam are also derived based on the modal data obtained from the mode superposition methodology and the orthogonal property between the normal mode shapes. Numerical results reveal that the spring mass of the absorber has the effect of suppressing the maximum dynamic responses of the main structural system and should be considered in the formulation to agree with the practical situation.

EVALUATION OF MULTIPLE DUAL TUNED MASS DAMPERS FOR STRUCTURES UNDER HARMONIC GROUND ACCELERATION

The multiple dual tuned mass dampers, referred to as the MDTMD, consisting of several units of dual tuned mass dampers (DTMD) are proposed for the first time herein, aimed at the effectiveness and robustness of the system for suppressing the undesirable vibrations of structures under the ground acceleration. The total number of dampers can be arbitrary and their natural frequencies are uniformly distributed. Ten typical types of the MDTMD can be devised by varying the system parameters. Employing the criteria chosen for optimum searching, parametric studies were carried out to evaluate the performance of the MDTMD of Type I-1 for its convenience in manufacturing, in which the larger mass blocks (LMBs) are assumed to have identical stiffness, but unequal masses, and the smaller mass blocks (SMBs) have identical stiffness and damping coefficient, but unequal masses and damping ratios. By adopting the maximum dynamic magnification factor (DMF) for each LMB and SMB used in estimating the stroke, an evaluation is also made for the stroke of the MDTMD. The numerical results indicate that the MDTMD (I-1) can provide better effectiveness and higher robustness in comparison with the dual tuned mass dampers (DTMD) and other MTMD systems of similar complexities. However, the stroke of the MDTMD is greater than that of the DTMD and the stroke of each SMB in the MDTMD is larger than that of the mass blocks (MBs) in the arbitrary integer and odd number based MTMD.