Modal Summation Technique Research Articles

A multibody dynamic model for predicting operational load spectra of dual clutch transmissions

In this paper, a generalized nonlinear time-varying multibody dynamic model of dual clutch transmissions (DCT) is presented. The model consists of clutches, shafts, gears and synchronizers, and can be used to model any DCT architecture. A nonlinear clutch model is used to determine the transmitted power to the transmission at any speed and clutch temperature. The clutch can be a single- or multi-plate clutch and can operate in a wet or dry-clutch configuration. A combined kinematic and powerflow simulation enables calculation of gear, shaft, bearing and clutch quasi-static loads as well as gear mesh frequencies following a duty cycle as the input. For the corresponding Linear-Time-Invariant (LTI) system model, natural frequencies and mode shapes are obtained by solving the eigenvalue problem. The modal summation technique is used to determine the steady state forced vibration response of the system. For the corresponding NTV system, Newmark's time-step marching based integration is used to determine both the steady state and transient forced vibration response of the system. The DCT model is exercised using a common transmission architecture operating at several different operating conditions. The resulting impact of changing operational conditions on gear and bearing loads as well as dynamic transmission error spectra are demonstrated.

Dynamic Response of Fractionally Damped Viscoelastic Plates Subjected to a Moving Point Load

AbstractThe dynamic response of fractionally damped viscoelastic plates subjected to a moving point load is investigated. In order to capture the viscoelastic dynamic behavior more accurately, the material is modeled using the fractionally damped Kelvin–Voigt model (rather than the integer-type viscoelastic model). The Riemann–Liouville fractional derivative of order 0 < α ≤ 1 is applied. Galerkin's method and Newton–Raphson technique are used to evaluate the natural frequencies and corresponding damping coefficients. The structure is subject to a moving point load, traveling at different speeds. The modal summation technique is applied to generate the dynamic response of the plate. The influence of the order of the fractional derivative on the free and transient vibrations is studied for different velocities of the moving load. The results are compared with those using the classical integer-type Kelvin–Voigt viscoelastic model. The results show that an increase in the order of the fractional derivative causes a significant decrease in the maximum dynamic amplification factor, especially in the “dynamic zone” of the normalized sweep time. The dynamic behavior of the plate is verified with ansys.

Realistic Shakemap M6.5 Pidie Jaya Earthquake 7 December 2016 Based on Modal Summation Technique

The shakemap were usually generated based on strong ground motion calculations on attenuation equations of the Ground Motion Prediction Equation (GMPE). However, the GMPE is a general equation which does not take into account the structural model of each layer of the subsurface. It considers only the structure of the surface layer known as Vs30 calculated as amplification in shakemap calculation. In this paper, the realistic calculation of shakemap for Pidie Jaya earthquake of 7 December 2016 with magnitude M6.5 is performed. Shakemap is generated from a number of synthetics seismogram calculations from source to site-based capital calculation summation Technique. Neo Deterministic Seismic Hazard Analysis (NDSHA) has included in the calculation of the synthetics seismograph. Before engaging the Vs30 effect, the realistic shakemap capable of producing effects of the angular distribution of a double-couple model in a realistic calculation. This could give an advantage of realistic shakemap while the GMPE based shakemap had not taken into account the double couple model. The realistic calculation of modal summation technique for shakemap can be further developed by involving local structural models in a more comprehensive manner.

Response spectra analysis of the modal summation technique verified by observed seismometer and accelerometer waveform data of the M6.5 Pidie Jaya Earthquake

Wave vibration confined in the boundary will produce stationary wave solution in discrete states called modes. There are many physics applications related to modal solutions such as air column resonance, string vibration, and emission spectrum of the atomic Hydrogen. Naturally, energy is distributed in several modes so that the complete calculation is obtained from the sum of the whole modes called modal summation. The modal summation technique was applied to simulate the surface wave propagation above crustal structure of the earth. The method is computational because it uses 1D structural model which is not necessary to calculate the overall wave propagation. The simulation results of the magnitude 6.5 Pidie Jaya earthquake show the response spectral of the Summation Technique has a good correlation to the observed seismometer and accelerometer waveform data, especially at the KCSI (Kotacane) station. On the other hand, at the LASI (Langsa) station shows the modal simulation result of response is relatively lower than observation. The lower value of the reaction spectral estimation is obtained because the station is located in the thick sedimentary basin causing the amplification effect. This is the limitation of modal summation technique, and therefore it should be combined with different finite simulation on the 2D local structural model of the basin.

Verification Modal Summation Technique for Synthetic and Observation Seismogram for Pidie Jaya Earthquake M6.5

Neo-Deterministic Seismic Hazard Assessment (NDSHA) method is a seismic hazard assessment method that has an advantage on realistic physical simulation of the source, propagation, and geological-geophysical structure. This simulation is capable on generating the synthetics seismograms at the sites that being observed. At the regional NDSHA scale, calculation of the strong ground motion is based on 1D modal summation technique because it is more efficient in computation. In this article, we verify the result of synthetic seismogram calculations with the result of field observations when Pidie Jaya earthquake on 7 December 2016 occurred with the moment magnitude of M6.5. Those data were recorded by broadband seismometers installed by BMKG (Indonesian Agency for Meteorology, Climatology and Geophysics). The result of the synthetic seismogram calculations verifies that some stations well show the suitability with observation while some other stations show the discrepancies with observation results. Based on the results of the observation of some stations, evidently 1D modal summation technique method has been well verified for thin sediment region (near the pre-tertiary basement), but less suitable for thick sediment region. The reason is that the 1D modal summation technique excludes the amplification effect of seismic wave occurring within thick sediment region. So, another approach is needed, e.g., 2D finite difference hybrid method, which is a part of local scale NDSHA method.

A Dynamic Model for Double-Planet Planetary Gearsets

In this study, a two-dimensional (2D), steady-state, discrete dynamic model of a double-planet planetary gearset is proposed. The dynamic model is generalized such that it can consist of N number of planet branches and can operate under any operating conditions (load and speed). The contact between each external to external and external to internal gear pair is modeled to obtain stiffnesses and mesh displacement excitations using a generalized load distribution model. The natural modes are computed by solving the corresponding eigenvalue problem. The forced vibration response to gear mesh excitations is obtained by applying the modal summation technique. The model is capable of predicting gear mesh dynamic load and dynamic transmission error spectra for each gear mesh, dynamic bearing load spectra for each bearing as well as gear body dynamic displacements. Forced vibration response of an example system that consists of three double-planet branches is studied to demonstrate the influence of some of the key design parameters.

Simulation of selected strong motion records of the 2003 MW = 6.6 Bam earthquake (SE Iran), the modal summation-ray tracing methods in the WKBJ approximation

A new analytical methodology for computing synthetic seismograms in 3-D anelastic media is applied to the modelling of the local records of the 2003 Bam Mw 6.6 earthquake, Southeastern Iran. The method is based on the combination of the modal summation technique with the asymptotic ray theory. The 3-D models are determined by a set of vertically heterogeneous sections (1-D structures) that are juxtaposed on a regular grid. The distribution of these sections in the grid is done in such a way to satisfy the condition of applicability of the WKBJ approximation, that is, the lateral variation of all the elastic parameters has to be small with respect to the prevailing wavelength. The procedure, described very briefly, has been applied for the area of the destructive 2003 December 26 Bam earthquake Mw 6.6, with a maximum cut-off frequency of 6 Hz. Synthetic seismograms show fine agreement with recorded signals when comparing shapes, amplitudes and spectra and evidence the relevance of 3-D effects.

Dynamic analysis of three-dimensional helical geared rotor system with geometric eccentricity

A dynamic model of a multi-shaft helical geared rotor system is presented. Rotating shafts of the system are modeled as Timoshenko beams. A general three-dimensional dynamic model of helical gear pairs with geometric eccentricity is developed for the gear mesh and bearing flexibility is included in the model as well. The transmission error and gear geometric eccentricity are simulated as excitations. Eigenvalue solution and the modal summation technique are used to predict the natural frequencies and forced responses of the system. Then two geared rotor system models are presented for validation of the gear dynamic model. It is demonstrated that the gear mesh model is effective for general geared rotor systems, spur and helical gears, one-stage and multi-stage systems. Finally, forced responses of an example system are analyzed to demonstrate the influences of the helical gear geometric eccentricity and the coupling between gear geometric eccentricity and rotor mass unbalance.

A dynamic model of a double-helical planetary gear set

A linear, time-invariant model of a double-helical planetary gear set is proposed in this paper. The model formulation allows the analysis of a gear set with any number of planets, any planet phasing and spacing configurations and any support conditions. Torsional, transverse, axial and rotational (rocking) motions of gears and the planet carrier are included in this three-dimensional model. Planets are allowed to be positioned equally or unequally spaced around the sun gear. The natural modes are computed by solving the corresponding Eigen value problem. The modal summation technique is used to find the forced response due to periodic gear mesh excitations under constant or proportional modal damping conditions. An example gear set consisting of four equally-spaced planets is considered at the end to demonstrate the influence of key design parameters including right-to-left stagger angle, and support conditions on the dynamic response of the double-helical planetary gear system. Numerous modes with dominant tilting motions are predicted to underline the essence of using a 3D formulation for double helical gears. Staggering of gear teeth is shown to impact the dynamic response of the gear system substantially. Different values of stagger are shown excite different types of modes, resulting in different dynamic response curves. It is also found that left and right sides of double helical gears can carry equal or different dynamic load amplitudes based on the right-to-left stagger values. Likewise the influence of the sun support conditions is shown to be more pronounced for the case of a non-zero stagger.

Neo-deterministic seismic hazard assessment in North Africa

North Africa is one of the most earthquake-prone areas of the Mediterranean. Many devastating earthquakes, some of them tsunami-triggering, inflicted heavy loss of life and considerable economic damage to the region. In order to mitigate the destructive impact of the earthquakes, the regional seismic hazard in North Africa is assessed using the neo-deterministic, multi-scenario methodology (NDSHA) based on the computation of synthetic seismograms, using the modal summation technique, at a regular grid of 0.2 × 0.2°. This is the first study aimed at producing NDSHA maps of North Africa including five countries: Morocco, Algeria, Tunisia, Libya, and Egypt. The key input data for the NDSHA algorithm are earthquake sources, seismotectonic zonation, and structural models. In the preparation of the input data, it has been really important to go beyond the national borders and to adopt a coherent strategy all over the area. Thanks to the collaborative efforts of the teams involved, it has been possible to properly merge the earthquake catalogues available for each country to define with homogeneous criteria the seismogenic zones, the characteristic focal mechanism associated with each of them, and the structural models used to model wave propagation from the sources to the sites. As a result, reliable seismic hazard maps are produced in terms of maximum displacement (D max), maximum velocity (V max), and design ground acceleration.

The SISMA prototype system: integrating Geophysical Modeling and Earth Observation for time-dependent seismic hazard assessment

An innovative approach to seismic hazard assessment is illustrated that, based on the available knowledge of the physical properties of the Earth structure and of seismic sources, on geodetic observations, as well as on the geophysical forward modeling, allows for a time-dependent definition of the seismic input. According to the proposed approach, a fully formalized system integrating Earth Observation data and new advanced methods in seismological and geophysical data analysis is currently under development in the framework of the Pilot Project SISMA, funded by the Italian Space Agency. The synergic use of geodetic Earth Observation data (EO) and Geophysical Forward Modeling deformation maps at the national scale complements the space- and time-dependent information provided by real-time monitoring of seismic flow (performed by means of the earthquake prediction algorithms CN and M8S) and permits the identification and routine updating of alerted areas. At the local spatial scale (tens of km) of the seismogenic nodes identified by pattern-recognition analysis, both GNSS (Global Navigation Satellite System) and SAR (Synthetic Aperture Radar) techniques, coupled with expressly developed models for interseismic phase, allow us to retrieve the deformation style and stress evolution within the seismogenic areas. The displacement fields obtained from EO data provide the input for the geophysical modeling, which eventually permits to indicate whether a specific fault is in a “critical state.” The scenarios of expected ground motion (shakemaps) associated with the alerted areas are then defined by means of full waveforms modeling, based on the possibility to compute synthetic seismograms by the modal summation technique (neo-deterministic hazard assessment). In this way, a set of deterministic scenarios of ground motion, which refer to the time interval when a strong event is likely to occur within the alerted area, can be defined both at national and at local scale. The considered integrated approach opens new routes in understanding the dynamics of fault zones as well as in modeling the expected ground motion. The SISMA system, in fact, provides tools for establishing warning criteria based on deterministic and rigorous forward geophysical models and hence allows for a well-controlled real-time prospective testing and validation of the proposed methodology over the Italian territory. The proposed approach complements the traditional probabilistic approach for seismic hazard estimates, since it supplies routinely updated information useful in assigning priorities for timely mitigation actions and hence it is particularly relevant to Civil Defense purposes.

Dynamic Modelling of Planetary Gears of Automatic Transmissions

In this paper, a generalized dynamic model for multi-stage planetary gear trains of automotive automatic transmissions is proposed. The planetary gear train is formed by N number of planetary gear sets of different types (single-planet, double-planet, or complex-compound), connected to each other in any given kinematic configuration. In addition, each planetary stage can have any number of parallel planet branches. A generalized power flow formulation and a gear mesh load distribution model are used to determine the stiffnesses, displacement excitations, and fundamental frequencies at the gear mesh interfaces. The natural modes are computed by solving the corresponding eigenvalue problem. The forced vibration response to gear mesh excitations is obtained by applying the modal summation technique. At the end, the model is applied to a three-stage planetary gear train representative of an automotive automatic transmission application to demonstrate the influence of coupling stiffnesses and the kinematic configurations on the natural modes and the dynamic response.

An approach of microzonation of the Sofia city

The city of Sofia has the densest population of Bulgaria — around 1.27 million people. The capital is exposed to high seismic risk. According to the Bulgarian seismic code, Sofia has been included in a seismic category characterized by intensity IX (MSK), which corresponds to a horizontal acceleration of 0.27 g for the anchoring of the elastic response spectrum. Because of the lack of instrumental seismic data, realistic synthetic strong motion waveforms for two opposite positions of seismic sources, have been generated for an expected earthquake scenarios (M = 7) along three geological profiles. A hybrid modeling method has been used for the modeling, which is based on the modal summation technique and finite difference scheme. The calculation has been done using an extended source model. The site effect is represented in terms of response spectra ratio (RSR), with respect to a bedrock model. The three components synthetic seismograms, computed in the domain of displacement, velocity and acceleration have been processed to extract some parameters very useful for engineering applications.

An earthquake scenario for the microzonation of Sofia and the vulnerability of structures designed by use of the Eurocodes

The study of the site effects and the microzonation of a part of the metropolitan Sofia, based on the modelling of seismic ground motion along three cross-sections are performed. Realistic synthetic strong motion waveforms are computed for scenario earthquakes ( M=7) applying a hybrid modelling method, based on the modal summation technique and finite differences scheme. The synthesized ground motion time histories are source and site specific. The site amplification is determined in terms of response spectra ratio (RSR). A suite of time histories and quantities of earthquake engineering interest are provided. The results of this study constitute a “database” that describes the ground shaking of the urban area. A case study of experiment-based assessment of vulnerability of a cast-in-situ single storey, industrial, reinforced concrete frame, designed according to Eurocodes 2 and 8 is presented. The main characteristics of damage index and storey drift are discussed for the purposes of microzonation.

Dynamic Analysis of a Multi-Shaft Helical Gear Transmission by Finite Elements: Model and Experiment

A dynamic model of a multi-shaft helical gear reduction unit formed by N flexible shafts is proposed in this study. The model consists of a finite element model of shaft structures combined with a three-dimensional discrete model of helical gear pairs. Bearing and housing flexibilities are included in the model as well. Eigenvalue solution and the Modal Summation Technique are used to predict the free and forced vibrations of the system. Results of experimental study on a helical gear-shaft-bearing system are also presented for validation of the model. It is demonstrated that the predictions match well with the experimental data in terms of excited modes and the forced response given in the form of the dynamic transmission error. Forced vibrations of an example system formed by three shafts are also studied to demonstrate the influence of some of the key system parameters.

Deterministic modelling for microzonation of Sofia - An expected earthquake scenario

The capital of Bulgaria, Sofia is a growing city with population around 1.22 million. The city is exposed to a high seismic risk since it is placed in the centre of Sofia seismic area. Over the centuries in the town of Sofia the macroseismic intensities have been larger than IX (MSK). A study of the site effects and the microzonation of a part of metropolitan Sofia, based on a modelling of seismic ground motion along three cross sections are presented. Realistic synthetic strong motion waveforms have been computed for an expected scenario earthquake (M = 7) applying a hybrid modelling method, based on the modal summation technique and finite differences scheme. The site amplification is determined in terms of response spectra ratio (RSR). A set of time histories and quantities of earthquake engineering interest are supplied, that allow the definition of six zones characterized by specific response spectra.

Deterministic Approach for the Seismic Microzonation of Bucharest

The mapping of the seismic ground motion in Bucharest, due to the strong Vrancea earthquakes is carried out using a complex hybrid waveform modeling method which combines the modal summation technique, valid for laterally homogeneous anelastic media, with finite-differences technique, and optimizes the advantages of both methods. For recent earthquakes, it is possible to validate the modeling by comparing the synthetic seismograms with the records. We consider for our computations the frequency range from 0.05 to 1.0 Hz and control the synthetic signals against the accelerograms of the Magurele station, low-pass filtered with a cut-off frequency of 1.0 Hz of the 3 last major strong (M w > 6) Vrancea earthquakes. Using the hybrid method with a double-couple seismic source approximation, scaled for the source dimensions and relatively simple regional (bedrock) and local structure models, we succeeded in reproducing the recorded ground motion in Bucharest at a satisfactory level for seismic engineering. Extending the modeling to the entire territory of the Bucharest area, we construct a new seismic microzonation map, where five different zones are identified by their characteristic response spectra.

Deterministic Earthquake Scenarios for the City of Sofia

— The city of Sofia is exposed to a high seismic risk. Macroseismic intensities in the range of VIII – X (MSK) can be expected in the city. The earthquakes that can influence the hazard in Sofia originate either beneath the city or are caused by seismic sources located within a radius of 40 km. The city of Sofia is also prone to the remote Vrancea seismic zone in Romania, and particularly vulnerable are the long-period elements of the built environment. The high seismic risk and the lack of instrumental recordings of the regional seismicity make the use of appropriate credible earthquake scenarios and ground-motion modelling approaches for defining the seismic input for the city of Sofia necessary. Complete synthetic seismic signals, due to several earthquake scenarios, were computed along chosen geological profiles crossing the city, applying a hybrid technique, which combines the modal summation technique and finite differences. The modelling takes into account simultaneously the geotechnical properties of the site, the position and geometry of the seismic source and the mechanical properties of the propagation medium. Acceleration, velocity and displacement time histories and related quantities of earthquake engineering interest (e.g., response spectra, ground-motion amplification along the profiles) have been supplied. The approach applied in this study allows us to obtain the definition of the seismic input at low cost, exploiting large quantities of existing data (e.g. geotechnical, geological, seismological). It may be efficiently used to estimate the ground motion for the purposes of microzonation, urban planning, retrofitting or insurance of the built environment, etc.

On the Incorporation of the Effect of Crustal Structure into Empirical Strong Ground Motion Estimation

This article has two purposes. Firstly, a validation exercise of the modal summation technique for the computation of synthetic strong-motion records is performed for two regions of Europe (Umbria-Marche and south Iceland), using a variety of region specific crustal structure models, by comparing the predicted ground motion amplitudes with observed motions. It is found that the rate of decay of ground motions is well predicted by the theoretical decay curves but that the absolute size of the ground motions is underpredicted by the synthetic time-histories. This is thought to be due to the presence of low-velocity surface layers that amplify the ground motions but are not included in the crustal structure models used to compute the synthetic time-histories. Secondly, a new distance metric based on the computed theoretical decay curves is introduced which should have the ability to model the complex decay of strong ground motions. The ability of this new distance metric to reduce the associated scatter in empirically derived equations for the estimation of strong ground motions is tested. It is found that it does not lead to a reduction in the scatter but this is thought to be due to the use of crustal structure models that are not accurate or detailed enough for the regions studied.

A deterministic seismic hazard map of India and adjacent areas

A seismic hazard map of the territory of India and adjacent areas has been prepared using a deterministic approach based on the computation of synthetic seismograms complete with all main phases. The input data set consists of structural models, seismogenic zones, focal mechanisms and earthquake catalogues. There are few probabilistic hazard maps available for the Indian subcontinent, however, this is the first study aimed at producing a deterministic seismic hazard map for the Indian region using realistic strong ground motion modelling with the knowledge of the physical process of earthquake generation, the level of seismicity and wave propagation in anelastic media. Synthetic seismograms at a frequency of 1 Hz have been generated at a regular grid of 0.2° x 0.2° by the modal summation technique. The seismic hazard, expressed in terms of maximum displacement (D m a x ), maximum velocity (V m a x ), and design ground acceleration (DGA), has been extracted from the synthetic signals and mapped on a regular grid over the studied territory. The estimated values of the peak ground acceleration are compared with the observed data available for the Himalayan region and are found to be in agreement. Many parts of the Himalayan region have DGA values exceeding 0.6 g. The epicentral areas of the great Assam earthquakes of 1897 and 1950 in northeast India represent the maximum hazard with DGA values reaching 1.2-1.3 g. The peak velocity and displacement in the same region is estimated as 120-177 cm s - 1 and 60-90 cm, respectively.