Effective Modal Mass Research Articles

Multi-Degrees-of-Freedom Energy Analysis for Identification of Failure Risk in Structural Components Subjected to Random Vibration and Shock Loading

AbstractWhen designing or analyzing a mechanical system, energy quantities provide insight into the severity of shock and vibration environments; however, the energy methods in the literature do not address localized behavior because energy quantities are usually computed for an entire structure. The main objective of this paper is to show how to compute the energy in the components of a mechanical system. The motivation for this work is that most systems fail functionally due to component failure, not because the primary structure was overloaded, and the ability to easily compute the spatial distribution of energy helps identify failure-sensitive components. The quantity of interest is input energy. That input energy can be decoupled modally is well known. What is less appreciated is that input energy can be computed at the component level exactly, using the component effective modal mass. We show that the steady-state input energy can be decomposed both spatially and modally and computed using input power spectra. A numerical example illustrates the spatial and modal decomposition of input energy and its utility in identifying components at risk of damage in random vibration and shock environments. We show that the modal properties of the structure and the spectral content of the input must be considered together to assess damage risk. Because input energy includes absorbed energy as well as relative kinetic energy and dissipated energy, it is the recommended energy quantity for assessing the severity of both random vibration and shock environments on a structure.

Dynamic analysis of aboveground open-top steel tanks subjected to wind loading

Aboveground steel tanks are prone to buckle under wind loading if they are empty. Geometrically nonlinear explicit dynamic analysis of empty open-top tanks subjected to wind loading are conducted using finite element analysis (FEA). Tanks with five different height to diameter aspect ratios, 4.0, 2.0, 1.0, 0.4, and 0.2, are adopted. Tanks are subjected to a gust of wind which fluctuates in magnitude over time to investigate whether a minor fluctuation, 10% of the average pressure, will trigger resonant behavior. Tanks with geometrical imperfections are also investigated using both dynamic analysis and static geometrically nonlinear analysis including imperfection (GNIA). It is found that, no matter if the tank does or does not have geometrical imperfections, resonance is not observed. Thus, a static analysis may be adequate for a practical tank design. The natural vibration frequencies of empty open-top tanks are studied as well, and comparisons with theoretical solutions and FEA results from the literature are presented. The effective modal mass is discussed to investigate the dynamic characteristics of the tanks examined. Insights for conducting an explicit dynamic analysis, including converged mesh size and time step limits, are also presented.

A Lumped Mass Model for Circular Micro-Resonators in Coriolis Vibratory Gyroscopes.

Coriolis vibratory gyroscopes (CVGs) with circular micro-resonators, such as hemispherical, ring, and disk resonators, exhibit excellent performances and have extraordinary potential. This paper discusses a generalized lumped mass model for both 3D and planar circular micro-resonators, establishing the relationship between the modal effective mass, the modal equivalent force, and the point displacement of the resonator. The point displacement description of a continuous circular resonator’s motion is defined from the view of capacitance measurement. The modal effective mass is, consequently, determined by the kinetic and the potential energy of the structure and is computed with numerical simulations. Moreover, the modal equivalent force, which can be theoretically calculated for any configuration of discrete electrodes, is deduced by using the concept of force density and the force distribution function. By utilizing the lumped mass model in this paper, the stiffness softening, the mode tuning, and the quadrature correction of the micro-resonators are investigated in detail. The theoretical model is verified by both the finite element method (FEM) and the experiments.

Identification of avian flapping motion from non-volant winged dinosaurs based on modal effective mass analysis.

The origin of avian flight is one of the most controversial debates in Paleontology. This paper investigates the wing performance of Caudipteryx, the most basal non-volant dinosaur with pennaceous feathered forelimbs by using modal effective mass theory. From a mechanical standpoint, the forced vibrations excited by hindlimb locomotion stimulate the movement of wings, creating a flapping-like motion in response. This shows that the origin of the avian flight stroke should lie in a completely natural process of active locomotion on the ground. In this regard, flapping in the history of evolution of avian flight should have already occurred when the dinosaurs were equipped with pennaceous remiges and rectrices. The forced vibrations provided the initial training for flapping the feathered wings of theropods similar to Caudipteryx.

Study on dynamic characteristics analysis of CNC pipe thread lathe based on the energy of modal effective mass

Owing to the modal analysis of complex structure system exist in the problems such as the local mode cannot be identified and the overall modal vibration mode cannot be extracted, this study deduces a formula of modal effective parameters based on the kinetic energy and introduces to the simulation of mechanical characteristics of the SCK230 CNC pipe thread repair the lathe bed. The simulation shows that this method can identify the local mode of three directions effectively, and determines whether the numbers of solved modalities satisfy the requirement of precision. In order to improve the natural frequencies of the bed and analyse the influence of the opening of the bed on its dynamic performance, three optimisation schemes are proposed based on this, and verified the feasibility of the optimisation scheme. Through structural optimisation, improve the dynamic characteristic of the lathe bed and reduce the manufacturing cost, and the optimisation of the structure can be achieved without experimental modal analysis.

Optimum earthquake-tuned TMDs: Seismic performance and new design concept of balance of split effective modal masses

The present paper, concerning the performance of optimum TMDs in seismic engineering, is composed of two parts. First, Part I deals with a comprehensive investigation on the effectiveness of an optimum seismic-tuned TMD towards reducing the earthquake response of linear frame structures, which is set up by a dedicated optimisation algorithm, allowing to achieve the best control device for a specific structural system and earthquake event. First, a representative set of 16 primary structures and 18 selected earthquakes has been considered, for a total of 288 cases. Second, additional trials based on a benchmark 10-storey frame and on a codified set of far-field earthquakes (FEMA P695 database), composed of 2×22=44 records, have been developed, for a further assessment. The presented outcomes provide extensive information about the TMD performance in the seismic engineering scenario, for a-priori known seismic input. Then, Part II reports a comprehensive post-processing and interpretation analysis of the results above. Specifically, connections between seismic TMD performance and modal structural properties are envisaged. Cross-comparisons, presented here through typical sample cases, analyse the changes of the structural characteristics after TMD insertion and monitor the associated effects on the seismic response reduction. The crucial role of achieving a balanced split of effective modal masses on the tuned mode is investigated, and highlighted as an essential ingredient to rely on a high TMD effectiveness in the seismic context. This criterion may be implemented within the tuning process, as a basic true structure-based concept for TMD design in Earthquake Engineering applications, thus for a-priori unknown seismic input.

Structural optimization with an automatic mode identification method for tracking global vibration mode

ABSTRACTThis article presents a mode identification method for structural optimization with global mode constraints to overcome the mode switching problem. In engineering design, the natural frequencies of global vibrations for a complex structure, the orders of which would not be constant in optimization loops, are usually very difficult to constrain. In this case, an incorrect constraint may lead to an unreliable design. A mode identification technique based on modal effective mass fraction is implemented to track the global modes such that the constraints will be updated subsequently and the optimizer can run correctly. A study case with comparison to traditional modal assurance criterion approaches demonstrates the advantages of this technique. An optimization framework has been developed with the new proposed mathematical model. Two numerical optimization examples, of a space truss and a simplified satellite structure, are presented to demonstrate the feasibility and applicability of this process.

A mixed model reduction method for preserving selected physical information

A new model reduction method in the frequency domain is presented. By mixedly using the model reduction techniques from both the time domain and the frequency domain, the dynamic model is condensed to selected physical coordinates, and the contribution of slave degrees of freedom is taken as a modification to the model in the form of effective modal mass of virtually constrained modes. The reduced model can preserve the physical information related to the selected physical coordinates such as physical parameters and physical space positions of corresponding structure components. For the cases of non-classical damping, the method is extended to the model reduction in the state space but still only contains the selected physical coordinates. Numerical results are presented to validate the method and show the effectiveness of the model reduction.

FE Model of the Fatih Sultan Mehmet Suspension Bridge Using Thin Shell Finite Elements

This paper presents the results of an eigenvalue analysis of the Fatih Sultan Mehmet Bridge. A high-resolution finite element model was created directly from the available design documents. All physical properties of the structural components were included in detail, so no calibration to the measured data was necessary. The deck and towers were modeled with shell elements. A nonlinear static analysis was performed before the eigenvalue calculation. The calculated natural frequencies and corresponding mode shapes showed good agreement with the available measured ambient vibration data. The calculation of the effective modal mass showed that nine modes had single contributions higher than 5 % of the total mass. They were in a frequency range up to 1.2 Hz. The comparison of the results for the torsional modes especially demonstrated the advantage of using thin shell finite elements over the beam modeling approach.

Optimal Spectral Acceleration-based Intensity Measure for Seismic Collapse Assessment of P-Delta Vulnerable Frame Structures

This study proposes an “optimal” spectral acceleration-based intensity measure (IM) to assess the collapse capacity of highly inelastic frame structures vulnerable to the P-delta effect. The IM is derived from the geometric mean of the spectral pseudo-acceleration over a certain period interval. The lower bound period of the averaging interval is related to the mode in which 95% of the effective modal mass is exceeded. The upper bound period is 1.6 times the fundamental period. This IM provides minimum, or close to the minimum, dispersion for frames with different fundamental periods of vibration, or number of stories.

An approach for minimum rotational response of medium-rise asymmetric structures under seismic excitations

Studied in this article is the elastic and inelastic rotational behavior of asymmetric multi-story buildings under unidirectional and bidirectional excitations in relation to the location of the modal center of rigidity. The concept of this point in mono-symmetric uniform buildings has been outlined by the author in earlier papers, and its significance lies on the property that when its location is within a close distance from the mass axis the torsional response is mitigated. At present, a generalized definition of this point is given for buildings with double eccentricity, using (1) the approach of the continuum medium and (2) the stiffness matrix methodology. To determine the location of modal center of rigidity in practical terms, a few modal quantities are required (element frequencies and effective modal masses), which may be easily obtained by commercial engineering software. To verify the aforesaid property of modal center of rigidity under unidirectional and bidirectional excitations, the rotational response of 10-story wall-frame structures is examined under the ground motions of Erzincan 1992 and Kobe 1995.

Optimal sensor placement for long-span cable-stayed bridge using a novel particle swarm optimization algorithm

In health monitoring of long-span structures, proper arrangement of sensors is a key point because of the need to acquire effective structural health information with limited testing resources. This study proposes a novel approach called dual-structure coding and mutation particle swarm optimization (DSC-MPSO) algorithm for the sensor placement. The cumulative effective modal mass participation factor is firstly derived to select the main contributions modes. A novel method combining dual-structure coding with the mutation operator is then utilized to determine the optimal sensors configurations. Finally, the feasibility of the DSC-MPSO algorithm is verified by optimizing the sensors locations for a long-span cable-stayed bridge. The effective independence method, genetic algorithm and standard particle swarm optimization algorithm are taken as contrast experiments. The simulation results show that the proposed algorithm in this paper could improve the convergence speed and precision. Accordingly, the method is effective in solving optimal sensor placement problems.

Force limited vibration testing: an evaluation of the computation of C<sup>2</sup> for real load and probabilistic source

To prevent over-testing of the test-item during random vibration testing Scharton proposed and discussed the force limited random vibration testing (FLVT) in a number of publications. Besides the random vibration specification, the total mass and the turn-over frequency of the load (test item), <TEX>$C^2$</TEX> is a very important parameter for FLVT. A number of computational methods to estimate <TEX>$C^2$</TEX> are described in the literature, i.e., the simple and the complex two degrees of freedom system, STDFS and CTDFS, respectively. The motivation of this work is to evaluate the method for the computation of a realistic value of <TEX>$C^2$</TEX> to perform a representative random vibration test based on force limitation, when the adjacent structure (source) description is more or less unknown. Marchand discussed the formal description of getting <TEX>$C^2$</TEX>, using the maximum PSD of the acceleration and maximum PSD of the force, both at the interface between load and source. Stevens presented the coupled systems modal approach (CSMA), where simplified asparagus patch models (parallel-oscillator representation) of load and source are connected, consisting of modal effective masses and the spring stiffness's associated with the natural frequencies. When the random acceleration vibration specification is given the CSMA method is suitable to compute the value of the parameter <TEX>$C^2$</TEX>. When no mathematical model of the source can be made available, estimations of the value <TEX>$C^2$</TEX> can be find in literature. In this paper a probabilistic mathematical representation of the unknown source is proposed, such that the asparagus patch model of the source can be approximated. The chosen probabilistic design parameters have a uniform distribution. The computation of the value <TEX>$C^2$</TEX> can be done in conjunction with the CSMA method, knowing the apparent mass of the load and the random acceleration specification at the interface between load and source, respectively. Data of two cases available from literature have been analyzed and discussed to get more knowledge about the applicability of the probabilistic method.

Force limited random vibration testing: the computation of the semi-empirical constant $$C^2$$ C 2 for a real test article and unknown supporting structure

To prevent over-testing of the test-item during random vibration testing Scharton proposed and discussed the force limited random vibration testing (FLVT) in a number of publications. Besides the random vibration specification, the total mass and the turn-over frequency of the test article (load), \(C^2\) is a very important parameter for FLVT. A number of computational methods to estimate \(C^2\) are described in the literature, i.e. the simple and the complex two degree of freedom system, STDFS and CTDFS, respectively. The motivation of this work is to evaluate the method for the computation of a realistic value of \(C^2\) to perform a representative random vibration test based on force limitation, when the description of the supporting structure (source) is more or less unknown. Marchand discussed the formal description of obtaining \(C^2\), using the maximum PSD of the acceleration and maximum PSD of the force, both at the interface between test article and supporting structure. Stevens presented the coupled systems modal approach (CSMA), where simplified asparagus patch models (parallel-oscillator representation) of load and source are connected. The asparagus patch model consists of modal effective masses and spring stiffnesses associated with the natural frequencies. When the random acceleration vibration specification is given the CSMA method is suitable to compute the value of the parameter \(C^2\). When no mathematical model of the source can be made available, estimations of the value \(C^2\) can be find in literature. In this paper a probabilistic mathematical representation of the unknown source is proposed, such that the asparagus patch model of the source can be approximated. The chosen probabilistic design parameters have a uniform distribution. The computation of the value \(C^2\) can be done in conjunction with the CSMA method, knowing the apparent mass of the load and the random acceleration specification at the interface between load and source, respectively. Data of two cases available from literature has been analyzed and discussed to get more knowledge about the applicability of the probabilistic method.

A single-run multi-mode pushover analysis to account for the effect of higher modes in estimating the seismic demands of tall buildings

In this paper, a simpler and more effective procedure, called the single-run multi-mode pushover (SMP) analysis, is proposed to account for the effect of higher modes in estimating the seismic demands of tall buildings. The main simplification of this procedure is that the effect of higher modes is concentrated into a single invariant lateral force distribution computed by algebraically adding the modal story forces whereby a single-run multi-mode pushover analysis is implemented. An important advantage of the proposed procedure is that the spectral pseudo-acceleration of ground motions (or design spectrum) as a weighting parameter as well as the effective modal mass ratio is incorporated into the modal story forces. Then, the effect of the frequency content of ground motions is considered in the modal lateral forces. Another advantage of the SMP procedure is that the reversal of sign in the story forces of higher modes is taken into account. In this procedure, the seismic demands are finally obtained by enveloping the results of single-run conventional and single-run multi-mode pushover analyses. To assess the accuracy of the proposed procedure, three special steel moment-resisting frames are considered. The seismic demands resulting from the SMP procedure are compared to those from the nonlinear response history analysis as a benchmark solution, as well as to those predicted from the modal pushover analysis and the consecutive modal pushover procedures. The results demonstrate that the SMP procedure can accurately estimate the seismic demands of tall buildings.

Prediction of equivalent static loads act on a micro satellite via modal analysis

Article history: Received October 6, 2014 Accepted 2 February 2015 Available online 2 February 2015 In this research, the influences of modal properties of a micro cubic satellite on equivalent static loads (due to combination of quasi-static and dynamic loads during launch time) have been studied. The study shows that the magnitude of equivalent static loads can be affected by satellite effective modes and effective mass distribution along natural frequencies. Besides, when to distinct launchers with different dynamic environments candidate for launch, analytical results illustrate that sometimes the equivalent static combined load value can be higher for the launcher with smaller quasi-static loads. This phenomenon is due to effective modal mass distribution of the satellite. Consequently, a higher combined load may be occurred during launch for smaller quasi-static conditions. Furthermore, in separation phase of satellite from the launcher, the satellite modal parameters influence the magnitude of applied shock load in both axial (launch) and lateral directions, importantly. Thus the present study yields reliable input values for separation shock test which considers dynamic properties of the satellite in comparison with some standards and launcher manuals, suggest a rough estimation of shock load, neglecting the satellite structure dynamic behavior. © 2015 Growing Science Ltd. All rights reserved.

Seismic Performance Analysis of Continuous Rigid Frame Bridge with High-Piers and Long-Span

The dynamic characteristics of Tukan Wujiang Bridge are analyzed numerically. The bridge, which is still under construction, is a continuous rigid frame bridge with high-piers and long-span in Wulong County, Chongqing, China. A spatial finite element model is established for the bridge with the finite element software ABAQUS. The natural vibration characteristics and effective modal participation mass of the bridge in the built-up case is obtained. The variation of dynamic characteristics of continuous rigid frame bridge subjected to the dynamic earthquake loading is studied, and the number of modes necessary to the vibration mode combination when applying the response spectrum method under the earthquake is obtained. In addition, the seismic response of the bridge is analyzed with both the response spectrum method and the time-history method, and the maximum response of structure under various probability of earthquake is obtained. The results of the two methods are compared. The comprehensive seismic performance of the bridge is analyzed and evaluated. The results show that the seismic performance of the rigid frame bridge satisfies the expected design performance.

Influence of structural system measures on the dynamic characteristics of a multi-span cable-stayed bridge

A three-dimensional finite element model for the Jiashao Bridge, the longest multi-span cable-stayed bridge in the world, is established using the commercial software package ANSYS. Dynamic characteristics of the bridge are analyzed and the effects of structural system measures including the rigid hinge, auxiliary piers and longitudinal constraints between the girders and side towers on the dynamic properties including modal frequency, mode shape and effective mass are studied by referring to the Jiashao Bridge. The analysis results reveal that: (i) the installation of the rigid hinge significantly reduces the modal frequency of the first symmetric lateral bending mode of bridge deck. Moreover, the rigid hinge significantly changes the mode shape and effective mass of the first symmetric torsional mode of bridge deck; (ii) the layout of the auxiliary piers in the side-spans has a limited effect on changing the modal frequencies, mode shapes and effective masses of global vibration modes; (iii) the employment of the longitudinal constraints significantly increases the modal frequencies of the vertical bending modes and lateral bending modes of bridge deck and have significant effects on changing the mode shapes of vertical bending modes and lateral bending modes of bridge deck. Moreover, the effective mass of the first anti-symmetric vertical bending of bridge deck in the longitudinal direction of the fully floating system is significantly larger than that of the partially constrained system and fully constrained system. The results obtained indicate that the structural system measures of the multi-span cable-stayed bridge have a great effect on the dynamic properties, which deserves special attention for seismic design and wind-resistant design of the multi-span cable-stayed bridge.

Optimal Sensor Placement for Latticed Shell Structure Based on an Improved Particle Swarm Optimization Algorithm

Optimal sensor placement is a key issue in the structural health monitoring of large-scale structures. However, some aspects in existing approaches require improvement, such as the empirical and unreliable selection of mode and sensor numbers and time-consuming computation. A novel improved particle swarm optimization (IPSO) algorithm is proposed to address these problems. The approach firstly employs the cumulative effective modal mass participation ratio to select mode number. Three strategies are then adopted to improve the PSO algorithm. Finally, the IPSO algorithm is utilized to determine the optimal sensors number and configurations. A case study of a latticed shell model is implemented to verify the feasibility of the proposed algorithm and four different PSO algorithms. The effective independence method is also taken as a contrast experiment. The comparison results show that the optimal placement schemes obtained by the PSO algorithms are valid, and the proposed IPSO algorithm has better enhancement in convergence speed and precision.

An adaptive pushover procedure based on effective modal mass combination rule

In order to overcome the major drawbacks of conventional pushover methods, researchers have recently been motivated to develop adaptive pushover procedures by which effect of higher modes as well as progressive damage accumulation are taken into account. In spite of their vigorous theory, these novel methods suffer from the quadratic modal combination rules, in which the sign reversals of load vectors in higher modes are neglected and consequently lead to a positive load pattern. In this paper, a displacement-based adaptive modal pushover method, called APAM, based on effective modal mass combination rule is developed in order to include the sign reversals in the load vectors. In this combination rule a modification factor associated to each mode of interest is determined and applied to the corresponding load vector. The modified modal load vectors are algebraically added and subtracted and result in a range of load pattern and thus, multiple pushover analysis is required. These load patterns are independently applied to the structure within an adaptive framework and the envelope of demand values is considered. These modification factors are updated proportional to the instantaneous dynamic characteristic of structure in each step. Another novel aspect of the proposed method is that the target displacement is estimated during the analysis by implementing the concept of capacity spectrum method recommended by ATC 40. In order to assess the accuracy of this method in predicting the seismic responses, the proposed methodology is applied to three different moment-frame buildings. The obtained results demonstrate that APAM procedure provides well estimation of important seismic demand parameters.