Modal Force Research Articles

A Novel Optimization Algorithm Based on Modal Force Information for Structural Damage Identification

This paper proposes a novel optimization algorithm called modal force information-based optimization (MFIBO) to identify the location and severity of damage in structures. The main idea behind the MFIBO is to take advantage of information captured from the modal force of structural elements to seek the optimum damage variables. The modal element force, defined as the internal element force caused by the action of mode shapes, allows the MFIBO to recognize promising directions in the search space and assists in accelerating the optimization process. Indeed, unlike meta-heuristic optimization algorithms, which disregard explicit information about the problem and rely only upon time-consuming stochastic search computations, the MFIBO employs an informed search strategy to perform optimization in a rational and directed manner. In order to assess the effectiveness and applicability of the proposed MFIBO algorithm, four benchmark damage identification examples of truss and frame structures are conducted under both noise-free and noisy conditions. In each example, the results of the MFIBO are also compared with those attained by two well-known meta-heuristic algorithms, namely the differential evolution and the teaching–learning-based optimization. The obtained results reveal that the MFIBO is able to accurately and reliably identify structural damage with a significantly lower computational burden compared to the meta-heuristic algorithms.

Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

Under certain conditions, the dynamics of a nonlinear mechanical system can be represented by a single nonlinear modal oscillator. This holds, in particular, under external excitation near primary resonance or under self-excitation by negative damping of the respective mode. The properties of the modal oscillator can be determined by computational or experimental nonlinear modal analysis. The simplification to a single-nonlinear-mode model facilitates qualitative and global analysis, and substantially reduces the computational effort required for probabilistic methods and design optimization. Important limitations of this theory are that only purely mechanical systems can be analyzed and that the respective nonlinear mode has to be recomputed when the system’s structural properties are varied. With the theoretical extension proposed in this work, it becomes feasible to attach linear subsystems to the primary mechanical system, and to approximate the dynamics of this coupled system using only the nonlinear mode of the primary mechanical system. The attachments must be described by linear ordinary or differential-algebraic equations with time-invariant coefficient matrices. The attachments do not need to be of purely mechanical nature, but may contain, for instance, electric, magnetic, acoustic, thermal or aerodynamic models. This considerably extends the range of utility of nonlinear modes to applications as diverse as model updating or vibration energy harvesting. As long as the attachments do not significantly deteriorate the host system’s modal deflection shape, it is shown that their effect can be reduced to a complex-valued modal impedance and an imposed modal forcing term. In the present work, the proposed approach is computationally assessed for the analysis of exciter-structure interaction. More specifically, the force drop typically encountered in frequency response testing is revisited. A cantilevered beam with cubic spring and an attached electro-dynamical shaker serves as benchmark. The proposed approach shows excellent accuracy. Mainly the already known limitations of single-nonlinear-mode theory reappear. In particular, higher harmonics should not be too pronounced. In the transient case, the time scales of vibration and amplitude-phase modulation should be well separated, and the attachment dynamics should be in quasi-steady state.

Cable tension identification based on near field radiated acoustic pressure signal

To obtain cable force quickly and accurately, and to develop a novel method for the construction of a large number of cable structures and models, this paper proposed a novel method-Acoustic Radiation Technology for measuring cable tension. This is the first study to measure cable force utilizing the radiated sound pressure signals captured at a fixed position near the structure. The experiment is carried out on a prestressed steel strand in the laboratory, and the effectiveness of ART for cable force identification is verified. At the same time, the application potential of this technology is discussed by combining intelligent mobile devices. The results show that the average error of the existing frequency data is within 0.2%, and the maximum average error of cable tension calculated by the first natural frequency is less than 2%. This method provides a simple, and effective way for obtaining structural modal and cable force.

Variable Modality in Pintupi-Luritja Purposive Clauses

This paper investigates the modal and non-modal uses and readings of the purposive suffix in the Western Desert (Pama-Nyungan) language Pintupi-Luritja. It is shown that the suffix is associated with a range of root-modal readings, with some variability in modal force. The modal readings are investigated in a variety of non-upward-entailing environments and compared with those of other variable modal force languages as described in the literature. I suggest that the purposive suffix does not behave in the same way as in these languages, which suggests that the typology of variable force modality is not uniform. I conclude by suggesting a connection to the modality described in non-finite and nominalised environments in a number of other languages.

Human-induced loading and dynamic response of footbridges in the vertical direction due to restricted pedestrian traffic

Despite extensive research in the field of human-induced vibration in the last twenty years, there is still lack of reliable load and response models to assess vibration serviceability of footbridges under human-induced excitation, especially with reference to crowded conditions. This paper aims to provide a reliable characterization of pedestrian-induced loading on footbridges in restricted traffic condition, and to compare the reliability of current guidelines and advanced spectral models in the serviceability assessment of footbridges with respect to vertical vibrations. Numerical simulations of restricted pedestrian traffic are carried out through an agent-based model, considering variable pedestrian densities and deck widths. The probability distribution of the step frequency and the power spectral density function of the modal force are obtained numerically. Based on the numerical results, a Modified Generalized Equivalent Spectral Model is proposed. Finally, two numerical examples are analyzed to assess the reliability of current guidelines and of the Modified Generalized Equivalent Spectral Model in the estimate of the maximum footbridge acceleration.

Flexible Structure Vibration Control Using Double-Gimbal Variable-Speed Control Moment Gyros

This paper studies the dynamics and control of a flexible space structure using double-gimbal variable-speed control moment gyros (DGVSCMGs). The DGVSCMGs are mounted on the flexible structure as actuators for attitude and vibration control. A flexible structure with a cluster of angular momentum devices is considered to be a gyroelastic body. The detailed dynamics of a single gyroelastic body under the influence of the actuators is derived using Kane’s method. A simple controller capable of vibration suppression is proposed for the attitude maneuver. A novel steering law is designed for the DGVSCMGs. Null motion of the gimbal angles and rotor rates is considered to compensate for the undesirable modal force in the vibration equation. Numerical simulations are performed to verify the effectiveness of the proposed controller and steering law.

Still going strong

In “Must ...stay ...strong!” (von Fintel and Gillies in Nat Lang Semant 18:351–383, 2010. https://doi.org/10.1007/s11050-010-9058-2), we set out to slay a dragon, or rather what we called The Mantra: that epistemic must has a modal force weaker than expected from standard modal logic, that it doesn’t entail its prejacent, and that the best explanation for the evidential feel of must is a pragmatic explanation. We argued that all three sub-mantras are wrong and offered an explanation according to which must is strong, entailing, and the felt indirectness is the product of an evidential presupposition carried by epistemic modals. Mantras being what they are, it is no surprise that each of the sub-mantras have been given new defenses. Here we offer them new problems and update our picture, concluding that must is (still) strong.

A conjugate modal force strategy for instability analysis of thin-walled structures: an unconstrained vector positional finite element approach

The importance of knowing critical loads and post-critical behavior of thin-walled structures motivates the development of several scientific and practical studies. Most references are concerned with stability analysis for small displacements (first order approach), or with second order stability analyzes, a less precise geometric nonlinear strategy. However, very flexible structures or ones that present small loss of stiffness after the first critical load need more careful analysis. Here we present a shell numerical formulation capable of carrying out stability analysis of thin-walled structures developing large displacements. This formulation uses generalized unconstrained vectors as nodal parameters instead of rotations. To make possible a complete stability analysis using unconstrained vectors, we present an original strategy that imposes a Conjugate Modal Force at the vicinity of critical points, allowing an accurate choice of post-critical paths. Non-conservative forces are also considered and results are compared with literature benchmarks, demonstrating the accuracy and capacity of the proposed formulation.

An efficient coupled-mode flutter analysis method for turbomachinery

An efficient aeroelastic eigenvalue method based on time domain system identification has been developed to analyze coupled-mode flutter in turbomachinery. The aeroelastic stability problem is transformed into a generalized eigenvalue problem (GEP) by incorporating the aeroelastic equation and an aerodynamic modal force model. The aerodynamic modal force model correlates the generalized aerodynamic modal force with generalized motion variables by influence coefficient matrices which are calculated through a time domain system identification method. The aerodynamic damping ratios and the corresponding eigenfrequencies for all interblade phase angles (IBPA) can be obtained conveniently from a single time domain aerodynamic computation. The proposed method has been validated and demonstrated by presenting results for NASA Rotor 67 flutter cases in this paper.

Optimal Vibration Control of Membrane Structures with In-Plane Polyvinylidene Fluoride Actuators

Different kinds of membrane structures have been proposed for future space exploration and earth observation. However, due to the low stiffness, high flexibility, and low damping properties, membrane structures are likely to generate large-amplitude (compared to the thickness) vibrations, which may lead to the degradation of their working performance. In this work, the governing equations are established at first, taking into account the modal control force induced by the polyvinylidene fluoride (PVDF) actuator. The optimal vibration control of the membrane structure is explored subsequently. A square PVDF actuator is attached on the membrane to achieve the vibration suppression. The influence of actuator position and control gains on the vibration control performance are studied. The optimal criteria for actuator placement and energy allocation are developed. Several case studies are numerically simulated to demonstrate the validity of the proposed optimization criteria. The analytical results in this study can serve as guidelines for optimal vibration control of membrane structures. Additionally, the proposed optimization criteria can be applied to active control of different flexible structures.

Dynamics and Isotropic Control of Parallel Mechanisms for Vibration Isolation

Parallel mechanisms have been employed as architectures of high-precision vibration isolation systems. However, their performances in all degrees of freedom (DOFs) are nonidentical. The conventional solution to this problem is isotropic mechanism design, which is laborious and can hardly be achieved. This article proposes a novel concept; namely, isotropic control, to solve this problem. Dynamic equations of parallel mechanisms with base excitation are established and analyzed. An isotropic control framework is then synthesized in modal space. We derive an explicit relationship between modal control force and actuation force in joint space, enabling implementation of the isotropic controller. The multi-DOF system is transformed into multiidentical single-DOF systems. Under the framework of isotropic control, parallel mechanisms obtain an identical frequency response for all modes. An identical corner frequency, active damping, and rate of low-frequency transmissibility are achieved for all modes using a combining proportional, integral, and double integral compensator as a subcontroller. A 6-U P S parallel mechanism is presented as an example to demonstrate effectiveness of the new approach.

A direct displacement-based seismic design method using a MDOF equivalent system: application to R/C framed structures

An improved version of the direct displacement-based design (DDBD) method for the seismic design of plane moment resisting frames in the framework of performance-based design approach is presented. The method employs a multi-degree-of-freedom equivalent system instead of the single-degree-of-freedom equivalent system used by the conventional DDBD method. Thus, the proposed method can take more rationally and with higher accuracy into account the higher mode and P-Δ effects than the conventional one. This is accomplished with the aid of the concept of deformation dependent equivalent modal damping ratios previously developed by the present authors for other purposes and the concept of the design modal displacements developed herein. These design modal displacements are determined on the basis of target inter-storey drift ratios for every performance level and the first few modes significantly contributing to the structural response. Thus, one can determine from the displacement spectrum with high amounts of viscous damping the required modal periods for known values of the design modal displacements. From those modal periods, the corresponding required modal stiffness and hence the modal base shear forces can be obtained. The final required design base shear can be obtained by a combination rule, like the SRSS rule. Numerical examples involving the seismic design of two moment resisting reinforced concrete plane frames are presented in detail for illustrating the proposed approach and demonstrating its merits over the conventional DDBD method and the force-based design method of Eurocode 8.

On metaphysically necessary laws from physics

How does metaphysical necessity relate to the modal force often associated with natural laws (natural necessity)? Fine (2002) argues that natural necessity can neither be obtained from metaphysical necessity via forms of restriction nor of relativization — and therefore pleads for modal pluralism concerning natural and metaphysical necessity. Wolff (Philosophy of Science, 80(5), 898–906, 2013) aims at providing illustrative examples in support of applying Fine’s view to the laws of nature with specific recourse to the laws of physics: On the one hand, Wolff takes it that equations of motion can count as examples of physical laws that are only naturally but not metaphysically necessary. On the other hand, Wolff argues that a certain conservation law obtainable via Noether’s second theorem is an instance of a metaphysically necessary physical law. I show how Wolff’s example for a putatively metaphysically necessary conservation law fails but argue that so-called topological currents can nevertheless count as metaphysically necessary conservation laws carrying physical content. I conclude with a remark on employing physics to answer questions in metaphysics.

Influence of infill walls on modal expansion of distribution of effective earthquake forces in RC frame structures

It is quite apparent that engineering concerns related to the influence of masonry infills on seismic behavior of reinforced concrete (RC) structures is likely to remain relevant in the long term, as infill walls maintain their functionalities in construction practice. Within this framework, the present paper mainly deals with the issue in terms of modal expansion of effective earthquake forces and the resultant modal responses. An adequate determination of spatial distribution of effective earthquake forces over the height of the building is highly essential for both seismic analysis and design. The possible influence of infill walls is investigated by means of modal analyses of two-, three-, and four-bay RC frames with a number of stories ranging from 3 to 8. Both uniformly and non-uniformly infilled frames are considered in numerical analyses, where infill walls are simulated by adopting the model of equivalent compression strut. Consequently, spatial distribution of effective earthquake forces, modal static base shear force response of frames, modal responses of story shears from external excitation vector and lateral floor displacements are obtained. It is found that, infill walls and their arrangement over the height of the frame structure affect the spatial distribution of modal inertia forces, as well as the considered response quantities. Moreover, the amount of influence varies in stories, but is not very dependent to bay number of frames.

Resonance Analysis of Cantilever Plates Subjected to Moving Forces by a Semi-Analytical Method

Cantilever plate structures are widely used in civil and aerospace engineering. Here, a semi-analytical method is proposed to calculate the dynamic responses of cantilever plates subjected to moving forces. The Rayleigh–Ritz method is used to obtain the semi-analytical modal frequencies and shapes of a thin, isotropic, and rectangular cantilever plate using the assumed mode shapes that fulfill the boundary conditions of the plate. The modal superposition method is used to decouple the motion equations of the cantilever plate to obtain a series of modal equations. Then, the generalized forces are transformed into a Fourier series in terms of discrete harmonic forces. The dynamic responses of the cantilever plate are obtained by superimposing the analytical responses of a number of single-degree-of-freedom modal systems under discrete harmonic forces. The proposed semi-analytical method is verified through comparison with the numerical method. Then, the vibration of the cantilever plate under the action of moving forces is investigated based on the semi-analytical results. It is found that the contribution of the high-order modes to the dynamic responses of the plate cannot be ignored. In addition, the wavelengths of the mode shapes not only affect the magnitude of the modal forces but also the dominant frequency of the modal forces. Resonant responses of the plate are produced by the moving forces when the load interval equals the wavelength of the mode shape of a high-order mode and the exciting frequency of the moving forces equals the natural frequency of this mode.

High-frequency disturbance force suppression mechanism of a flywheel equipped with a flexible dynamic vibration absorber

Flywheels generate speed-related disturbances and induce micro-vibrations that influence the performance of high-sensitivity instruments on board. This study addresses dynamic modeling and disturbance force suppression of the flywheels due to its inherent characteristic structural modes. The disturbance force transmission of a rotating flywheel due to a unit radial force applied at the rim of the wheel body is reported using the three-dimensional finite element method and frequency response function–based substructuring method. The characteristic structural modes for the radial and axial disturbance forces are identified. The axial deformation–dominated flapping mode and the radial deformation–dominated transverse mode will contribute most to the axial and radial disturbance forces, respectively. A flexible ring structure, which is rested on the arms of the wheel body through independent viscoelastic pads and simultaneously in contact with the inner rim of the wheel body by independent resilient cushion members, is proposed to function as a damped dynamic vibration absorber. The modes of the dynamic vibration absorber and the modes of the flywheel equipped with the dynamic vibration absorber are analyzed. It is shown that the dynamic vibration absorber is effective to suppress both the radial and axial disturbance forces at the characteristic structural modes under different rotational speeds, provided that the loss factor of the complex elastic modulus for the viscoelastic pads is larger than 0.2 and the proportional damping constant for the stiffness matrix is larger than 6 Ns/m. Experimental analyses are conducted on a flywheel with a well-designed dynamic vibration absorber to validate the theoretical findings. Modal tests and disturbance force measurements are carried out for the flywheel with/without the dynamic vibration absorber. It is shown that the identified characteristic structural modes will contribute remarkable peaks for the radial and axial disturbance forces. The proposed dynamic vibration absorber is capable to suppress the high-frequency disturbance forces efficiently under different rotational speeds.

Modal analysis of a spinning disk in a dense fluid as a model for high head hydraulic turbines

In high head Francis turbines and pump-turbines in particular, Rotor Stator Interaction (RSI) is an unavoidable source of excitation that needs to be predicted accurately. Precise knowledge of turbine dynamic characteristics, notably the variation of the rotor natural frequencies with rotation speed and added mass of the surrounding water, is essential to assess potential resonance and resulting amplification of vibrations. In these machines, the disk-like structures of the runner crown and band as well as the head cover and bottom ring give rise to the emergence of diametrical modes and a mode split phenomenon for which no efficient prediction method exists to date. Fully coupled Fluid–Structure Interaction (FSI) methods are too computationally expensive; hence, we seek a simplified modeling tool for the design and the expected-life prediction of these turbines.We present the development of both an analytical modal analysis based on the assumed mode approach and potential flow theory, and a modal force Computational Fluid Dynamics (CFD) approach for rotating disks in dense fluid. Both methods accurately predict the natural frequency split as well as the natural frequency drift within 7.9% of the values measured experimentally. The analytical model explains how mode split and drift are respectively caused by linear and quadratic dependence of the added mass with relative circumferential velocity between flexural waves and fluid rotation.

Nonlinear vibration control effects of membrane structures with in-plane PVDF actuators: A parametric study

In recent years, with the increasing use of membrane structures in space applications, the problems of large-amplitude (compared to the thickness) vibrations arise; leading to the degradation of their working performance. It is challenging to control the large-amplitude vibrations of membranes due to the high flexibility, low damping, and strong geometrical nonlinearity. In this study, polyvinylidene fluoride (PVDF) actuators are considered to develop the active vibration control of a pre-tensioned Kapton membrane. The governing equations are established based on the geometrically nonlinear theory of plates, taking into account the modal control force induced by the actuators. To analyze the nonlinear dynamic characteristics of the membrane, the theoretical and approximate solutions of nonlinear frequencies of the membrane are obtained, and the discrepancies between the two solutions under different vibration amplitudes are discussed. Control performance for both free vibration and harmonic forced vibration of the membrane is numerically studied. A comprehensive parametric study is carried out to investigate the influences of different system parameters, including the pretension and the size of the membrane, and the position and number of the actuators, on the control performance. Analytical results indicate that the vibration of the membrane can be effectively suppressed with the appropriately laminated PVDF actuators. The results of this research can be extended to the active control of different gossamer space structures.

Generalized algorithms for the identification of seismic ground excitations to building structures based on generalized Kalman filtering under unknown input

The exact information of seismic excitation and structural state is a prerequisite for structural seismic safety assessment and vibration control. When the seismic excitation to a structure is not measured, the seismic excitation can be identified as an inversed problem from measured structural responses. Although some relevant approaches have been developed, there are certain limitations or drawbacks in the existing approaches. To circumvent these problems, two generalized algorithms are proposed for the identification of seismic ground excitation to multi-story and tall buildings, respectively. When the seismic ground excitation to a structure is not measured, the data measured by a structural health monitoring system are structural absolute responses. So the structural motion equation in the absolute coordinate system is derived, in which the unknown seismic ground excitation is treated as unknown external force acting on the structure. First, the identification of unknown seismic excitations to multi-story building structures is studied. A generalized Kalman filtering under unknown input is proposed for the identification of structural state and unknown seismic excitation without the observation of structural absolute acceleration responses at the location of unknown external force. The derivation of the proposed generalized Kalman filtering under unknown input is based on the classical Kalman filter, but is more general than the existing identification approaches based on Kalman filter with unknown input in the deployments of accelerometers in the building structure. Then, it is extended to explore the identification of unknown seismic excitations to tall building structures. To avoid substructural identification from the top to bottom in a sequential manner, the motion equation in absolute coordinate system is reduced by modal expansion. Moreover, instead of the identification of unknown modal forces in previous approaches, the seismic excitation is directly identified without increasing the number of unknown forces. To demonstrate the proposed algorithms, numerical examples of identifying seismic excitations to a 6-story shear building and an 18-story tall building are investigated.

Study on Radiated Noise of a Panel under Fluctuating Surface Pressure Due to an Idealized Side Mirror

As traditional automobiles develop towards new energy vehicles, the noise, vibration and harshness (NVH) performance of automobiles is facing new challenges. Without the cover of the traditional engine noise and inlet and exhaust noise, the high-speed wind noise becomes more prominent. Thus, research on the calculation method of vehicle interior noise in high-speed driving condition is needed. However, vehicle body structure is complex, and the external excitation components are complicated. In order to analyze the method of predicting the vehicle interior noise at high speed, an idealized side mirror model is taken as the research object in this paper and the radiated noise of a panel under the fluctuating surface pressure (FSP) due to the idealized side mirror is studied. The FSP of the panel is first studied by the numerical simulations of incompressible and compressible flow field. For the incompressible flow field, the Corcos turbulent boundary layer (TBL) model is established to simulate the convective component and the boundary element method (BEM) is used to extract the acoustic component. Subsequently, the Corcos model coupling BEM method, the random modal force coupling BEM method and the deterministic modal force coupling BEM method are used separately to calculate the noise of the panel under the FSP. For the compressible flow field, the convective and acoustic component in the fluctuating pressure are separated by the wavenumber-frequency spectrum (WFS) method. The radiated noise of the panel under the FSP is calculated again by using the WFS, the method of random modal force and the method of deterministic modal force, respectively. Then, the computational time of the six methods of incompressible and compressible calculation is compared. Finally, a fast and accurate method of calculating the panel radiated noise under FSP is obtained by comparing the computational accuracy with the experimental results and combining the computational time: the method of incompressible random modal force. This method can be used to quickly and accurately analyze the vehicle interior noise at high speed, and to optimize the exterior protrusions and the vehicle sound package for improving the vehicle NVH performance at high speed.