Displacement Frequency Response Functions Research Articles

A novel lever-arm tuned mass damper inerter (LTMDI) for vibration control of long-span bridges

In this study, the lever mechanism is incorporated with the tuned mass damper inerter (TMDI) to further promote the practical application of TMDI in long-span bridges. Four potential configurations of the lever-arm tuned mass damper inerter (LTMDI) are first established, and the corresponding governing equations are derived. Then, the optimum configuration of LTMDI is identified by minimizing the peak of non-dimensional displacement frequency response function (FRF). Subsequently, the influences design parameters, such as mass ratio, inertance ratio, lever mass ratio, and spanning distance of the inerter are investigated. Based on the results, empirical design formulas of design parameters are proposed and verified. Finally, a cable-stayed bridge is adopted as a numerical case to further illustrate the performance of LTMDI in terms of response reduction and robustness. The results show that the involving of the lever into TMDI is an effective mechanism for implementing the inerter’s spanning in long-span bridges. With this mechanism, it is practically feasible to achieve a high effectiveness, low space requirement and more robustness control device for long-span bridges. The developed design formulas have a high accuracy, which can provide reliable support for the design of the LTMDI in engineering applications.

Design, modeling, and experiment of a multi-bifurcated cantilever piezoelectric energy harvester

A multi-bifurcated cantilever piezoelectric energy harvester (BCPEH) is designed and verified to achieve a wide and adjustable response frequency band. The theoretical model is derived based on the Euler-Bernoulli beam theory and continuity boundary conditions to investigate the dynamic response of the BCPEH. The displacement frequency response function and the voltage frequency response function of the BCPEH are deduced based on the Galerkin method, and the theoretical results of a typical multi-bifurcated cantilever piezoelectric energy harvester, the Y-shaped BCPEH, are verified by the finite element method (FEM) and experiments. In addition, by comparing experimental output power of the Y-shaped BCPEH with that of the traditional cantilever-based piezoelectric energy harvester with the same mass of the bifurcated part at the beam-tip, it demonstrates that the Y-shaped BCPEH has a wider operational frequency band. Moreover, it is found that the Y-shaped BCPEH can be designed with an asymmetric configuration to adjust its response frequency distribution. The number of resonant frequencies and the output power of the asymmetric Y-shaped BCPEH are higher than that of the symmetric Y-shaped BCPEH. And the Y-shaped BCPEH has even better performance than L-shaped BCPEH. This study provides a new design concept for enhanced energy harvester.

Wave reflection characteristics at the pile tip-soil interface under low strain dynamic condition

The wave reflection characteristics of the pile tip-soil interface under the low-strain condition were studied for exploring the possible quantitative method of detecting and judging the support conditions of the pile base. Based on the one-dimensional wave theory, a new theoretical method was proposed for studying the relation between the upward stress wave (UW) and downward stress wave (DW) under different support stiffness (k′) of the base soil. The frequency response functions (FRFs) of the displacement, velocity, and axial force were derived, simulating the soil resistance with the viscoelastic model. When the downward velocity (vd) was a semi-sinusoidal function, the upward velocity (vu) under different support stiffness was calculated. It was shown that the FRFs reflects the reflection characteristics in the frequency domain. The FRF of velocity is the same as FRF of displacement but opposite to that of force. The relation between vu/vd and k′ is antisymmetric about the horizontal axis of logarithmic coordinate, and vu/vd decreases from 1 to −1 as k′ increases from 0 to +∞. The model experiment verified the theoretical calculation results. This research is useful for the theoretical study of the reflection characteristics at the pile tip and engineering applications.

Investigation of 2D Rainbow Metamaterials for Broadband Vibration Attenuation.

Phononic crystals (PnCs) and metamaterials are widely investigated for vibration suppression owing to the bandgaps, within which, wave propagation is prohibited or the attenuation level is above requirements. The application of PnCs and metamaterials is, however, limited by the widths of bandgaps. The recently developed rainbow structures consisting of spatially varied profiles have been shown to generate wider bandgaps than periodic structures. Inspired by this design strategy, rainbow metamaterials composed of nonperiodic mass blocks in two-dimensional (2D) space were proposed in the present study. The blocks were connected by curved beams and tessellated with internal voids to adjust their masses. In order to demonstrate the effects of the rainbow design, two 2D metamaterials, with periodic and nonperiodic units, respectively, were investigated and manufactured using additive manufacturing technologies. Receptance functions, i.e., displacement frequency response functions, of the manufactured metamaterials were calculated with finite element models and measured with a testing system containing a mechanical shaker, an impedance head, and a laser Doppler vibrometer. The obtained numerical and experimental results showed that the metamaterial with rainbow blocks has extended bandgaps compared with the periodic metamaterial.

Study and Comparison of Seismic Behaviour of Isolator-Damper Hybrid Control System with Conventional Structural Systems

In this paper, in addition to introduce a hybrid structural system contained local isolators and dampers, its behavior and functional capabilities were studied on a conventional structure. For this purpose, an RC frame building with six-story was designed based on valid codes and then, in four cases based on the number of spans, it was split into two separate adjacent frames. Base isolation was done underneath the columns of one frame, while the bottom connections of the other frame’s columns were remained fixed and viscous dampers provided the connection of two adjacent frames on the same floors. Nonlinear time history analysis (NTHA) under three near-fault and three far-fault earthquakes and frequency-domain analysis are performed. Displacement, drift, acceleration and shear forces of the stories in the four proposed hybrid cases with two limited cases, full base-isolated and full base-fixed frames, as well as nonlinear hysteresis behavior of a damper and an isolator are assessed. The results showed that using the novel hybrid control method in most cases can mitigate deteriorating effects of all types of seismic motions observed in the conventional structural systems. However, among them, two cases (2 isolated columns -5 fixed columns and 3 isolated columns -4 fixed columns) had the best significant influence on seismic performance and structural response reduction. Furthermore, frequency response functions of displacement and acceleration with respect to ground acceleration demonstrated that the two proposed cases further suppress the responses of the limited cases, over a wide range of frequencies including all natural frequencies. Due to decrease about 50-70% in the number of base isolators (compared to full isolation) lead to considerable construction cost savings. In spite of the limitation of ASCE7-10 code on separately using base isolators and dampers on structure, applying the proposed combination technique of these two dissipating devices can overcome the limitation.

Coupled Dynamics of Cable-Harnessed Structures: Experimental Validation

The experimental study and model validations for the coupled dynamics of a cable-harnessed beam structure are presented. The system under consideration consists of multiple pretensioned cables attached along the length of the host beam structure positioned at an offset distance from the beam centerline. Analytical model presented by the coupled partial differential equations (PDEs) for various coordinates of vibrations is found, and the displacement frequency response functions (FRFs) obtained for both Euler–Bernoulli and Timoshenko-based models are compared to those from the experiments for validation. The results are shown to be in very good agreement with the experiments.

Identification of weak nonlinearities in MDOF systems based on reconstructed constant response tests

A novel strategy to characterize and identify structural nonlinearities in MDOF systems based on reconstructing constant response tests from constant excitation tests is developed in this paper. Constant displacement frequency response functions (FRFs) can be measured by a stepped sine test where the displacement is controlled at every frequency of interest. In these FRFs, the nonlinear restoring force is effectively linearized and natural frequencies can be estimated by linear modal analysis. Using a series of constant displacement tests, the relationship of equivalent stiffness versus displacement can be established by curve fitting, and hence, the nonlinear stiffness characterized. This paper proposes a method to reconstruct the constant displacement FRFs from stepped sine tests with constant excitation; this avoids the requirement to control either the response or force amplitude leads to a faster and more stable testing program. Similarly, damping nonlinearities in structures can be characterized and identified by constant velocity tests reconstructed in a similar way. This approach of FRF reconstruction is mathematically simple and suitable for structures with weak nonlinearities. The method is demonstrated on a framed structure with unknown weak nonlinearities, and the nonlinear stiffness and damping parameters of the structure are identified and validated. The results demonstrate the feasibility and effectiveness of the approach and also show the potential for practical applications in engineering.

Frequency Response-Based Indirect Load Identification Using Optimum Placement of Strain Gages and Accelerometers

This paper presents an approach for indirect identification of dynamic loads acting on a structure through measurement of structural response at a finite number of optimally selected locations. Using the concept of frequency response function (FRF), the structure itself is considered as a load transducer. Two different types of sensors are investigated to measure the structural response. These include a use of accelerometers that leads to the identification of the displacement mode shapes. The second measurement approach involves a use of strain gages since strain measurements are directly related to imposed loads. A use of mixed strain-acceleration measurements is also considered in this work. Optimum sensor locations are determined herein using the D-optimal design algorithm that provides most precise load estimates. The concepts of indirect load identification, strain frequency response function (SFRF), displacement frequency response function (DFRF), along with the optimal locations for sensors are used in this paper. The fundamental theory for strain-based modal analysis is applied to help estimate imposed harmonic loads. The similarities and differences between acceleration-based load identification and strain-based load identification are discussed through numerical examples.

A damping estimation method based on power ratio

A bandwidth method based on power ratio is proposed to evaluate the system damping by using frequency response functions. For single-degree freedom systems, exact formula for calculating damping ratio from displacement frequency response function is established. Additionally, an approximate formula to estimate the damping ratio from acceleration frequency response function is also derived. Both are represented in terms of the power ratio and the bandwidths relative to the corresponding peak frequencies. In contrast to the well-known half-power method, the proposed method can be used for relatively large damping ratios by selecting a corresponding high power ratio. The accuracy of the proposed formulas in damping estimation is investigated for a four-degree of freedom system by numerical experiments. The results show that by increasing the power ratio, the estimation errors in damping ratios for the four-degree of freedom system can be significantly reduced.

A Comparison of the Improved and Classic Half-Power Bandwidth Methods in Estimating Damping for Multi-DOF Systems

It is significant to estimate the damping ratios for stability of some engineering structures and micro-structures. This note will evaluate Multi-DOF (MDOF) linear system damping ratios. This note is concerned with presenting improved approximate formulas of half-power bandwidth method (HPBM) for evaluating MDOF linear system damping ratios based on their DFRF and AFRF curve. A two-DOF (TDOF) linear vibration system is specially given to establish numeric experiments. The classic, third-order corrections and improved formulas of HPBM are listed. More specifically, using its displacement frequency response function (DFRF) and acceleration frequency response function (AFRF), the damping ratios of a linear system with TDOF are evaluated for a large range of significant factors. It is observed that the present improved formulas of evaluating damping ratios with HPBM are explicit and precise for a certain range.

Comparison of harmonic balance and multi-scale method in characterizing the response of monostable energy harvesters

Abstract Harmonic balance and multi-scale methods have been widely applied to investigate the nonlinear dynamics and performance enhancement methods of nonlinear vibration energy harvesters. However, there are rare works considering the relative accuracy of the two methods in characterizing the response of nonlinear harvesters. Therefore, new insights into the analytical accuracy of harmonic balance and multi-scale methods are provided based on a monostable energy harvester in this paper. Frequency response functions of displacement and voltage for a monostable harvester are derived according to harmonic balance expansion and multi-scale method respectively. Monostable system parameters identified from experimental measurements are obtained for numerical investigation of the theoretical solution from harmonic balance and multi-scale methods. Results demonstrate that harmonic balance method is more accurate to characterize the frequency response under different acceleration levels, as well as the amplitude response for various excitation frequencies. While for multi-scale method, it is of great importance to choose proper keeping parameter to describe the nearness of excitation frequency to the resonance frequency.

Theoretical Research on Two Improved Optimal Sensor Placement Methods

Optimal sensor placement technique plays a key role in structural health monitoring and structural vibration control. Based on the advantages and disadvantages of effective independence (EI) and modal kinetic energy (MKE) methods, two improved optimal sensor placement methods which are Effective Independence - Average Acceleration Amplitude (EI-AAA) method and Effective Independence - Modal Kinetic Energy (EI-MKE) method are proposed in this paper. Firstly the formulas are deduced from modal expansion of multiple-input-multiple-output (MIMO) displacement frequency response function matrix. Then a computational simulation of steel cross beam structure has been implemented to demonstrate the feasibility of the two improved methods above. The obtained optimal sensor locations using the two improved methods are compared with those gained by EI method and MKE method. Finally six classical comparison criteria are employed to demonstrate the advantage and disadvantage of these four methods. The results showed that some innovations proposed in this paper are effective and reliable. The two improved optimal sensor placement methods (EI-AAA method and EI-MKE method) can not only make the truncated mode shapes as linearly independent as possible but also enable the measured modal kinetic energy to maintain the maximum value.

Damage detection by mixed measurements using accelerometers and strain gages

This work investigates the relationship between the FRFs (frequency response functions) measured by accelerometers and strain gages utilized widely for investigating structural performance. Modifying the GDM (global-deviation method), this study examines the possibility of damage detection in utilizing both sensors together. The experimental results on the mixed utilization of two sensors show that the SFRF (strain frequency response function) data measured by strain gages in the neighborhood of end supports are more practical in establishing the baseline curve than the DFRF (displacement frequency response function) data measured by accelerometers. It is shown that the modified GDM can be effectively utilized in detecting damage based on the mixed measurements of accelerometers and strain gages.

Prediction of atomic force microscope probe dynamics through the receptance coupling method

The increased growth in the use of tip-based sensing, manipulations, and fabrication of devices in atomic force microscopy (AFM) necessitates the accurate prediction of the dynamic behavior of the AFM probe. The chip holder, to which the micro-sensing device is attached, and the rest of the AFM system can affect the overall dynamics of the probe. In order to consider these boundary effects, we propose a novel receptance coupling method to mathematically combine the dynamics of the AFM setup and probe, based on the equilibrium and compatibility conditions at the joint. Once the frequency response functions of displacement over force at the tool tip are obtained, the dynamic interaction forces between the tip and the sample in nanoscale can be determined by measuring the probe tip displacement.

THEORETICAL AND EXPERIMENTAL STUDY OF MODAL STRAIN ANALYSIS

The relationship between the strain mode and the displacement mode for vibrating elastic structures is derived. It is based on the idea that when a structure is subjected to dynamic loading, its strain response can be expressed by the superposition of contributions from the “natural strain modes ”. Both the natural strain and natural displacement modes represent the dynamic characteristics of a vibrating structure. By using the finite element method, it has been possible to relate the Strain Frequency Response Function (SFRF) to the Displacement Frequency Response Function (DFRF). The technique used in the modal strain test and the method used for parameter identification are described. An experimental study of the modal strain analysis of cantilever rectangular thin plates has been performed. This shows that the strain mode is more sensitive to local changes of the structure than the displacement mode.