Natural Frequency Shift Research Articles

Mass sensing using a virtual cantilever virtually coupled with a real cantilever

Mass measurements based on the eigenmode shift of coupled cantilevers provide much higher sensitivity than the method based on the natural frequency shift of a single cantilever. The essence of the former method is that the two cantilevers are identical and are weakly coupled. However, in practical coupled cantilevers, these requirements cannot be satisfied completely because of the limitations of machining accuracy. To realize the advantages of this method fully, we propose the concepts of a “virtual cantilever” and “virtual coupling.” In the proposed method, one of the real cantilevers is replaced with a virtual cantilever that is virtually coupled with the remaining real cantilever. The virtual cantilever dynamics and the effect of coupling on the real cantilever are thus calculated in real-time using a digital computer, and the real cantilever is also actuated in real-time based on the coupling effect obtained. By adjusting the virtual cantilever's physical parameters to match those of the real cantilever while tuning the virtual coupling stiffness to be low in the digital scheme, identical real and virtual cantilevers and the weak coupling required can be realized. Additionally, to produce self-excitation with a steady-state amplitude, we apply the linear and nonlinear velocity feedback control technique that was proposed in our previous study for the above system. We confirm the validity of the virtual cantilever and virtual coupling concepts proposed for mass sensing experimentally using a real single macrocantilever and a virtual cantilever virtually coupled with the real cantilever as a prototype.

Mode shape curvature squares method for crack detection in railway prestressed concrete sleepers

This unprecedented study aims to demonstrate the potential to locate crack damage and severity in full-scale railway prestressed concrete sleepers. The dynamic mode shape curvature squares method has been utilised. This approach adopts structural deflection shapes extracted from experimental modal analysis using an impact excitation. The railway concrete sleepers are exposed to the first crack and 1.5 times above the first-crack loads using experimental setup in accordance with the British Standard (for benchmarking purpose). It is found that the first cracking of the sleepers has caused non-significant stiffness reduction of the structural component evidenced from the fact that the natural frequency shifts are only few percent and no visible differences in deflection mode shapes are observed. In addition, the mode shape curvature alone cannot indicate the crack establishment in the railway sleepers. However, a relatively sparse wireframe for deflection shapes enable the plausible density of sensor grids that can be further analysed for damage detection. In this paper, the curvature square approach has been developed to quantify and locate the damage in the railway sleepers. The standardized damage index distributions and subsequent thresholding with different levels of confidence of damage localization have been firstly developed to enhance the accuracy of crack localization. It is found that accurate locations of cracks mostly in the mid span of the sleepers can be reasonably detected in accordance with the actual crack measurements from the experimental data. The novel insight into this approach can enhance the new development of on-board technology for railway sleeper crack detection using operational modal signals.

Review of applications of self‐excited oscillations to highly sensitive vibrational sensors

AbstractVibrational sensors with resonators are suited for online monitoring because of their fast response and ability to measure instantly and continuously. Also, their miniaturization realizes much higher resolution. In this review, we begin by discussing sensor resolution based on the natural frequency shift of the resonator. For mass and stiffness sensing, the detection method for the natural frequency shift by self‐excited oscillation is characterized comparing with that by external excitation. Self‐excited oscillation automatically compensates for the viscous damping effects of the environment on the resonators, thus ensuring direct and accurate detection of the natural frequency shift. Another application of self‐excited oscillation is to viscometers. Instead of detecting the natural frequency shift, this system detects the critical feedback gain to generate self‐excited oscillation. This is important for measuring very high viscosity, where the peak of the frequency response curve is ambiguous or does not exist, meaning the Q value cannot be estimated from such curves. Another method, introduced for ultra‐sensitive mass sensing, is based on the eigenmode shift in multiple weakly coupled resonators. In this system, the self‐excitation compensates for the viscous damping effects to enable direct detection of the eigenmode shift.

Anomalous Elastic Properties of Attraction-Dominated DNA Self-Assembled 2D Films and the Resultant Dynamic Biodetection Signals of Microbeam Sensors.

The condensation of DNA helices has been regularly found in cell nucleus, bacterial nucleoids, and viral capsids, and during its relevant biodetections the attractive interactions between DNA helices could not be neglected. In this letter, we theoretically characterize the elastic properties of double-stranded DNA (dsDNA) self-assembled 2D films and their multiscale correlations with the dynamic detection signals of DNA-microbeams. The comparison of attraction- and repulsion-dominated DNA films shows that the competition between attractive and repulsive micro-interactions endows dsDNA films in multivalent salt solutions with anomalous elastic properties such as tensile surface stresses and negative moduli; the occurrence of the tensile surface stress for the attraction-dominated DNA self-assembled film reveals the possible physical mechanism of the condensation found in organism. Furthermore, dynamic analyses of a hinged–hinged DNA-microbeam reveal non-monotonous frequency shifts due to attraction- or repulsion-dominated dsDNA adsorptions and dynamic instability occurrence during the detections of repulsion-dominated DNA films. This dynamic instability implies the existence of a sensitive interval of material parameters in which DNA adsorptions will induce a drastic natural frequency shift or a jump of vibration mode even with a tiny variation of the detection conditions. These new insights might provide us some potential guidance to achieve an ultra-highly sensitive biodetection method in the future.

Geotechnical aspects of offshore wind turbine dynamics from 3D non-linear soil-structure simulations

The development of the offshore wind industry is motivating substantial research efforts worldwide, where offshore wind turbines (OWTs) of increasing size are being installed in deeper water depths. Foundation design is a major factor affecting the structural performance of OWTs, with most installations founded to date on large-diameter monopiles.This work promotes advanced 3D finite element (FE) modelling for the dynamic analysis of OWT-monopile-soil systems. A detailed FE model of a state-of-the-art 8 MW OWT is analysed by accounting for dynamic soil-monopile interaction in presence of pore pressure effects. For this purpose, the critical-state, bounding surface SANISAND model is adopted to reproduce the hydro-mechanical cyclic response of the sand deposit. The response to realistic environmental loading histories (10 min duration) are simulated, then followed by numerical rotor-stop tests for global damping estimation.While linking to existing literature, all FE results are critically inspected to gain insight relevant to geotechnical design. The modelling tools adopted (i) support the robustness of ‘soft-stiff’ foundation design with respect to natural frequency shifts, even during severe storm events; (ii) provide values of foundation damping in line with field measurements; (iii) suggest that pore pressure effects might more likely affect soil-monopile interaction under weak-to-moderate environmental loading.

Dynamic Modification and Damage Propagation of a Two‐Storey Full‐Scale Masonry Building

The progressive change of modal characteristics due to accumulated damage on an unreinforced masonry (URM) building is investigated. The stone URM building, submitted to five consecutive shakings, has been experimentally studied on the shaking table of EUCENTRE laboratory (Pavia, Italy). The dynamic characteristics of the test specimen are analytically estimated using frequency and state‐space modal identification from ambient vibration stationary tests carried out before the strong motion transient tests at various levels of damage. A singular value (SV) decomposition of the cross‐correlation matrix of the acceleration response in the frequency domain is applied to determine the modal characteristics. In the time domain, the subspace state‐space system identification is performed. Modal characteristics evolve from the initial linear state up to the ultimate collapse state in correlation with accumulated damage. Modal frequencies shorten with increasing intensity, whereas modal damping ratios are enhanced. Modal shapes also change with increasing level of accumulated damage. Comparing the evolution of modal characteristics, it is concluded that modal damping ratio shift can be better correlated with the system’s actual performance giving a better representation of damage than that of natural frequency shift ratio or the modes difference.

Sequential combined thermal cycling and vibration test and simulation of printed circuit board

Automotive environment generates vibration and temperature fluctuation, which can be damaging for electronic boards. Furthermore, combined temperature and vibration is a type of testing that recreates environments approximating more closely operating environments. In this paper, vibration tests are performed during thermal cycling to investigate the influence of temperature on Printed Circuit Board (PCB) responses. A combined test of temperature and vibration is designed to evaluate the PCB response after each step of one thermal cycle [−40/125] °C. The frequency and displacement can be linked to the strain and strain rate during a board level vibration test. They are obtained from the acceleration during a sine sweep application on a shaker, which is measured using a light weight accelerometer. Experimental results show that temperature significantly affects the PCB responses. Temperature variation leads to striking differences in vibration loading intensity. The PCB first natural frequency shifts from 328 Hz to 313 Hz with temperature increase from 25 °C to 125 °C. A complex PCB is characterized using tension and flexure tests at different temperatures, to extract user input parameter for simulations. An accurate numerical model for sequential combined thermal cycling and vibration simulation is developed and validated using experiments. Thanks to these simulations, the displacement evolution due to combined loading is possible.

A novel method of vibration modes selection for improving accuracy of frequency-based damage detection

Frequency-based damage detection techniques have been widely applied to structural health monitoring. By analysing the changes (shifts) in natural frequencies in a structure with and without damage, these techniques solve the inverse problem of determining size and location of damage. In the existing literature, the first few or random modes of frequency shifts are given to the inverse algorithms as inputs in order to predict the damage parameters. These frequency shifts can be either numerical or measured. While the accuracy of prediction in the former (numerical) case has been found to be satisfactory, the use of measured frequencies has often shown large errors. This can be attributed to unavoidable noise in frequencies, including the mismatch between FEM model and real structure, as well as the noise in the measurement itself. Previous research has shown that the noise in frequency will actually be magnified in the discrepancy of frequency shifts, and thus affect the damage prediction accuracy. And moreover, the same noise added to different modes of frequency of a damaged case will lead to the different levels of deviation in different modes of frequency shifts. This observation indicates that potentially some modes of frequency shifts are less affected by the noise than others for a given case. However, so far, there has been no studies that attempt to identify particular vibration modes of frequency shifts that are (a) less affected by the noise for all damage cases and (b) result in a more accurate prediction of damage. In this study, a novel concept of Noise Response Rate (NRR) is proposed to evaluate the sensitivity of each mode of the frequency shift to noise. Further, it is shown that selecting the vibration modes with low NRR values improves the prediction accuracy of frequency-based damage detection. The efficacy of NRR is demonstrated through a case study on a composite curved plate compared with the conventional method for damage detection.

Frequency Invariability of (Pb,La)(Zr,Ti)O₃ Antiferroelectric Thick-Film Micro-Cantilevers.

Micro-electromechanical systems comprising antiferroelectric layers can offer both actuation and transduction to integrated technologies. Micro-cantilevers based on the (Pb0.97La0.02)(Zr0.95Ti0.05)O3 (PLZT) antiferroelectric thick film are fabricated by the micro-nano manufacturing process, to utilize the effect of phase transition induced strain and sharp phase switch of antiferroelectric materials. When micro-cantilevers made of antiferroelectric thick films were driven by sweep voltages, there were two resonant peaks corresponding to the natural frequency shift from 27.8 to 27.0 kHz, before and after phase transition. This is the compensation principle for the PLZT micro-cantilever to tune the natural frequency by the amplitude modulation of driving voltage, rather than of frequency modulation. Considering the natural frequency shift about 0.8 kHz and the frequency tuning ability about 156 Hz/V before the phase transition, this can compensate the frequency shift caused by increasing temperature by tuning only the amplitude of driving voltage, when the ultrasonic micro-transducer made of antiferroelectric thick films works for such a long period. Therefore, antiferroelectric thick films with hetero-structures incorporated into PLZT micro-cantilevers not only require a lower driving voltage (no more than 40 V) than rival bulk piezoelectric ceramics, but also exhibit better performance of frequency invariability, based on the amplitude modulation.

Vibration Control of a Flexible Beam with Embedded Shape Memory Alloy Wire

Structural vibrations can damage systems and lead to high cycle fatigue, excess noise or even catastrophic failure. The research presented herein aims to quantify the effect of embedded shape memory alloy (SMA) wire as an active suppression system to mitigate such vibrations in a flexible clamped-free beam. A silicone beam with embedded SMA wire was designed. Natural frequencies were calculated using a frequency response function (FRF) at varying temperatures. Experiments resulted in an average natural frequency shift of 44.7% and an amplitude decrease of 34% at the tip of the beam. Control samples were created using the same dimensions and silicone material. One control sample contained embedded aluminum wire while the other sample contained no embedded material. The addition of the unactuated SMA wire resulted in an average natural frequency shift of 160% and amplitude decrease of 44.6% while the actuated SMA wire resulted in a shift of 258% and amplitude decrease of 63.6%. The natural frequency of the embedded aluminum wire sample remained the same at low and high temperatures, leading to the conclusion that the frequency shift of the SMA sample was a result of the shape memory effect.

Cell Network-Based Formulas for Fast Determination of Flexural Stiffness Reduction of U-Shaped Core Shear Walls

Reinforced concrete (RC) core shear wall is one of the most widely used earthquake-resisting systems. Degradation of a core wall's flexural stiffness is vital for understanding the natural frequency shift of the damaged building. But it is hard to capture, often necessitating complex finite element analyses (FEAs). This study seeks to provide an efficient tool to quickly determine the remaining flexural stiffness of U-shaped core walls. Importantly, the tool is designed to require only the easy-to-collect observational damage information. Of primary novelty is a network of microscopic unit cells, each consisting of nonlinear concrete and steel springs along with a compression-only gap. Validations with three U-shaped walls tested under complex and multidirectional loading paths show that the proposed formulas appear promising in quickly determining the trend of degrading flexural stiffness compared with a high-precision multiscale FEA program. All the formulas written in Matlab codes are made publicly available. Using the portable formulas running on a laptop, practicing engineers and researchers will be able to swiftly diagnose core U-shaped walls after quick on-site or laboratory observations.

A megahertz-frequency tunable piecewise-linear electromechanical resonator realized via nonlinear feedback

Vibration-based sensing modalities traditionally have relied upon monitoring small shifts in natural frequency in order to detect structural changes (such as those in mass or stiffness). In contrast, bifurcation-based sensing schemes rely on the detection of a qualitative change in the behavior of a system as a parameter is varied. This can produce easy-to-detect changes in response amplitude with high sensitivity to structural change, but requires resonant devices with specific dynamic behavior which is not always easily reproduced. Desirable behavior for such devices can be produced reliably via nonlinear feedback circuitry, but has in past efforts been largely limited to sub-MHz operation, partially due to the time delay limitations present in certain nonlinear feedback circuits, such as multipliers. This work demonstrates the design and implementation of a piecewise-linear resonator realized via diode- and integrated circuit-based feedback electronics and a quartz crystal resonator. The proposed system is fabricated and characterized, and the creation and selective placement of the bifurcation points of the overall electromechanical system is demonstrated by tuning the circuit gains. The demonstrated circuit operates at 16 MHz. Preliminary modeling and analysis is presented that qualitatively agrees with the experimentally-observed behavior.

Vibration-based assessment of delaminations in FRP composite plates

Delamination is a frequently occurring type of damage in laminated fibre reinforced polymer (FRP) composites and causes substantial loss in structural stiffness and usable service life. The detection of delaminations in FRP composites is critical for the safe and reliable use of these materials in aeronautical and other industries. Structural Health Monitoring (SHM) techniques based on vibration measurements have proven to be promising towards this end. There have been comprehensive studies of FRP beams with through-width delaminations, but the damage assessment of FRP plates with embedded delaminations using frequency-based detection has not been extensively studied. To solve the inverse problem of determining size and location of delamination from changes in the natural frequencies, this paper presents a new surrogate assisted optimisation (SAO) method for predicting the location and size of delaminations in fibre reinforced composite plates using natural frequency shifts as indicative parameters. The proposed frequency-based delamination assessment method is validated using finite element models of FRP plates with embedded delaminations and by experimental modal analysis. Modal testing was conducted using scanning laser vibrometer on carbon/epoxy and glass/epoxy FRP plates that were manufactured with artificially induced delaminations. The proposed SAO algorithm was compared to an Artificial Neural Network (ANN) method in terms of database size, prediction accuracy and sensitivity to noisy data. The results show that the proposed inverse algorithm can predict the delamination parameters of location and size with good accuracy for numerically simulated frequency shift data but the prediction accuracy was reduced with experimental data. A comparison of the two inverse algorithms show that the SAO method has significant advantages compared to the ANN algorithm for delamination prediction.

Mitigation of flutter vibration using embedded shape memory alloys

Aeroelastic flutter is a major concern for designers of planes, boats, turbines and other systems that experience aerodynamic loading. The typically undesirable phenomenon can cause aircraft instability, high cycle fatigue, excess noise, and in some cases catastrophic failure. This research aims to quantify the effect of embedded shape memory alloys (SMA) as an active suppression system to mitigate vibrations induced by aeroelastic flutter through an experimental investigation. A silicone plate with embedded SMA wire was designed. A subsonic wind tunnel was utilized to actuate flutter vibration. Control samples were created using the same dimensions and silicone material. One control sample contained embedded aluminum wire while the other sample contained no embedded material. Frequencies at the tip of the fluttering plates were calculated using a Fast Fourier Transfer (FFT) at varying temperatures. Experiments resulted in an average natural frequency shift of 44.6% at the sample tip upon actuation of the SMA material. Two dimensional vibrational scanning tests revealed three vibration modes in the fluttering flags. A first bending mode revealed a tip amplitude decrease of 34.4% upon actuation of the embedded SMA material. A second bending mode revealed a tip amplitude decrease of 33.6%. A torsional mode revealed a tip amplitude decrease of 60.7%. The frequency of the embedded aluminum wire sample remained relatively constant at low and high temperatures, leading to the conclusion that the frequency shift of the SMA sample was a result of the shape memory effect.

Eigenfrequency shift mechanism due to variation in the cross sectional area of a conduit

ABSTRACTEigenfrequency (i.e. natural resonant frequency) shift due to variation in the cross sectional area of a conduit is observed in different applications such as vocal tracts, musical instruments and water supply systems. In the context of water supply systems, past research focused on developing inverse problem techniques that use shifts in eigenfrequencies to identify blockages. The goal of this paper is to understand the mechanisms that cause this eigenfrequency shift. Application of the principle of action invariance to a small blockage shows that the eigenfrequency shift is directly related to the change in energy. For a severe blockage, the pipe system is decoupled into two independent subsystems: a subsystem that involves the blockage and another that involves the intact pipe section. The decoupling is lost when the blockage length is such that the fundamental frequencies of the two subsystems are close or equal, resulting in resonance where perturbation theory is successfully used to derive a simple frequency relation.

Fabrication of a resonant PEMC sensor using hybrid $$\upvarepsilon $$ ε -constraint lexicographical ranking DE-MSBA with Jacobean approximation

In the current research, a novel algorithm is proposed for the optimal fabrication of a resonant piezoelectric excited millimeter-sized cantilever (PEMC) sensor. PEMC sensor is one of the most well-known types of sensors with a wide range of applications in today’s industry. The proposed design mechanism is general, and can be used for the fabrication of PEMCs for different applications. A physics-based analytical model based on Euler–Bernoulli theory is adopted for performance analysis. A novel hybrid nature-inspired optimizer, called $$\upvarepsilon $$ -constraint lexicographical ranking differential evolution mutable smart bee algorithm ( $$\upvarepsilon $$ lr-DEMSBA), is developed which can effectively search the non-convex and multimodal solution domain. The optimal values of PEMC design parameters are obtained using $$\upvarepsilon $$ lr-DEMSBA. To ensure the veracity of the parameters suggested by $$\upvarepsilon $$ lr-DEMSBA, a PEMC is fabricated using the suggested optimal values, and some post analysis are carried out. By investigating the relation between the frequency shift of natural frequency and increasing of immersion depth, it is observed that the linear performance zone of optimized PEMC sensor is increased by 2.5 mm which is a favorable property. Also, from numerical viewpoint, it was observed that $$\upvarepsilon $$ lr-DEMSBA surpasses different variants of optimization techniques for the current case study.

Dynamic behavior of an electrostatic MEMS resonator with repulsive actuation

Static and dynamic analyses of an electrostatic microbeam under repulsive force actuation are presented. The repulsive force, created through a specific electrode configuration, generates a net electrostatic force on the beam pushing it away from the substrate. This allows large out-of-plane actuation and eliminates the pull-in instability. For example, a dynamic amplitude of 15 $$\upmu $$ m was recorded for a 500- $$\upmu $$ m-long cantilever at a DC voltage of 195 V and an AC voltage of 1 V, while the initial gap was only 2 $$\upmu $$ m. This study includes mathematical modeling and simulations for a cantilever and a clamped–clamped beam, as well as experimental validation. The beam is modeled using Euler–Bernoulli beam theory and electromechanical coupling effects. Cantilever tip displacement, clamped–clamped midpoint deflection, and natural frequency shifts are reported. Governing equations are solved numerically using the shooting method, which provides a complete picture of the beam dynamics. The numerical results are verified with experimental data from fabricated beams using PolyMUMPs standard fabrication. Frequency response results reveal a mixed softening and hardening behavior and secondary resonances originating from quadratic and cubic nonlinearities in the governing equations. The analysis provides insight for applications in optical and gas sensors where a large signal-to-noise ratio and, sometimes, a wide frequency bandwidth are desired.

Probabilistic reliability of micro-resonators with thermoelastic coupling

Analyzing the probabilistic reliability of micro-resonators is essential for the design of microelectromechanical systems (MEMS). In order to investigate more complex structures and take into account the random variation of the design parameters, an efficient thermoelastic Finite Element (FE) formulation is derived from Hamilton’s variational principle. A generalized eigenvalue equation can be obtained through the formulation. The effects of thermoelastic coupling and the randomness of parameters on the natural frequency are studied by using a non-symmetric block Lanczos algorithm combined with the right and left eigenproblem. Considering the influence of thermoelastic coupling on temperature is little, the thermoelastic coupling term in heat conduction equation can be negligible, thus the heat conduction equation is simplified based on linearization. Depending on the eigenproblem for temperature field and thermal stress in micro-resonators, further analysis is conducted on the randomness of temperature and stress in micro-resonators. For reliability analysis, a method which considering both strength and frequency is proposed. The strength failure criterion is set based on fracture mechanics, and frequency reliability is taken into account the probabilistic of resonators when the natural frequency is close to the excitation frequency. With a simply supported model of micro-resonators as an example, the numerical solutions of quality factor have shown little difference with the analytical solutions proposed by Zener and Lifshitz, verifying the correctness of the model calculations. Finally, the natural frequency shift and the probabilistic reliability with temperature are calculated.

Establishing detection maps for carbon nanotube mass sensors: molecular versus continuum mechanics

In this work, the mass-sensing ability and the corresponding detection efficiency of a fully clamped single-walled carbon nanotube (SWCNT) are investigated through a free vibration analysis, by utilizing mainly a molecular mechanics (MM) formulation and secondarily a continuum mechanics (CM) analytical approximation. The MM method is based on representing the SWCNT as a three-dimensional (3D) finite element frame of point masses and linear springs, while the CM one is grounded on the Euler–Bernoulli beam theory. The overall effort is focused on obtaining detection maps, based on natural frequency data which are capable of leading to the straightforward identification of an unknown mass attached to the external surface of the candidate SWCNT sensor. For this reason, the SWCNT relative natural frequency shifts due to the mass addition, regarding specific modes of vibration and for a variety of mass magnitude and position combinations, are calculated beforehand. Then, the magnitude as well as the position of the added mass may be graphically found by superposing the arisen relative natural frequency shift variations in common contour diagrams. Some results from the open literature are utilized to confirm the predictive performance of the proposed MM method concerning the free vibration of pure SWCNTs, while the MM and CM established detection maps are set in contrast to examine whether a mass nanosensor may be satisfactorily treated in a continuum manner.

Mass sensing using self-excited oscillation in viscous environments

A mass measurement method in viscous environments using self-excited oscillation is proposed theoretically and experimentally. The utilization of the natural frequency shift in a resonator has attracted many researchers due to the high sensitivity. Small mass measurement with the order of fg and ag has been reported. In the conventional methods, the natural frequency shift is detected from the shift of the excitation frequency at which the frequency response curve takes the peak value. However, it is not applicable to the measurement in high viscous environments, i.e., the observation of the chemical reaction and the mass sensing for biological samples because the peak is ambiguous or does not exist. In this research, as a mass measurement method applicable in viscous environments, we propose the utilization of self-excited oscillation to measure the shift of the natural frequency caused by an sample added to the resonator. The resonator is actuated by positive velocity linear feedback. Self-excited oscillation occurs when the linear feedback gain exceeds a critical value depending on the viscosity in measurement environments. Also, by applying the nonlinear feedback the divergence of the self-excited oscillation is suppressed and the response amplitude is kept constant. The self-excited resonator vibrates at the natural frequency regardless of the viscosity in measurement environments because the velocity feedback compensates for the viscous effect. Then, not relying on a frequency response curve, the method using natural frequency shift becomes applicable even in high viscous environments. We fabricated a theoretically proposed self-excited mass sensor and conducted mass measurements experimentally in air, in water, and in a liquid with 1.17 x 104 mPa . s viscosity. As a result, the mass of 0.0091 kg was detected in the respective viscous environments. The ratio of the additional mass to the mass of the sensor is 0.59 %.