Natural Frequency Sensitivity Research Articles

Natural Characteristics of a Marine Two-Stage Tandem Hybrid Planetary System

This study focuses on a marine two-stage tandem hybrid planetary system. Natural frequencies and vibration modes are determined using a translational–torsional coupled dynamic model. Based on the motion characteristics of the transmission system, free vibration is categorized into three typical modes. The parameter sensitivity of natural frequencies is computed, and the effects of structural parameters such as unequally spaced planet, mesh stiffness, planet mass and rotational inertia on the natural frequencies are analyzed. Utilizing the coupling factor, the mode transition criterion for the natural frequencies response to parameters is formulated. The results demonstrate that the vibration modes of the two-stage tandem hybrid planetary system can be classified as the fixed-axis train vibration mode, the differential train vibration mode, and the coupled vibration mode. Unequally spaced planet primarily disrupts vibration modes without significantly affecting natural frequencies. In contrast, the effects of mesh stiffness, planet mass and rotational inertia on the modes are opposite to those of unequally spaced planets. The validity of the parameter sensitivity and mode transition criterion is substantiated through illustrative examples.

A novel analytical-based viewpoint for MEMS mass spring system: systematic design guideline for dimensional tolerance and footprint reduction powered by dimensionless analysis

Mass-spring is one of the most used structures in sensors and actuators of MEMS technology, including accelerometers, gyroscopes, harvesters, etc. Some of the major design parameters of the structure are natural frequency and displacement sensitivity. The limitations of microfabrication processes should be considered in the design process. The optimized geometry topology can reduce the area occupied by the structure in the fabrication and increases the number of possible structures per wafer. This study aims to provide a novel systematic design and optimization of a MEMS mass-spring system by using a dimensionless design flowchart. Based on the dimensionless design, this algorithm provides the designer with the optimized dimensions of the mass-spring by following the provided dimensionless diagrams, taking into account stress, displacement sensitivity, natural frequency, and fabrication limitations. Also, the optimized dimensions of the mass-spring system are obtained using the dimensionless design flowchart with respect to the desired natural frequency and displacement sensitivity. In addition, genetic algorithms and dimensionless diagrams present a procedure to minimize the footprint of the structure while keeping the desired natural frequency and displacement sensitivity. Results demonstrate that up to a 3% change in both natural frequency and displacement sensitivity, leads to a 45% reduction in the structure area compared to the related literature of MEMS accelerometers.

Vibration Characteristics of Mistuned Blisk with Chordwise Variable Thickness Blades Subjected to Aerodynamic Prestress

In this paper, the influence of aerodynamic prestress on vibration characteristics of mistuned blisk is investigated by comparative study considering one-way fluid–structure interaction. It is found that the aerodynamic prestress changes twisted angle of the blade, increases natural frequencies of the mistuned blisk and affects the sensitivity of natural frequency to rotational velocity. To facilitate the mechanism study of these phenomena, a simplified blade model with chordwise variable thickness is established. The effects of Young’s modulus, rotational velocity and pre-twisted angle on natural frequency are investigated by the first-order shear deformation plate theory and Chebyshev–Ritz method. The results show that the natural frequency of the blade is positively correlated with the rotational velocity and pre-twisted angle. For different twisted angles, the same rotational velocity change can cause different natural frequency increments. In addition, the aerodynamic prestress can suppress the effect of Young’s modulus mistuning on natural frequency.

Dynamic damage characterization of slender masonry structures by radar interferometry: a case study

The work shows an experimental and numerical investigation on the dynamic identification of damage in slender masonry structures using the innovative remote sensing technique of ground-based radar interferometry. The case study concerns San Cataldo’s masonry lighthouse, which is located in Puglia (Italy). Here, we experimentally determine the natural frequencies of the lighthouse by carrying out Ambient Vibration Tests (AVT) using an interferometric radar system. Starting from the experimental results, we calibrate a numerical model of the structure and study the sensitivity of the first natural frequencies of the model to five possible damage scenarios involving different parts of the structure and different degrees of damage. Comparing numerical and experimental results, we show that the interferometric radar can be effectively applied to detect damage involving the lower part of the structure.

Design optimization of an optical fiber air-backed concentric hybrid mandrel hydrophone by an iterative design and finite element analysis for submarine passive array applications

The optical fiber hydrophones are well suited for the submarine passive sonar applications taking into consideration of their high sensitivity, fully electrically passive characteristics etc. and are definitely an equivalent replacement for Lead Zirconate and Lead Titanate ceramic hydrophones. In this paper, finite element analysis of several sets of cases of concentric hybrid material-based air-backed OF mandrel hydrophones are analysed and tuned to their geometric dimensions and their mechanical properties for design optimization, suitable for submarine sonar applications. A finite element method is adopted for analysis, strain and sensitivity calculations using ANSYS software. An inverse dependency of the sensitivity and the natural frequency were observed and was overcome by an innovative method where a hybrid inner hollow tube, combining both Aluminium and high-density polyethylene materials, were used, improving both the stiffness property and compliance of the mandrel and thereby improving both the natural frequency and pressure sensitivity. The harmonic analysis was carried out at 5 KHz acoustic pressure, by adding hydrostatic pressures of 2 MPa, 3 MPa and 4 MPa and obtained satisfying results. The acoustic directivity of this OF mandrel hydrophone was estimated, plotted and an omni directional pattern was obtained.

Influence of the blade size on the dynamic characteristic damage identification of wind turbine blades

Abstract In this article, the sensitivity coefficients of dynamic characteristic damage identification of blades with different sizes were investigated. The results show that the first third-order vibration modes of the blade before and after damage are consistent, and the wind turbine blade size has no significant effect on the vibration mode; with the increase of the blade size, the first-, third- and fourth-order natural frequency sensitivity coefficients decrease gradually, while the second-, fifth- and sixth-order natural frequency sensitivity coefficients increase gradually; as the blade size increases, the third-order displacement mode sensitivity coefficient of the blade gradually increases, which indicates that the displacement modes identification effect is better with the increase of the blade size. With the increase of the blade size, the first- and third-order strain modal sensitivity coefficients increase gradually, which indicates that when using first- and third-order strain modes for damage identification, the larger the blade size, the better the identification effect; with the increase of the blade size, the second-order strain modal sensitivity coefficient decreases gradually, which indicates that when using second-order strain mode for damage identification, the larger the blade size, the worse the identification effect. This research could provide a theoretical basis for the application of the dynamic characteristic damage identification method in the damage identification of wind turbine blades of different sizes.

Natural frequencies of porous orthotropic two-layered plates within the shear deformation theory

This paper analyzes the natural frequencies of porous orthotropic laminated composite plates with two different porosity models based on the higher-order shear deformation theory. The fundamental relations of natural frequency analysis are derived by using the virtual work principle and hyperbolical shear deformation theory. The obtained partial differential equations system is reduced to an ordinary differential equations system via approximation functions suitable for simply supported boundary conditions and the Galerkin method. After some mathematical operations, the natural frequency equation of porous orthotropic laminated composite plates is obtained in the framework of hyperbolical shear deformation theory. The natural frequency equation based on the classical laminated plate theory can be determined by ignoring the shear strains in the theoretical formulations. After two validation studies by using appropriate results in the literature, parametric analyses are performed to show the sensitivity of natural frequencies to shear deformation, porosity model, orthotropy, layer sequence, and geometric properties.

Sensitivity Analysis and Robust Assignment of Natural Frequencies for Vibrating Systems Using the Receptance Method

This paper presents a new receptance-based method for the robust assignment of natural frequencies. This method aims to address the deviation of the actually achieved natural frequencies for target natural frequencies when assigning natural frequencies, which usually arises due to uncertainties in passive structural modifications (i.e. mass and stiffness modifications). The mathematical expression is derived for the sensitivity of the target natural frequency to the uncertainties in modification parameters, which are subsequently employed to characterize the potential deviations of the target natural frequencies in the total differential form. The obtained deviation expression is integrated into the optimization calculation of the natural frequency assignment as an extra penalty term to boost the acquisition of highly robust modification parameters. The proposed robust assignment method, as a non-probabilistic technique, can be employed without any knowledge of the probabilistic distribution of uncertainty in modification parameters. Moreover, it uses only the measured frequency response functions (FRFs), preserving the advantage of the receptance method not needing a numerical model. Two numerical examples and a comparison with a typical eigenstructure assignment method (reported in the literature) demonstrate that the proposed method can provide highly robust modification parameter solutions. The application of the proposed method to the natural frequency assignment of a laboratory water-filled pipeline system gives experimental evidence of its effectiveness.

Structural health monitoring of South America's first 6-Story experimental light-frame timber-building by using a low-cost RaspberryShake seismic instrumentation

A low-cost seismic instrumentation system (LCSIS) has been implemented and validated for the structural health monitoring of South America's first experimental 6-story light-frame timber building (Peñuelas Tower). Furthermore, coupling the LCSIS with relative humidity and temperature measuring system has allowed for investigating the ambient-induced variations of the building's dynamic properties. The LCSIS has been accomplished using six accelerographs RaspberryShake® R4SD (R4SD), which were interconnected using a local network controlled by an auxiliary Raspberry Pi3 (RPi3). The RPi3 performs the following activities: (i) it works like an NTP server that allows the synchronization between the seismic instruments; (ii) it is connected to the internet to provide remote control and communication; (iii) it has internal subroutines to periodically estimate the dynamic properties of the building by using the FDD method from hourly ambient vibration tests based on the velocity data obtained from the vertical geophones included in R4SD instruments. The LCSIS was electrically supplied using a 55A battery and a battery charger to ensure operability during power outages. The building's natural frequencies and mode shapes were identified from operational modal analyses of five integrated ambient vibration tests, measured by a conventional seismic instrumentation system (Episensors ES-U2), showing consistent dynamic properties. The LCSIS was validated from shake table tests, observing a good agreement for signal frequencies smaller than 25–30 Hz in comparison with conventional measuring systems as well as measuring the response during an earthquake of low-intensity Mw = 4.6. Once the LCSIS was validated, a BME280 sensor was added to continuously monitor the temperature and relative humidity inside the building, which allowed for the implementation of a structural health monitoring system to track the ambient-induced variations of three principal natural frequencies. It was found a high sensitivity of the building's natural frequencies due to ambient variations, exhibiting peak-to-peak daily variations of 9.5–10.9 % and overall variations of 24.7–29.2 % during an eleven-month period, thus evidencing that timber structures could be more susceptible to temperature and relative humidity variations than other structural systems. Unexpectedly, the timber structure was found to be stiffer under moist conditions, which may be attributed to the tightening of timber assemblies due to wood swelling. A 48-hour state-space model was built to accurately compute the estimated natural frequencies as a function of the recorded temperature and relative humidity, showing that this procedure could be used to effectively disaggregate the ambient-induced variations from changes generated by damage or other unknown sources.

Topology optimization of shell-infill structures for natural frequencies

PurposeThe purpose of this paper is to develop a topology optimization algorithm considering natural frequencies.Design/methodology/approachTo incorporate natural frequency as design criteria of shell-infill structures, two types of design models are formulated: (1) type I model: frequency objective with mass constraint; (2) type II model: mass objective with frequency constraint. The interpolation functions are constructed by the two-step density filtering approach to describe the fundamental topology of shell-infill structure. Sensitivities of natural frequencies and mass with respect to the original element densities are derived, which will be used for both type I model and type II model. The method of moving asymptotes is used to solve both models in combination with derived sensitivities.FindingsMode switching is one of the challenges faced in eigenfrequency optimization problems, which can be overcome by the modal-assurance-criterion-based mode-tracking strategy. Furthermore, a shifting-frequency-constraint strategy is recommended for type II model to deal with the unsatisfactory topology obtained under direct frequency constraint. Numerical examples are systematically investigated to demonstrate the effectiveness of the proposed method.Originality/valueIn this paper, a topology optimization method considering natural frequencies is proposed by the author, which is useful for the design of shell-infill structures to avoid the occurrence of resonance in dynamic conditions.

The approximate calculation of the natural frequencies of a Stockbridge type vibration damper and analysis of natural frequencies' sensitivity to the structural parameters

Abstract. Vibration damper is widely used in overhead transmission lines to alleviate aeolian vibration. Its natural frequencies are important parameters for a vibration damper. In this paper, the approximate calculation formulas of natural frequencies of the one-side subsystem of a Stockbridge type vibration damper were derived and the design sensitivity analysis of the natural frequencies was studied using partial differential equations with respect to each concerned parameter including the length of the steel strand, the mass of the counterweight, the eccentric distance, and the radius of gyration of the counterweight. Through a case study that considered a variation of up to ±30 % in the values of the design parameters, the exact calculation and approximate calculation results of the natural frequencies were analysed, and the sensitivity of the vibration damper's natural frequencies to the design parameters was studied. The results show that, within the range of the parameters used in this study, the approximately calculated first-order frequency is lower than the exact values, whereas the approximately calculated second-order frequency is larger than the exact values. The sensitivity analysis indicates that the first-order frequency is highly sensitive to the steel strand's length, whereas it is moderately sensitive to the counterweight's mass and slightly sensitive to the eccentric distance and the gyration radius of the counterweight; the second-order frequency is highly sensitive to the steel strand's length and the gyration radius of the counterweight, moderately sensitive to the counterweight's mass, and slightly sensitive to the eccentric distance. It will provide theoretical guidance and approximate analysis method in engineering for the design of the vibration damper.

Vibration and sensitivity analysis of double-layered non-uniform piezoelectric microcantilever as a self-sensing mass sensor

Piezoelectric microcantilevers are specially popular in microelectromechanical system (MEMS) devices due to their high accuracy, environmental friendliness, low response time, and compatibility with compact designs. Since mass sensors are an important application of piezoelectric microcantilevers (PMCs), it is particularly important to determine PMCs sensitivity to nanoparticle absorption. For this purpose, a dynamic analysis was carried out to derive the equation governing vibrations of a PMC in air and water environments. A sphere chain model was used to model hydrodynamic forces acting on the PMC. In order to utilize the bending of the PMC for determining the output current from the piezoelectric layer, the resulting equation was solved by taking into account how the PMC geometrical discontinuities affected eigenvalues. In addition to the sensitivity of natural frequency and amplitude of these sensors, sensitivity of output electric current from their piezoelectric layer is also important. Therefore, the sensitivity of these parameters to nanoparticle absorption was analyzed and compared so as to select the suitable parameter for achieving satisfactory performance of PMC mass sensors in air and liquid working environments. The importance of selecting proper geometrical dimensions in the design of PMC mass sensors is realized by considering the fact that their sensitivity can be enhanced by properly selecting their dimensions. Accordingly, the effects of geometrical dimensions of the sensor on its sensitivity were determined through Sobol sensitivity analysis. The simulation results indicated that the sensitivity of output current from the piezoelectric layer and that of the vibration amplitude in the first three modes were the suitable parameters determining the performance of the PMC mass sensor in air and fluid environments, respectively.

Development of a Modal-Based Turbomachine Blade-Disk Attachment Inspection Technique

Turbine blade failures are among the leading causes of steam turbine failure. Failure types typically include cracking, rubbing, blade fouling, and foreign object damage. There is currently a range of non-destructive testing methods used to detect damage at the blade-disk attachment zone, all of which involve disassembling of the blade from the disk for periodic inspection. Evidence indicate that a method to detect damage at the blade-disk attachment zone using a non-contact, non-destructive in-situ off-line modal-based structural health monitoring technique could be useful under some circumstances. Such a technique would have the advantage of eliminating the necessity to disassemble blades during inspection. This would result in significant cost savings. Also, defects associated with the disassembly and reassembly of blades would be avoided. Thus, the aim of this study was to develop a modal-based turbomachinery blade disk attachment inspection technique. Modal parameters were acquired from a robust experimental modal analysis of freely supported low-pressure steam turbine blade-disk segment assemblies. Artificial single-location cracks were intentionally introduced into the turbine blades by cutting a 1 mm thickness notch at three probable damage locations, namely, at the upper pinhole on the leading-edge pressure side, above the root at the base of the aerofoil on the leading-edge and on the trailing-edge. In this work, a finite element analysis of the bladed disk segment assemblies was carried out with and without damage. To validate the reliability of the numerical models, the numerical results were correlated with the measured values, the results of which showed a strong correlation. Finally, a parametric study was conducted in which various healthy and damaged blade-disk cases were systematically investigated. This was done to examine the sensitivity of the blade natural frequency to damage. The artificial damage above the root was found to cause the largest changes in natural frequency. These changes were even more pronounced for assemblies with two blades. Receiver operating characteristic curves were used to assess the discriminatory ability of the results. Each damage case was found to be unique and therefore identifiable from its corresponding healthy case.

On the theoretical and molecular dynamic methods for natural frequencies of multilayer graphene nanosheets incorporating nonlocality and interlayer shear effects

In this paper, a multiplate nonlocal shear model and molecular dynamic simulations are presented to investigate the effects of interlayer shear and nonlocality on the natural frequencies of multilayer graphene sheets (MLGSs). From one aspect in the optimal design of such structures, the interaction between graphene layers, which can significantly vary the static and dynamic behavior due to lack of solidity of layers stack, should be considered. On the other hand, it is requied that the nonlocality phenomenon which has an effective role in the mechanical analysis of nanostructures is taken into account. To this aim, the equation of motion along with corresponding boundary conditions is derived by using the Kirchhoff plate model and nonlocal continuum theory to capture these effects. For a case study, the free vibration problem of simply supported MLGSs is studied by presenting the closed-form Navier’s solution of natural frequencies and carrying out Molecular dynamics (MD) simulations to estimate the value of nonlocal parameter. Finally, the influences of interlayer shear modulus, number of layers, foundation stiffness, and nonlocal parameter on the natural frequencies of such MLGSs are examined in detail. Findings show that natural frequencies including nonlocality effect become smaller than ones of the classical continuum theory and also have good agreements with MD simulations. Moreover, it is seen that the sensitivity of natural frequency to the nonlocal parameter becomes more significant when the number of layers or interlayer shear modulus increases as well as the aspect ratio or width of sheets decreases.

Mathematical modeling and dynamic characteristic analysis of a novel parallel tracking mechanism for inter-satellite link antenna

Inter-satellite link antenna tracking mechanism is an important enabling component of relay satellite for continuous real-time tracking of medium and low orbit target spacecraft. Facing the major demand of advanced technology development in aerospace engineering, this paper mainly tackles the problem on the rigid-flexible coupling dynamic modeling and dynamic characteristics analysis of a novel parallel tracking mechanism. Firstly, adopting a novel shape function of three-dimensional isoparametric beam element with curvature, the elasto-dynamic model is derived with floating coordinate method and Lagrangian equation. Then the natural frequencies of the system are investigated and ANSYS software is utilized to verify the model. An efficient numerical solution strategy consisting of the implicit integration algorithm and Baumgarte stabilization technology is employed to precisely evaluate the dynamic response of the system. Meanwhile, trajectory planning is carried out for high tracking precision. Furthermore, the validity of the numerical results of dynamic response is illustrated by simulations with the ADAMS software. Finally, the natural frequency sensitivity analysis is implemented by direct differential method with a novel sensitivity index to identify the structure weakness of the mechanism. A novel design idea for the cross-sectional parameters of each flexible beam is proposed with the sensitivity index. Above mentioned research results enrich the basic theoretical researches of parallel mechanism and lay a theoretical foundation for the practical application of tracking mechanism.

Design and analysis of high-frequency fiber Bragg grating vibration sensor

The fiber Bragg grating vibration sensor has received a lot of attention due to its unique performance. However, the natural frequency of the sensor limits its development and application. This paper presents an acceleration sensor with an isosceles triangular cantilever and mass block structure, which has high natural frequency and high sensitivity. The optical fiber sensing system consists of two fiber gratings. One section of the fiber grating is pasted on the cantilever and serves as an accelerometer; the other is used as a reference grating. The first-order natural frequency of the accelerometer is 8356 Hz, and its sensitivity is 0.46 pm g−1. A vibration test confirms the reliability of the sensor, and the resonant frequency of the sensor is 8193 Hz. The principle of light intensity demodulation is adopted in the sensor system and its sensitivity is 45.82 mV g−1 with a linearity of 0.997. The sensor has the advantages of high frequency and sensitivity, and thus can be applied to the rail transit bearing systems

Uncertainty propagation in dynamics of composite plates: A semi-analytical non-sampling-based approach

In this study, the influences of spatially varying stochastic properties on free vibration analysis of composite plates were investigated via development of a new approach named the deterministic-stochastic Galerkin-based semi-analytical method. The material properties including tensile modulus, shear modulus, and density of the plate were assumed to be spatially varying and uncertain. Gaussian fields with first-order Markov kernels were utilized to define the aforementioned material properties. The stochastic fields were decomposed via application of the Karhunen-Loeve theorem. A first-order shear deformation theory was assumed, following which the displacement field was defined using admissible trigonometric modes to derive the potential and kinetic energies. The stochastic equations of motion of the plate were obtained using the variational principle. The deterministic-stochastic Galerkin-based method was utilized to find the probability space of natural frequencies, and the corresponding mode shapes of the plate were determined using a polynomial chaos approach. The proposed method significantly reduced the size of the mathematical models of the structure, which is very useful for enhancing the computational efficiency of stochastic simulations. The methodology was verified using a stochastic finite element method and the available results in literature. The sensitivity of natural frequencies and corresponding mode shapes due to the uncertainty of material properties was investigated, and the results indicated that the higher-order modes are more sensitive to uncertainty propagation in spatially varying properties.

Assessment of bridge natural frequency as an indicator of scour using centrifuge modelling

One of the most prevalent causes of bridge failure around the world is “scour”—the gradual erosion of soil around a bridge foundation due to fast-flowing water. A reliable technique for monitoring scour would help bridge engineers take timely countermeasures to safeguard against failure. Although vibration-based techniques for monitoring structural damage have had limited success, primarily due to insufficient sensitivity, these have tended to focus on the detection of local damage. High natural frequency sensitivity has recently been reported for scour damage. Previous experiments to investigate this have been limited as a result of the cost of full-scale testing and the fact that scaled-down soil-structure models tested outside a centrifuge do not adequately simulate full-scale behaviour. This paper describes the development of what is believed to be the first-ever centrifuge-testing programme to establish the sensitivity of bridge natural frequency to scour. A 1/60 scale model of a two-span integral bridge with 15 m spans was tested at varying levels of scour. For the fundamental mode of vibration, these tests found up to a 40% variation in natural frequency for 30% loss of embedment. Models of three other types of foundation, which represent a shallow pad foundation, a deep pile bent and a deep monopile, were also tested in the centrifuge at different scour levels. The shallow foundation model showed lower frequency sensitivity to scour than the deep foundation models. Another important finding is that the frequency sensitivity to “global scour” is slightly higher than the sensitivity to “local scour”, for all foundation types. The level of frequency sensitivity (3.1–44% per scour depth equivalent to 30% of embedment of scour) detected in this experiment demonstrates the potential for using natural frequency as an indicator of both local and global scour of bridges, particularly those with deep foundations.

Symmetry properties of natural frequency and mode shape sensitivities in symmetric structures

When updating a finite element (FE) model to match the measured properties of its corresponding structure, the sensitivities of FE model outputs to parameter changes are of significant interest. These sensitivities form the core of sensitivity-based model updating algorithms, but they are also used for developing reduced parametrizations, such as in subset selection and clustering. In this work, the sensitivities of natural frequencies and mode shapes are studied for structures having at least one plane of reflectional symmetry and distinct natural frequencies. It is first shown that the mode shapes of these structures are either symmetric and anti-symmetric, which is used to prove that natural frequency sensitivities are equal for symmetric parameters. Conversely, mode shape sensitivities are shown to be unequal for symmetric parameters, as measured by cosine distance. These topics are explored with a small numerical example, where it is noted that mode shape sensitivities for symmetric parameters exhibit similar properties to asymmetric parameters.

Numerical estimation of the influence of joint stiffness on free vibrations of frame structures via the scattering of waves at elastic joints

In the structural design of mechanical products, natural frequencies must be controlled to reduce noise and vibration. In particular, the stiffness of the joints which assemble the structural components affects the natural frequencies. Therefore, it is important to predict the influence of joint stiffness on natural frequencies. Generally, these effects are determined by iterative finite element analyses of assembled structural models. Because this results in high computational costs, the sensitivity of natural frequencies to joint stiffness should be determined by a different approach to make the structural design process more efficient. Therefore, this paper proposes the use of reflection and transmission coefficients of elastic joints to predict the dependency of natural frequencies on joint stiffness. First, we formulate the reflection and transmission coefficients of joint stiffness, and then organize the coefficients using a ray tracing method. These formulations enable us to discuss the mechanisms which determine the natural frequency of a structure based on a wave approach using the phase-closure principle. Therefore, by applying the phase-closure principle to the frame structure, we investigate the formation of bending modes, which suggests that the effects of joint stiffness on natural frequencies correspond to the dependence of the reflection and transmission coefficients on joint stiffness. Therefore, these coefficients are useful indicators for estimating the influence of joint stiffness.