Abstract
Purpose
This paper aims to provide a new method to study and improve the dynamic characteristics of the four-column resistance strain force sensor through the elastomer structure design and optimization.
Design/methodology/approach
Based on the mechanism analysis method, the authors first present a dynamic characteristic model of the four-column resistance strain force sensors’ elastomer. Then, the authors verified and modified the model according to the Solidworks finite element simulation results. Finally, the authors designed and optimized two types of four-column elastomers based on the dynamic characteristic model and verified the improvement of sensor dynamic performance through a hammer knock dynamic experiment.
Findings
The Solidworks finite element simulation and hammer knock dynamic experiment results show that the relative error of the model is less than 10%, which confirms the accuracy of the model. The dynamic performance of the sensors based on the model can be improved by more than 30%, which is a great improvement in sensor dynamic performance.
Originality/value
The authors first present a dynamic characteristic model of the four-column elastomer and optimize the four-column sensors successfully based on the mechanism analysis method. And a new method to study and improve the dynamic characteristics of the resistance is provided.
Keywords
Citation
Chen, S., Lv, H., Zhao, Y. and Wang, M. (2024), "Dynamic characteristics optimization of elastomer for resistance strain force sensor", Sensor Review, Vol. 44 No. 4, pp. 405-413. https://doi.org/10.1108/SR-02-2022-0067
Publisher
:Emerald Publishing Limited
Copyright © 2024, Si Chen, Haoran Lv, Yinming Zhao and Minning Wang.
License
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial & non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode
1 Introduction
Modern sensor technology includes the development of sensitive materials (Tang et al., 2011; Ye et al., 2001), sensor design (De Vries, 2018; Zhou et al., 2013; Zhou and Luo, 2015; Ding, 2011), manufacture (XXXX, 2019; De Vries, 2018; Tang, 2016) and applications (Mu et al., 2020; Tang and Huo, 2020; Zhao, 2020), sensor technology has been widely used in industrial production (Cai, 2019; Sheng and Huang, 2014), environmental protection (Ma et al., 2018), resource investigation, medical diagnosis (Guo and Zhang, 2017), bioengineering, ocean exploration (Wu et al., 2020), aerospace, space exploration and other fields.
The column sensor often has the best comprehensive performance among the resistance strain force sensors (Zhang et al., 2012). Therefore, the four-column sensor is selected as the representative of the resistance strain force sensor to study the dynamic characteristic optimization method and related technology. The elastomer of four-column resistance strain force sensor is called four-column elastomer. The four-column elastomer has the greatest impact on the dynamic characteristics (Zhao et al., 2017). Natural frequency is the primary index for the study of system dynamic characteristics, which is generally accepted in the international research field (Xu et al., 2014; Li et al., 2017; Yang et al., 2017). Establishing a complete and accurate natural frequency model of four-column elastomer is the core and foundation of optimizing the dynamic characteristics of four-column sensor based on the above analysis.
The research of this subject is conducive to the development of four-column sensors in the dynamic measurement field and fills the gap in the dynamic characteristics design theory of resistance strain force sensors.
Researchers have proposed and applied the principle of force sensor optimization design to solve the optimal structural parameters. Chen improved the structural stiffness of the sensor to 3,000 N/m (Chen and Yuan, 1997); Wu improved the sensor sensitivity to 1.64 × 10−3 (Wu et al., 2011); Xie changed the six-dimensional force sensor to integral pretightening double-layer parallel structure, improved the static and dynamic performance of sensor (Liu, 2016); Liu obtained the minimum range of the measurement branch and the corresponding parameters according to the working condition function model (Xie, 2011). The researchers have established a physical model of the sensor, by changing the structural parameters, using the optimal method to find the limit value and change trend of the observed target. At the same time, some researchers have found that the four-column elastomer has stronger antibias load capacity and better dynamic characteristics, which lays a foundation for the study of this paper. However, there is no quantitative analysis of the dynamic performance of the parameters of the structure.
The research content can be summarized as follows:
We analyzed and studied the structural mechanism of the four-column elastomer and carried out the finite element simulation analysis by using solid works simulation software.
We established the natural frequency model and modified the model according to the results of theoretical research.
We carried out the optimal design of the four-column sensor and the sensor experiment, verified the accuracy of the established model and the improvement effect, completed the closed loop research of theory, simulation and experiment.
2 Modeling method
The modeling method is mechanism analysis method (Xu and Wen, 1990; Chen and Yuan, 1997), and the process is as follows:
We analyzed the four-column elastomer and get the equivalent mode of the main structure.
We studied the natural frequency of the equivalent model, stiffness distribution and mass distribution of four-column elastomer based on mechanism analysis method.
We deduced and calculated the equivalent beam model, and combined the analysis of stiffness distribution and mass distribution of four-column elastomer, so as to established the natural frequency model of four-column elastomer.
2.1 Natural frequency theory
Natural frequency is defined as the specific frequency determined by the system itself when the structural system moves under external excitation (Xu et al., 2014; Li et al., 2017; Yang et al., 2017). The natural frequency theory contains the following two important conclusions (Li et al., 2017; Yang et al., 2017), which are important theoretical basis for dynamic modeling of this subject:
The natural frequency of an object is the inherent property of the object, which is determined by its own properties (such as quality and material) and unconcerned with external conditions (such as force state, constraint state and space state).
The natural frequency of an object is only related to the stiffness distribution and mass distribution of the object, which is expressed as follows:
(1)
ω is the natural frequency of the object, kp is the equivalent stiffness distribution of the object and mp is the equivalent mass distribution of the object.
2.2 Equivalent model analysis
When modeling a four-column elastomer, the four columns, the upper-end faces and the lower-end faces are integrated structures. The lower-end face can be constrained as a fixed end, and the upper-end face can be stressed as a free end. Each column can be simplified as one end supported and the other end free beam as Figure 1 shows. “A” indicates the lower face and “B” the upper face.
From the modal simulation image of the four-column elastomer in Figure 2, it can be seen that the modal analysis results of the lower end face of the four-column elastomer are displayed in blue, indicating that the lower end face of the elastomer does not participate in the frequency response process. The results of the modal analysis of the upper-end face are displayed in red, indicating that the frequency response of the upper-end face of the elastomer is complex and unstable, and cannot be used as the main object for the establishment of the natural frequency model of the four-column elastomer. The results of the four-column modal analysis are shown in green, indicating that the natural frequency response of the four columns of the elastomer is concentrated and stable, which dominates the frequency response analysis and can be used as the main object for the establishment of the natural frequency model of the four-column elastomer. As the final structure which applied the external load, the dynamic performance of the four-column elastomer is mainly related to the dynamic performance of the columns and the columns can be used as the main object for the establishment of the four-column elastomer natural frequency model.
We four identical columns are integrated with the upper and lower end faces by studying the overall structure form and component distribution state of the four-column elastomer. Under this structure form, the four columns are equivalent to “parallel” with each other. Refer to the structure parallel problem in Hooke’s law (Zeng and Yang, 2024; Luo et al., 2015; Ma, 1965), the stiffness of the overall parallel system is the sum of the stiffness of each parallel structure (Zeng and Yang, 2024; Luo et al., 2015; Ma, 1965), the mass distribution of four-column elastomer can be regarded as the mass distribution of each column.
Based on the above analysis of stiffness distribution and mass distribution, it can be seen from the definition formula of natural frequency that the relationship between the overall natural frequency of four-column elastomer and each column is as follow:
The natural frequency of the whole four-column elastomer is about twice that of each column.
ωn is the natural frequency of the whole four-column elastomer, ω0 is the natural frequency of each column, kp is the overall stiffness distribution of the four-column elastomer, mp is the overall mass distribution of the four-column elastomer, k0 is the stiffness distribution of each column and m0 is the mass distribution of each column.
2.3 Establish the natural frequency model
The modeling method refers from the variational method of Hu Hai-chang (Hu, 1957), and is applied to the study of the natural frequency model of four-column elastomer column.
Make a linear elastic body vibrate naturally without external force, and the amplitude of the stress component is σx, σy,…, σz. The amplitude of the displacement component is u,v,w. The circumferential frequency is ω, and the expression of motion is as follows:
For the allowable state specified above, the extreme value of the formula in various allowable states corresponds to each order of natural frequency.
The minimum extreme value of the above formula is the first-order natural frequency of the object (Hu, 1957). And the natural frequencies of each order are the extreme values arranged from small to large (Hu, 1957).
After obtaining the natural frequency formula of general linear elastomer, we added the constraints of the columns’ equivalent beam model, the natural frequency model of the four-column elastomer is obtained as follows:
ρ is the material density of four-column elastomer (unit: kg/m3), E is the elastic modulus (unit: Pa), I is the section moment of inertia (unit: m4) and l is the height parameter of the elastomer.
According to the natural frequency definition, the overall natural frequency of the four-column elastomer body is about two times the natural frequency of each column. The formula of intrinsic angular frequency (unit: rad/s) is converted into the formula of intrinsic circular frequency (unit: Hz), and the rectangle of inertia of the section is expanded to obtain the natural frequency model of the four-column elastomer, which is shown in formula (6).
Here f is the natural frequency of the elastomer model; b is the length parameter of the elastomer model; h is the width parameter of the elastomer model.
3 Simulation and modified
We took the four-column elastomer with a measuring range of 20t as the object of initial simulation research and comparison. The initial structural dimensions are shown in Table 1.
The side parameter along the radial direction is the length parameter b, the side length parameter along the circumferential direction is the width parameter b the column spacing d is defined as the spacing between the centers of the columns on the opposite side. The struct parameters are shown in Figure 3.
3.1 Modify structure parameter
When some specific structure parameter changes, there is a great difference between the natural frequencies calculated by the model and simulation results, as Figure 4 shows.
Through the analysis of the stiffness distribution and the study on the effective structure parameter in the model. There is a correction formula under the condition when the width parameter and height parameter change.
Expression (7) means that when the widening of the stiffness distribution changes, the length and width of the four columns are exchanged. After the change of the width parameter h occurs, the stiffness distribution of the four columns of the four-column elastomer changes, that is, the widened column does not cause the change of the width parameter h but causes the change of the length parameter b.
Figure 4 shows that the results of the uncorrected four-column elastomer natural frequency model are very different from the simulation results. With a lot of calculation and simulation results, it is found that there is a correction coefficient between the natural frequency calculated by the natural frequency model of the four-column elastomer and the natural frequency obtained by the simulation, and the coefficient changes with the height change value. The coefficient is formally as shown in expression (8).
The results in Figure 5 show that the calculation results of the dynamic mathematical model are more accurate and the correction effect of the model is good.
We studied the change of natural frequency when the column spacing d changes. Combining simulation and theory, it can be seen that when the four columns move toward the center, the symmetrical structure of the four-column elastomer causes the stiffness distribution to change, and the effective length b decreases by half the distance approaching. A corrective formula that reflects the change in the spacing between four columns of a four-column elastomer such as Shown in expression (9).
Here b is the effective length, d is the column spacing, Δd is the changed spacing.
We studied the change of natural frequency when the column spacing d, the diameter R and the height H of the upper end face changes. The relationship between the natural frequency of the four-column elastomer and the stiffness distribution and mass is as follows:
Here ω1 and ω2 are the original natural frequencies and the natural frequencies after the structural parameters of the upper face are changed, respectively. kp is the stiffness distribution of the four-column elastomer before and after the upper-end face structural parameters changing. m1 and m2 are, respectively, the mass of the original elastomer and the mass of the elastomer after the structural parameters of the upper face changing.
Therefore, the natural frequency relationship of the four-column elastomer before and after the structural parameters of the upper-end face change is as follows:
According to the research of theory and simulation results, the correction expression reflecting the change of column spacing d, the diameter R and the height H of the upper-end face is as follow:
Here R and H are respectively the radius and thickness of the upper end face of the elastomer. b0, h0, l0, R0 and H0 are the initial structural dimensions.
The correction results are showed is Figure 6.
The results in Figure 6 show that the proposed correction methods make the calculation results of the model more accurate and verify the correction theory.
3.2 Summary
After the theory and simulation analysis, the changes of structural parameters related to the increase of natural frequency are as follows:
lifting the column length parameter b;
lifting the column width parameter h;
reduce the column height parameter l;
reduce the radius parameter R of the upper-end face; and
reduce the height parameter H f the upper-end face.
The theoretical research conclusion is consistent with the simulation research conclusion.
According to the four-column elastomer natural frequency model analysis and correction, the complete four-column elastomer natural frequency model is obtained:
The meaning of each parameter in the formula
ρ is the material density of four-column elastomer (unit: kg/m3);
E is the elastic modulus (unit: Pa);
b is the column length parameter (unit: m);
h is the column width parameter (unit: m);
l is the column height parameter (unit: m);
R is the upper-end face radius parameter (unit: m);
H is the upper-end face height parameter (unit: m); and
b0, h0, l0, R0 and H0 are the initial structural dimensions (unit: m).
Under general conditions, expression (13) is the first three-order natural frequency model function expression. For the high-order conditions, the coefficients can also be taken as 2.232, 3.77, 6.63 and 12.25 (Zeng and Yang, 2024; Luo et al., 2015; Ma, 1965). The equivalent beam mode is consistent with the column mode in the simulation study, which shows the correctness of the theoretical research.
4 Dynamic experiment
4.1 Purpose and steps
The aim of the experiment is to completed the closed loop research of theory, simulation and experiment.
Strictly speaking, the experimental object should be in a completely free state in the dynamic test and cannot be connected with other objects on any spatial coordinate axis, especially cannot have direct contact with the earth (AI and Cao, 2015). Under the existing experimental conditions, the experimental method can be realized by equivalent experiment, we placed the sensor system on the very soft and thick foamed plastic pads, which can be regarded as free state. This experimental method can realize the ideal dynamic test under the actual support conditions.
The experimental steps of four-column elastomer dynamic test are roughly divided into the following steps.
We patched and encapsulated the existing or optimized four-column elastomer, and connected to the test instrument system.
We placed the four-column elastomers on a thick foam plastic pad and placed the test instrument in the exclusive instrument box. Indirectly isolate the whole four-column sensor system from the earth.
We keep the position and direction of knocks consistent, knock three to five–5 times, record the cycle time of its response waveform curve.
We compared and analyzed the results of the model, simulation and experiment, analyzed the relative error, error sources and the limitations of research method.
4.2 Experiment and optimization design
The objects of experiment and structural optimization design are FC-10t sensor and FC-20t sensor. The optimal design schemes for the four-column elastomer are:
We keep the length parameter b and width parameter h unchanged so as to keep the coupling characteristics, flexural modulus and sensitivity of four-column elastomer.
We adjusted the column spacing to maximize the natural frequency of the four-column elastomer through simulation research.
We reduced the height parameter l, reduced the height parameter H of the upper end face and reduced the radius parameter R of the upper end face. These three methods to improve the dynamic characteristics of the four-column sensor are constrained by engineering practice and boundary conditions.
Based on the above analysis, the structural parameters of FC-10t and FC-20t four-column elastomer before and after optimization design are shown in Table 2.
During the test, we connected the four-column sensor to the test instrument FD-3,000 dynamic and static load measuring instrument. The dynamic test performance of the test instrument is more than ten times that of the sensor and can collect the dynamic information of the sensor perfectly. Its shape and technical indicators are shown in Figure 7 and Table 3.
The natural frequency and analysis of FC-10t four-column sensor before and after structural optimization are shown in Figure 8 and Tables 4 and 5.
The experiment results showed that, after the optimization, the natural frequency of FC-10t four-column elastomer increased 4081.2 Hz, the increase range is 49.38%.
We analyzed the experiment dynamic response waveform and extract the dynamic parameters, the rise time decreases, the peak time decreases slightly, the adjustment time remains basically unchanged and the overshoot time increases. The results showed that the rapidity and the amplitude frequency characteristics of the four-column elastomer FC-10t improve.
The natural frequency and analysis of FC-20t four-column sensor before and after structural optimization are shown in Figure 9 and Tables 6 and 7.
After optimized the FC-20t four-column sensor structural parameters, the experiment results showed that its natural frequency increased 3416.1 Hz and the increase range is 31.43%.
After optimization, the rise time decreases, the peak time decreases slightly, the adjustment time is basically unchanged and the overshoot increase. The dynamic performance improve, but its rapidity is lost. Overall, the amplitude frequency characteristics of the four-column elastomer FC-20t also improved.
5 Conclusions
We used the mechanism analysis method, the static and dynamic simulation analysis of finite element method and dynamic knocking method, studied and optimized the natural frequency and other dynamic properties of the four-column strain force sensor. We proposed the design theory of the four-column strain force sensor and researched the dynamic characteristics of four-column strain force sensor. We optimized the four-column sensors’ structure and improved the dynamic performance based on the research theory. On the basis of not having a great impact on its static performance, the natural frequency of FC-10t sensor improved from 8264.5–12345.7 Hz, and the rapidity of the sensor improved slightly. The natural frequency of FC-20t sensor improved from 10869.6– 14285.7 Hz, and the stability improved slightly. In general, the natural frequency and related dynamic performance improved more than 30%, and the optimization effect is verified by simulation and experiment, which completely verifies the research results and conclusions of the subject.
Figures
Figure 1
Schematic diagram of equivalent beam model
Figure 2
Modal and frequency analysis diagram of four-column elastomer
Figure 3
Structural parameters of four-column elastomer
Figure 4
(a) Comparison of uncorrected model when weight parameter change and (b) comparison of uncorrected model when height parameter change
Figure 5
(a) Comparison of corrected model when weight parameter change and (b) comparison of corrected model when height parameter change
Figure 6
(a) Comparison of corrected model when column spacing change; (b) Comparison of the upper-end face diameter change and (c) Comparison of the upper-end face height change
Figure 7
FD-3,000 dynamic and static load measuring instrument
Figure 8
(a) FC-10t four-column sensor and (b) FC-10t four-column sensor after structural optimization
Figure 9
(a) FC-20t four-column sensor and (b) FC-20t four-column sensor after structural optimization
Initial structural parameters
| Structural parameters | 20t |
|---|---|
| Cross section b × h (mm) | 15 × 15 |
| The column height l (mm) | 20 |
| Radius of upper-end face R (mm) | 52.5 |
| Height of upper-end face H (mm) | 20 |
| The column spacing d(mm) | 65 |
Table by author
Structural parameters of FC-10t and FC-20t
| Structural parameters | FC-10t | FC-20t | FC-10t (after optimization) | FC-20t (after optimization) |
|---|---|---|---|---|
| Cross section b × h (mm) | 8 × 8 | 11 × 11 | 8 × 8 | 11 × 11 |
| The column height l (mm) | 30 | 30 | 15 | 15 |
| Radius of upper-end face R (mm) | 27.5 | 27.5 | 23 | 23 |
| Height of upper-end face H (mm) | 18 | 18 | 7 | 7 |
| The column spacing d (mm) | 27 | 27 | 18 | 21 |
Table by author
FD-3,000 technical indicators
| Type specification | FD-3,000 |
|---|---|
| Channel | 4 |
| Accuracy level | Static force: 0.3 dynamic force 1 |
| Unit select | Force value, torque and other units |
| Sampling frequency | 500 kHz |
| Input signal | ±3.0 mV/V |
| Working temperature | −10 ∼ 50°C |
| Function | Real time loading force curve display and data analysis |
| Application | Fatigue testing machine verification, static force verification, impact force test and electric power wrench calibration |
| Overall dimension | 320 × 280 × 140mm |
| Weight | 2.5 kg |
Table by author
Result and analysis of FC-10t before and after structural optimization
| Sensor type | Experimental result (Hz) | Simulation result (Hz) | Model results (Hz) | Simulation relative error (%) | Model relative error (%) |
|---|---|---|---|---|---|
| Before optimization | 8264.5 | 9067.7 | 10,971 | 9.71 | 32.7 |
| After optimization | 12345.7 | 17,606 | 11637.4 | 42.6 | 6.09 |
Table by author
Dynamic characteristic parameters of FC-10t before and after structural optimization
| Dynamic characteristic parameters | Before optimization | After optimization |
|---|---|---|
| Rise time | 1.00 s | 0.01 s |
| Peak time | 1.00 s | 0.93 s |
| Adjustment time | 2.99 s | 2.99 s |
| Overshoot | 0 | 947.2% |
Table by author
Result and analysis of FC-20t before and after structural optimization
| Sensor type | Experimental result (Hz) | Simulation result (Hz) | Model results (Hz) | Simulation relative error (%) | Model relative error (%) |
|---|---|---|---|---|---|
| Before optimization | 10869.6 | 10,746 | 10177.5 | 1.04 | 6.36 |
| After optimization | 14285.7 | 22,076 | 13875.5 | 54.5 | 2.87 |
Table by author
Dynamic characteristic parameters of FC-20t before and after structural optimization
| Structural parameters | Before optimization | After optimization |
|---|---|---|
| Rise time | 0.05 s | 0.02 s |
| Peak time | 5.51 s | 5.00 s |
| Adjustment time | 5.99 s | 5.99 s |
| Overshoot | 10.54% | 189.4% |
Table by author
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Further reading
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Wei, R. Research on Dynamic Test Excitation Method of Multi-Dimensional Force Sensor, Anhui University of engineering.
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