Equivalent damping characteristics of Fe-Mn alloy helical spring

Abstract

Aiming at the low damping energy dissipation characteristics of ordinary helical spring, the helical spring was prepared by Fe-Mn damping alloy with energy dissipation effect, and its equivalent damping effect was studied. Firstly, the influencing factors of damping energy dissipation of Fe-Mn alloy spiral spring are analyzed by orthogonal cyclic loading experiment and digital image recognition (DIC) technology. Based on the principle of energy conservation, the mechanical model of damping spring is established, and the equivalent damping ratio of Fe-Mn alloy spiral spring is deduced to follow the law of nonlinear exponential change. At the same time, the mechanical properties and energy dissipation characteristics of Fe-Mn helical spring are simulated and compared by finite element method. The results show that the equivalent damping ratio of the Fe-Mn alloy spiral spring is not only related to the geometric parameters of the damping alloy material and the spring, but also positively correlated with the equivalent strain of the spring under the external load. These studies have laid a theoretical foundation for the further research and application of Fe-Mn spiral spring.

Keywords

Fe-Mn alloy damping performance Fe-Mn alloy helical spring energy consumption characteristics

Introduction

As one of the most common engineering components, helical spring is widely used in the field of vibration isolation such as mechanical equipment, aerospace, and building structure because of its vibration reduction, buffering, and high rebound characteristics.¹ However, because ordinary helical springs do not have damping characteristics, they usually need to be compensated by combining external damping devices, which not only increases the complexity and cost of the product, but also makes it difficult to meet the application requirements in some scenes with limited use space and harsh environment. Therefore, it is necessary to develop a helical spring with damping effect.

Damping alloy is a new type of metal functional material that utilizes the internal friction mechanism of the material itself to convert external vibration energy into heat energy for dissipation.² Using its prepared spring as a vibration isolation unit has gradually become a research hotspot. Wang et al.³ designed a six-degree-of-freedom spring vibration isolation system using a rectangular cross-section helical spring made of manganese-copper damping alloy. The vibration isolation effect was verified by simulation analysis and experimental test, which effectively isolated the influence of the micro-vibration of the reaction wheel on the satellite imaging quality. Zhu et al.⁴ designed a manganese-copper damping alloy disc spring buffer using a disc spring made of manganese-copper damping alloy. The vibration isolation effect was verified by simulation analysis and experimental test. The results show that the manganese copper damping alloy disc spring buffer has better vibration isolation effect than the original spring buffer. Xintai et al.⁵ used Fe-Mn damping alloy wire to make a kind of Fe-Mn metal rubber, and analyzed the influence of different heat treatment states on the mechanical properties of Fe-Mn metal rubber, and compared it with 304 stainless steel metal rubber. The results show that Fe-Mn metal rubber has the dual effects of metal wire dry friction energy dissipation and material microscopic energy dissipation, and its damping performance is better than that of 304 stainless steel metal rubber. In order to reduce the adverse response of the structure, Lu et al.⁶ proposed an improved form of traditional tuned vibration absorber based on shape memory alloy springs, called shape memory alloy-spring tuned vibration absorber. The nonlinear behavior of shape memory alloy is verified by experiments, and the vibration reduction excitation under seismic excitation and the nonlinear hysteresis performance of shape memory alloy are analyzed. It can be seen that the damping alloy spring has significant energy dissipation effect and wide application prospect in practical engineering.

Based on the research foundation of the application of damping alloy spring in vibration reduction and isolation, the preparation of energy-dissipating spiral spring by Fe-Mn damping alloy shows significant innovation, technical advantages, and engineering necessity. Its core innovation lies in the integration of the intrinsic high damping characteristics of the material (derived from the micro-internal friction mechanism such as twin boundary movement and dislocation pinning) and the inherent friction energy consumption of the helical spring structure, forming a synergistic dual energy consumption mechanism, breaking through the traditional stainless steel spring. The limitation of single structure energy consumption. The material system can achieve precise design of stiffness, damping characteristics, and energy dissipation efficiency by adjusting alloy composition, heat treatment process (such as optimizing phase transformation and twin density), and spring geometric parameters (wire diameter, number of turns, pitch), and meet the customized vibration isolation requirements of high-frequency micro-vibration. Its advantages are reflected in the fact that the Fe-Mn damping alloy helical spring provides a promising integrated technical approach for high-performance vibration isolation in the field of precision engineering by virtue of the core innovation of material-structure collaborative energy consumption.

Therefore, the helical spring is prepared by Fe-Mn damping alloy, and its damping characteristics are studied. Because Fe-Mn alloy material has energy dissipation characteristics, it exhibits super-viscoelastic characteristics under certain strain, and its constitutive model needs to be established. In this paper, a series of mechanical properties tests are carried out to test its mechanical properties, and the material characteristic parameters are extracted according to the relevant numerical analysis theory. The DMA test simulation model is established to obtain more accurate simulation model parameters. Currently, in numerical simulation analysis of damping alloys, most of the linear viscoelastic models or simple viscosity models are used to simulate the damping characteristics of damping alloys. Huo et al.⁷ used the Nastran finite element software to establish a numerical control tool model, and defined the damping loss factor of M2052 alloy as a constant for numerical simulation. Sánchez Iglesias and Fernández López⁸ used Rayleigh damping to define the damping characteristics of materials, and verified the rationality of the definition of material parameters through hammer test and simulation analysis of corresponding working conditions. In related experiments and theoretical studies, damping alloys have obvious strain amplitude correlation characteristics, that is, under the same excitation frequency, the loss factors measured by different strain amplitude values show a certain regular change.^9–12 Therefore, linear viscoelastic models obviously cannot define the amplitude nonlinear characteristics of damping alloys. Generally, the viscoelastic material constitutive uses Maxwell fluid model, Kelvin solid model, three-parameter solid model, and generalized Maxwell model. This paper defines it as a nonlinear viscoelastic material, and uses the generalized Maxwell model to define the material parameters of the damping alloy.¹³

Previous studies have focused on the basic mechanical properties and applications of ordinary helical springs, but there are relatively few studies on the energy loss mechanism and mechanical properties of damping helical springs. Therefore, this paper focuses on the equivalent damping characteristics of Fe-Mn alloy spiral spring with energy dissipation effect.

Orthogonal test and analysis of equivalent damping ratio of Fe-Mn helical spring

The key influencing factors of the equivalent damping coefficient of Fe-Mn damping alloy helical spring are studied. The relationship between the geometric characteristic parameters of Fe-Mn helical spring, such as wire diameter, middle diameter, and deformation, and its equivalent stiffness, energy loss coefficient, and equivalent damping ratio was studied by orthogonal test. As shown in Figure 1, the Fe-Mn spiral spring loading and unloading cycle test diagram. The damping spring was subjected to five cycles of cyclic loading and unloading at a constant loading rate of 10 mm/min and displacement amplitudes of 10, 15, 20, and 25 mm to study the mechanical properties and energy dissipation characteristics. After preloading deformation, the energy dissipation area of each ring of the damping spring increases significantly with the increase of the spring deformation, but the slope of the spring loading curve keeps coincident, indicating that the energy dissipation characteristics of the damping spring are positively correlated with its deformation, while the spring loading stiffness is approximately linear.

Figure 1.

Fe-Mn helical spring loading and unloading cycle test diagram.

Orthogonal test can significantly reduce the difficulty and frequency of experimental design, and avoid the interaction of multiple factors in the experimental results. The orthogonal design in the test can make the test more reliable and improve the scientific and reliability of the data. In the study of orthogonal test, the range analysis method is often used to explore the influence of various factors on the test results. Through this method, we can intuitively determine the primary and secondary order of the influence of each factor, and find the best level combination of each factor, so that the test results can be optimized. At the same time, the analysis of variance can effectively distinguish the significance of each factor on the target value.

The influence factors of equivalent damping ratio of Fe-Mn damping alloy helical spring were studied by L₉(3⁴) test with 4 factors and 3 levels of orthogonal design. The equivalent damping coefficient and the spring wire diameter d, the middle diameter D, and the deformation S are selected as the investigation factors. The key factors affecting the optimal equivalent damping ratio of Fe-Mn damping alloy helical spring were determined by visual analysis and variance analysis.

Factor level table: Taking the spring wire diameter d (factor A), the middle diameter D (factor B), and the deformation S (factor C) as the influencing factors, the experiment was designed with 4 factors and 3 levels. The factor level table is shown in Table 1.

Table 1.

Orthogonal test factor level table.

Level	Factors
	A wirediametermm	B mediumdiametermm	C deformationmm	D(blank)
1	10	60	15
2	12	70	20
3	14	80	25

The experimental design and results, with the equivalent damping coefficient of the damping spiral spring as the index, the orthogonal experimental design is carried out, and the results are shown in Table 2.

Table 2.

Orthogonal test design table and result analysis.

Test number	A	B	C	D	Equivalent damping coefficient (%)
1	1	1	1	1	2.305
2	1	2	2	2	2.322
3	1	3	3	3	2.079
4	2	1	2	3	2.716
5	2	2	3	1	3.142
6	2	3	1	2	1.739
7	3	1	3	2	4.022
8	3	2	1	3	2.147
9	3	3	2	1	2.200
k1	2.2351	3.0145	2.0635	2.5488
k2	2.5323	2.5367	2.4127	2.6944
k3	2.7898	2.0062	3.0810	2.3141
R	0.5547	1.0083	1.0175	0.3803

Intuitive analysis: Intuitive analysis of the sum of the equivalent damping coefficients of each factor at different levels. From Table 3, it can be obtained that R_C > R_B > R_A > R_D, so the influence of factors on the equivalent damping coefficient of damping helical spring is C > B > A. Factor C has a great influence on the equivalent damping ratio, and the influence of factor A is relatively small. The spring model with the maximum equivalent damping ratio is A₃B₁C₃, that is, the damping helical spring with 14 mm wire diameter, 60 mm medium diameter, and 25 mm deformation has the best equivalent damping coefficient.

Table 3.

Variance analysis table.

Source of variance	The sum of squared deviations	Degree of freedom	Mean square	F value	p Value	Prominence
A	0.462	2	0.231	2.093	0.323	p > 0.05
B	1.526	2	0.763	6.909	0.126	p > 0.05
C	1.604	2	0.802	7.260	0.121	p > 0.05
D (error)	0.221	2	0.110

The variance analysis is shown in Table 3: The results of variance analysis show that factors A, B, and C have significant differences in equivalent damping ratio. The sum of squared deviations of the equivalent damping ratio of the Fe-Mn damping alloy spiral spring is C > B > A, indicating that the deformation, pitch diameter, and wire diameter of the spring are gradually weakened in terms of the importance of affecting the equivalent damping ratio. However, the deviation probability p values are all greater than 0.05, indicating that the deformation, middle diameter, and wire diameter of the spring are significant factors for its equivalent damping ratio.

The influence of different pitch diameters on the performance of Fe-Mn helical spring is shown in Figure 2. The loading and unloading mechanical behaviors of three groups of helical spring specimens with a wire diameter of 12 mm and a middle diameter of 60, 70, and 80 mm were compared. The results show that the middle diameter significantly affects the stiffness and energy dissipation characteristics of the steel wire: the stiffness increases with the decrease of the middle diameter (when the middle diameter is 60 mm, the initial elastic slope is the largest), and the high curvature of the small middle diameter enhances the ability of the material to resist deformation. In addition, because the damping characteristics of Fe-Mn alloy increase with the strain applied to the material, the smaller diameter helical spring bears greater strain under the same deformation, thus improving the damping performance of the alloy.

Figure 2.

Effect of different pitch diameters on the performance of Fe-Mn helical spring: (a) characteristic curve of 12 mm wire diameter under 20 mm deformation and (b) displacement-force histogram of 12 mm wire diameter spring.

Under the same displacement condition, the energy dissipation capacity of the Fe-Mn spring increases with the increase of the wire diameter, but decreases with the increase of the pitch diameter. This shows that the energy dissipation characteristics of the Fe-Mn spring are affected by the damping characteristics of the material and the geometric parameters of the spring. The damping performance of Fe-Mn spring can be improved by enhancing the damping characteristics of the material, increasing the wire diameter of the spring or reducing the middle diameter of the spring. When the deformation of the iron-manganese alloy helical spring with a wire diameter of 14 mm and a pitch circle diameter of 60 mm increases from 20 to 25 mm, the external work increases from 36,683.31 to 56,038.27 N mm, the dissipated energy increases from 6041.63 to 11,092.78 N mm, and the energy dissipation coefficient increases from 0.057 to 0.07, showing good energy dissipation capacity.

Figure 3 is the influence of different steel wire diameters on the performance of Fe-Mn spiral spring. The force-displacement responses of three groups of steel wire specimens with fixed middle diameter of 60 mm and wire diameter of 10, 12, and 14 mm during tensile process are compared. The results show that the increase of wire diameter significantly improves the strength and stiffness. The ultimate load of the spring sample with a wire diameter of 14 mm is the highest (≈3300 N), which is 83% higher than that of the spring sample with a wire diameter of 10 mm (≈1800 N). The initial elastic slope is the steepest (maximum stiffness), which increases the cross-sectional area and enhances the anti-deformation ability.

Figure 3.

Effect of different wire diameters on the performance of Fe-Mn helical spring: (a) characteristic curve of 60 mm diameter under 20 mm deformation and (b) displacement-force histogram of 60 mm medium diameter spring.

The energy consumption characteristics increase with the increase of wire diameter. Due to the same diameter and deformation, the energy dissipation performance of the Fe-Mn helical spring is positively correlated with the volume of the material participating in the effective deformation. Under larger wire diameter, the effective volume of Fe-Mn damping alloy wire diameter participating in deformation is larger, resulting in greater energy dissipation effect of Fe-Mn helical spring. Under the condition of fixed diameter (60 mm), the selection of wire diameter needs to balance the strength and deformation requirements. The wire diameter of 14 mm is suitable for high-load static structures (such as heavy-duty springs), while the wire diameter of 10 mm is more suitable for dynamic scenarios that require large deformation and energy absorption (such as shock absorbers), but the latter needs to be alert to the risk of premature failure under low load.

Figure 4 shows the relationship between the equivalent damping coefficient and deformation of Fe-Mn alloy helical springs with different specifications. The results show that the combination of the wire diameter (d) and the middle diameter (D) of the Fe-Mn helical spring significantly changes the damping characteristics of the spring. Figure 4(a) shows that when d = 10 mm, increasing the middle diameter (60 mm→80 mm) leads to an overall decrease in the equivalent damping ratio, especially in the displacement range of 4–10 mm. The damping ratio of the D = 60 mm spring (0.035) is 40% higher than that of the D = 80 mm spring (0.025), indicating that the smaller diameter enhances the energy dissipation capacity.

Figure 4.

Relationship between equivalent damping coefficient and deformation of Fe-Mn alloy helical spring with different specifications: (a) wire diameter of 10 mm, (b) wire diameter of 12 mm, and (c) wire diameter of 14 mm.

Figure 4(b) further verifies this rule: when d = 12 mm, the damping peak value of D = 60 mm spring (0.020) is still higher than that of D = 80 mm spring (0.015), but all curves tend to converge when the displacement is greater than 20 mm, indicating that the influence of geometric parameters is weakened under large deformation. The energy dissipation advantage of 60 mm diameter relative to 70 mm diameter is not significant in the whole deformation range, even below 70 mm diameter. This may be related to the consistency of spring sample processing. When d increases to 14 mm in Figure 4(c), the damping ratios corresponding to all pitch diameters decrease significantly (the peak value is only 0.010–0.015), and the gain effect of increasing pitch diameter on damping is weakened. In general, it can be concluded that the combination of larger steel wire diameter and smaller pitch diameter can optimize the damping performance, which has guiding significance for the design of damping spring.

Figure 5 shows the energy dissipation curve of the single cycle of the energy-consuming helical spring. In this curve, the closed-loop area enclosed by the OA loading curve and the unloading curve represents the loss energy during a loading and unloading cycle, which can be calculated by integral. The area below the OA loading curve represents the maximum deformation energy stored in one cycle, while the triangle area of the OA equivalent stiffness curve around the x-axis is represented by the red shadow area in the Figure, which represents the maximum elastic potential energy stored in one cycle. The hysteresis curve intuitively shows the energy loss of the damping alloy material under cyclic loading and unloading.

Figure 5.

Energy consumption spring loading and unloading characteristic curve.

When the helical spring is subjected to axial load, the spring produces axial deformation. The initial helix angle, middle diameter, and height of the helical spring before deformation are $α_{0}, D_{0} and h_{0}$ respectively. The lift angle, average middle diameter, and length of the spring after deformation are $α, D and h$ respectively. The wire diameter, effective number of turns, axial force, torque, and bending moment of the cross section of the spring are expressed as $d, n, F, T and M$ . The axial force $F$ of the spring along the loading direction can be obtained by the vector sum of torque and bending moment. The relationship between the axial deformation $Δ x$ of the helical spring and the helix angle $α$ can be calculated. The shear strain energy and bending normal strain energy of the spring material are obtained from the shear strain and normal strain equations of the helical spring wire section deduced by Wahl, which are the elastic strain energy $V_{(ε)}$ of the material.

γ (r) = \frac{2 r}{D_{0}} \cos α_{0} (\sin α - \sin α_{0})

(1)

ε (y) = \frac{2 y}{D_{0}} \cos α_{0} (\cos α_{0} - \cos α)

(2)

V_{(ε)} = \frac{1}{2} G (γ (r))^{2} + \frac{1}{2} E (ε (y))^{2}

(3)

In the formula, the shear modulus of the spring material, the spring modulus of the spring material, the polar moment of inertia of the spring material section, and the moment of inertia of the spring material section are expressed $G, E, I_{P} and I$ respectively. In order to simplify the calculation, the rise angle of the Fe-Mn helical spring is very small $α$ < 9°. When subjected to axial load, the bending deformation is small and can be ignored. The elastic strain energy of the spring material can be equivalent to the torque strain energy of the material. On the cross section of the spring, the torsional shear stress $τ_{(ρ)}$ at any point from the center of the circle is $ρ$ , and the spring strain energy per unit volume $v_{(ε)}$ is:

v_{(ε)} = \frac{{(τ_{(ρ)})}^{2}}{2 G} = \frac{128 F^{2} D^{2} ρ^{2}}{G π^{2} d^{8}}

(4)

$V$ represents the volume of the spring, $l$ represents the length along the spring line, $θ$ represents the polar angle of the spiral line, and takes the infinitesimal segment to integrate, then the strain energy $V_{(ε)}$ of the spring can be obtained:

V_{(ε)} = \int_{V} v_{(ε)} dV = \frac{4 F^{2} D^{3} n}{G d^{4}}

(5)

According to the analytical model and energy conservation of helical spring, the external force work W of helical spring is approximately equal to the elastic strain energy of spring material. For the Fe-Mn helical spring with energy dissipation characteristics, the work done by the external force should be equal to the sum of the elastic strain energy and dissipation energy $Δ W$ of the spring.

W = V_{(ε)} + Δ W = \frac{4 F^{2} D^{3} n}{kwG d^{4}} + Δ W

(6)

In the formula $kw$ , $F$ , d, $D$ , $E$ , $G$ , $n$ , and $x$ represent the curvature coefficient, loading equivalent force, wire diameter, middle diameter, elastic modulus, shear modulus, effective number of turns, and deformation of the spring, respectively. The elastic modulus of the Fe-Mn material was measured to be 182 GPa and the Poisson’s ratio was 0.25. By analyzing the geometric parameters and energy curve of Fe-Mn spring with wire diameter of 14 mm and middle diameter of 60 mm, the equivalent damping ratio of Fe-Mn alloy spiral spring follows the law of nonlinear exponential change, as shown in formula (8):

ζ = \frac{Δ W}{2 π U}

(7)

ζ = 877 \times \exp (\frac{ε}{12.3} \sqrt{\frac{2 E π^{2} D^{4} n^{2}}{k_{w}^{2} G d^{2}}}) - 1457.35

(8)

The finite element simulation analysis of the equivalent damping of Fe-Mn helical spring

Hyperelastic constitutive definition of Fe-Mn alloy material

In this paper, the alloy with Mn mass fraction of 17.77% and C mass fraction of 0.003% is selected, which is Fe-17.77Mn-0.003C. The main heat treatment process is to prepare steel ingot by vacuum induction furnace smelting. The hot forging temperature of the ingot is 1150°C, and the final forging temperature is controlled at 880°C. Then the forged alloy was hot rolled to a thickness of 10 mm. The hot-rolled alloy was heated at 900°C for 10 h and then cooled in water.

Three identical tensile specimens were used, with a diameter of (φ6 ± 0.1) mm, a total length of (150 ± 0.2) mm, and an extension spacing of (10 ± 0.1) mm. All specimens were placed in the test room 48 h before the start of the test, and the room temperature was maintained at about 23°C. As shown in Figure 6, the test was carried out with reference to the ISO 6892-1: 2019 standard, using an electronic universal tensile testing machine, and the tensile rate was controlled by strain. The tensile strain rate was 0.00025 s⁻¹, and the three tests were averaged. The axial tensile stress-strain data of the whole test were recorded.

Figure 6.

Tensile test of Fe-Mn damping alloy.

The uniaxial tensile test results are shown in Figure 7. The stress-strain curve before the yield of the damping alloy material is extracted, that is, the strain range in the Figure is less than 0.5%. It can be seen that the damping alloy exhibits certain nonlinear elastic characteristics in the elastic deformation stage. As the strain gradually increases, the increasing trend of stress gradually slows down.

Figure 7.

Stress-strain fitting curve.

Based on the uniaxial tensile test results, the elastic section test data of the stress-strain relationship are obtained and input into the uniaxial tensile test result table defined by Abaqus hyperelasticity. The existing hyperelastic model is used for parameter fitting. The Mooney-Rivlin model, Ogden model, Neo Hooke model, and Yeoh model were used to fit the experimental data. The results show that the root mean square fitting error of the Ogden model is the smallest. It can be seen from the fitting results that the fitting curve of the Ogden model (N = 4) is the closest to the stress-strain curve obtained by the test.

U = \sum_{i = 1}^{N} \frac{2 μ_{i}}{a_{i}^{2}} ({\bar{λ}}_{1}^{ai} + {\bar{λ}}_{2}^{ai} + {\bar{λ}}_{3}^{ai} - 3) + \sum_{i = 1}^{N} \frac{1}{D_{i}} {(J^{el} - 1)}^{2 i}

(9)

Where: i is the order, $μ_{i}$ , $α_{i}$ , and $D_{i}$ is the material parameter of the corresponding order, the hyperelastic nonlinear characteristics are defined. The Ogden model parameters obtained by fitting are shown in Table 4:

Table 4.

Ogden model parameter table.

Order(i)	μ(i)	α(i)
1	−6.54e+08	−6.4427
2	3.33e+08	−0.63618
3	6.3e+08	−18.5476
4	−3e+09	5.17e-05

Viscoelastic constitutive definition of Fe-Mn alloy material

Figure 8 is the DMA three-point bending test. The test sample is a strip cuboid. The effective size of the sample test is 20 mm long, (3.99 ± 0.01) mm wide and (0.77 ± 0.01) mm thick. The test was carried out with reference to the standard ISO 6721-5, and the variation of the loss factor and complex modulus of the damping alloy with strain and frequency was tested. The displacement amplitude is selected to be 5–100 µm (strain range: 0.006%–0.115%), and the step size is 5. For each fixed amplitude, frequency scanning was performed. The frequency scanning range was 0.5–100 Hz. The test was repeated three times and the results were averaged.

Figure 8.

DMA three-point bend mode test.

It can be seen from Figure 9 that the storage modulus of the damping alloy gradually decreases with the increase of frequency. When the frequency is lower than 100 Hz, the modulus of the damping alloy is less than 184 GPa. When the frequency is higher, the modulus increases more obviously. When the frequency is close to 10,000 Hz, the storage modulus is close to 182 GPa. The loss modulus of the damping alloy gradually decreases with the increase of frequency. When the frequency is low, the loss modulus is as low as 9.4 GPa. With the increase of frequency, when the frequency is 10,000 Hz, the loss modulus is about 7.4 GPa.

Figure 9.

Main curves of loss factor and modulus of damping alloy.

In this paper, the nonlinear viscoelastic Maxwell model in Abaqus is used to define the parameters of the damping alloy material. The material parameters of the Maxwell model cannot be directly obtained from the mechanical test results, but need to be based on the material parameters of the linear viscoelastic model. The initial Maxwell model parameters are obtained by parameter conversion, and the material parameters that define the nonlinear mechanical properties of the damping alloy are more accurate.^14,15 The parameters of linear viscoelasticity defined by Prony series can be obtained by the master curve of modulus with frequency obtained by DMA time-temperature equivalent test.^16,17 The mechanical model of the Prony series is the generalized Maxwell model, which is composed of a spring and multiple Maxwell models in parallel. The mechanical model is shown in Figure 10. The expression of relaxation modulus Prony coefficient is:

E (t) = E_{\infty} + \sum_{m = 1}^{M} E_{m} \exp (- t / ρ_{m})

(10)

In the formula, $E_{\infty}$ is the parallel spring modulus, $E_{m}$ is the spring modulus of the m-th Maxwell model, that is, the relaxation modulus of the corresponding order, $η_{m}$ is the spring viscosity of the m-th Maxwell model, M is the number of parallel Maxwell models, $ρ_{m}$ is the relaxation time.

Figure 10.

Generalized Maxwell model.

The relationship between the relaxation modulus $E_{m}$ of the material and the corresponding storage modulus $E' (ω)$ , loss modulus $E ″ (ω)$ , and complex modulus $E * (ω)$ is¹⁸:

E' (ω) = E_{\infty} + \sum_{m = 1}^{M} \frac{ω^{2} ρ_{m}^{2} E_{m}}{ω^{2} ρ_{m}^{2} + 1}, m = 1, \dots, M

(11)

E ″ (ω) = \sum_{m = 1}^{M} \frac{ω ρ_{m} E_{m}}{ω^{2} ρ_{m}^{2} + 1}, m = 1, \dots, M

(12)

E * (ω) = E' (ω) + iE ″ (ω)

(13)

In the formula: $ω$ represents the angular frequency, $E^{'} (ω)$ represents the storage modulus, $E_{\infty} = E^{'} (ω) |_{ω \to 0^{+}}$ , when the angular frequency (ω) approaches 0. According to the main curve results of the storage modulus $E^{'} (ω)$ and loss modulus $E^{″} (ω)$ predicted by the time-temperature equivalence principle, through the formulas (3) and (4), the configuration method is used to make the test value and the calculated value equal at the preset configuration point to calculate the relaxation modulus. The distance between the configuration points is usually selected on the logarithmic coordinate axis with a difference of 1.¹⁹ The angular frequency and relaxation time are generally determined by the following formula:

ω_{m} = 10^{(m - c)}, m = 1, 2, 3 \dots M

(14)

ρ_{m} = k \times 10^{(m - c)}, m = 1, 2, 3 \dots M

(15)

In the formula: k is a constant, usually 2, c is the order of Prony series, this paper takes five order. The coefficients of prony series at room temperature are fitted by MATLAB least square method, and the objective function is taken.

\min {F (g, τ)} = \sum_{i = 1}^{n} ({(\frac{{E^{'}}_{c a l}}{{E^{'}}_{D M A}} - 1)}^{2} + {(\frac{{E^{″}}_{c a l}}{{E^{″}}_{D M A}} - 1)}^{2})

(16)

When the frequency is $ω_{i}$ , the calculated values of the storage modulus and the loss modulus are $E'_{cal}$ and $E''_{cal}$ , respectively, and the measured values are $E'_{DMA}$ and $E''_{DMA}$ , respectively. n is the number of experimental data points. The fourth-order prony series of the damping alloy is obtained by fitting, as shown in Table 5:

Table 5.

Definition of initial linear viscoelastic parameters.

Order (I)	G (I)	τ (I)
1	0.3	0.01
2	0.15	1
3	0.036	10
4	0.02	100
5	0.11	100,000

Modeling and equivalent damping simulation analysis of Fe-Mn helical spring

An equal-size simulation model of iron-manganese alloy helical spring (wire diameter 12 mm, medium diameter 70 mm, effective turns 5, support ring 2, free height 130 mm) test was established. The establishment of the finite element model includes the steps of defining the element, material properties, meshing, and setting boundary conditions. In order to ensure the accuracy of the calculation results, the solid element with higher calculation accuracy is used to mesh the spring model, as shown in Figure 11. The irregular sections at both ends of the spring are divided by tetrahedral element type C3D4 with a length of 5 mm, and the regular section in the middle is divided by C3D8 hexahedron grid.

Figure 11.

Meshing of Fe-Mn alloy helical spring.

The upper and lower grinding flat ends of the Fe-Mn alloy helical spring are coupled at the center points P1 and P2, respectively. The lower endpoint P2 is fixed, and the upper endpoint P1 is loaded with a certain frequency and displacement amplitude. In addition, a frictionless hard contact is set on the cylindrical surface of the spring to prevent the spring wire from being pressed into and embedded, as shown in Figure 12. The harmonic excitation causes the corresponding deformation of the spring. After stabilizing the amplitude of the excitation force, the reaction force of the spring P2 and the displacement data of P1 are extracted, and the corresponding working conditions are simulated and analyzed by using different frequencies and displacement amplitudes.

Figure 12.

Coupling point of Fe-Mn alloy helical spring.

According to the vertical displacement diagram of the Fe-Mn alloy spiral spring in Figure 13, the deformation of the damping spring decreases with the increase of the effective number of turns from top to bottom. Because the upper and lower end planes are the supporting rings and do not participate in the deformation, it can be seen that the maximum deformation of the spring occurs on the inner side of the second ring, and the corresponding equivalent maximum strain is located in this ring. The wire diameter section of the helical spring has a certain torsional curvature with the change of the helix angle. It can be seen that the effective strain of the spring section shows a nonlinear decreasing trend from the inside to the outside.

Figure 13.

Vertical displacement diagram of Fe-Mn alloy helical spring.

From the equivalent strain diagram of the Fe-Mn alloy helical spring in Figure 14, the maximum strain measured in the maximum deformation position can be estimated by the probe to be 0.391%, and the corresponding outer surface strain of the spring is 0.148%. The energy dissipation characteristics of the damping spring are positively correlated with its strain amplitude. For the Fe-Mn damping spring of the simulation model, the DIC digital image recognition method is used to measure the indicated field strain of the spring surface under the corresponding deformation to verify the accuracy of the spring simulation definition. At the same time, in order to further analyze the hysteresis characteristic curve of the spring at different frequencies, the equivalent damping characteristic law is obtained.

Figure 14.

Equivalent strain diagram of Fe-Mn alloy spiral spring.

The main operation process of DIC is as follows: (1) Debug the camera equipment and lighting, accurately connect each line, and set the camera angle between 25° and 60° (25°–30° is the best). (2) When installing a 50 mm fixed-focus lens, the lens cover is removed at a distance of 0.5–1 m to ensure that the measured object occupies 80% of the total size in the image to obtain accurate pixels. (3) Focusing: adjust the manual focusing of the camera lens, use the focusing function in the software to calibrate, if the measured object is mostly purple, the focusing is successful. (4) FPS frame rate adjustment: The maximum frame rate is around 7500, that is, the number of photos that can be taken per second.

Speckle production needs to be reasonably arranged according to the material and size. Due to the small diameter of the spring wire, the speckle spraying is used, and the black and white coverage is about 50%, which is randomly distributed. Calibration: Before starting the experiment, you need to select the appropriate calibration plate and take about 20 sets of photos for calibration. These calibration photos need to be imported into VIC 3D, and the calibration requirements should be less than 0.01. Set the subset size to 25 and the step size to 2. After the calibration is successful, the acquisition and post-processing can be completed.

As shown in Figure 15, the DIC test of Fe-Mn alloy spiral spring, first of all, a high contrast random speckle field is made on the surface of the spring sample, and the same area of the specific speckle field image is collected by the double camera synchronously. After the acquisition is completed, the left and right camera image data are grouped according to the time series, and the same target point in each group (before and after the two-spoke deformation) image is correlated and analyzed. The DIC analysis area map of the Fe-Mn alloy spiral spring is selected, as shown in Figure 16 rectangular surface. The digital speckle images of the surface of the measured object before and after deformation are taken by the camera. By matching the pixel set and the gray value of the corresponding image in the speckle image before and after deformation, the displacement of each point on the surface of the measured object is obtained. Based on the obtained displacement information of the surface of the object, combined with the finite element analysis theory, the strain information of each point on the surface of the object is obtained.

Figure 15.

The DIC test image of Fe-Mn alloy helical spring.

Figure 16.

The DIC displacement image of Fe-Mn alloy helical spring.

Figure 17 is the displacement loading diagram of Fe-Mn alloy spiral spring. The cyclic loading and unloading tests with displacement amplitudes of 5, 10, 15, 20, and 25 mm were carried out at a loading rate of 20 mm/min. Because the strain of the inner ring of the spiral spring is large and the strain of the outer ring is small, the DIC equipment can only identify the surface strain of the spiral spring, and the rectangular area is taken from the second to the fourth circle from top to bottom for testing. Due to the large deformation of the upper effective ring, the strain and vertical deformation of the outer surface of the second ring area are analyzed. From the relationship between the surface strain and deformation of the Fe-Mn alloy helical spring in Figure 15, it can be seen that the deformation of the selected area of the second ring of the spring is 2.7, 5.6, 8.3, 11, and 13.7 mm with the loading step of 5–25 mm. As the spring deformation increases, the corresponding regional strain gradually increases, and the corresponding strains are 0.026%, 0.058%, 0.089%, 0.118%, and 0.151%, respectively.

Figure 17.

The change of main strain and deformation ofFe-Mn alloy helical spring.

Figure 18 is the DIC strain image of Fe-Mn alloy helical spring. When the vertical deformation of the spring is 20 mm, the strain amplitude on the surface of the spring is about 0.151%, which is close to the strain of 0.148% at the corresponding position of the simulation model, indicating the accuracy of the simulation model and the definition of material properties.

Figure 18.

The DIC strain image of Fe-Mn alloy helical spring.

Figure 19 shows the simulation and experimental comparison of the force-displacement hysteresis characteristics under the displacement amplitude of 15 and 20 mm, respectively. The linear marker takes 14-60-15 (14 is the wire diameter-60 is the middle diameter-15 is the displacement) as an example. The number represents that the wire diameter of the spring is 14 mm, the middle diameter is 60 mm, and the deformation is 15 mm. It can be seen that the loading and unloading curves of Fe-Mn damping spring do not coincide, and the simulation and test curves have good coincidence under small deformation. With the increase of deformation, there is a slight difference in the hysteresis angle between the two.

Figure 19.

Force-displacement hysteresis characteristic curve.

With the increase of deformation, the hysteresis angle of the two is slightly different. The hysteresis angle of the simulation model increases and the equivalent stiffness increases. This is mainly because Fe-Mn damping alloy is a viscoelastic material, and its hysteresis angle is not only related to the strain amplitude, but also to the strain rate, temperature, and other factors. Therefore, further research is needed in this area.

When the deformation is 20 mm, the dissipation energy of the comparison test curve is 6041.63 N mm, while the dissipation energy of the simulation curve is 6038.92 N mm, the error is less than 1%, the energy dissipation coefficient is 0.057, and the equivalent damping ratio is 0.029. Through the theoretical formula of the equivalent damping ratio of the iron-manganese alloy spiral spring, the equivalent damping ratio is calculated to be 0.03 when the deformation is 20 mm, which is consistent with the simulation results. The simulation results of the equivalent damping ratio of the iron-manganese alloy helical spring under different loading displacements are consistent with the experimental results, indicating its accuracy.

Conclusion

The equivalent damping ratio of Fe-Mn alloy helical spring (wire diameter is 14, middle diameter is 60) is studied, and the law is as follows:

$ζ = 877 \times \exp (\frac{ε}{12.3} \sqrt{\frac{2 E π^{2} D^{4} n^{2}}{k_{w}^{2} G d^{2}}}) - 1457.35$ This indicates that the damping characteristics of the Fe-Mn helical spring are related to the material modulus and geometric size, and the equivalent damping ratio follows the law of nonlinear exponential change, which can further guide the design of the spring.

The DIC digital image correlation technology and finite element simulation analysis were combined to compare the Fe-Mn helical spring. The results show that the equivalent damping ratio of the Fe-Mn alloy spiral spring is not only related to the geometric parameters of the damping alloy material and the spring, but also positively correlated with the equivalent strain of the spring under the external load. The equivalent strain obtained by the simulation analysis is consistent with the theoretical derivation, which shows the accuracy of the derivation.

Supplemental Material

sj-xlsx-1-ade-10.1177_16878132251366135 – Supplemental material for Equivalent damping characteristics of Fe-Mn alloy helical spring

Supplemental material, sj-xlsx-1-ade-10.1177_16878132251366135 for Equivalent damping characteristics of Fe-Mn alloy helical spring by Tu Tiangang, Yang Weitao, Yang Qi and Xu Bin in Advances in Mechanical Engineering

Footnotes

Handling Editor: Sharmili Pandian

ORCID iD

Tu Tiangang

Consent to participate

Informed consent was obtained from all individual participants included in the study.

Author contributions

Conceptualization: Yang Qi. Data curation: Yang Weitao. Formal analysis: Tu Tiangang. Methodology: Xu Bin. Writing – original draft: Tu Tiangang. Writing – review & editing: Yang Qi, Xu Bin.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Shanghai central guide local science and technology development funds (phase III) (YDZX20233100002004) funded projects.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

All data that support the findings of this study are included in this manuscript and its supplementary information files.

Supplemental material

Supplemental material for this article is available online.

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