Viscoelastic materials and dampers are a kind of passive energy dissipation techniques, which are widely used for vibration isolation and suppression in the fields of aerospace, mechanical engineering, precision instruments, and civil engineering. Rao (2003) introduced the noise control and vibration isolation technology with special treated viscoelastic laminates and spray paints and its application in vehicles and commercial airplanes. Rashid and Nicolescu (2008) developed a tuned viscoelastic damper for the unwanted vibration control of a workpiece on a palletized workholding system in milling operations. The tuned viscoelastic damper has high damping performance over a wide range of excitation frequencies, and can effectively reducethe vibration amplitudes during the milling process. Xu Z. D. et al. (2019) utilized a new kind of vibration isolation and mitigation system with high damping viscoelastic materials for reducing dynamic responses of a platform structure. The simulation results show that the system can significantly reduce the dynamic responses of the platform. Xu (2007) and Tsai and Lee (1993) applied the viscoelastic dampers in civil engineering to control the seismic behaviors of the reinforced concrete frame structures and high-rise buildings, respectively. The mathematical models for viscoelastic materials are investigated, and the effectiveness of the viscoelastic dampers are verified with dynamic experiments.

In practical applications, viscoelastic dampers are always added to the building structures or equipment, and work together to reduce the vibration responses. The dynamic properties, structure design and applications of viscoelastic dampers have been extensively studied by scholars. Min et al. (2004) carried out the seismic experiments of a 5-story full-scale steel structure model with added viscoelastic dampers, and presented a design process of the viscoelastic dampers by using the modal strain energy method. Samali and Kwok (1995) summarized the usage of viscoelastic dampers in building structures, and identified the factors affecting the dynamic performance and design procedure of viscoelastic dampers. Matsagar and Jangid (2005) investigated the seismic behaviors of multi-storied base-isolated structures with various types of isolation systems. The structures were also connected to the adjacent base-isolated or base-fixed structures by using viscoelastic dampers. The governing equations of motions for the structures were derived and solved with the Newmark's step-by-step method. The viscoelastic damper connection is found to be effective and useful in upgrading the seismic performance of the combined structures. Xu et al. (2003, 2004) introduced the simplex method to optimize the design parameters and locations of viscoelastic dampers, and conducted shaking table tests about reinforced concrete structures with viscoelastic dampers to validate the efficiency of the simplex method.

Viscoelastic materials are the main components of the viscoelastic damper, and the mechanical properties of the viscoelastic damper are greatly affected by the damping performance of the viscoelastic materials. Therefore, it is necessary to study the dynamic properties of viscoelastic materials. The appropriate selection of mathematical models is the basis for the investigation of energy dissipation and material application of viscoelastic materials. The classical models, such as the Kelvin model, Maxwell model, generalized viscoelastic models et al. (Christensen, 1971), can well describe the dynamic properties of the viscoelastic materials with varying frequencies. Based on classical models, Payne (1963) found that the amplitude of loading displacement had a significant influence on the dynamic properties of viscoelastic materials and proposed the Krous model to capture the displacement effect. Drozdov and Dorfmann (2002) studied the fracture and reformation phenomenon of the polymer molecular chains and formulated the viscoelastic constitutive relations of the rubber polymers considering the temperature influence.

The theory of fractional derivative is a concept from mathematical field, which unifies and generalizes classical calculus for non-integer order of derivation. After the first introduction by Abel in the last decades of nineteenth century, the fractional derivative was successfully applied in many areas such as heat conduction, diffusion, viscoelasticity, and mechanics of solids, control theory, and electricity (Caputo, 1974; Bagley and Torvik, 1983; Koeller, 1984; Rossikhin and Shitikova, 1997). Pritz (2003) and Schiessel et al. (1995) utilized the fractional derivative to study the viscoelastic materials and obtained the five-parameter fractional derivative model and generalized fractional derivative mathematical model. Poojary and Gangadharan (2018) introduced fractional calculation to modify the traditional viscoelastic theories and used the fractional Maxwell model to describe the viscous behavior of magnetorheological elastomers. Liu and Xu (2006) adopt the higher-order fractional derivative model to discuss the rheological properties of human bones, and the test and numerical results show that the higher-order fractional derivative model is successful and efficient in describing the viscoelasticity of human tissues. Xu et al. (2014, 2015, 2016); Xu Y. S. et al. (2019) combined the fractional derivative mathematical models and the temperature-frequency equivalent theory to characterize the effects of ambient temperature and frequency on dynamic performance of viscoelastic dampers.

It can be seen that there are few studies on the dynamic properties of viscoelastic materials and dampers at extreme low temperature with fractional derivative. In the present paper, several dynamic properties tests of the viscoelastic damper at −5°C are conducted under different frequencies and displacements. The seven-parameter fractional derivative model is modified with the temperature-frequency equivalent principle and utilized to describe the dynamic behaviors of the viscoelastic damper. The 9050A and ZN22 viscoelastic materials are used to verify the modified seven-parameter fractional derivative model. The experimental and numerical results show that the viscoelastic damper has perfect energy dissipation at low temperature (−5°C), and the modified seven-parameter fractional derivative model can well capture the dynamic behavior of viscoelastic materials and dampers under different frequencies and temperatures.

To investigate the mechanical properties of the viscoelastic damper at low temperature, the dynamic properties tests under different excitation frequencies and displacement amplitudes with −5°C are carried out and analyzed. The results show that the viscoelastic damper has perfect energy dissipation capacity, and the dynamic properties are increasing with excitation frequency and reduces when the displacement amplitude increases.

The seven-parameter fractional derivative model include three parallel elements, one Hook spring element and two fractional Maxwell models (Müller et al., 2011), as shown in Figure 6. The stress-strain relation of the Hook spring can be given as

where μ₀ presents the modulus of the Hook spring element. For the two fractional Maxwell models

where μ_i and η_i present the modulus of the spring and the damping coefficient of the fractional dashpot for the i-th fractional Maxwell model, respectively, i = 1, 2; Dαi denotes the α_i-order fractional derivative, and 0 < α_i < 1; σ_i and ε_i denote the stress and strain of i-th fractional Maxwell model, respectively; ε_is andε_id present the strain of the spring and dashpot, respectively. Then we have

where σ_t and ε_t are the stress and strain of the seven-parameter fractional derivative model.

From Equations (5) and (6), we can obtain

By performing the Fourier transform on Equation (9), we can get

where ω is the angular frequency of the loading stress or strain; and j is the unit complex number; and the star symbol denotes that the strain and stress are in complex form. Then the modulus of each fractional Maxwell model in complex form, expressed in the frequency domain, can be obtained as

Together with Equations (4), (7), (8), and (11), we can obtain the complex modulus of the seven-parameter fractional derivative model

By applying the relation jαi=cos(αiπ2)+sin(αiπ2)j into Equation (12), the complex modulus can be decomposed into two parts, the real part and the imaginary part, which are defined as the storage modulus and loss modulus of viscoelastic materials, and the ratio of the loss modulus to the storage modulus is taken as the loss factor, then

Where G₁, G₂, and η denote the storage modulus, loss modulus and loss factor of viscoelastic materials.

There is an equivalent relationship between high temperature and low frequency for most viscoelastic materials when the temperature is from the glass transition temperature Tg to Tg⁺ 100°C, which can be described by the temperature-frequency equivalent theory (Xu et al., 2014, 2015, 2016; Xu Y. S. et al., 2019), as shown in Equation (16)

where T₀ is the reference temperature, and α_T is the function of temperature and has the form

when this theory is used to describe the dynamic properties of the viscoelastic damper with different temperatures, the dynamic parameters of the viscoelastic damper in Equations (13)–(15) can be changed as

Equations (17)–(20) are the formulations of the modified seven-parameter fractional derivative model. The advantage of this modified model is that it is more efficient in describing the nonlinear behavior of viscoelastic materials (Müller et al., 2011), and it can reveal the influence of different temperatures and frequencies on the mechanical properties of viscoelastic materials at the same time.

In this section, in order to deeply investigate the influence of frequency on the dynamic properties of the viscoelastic damper at low temperature (−5°C), the abovementioned mathematical model, the modified seven-parameter fractional derivative model, and the equivalent fractional Kelvin model (Xu et al., 2015) are employed to numerically calculate the storage modulus and loss factor of the viscoelastic damper. The expression of the equivalent fractional Kelvin model has the form

where q₀ and q₁ are the coefficients related to the viscoelastic materials, γ is the order of fractional derivative, α_T has been given in Equation (17).

Because Equations (17)–(22) in the two models could not describe the displacement amplitude influence, the impact of displacement amplitude on dynamic properties of viscoelastic dampers is ignored, and only parts of the test data (with displacement 0.2 and 1.5 mm) in Table 2 are used to determine the parameters of the modified seven-parameter fractional derivative model and the equivalent fractional Kelvin model with the least squares method. Then for the modified seven-parameter fractional derivative model, the parameters can be obtained as, μ0=2.8954×105, μ1=4.4113×107, η1=1.5075×106, α₁ = 0.2054, μ2=2.1123×108, η2=1.0714×106, α₂ = 0.7104, and T₀ = −16.89°C. For the equivalent fractional Kelvin model, q0=9.3111×105, q1=1.831×107, γ = 0.5516, and T₀ = −114.15°C. The test data with the displacement 1.0 mm (which are not used for the parameters determination) are used to verify the numerical results of the proposed model. The experimental and numerical results comparisons of the modified seven-parameter fractional derivative model and the equivalent fractional Kelvin model of the viscoelastic damper are summarized in Table 3 and graphed in Figure 7.

It can be obviously seen in Figure 7 that both the modified seven-parameter fractional derivative model and equivalent fractional Kelvin model have perfect accuracy in describing the characteristic parameters of the viscoelastic damper with different frequencies. The errors of both models for storage modulus and loss factor are <10%, but the errors of the modified seven-parameter fractional derivative model always have smaller values. For example, when the frequency is 1 Hz, the storage modulus and loss factor of the viscoelastic damper in test data are 3.0626 MPa and 0.9186; while for the numerical results from the modified seven-parameter fractional derivative model, the storage modulus and loss factor are 3.2062 MPa and 0.8836, and the errors are 4.69 and 3.52%, respectively; for equivalent fractional Kelvin model, the storage modulus and loss factor are 3.3053 MPa and 0.845, the errors are 7.83 and 8.02%, respectively.

To further verify the accuracy of the modified seven-parameter fractional derivative model with different temperatures, the experimental data (Xu et al., 2015) for 9050 A and ZN22 viscoelastic materials under different frequencies and temperatures are compared with the numerical results calculated from the modified seven-parameter fractional derivative model and the equivalent fractional Kelvin model.

Some parts of the experimental data are used to evaluate the model parameters, and the numerical results are compared with the whole test results. With the least squares method, the model parameters of 9050A materials can be determined as, μ0=1.0563×106, μ1=4.6305×107, η₁ = 6.1554, α₁ = 0.9741, μ2=7.5296×108, η2=4.685×104, α₂ = 0.3144, and T₀ = 153.85°C for the modified seven-parameter fractional derivative model; and q0=1.9032×106, q₁ = 229.7502, γ = 0.6888, and T₀ = 164.57°C for the equivalent fractional Kelvin model. The model parameters of ZN22 materials can also be obtained as, μ0=2.1575×106, μ1=4.6317×108, η₁ = 0.0547, α₁ = 0.6682, μ2=3.9676×108, η₂ = 0.0733, α₂ = 0.6326, and T₀ = 272.2°C for the modified seven-parameter fractional derivative model; and q0=2.1972×106, q₁ = 0.1322, γ = 0.6575, and T₀ = 270.72°C for the equivalent fractional Kelvin model. The experimental and numerical results comparisons of 9050A and Zn22 viscoelastic materials are given in Tables 4, 5, and vividly graphed in Figures 8, 9. Figure 8 shows the experimental and numerical results comparisons of 9050A material.

It can be concluded from Figure 8A that the modified seven-parameter fractional derivative model is more accurate than the equivalent fractional Kelvin model when describing the storage modulus with different frequencies. Take 0.1 Hz as an example, the test data for storage modulus is 2.5 MPa; and the numerical results from the modified seven-parameter fractional derivative model is 2.3737 Mpa with error 5.05%; and the numerical results from the equivalent fractional Kelvin model are 2.369 Mpa with error 5.24%. It can be seen in Figure 8B that with the frequencies 0.2–1.0 Hz, the errors of the modified seven-parameter fractional derivative model is a little larger than the equivalent fractional Kelvin model when describing the loss factor with different frequencies, this may due to the complexity of viscoelastic materials at low temperatures, test errors, or the local distortion when determining the model parameters.

It also can be revealed from Figures 8C,D that the modified seven-parameter fractional derivative model is better than the equivalent fractional Kelvin model in capturing the variation trends of storage modulus and loss factor with different temperatures. Take −20°C as an example, the test data for storage modulus and loss factor are 17 MPa and 1.38; and the numerical results from the modified seven-parameter fractional derivative model are 15.8014 Mpa and 1.3002, with errors 7.05 and 5.78%; the numerical results from the equivalent fractional Kelvin model are 13.5023 Mpa and 1.6153, with errors 20.57 and 17.05%. It should be emphasized in Figure 8D that the numerical loss factor curve from the modified seven-parameter fractional derivative model has more consistency with the experimental loss factor curve than that from the equivalent fractional Kelvin model, which proves the advantage of the modified seven-parameter fractional derivative model in describing the nonlinear dynamic behaviors of viscoelastic materials, especially at low temperature. The same conclusion can be obtained from Figure 9 which compares the experimental and numerical results of ZN22 viscoelastic material.

In present paper, the dynamic properties tests of the viscoelastic damper are carried out at low temperature (−5°C). The influence of frequency and displacement on the dynamic properties of the viscoelastic damper are discussed. The seven-parameter fractional derivative model is modified and applied to describe the dynamic behaviors of the viscoelastic damper. The experimental data for 9050 A and ZN22 viscoelastic materials under different frequencies and temperatures are utilized to validate the modified seven-parameter fractional derivative model. Finally, some notable conclusions can be obtained as follows:

Publicly available datasets were analyzed in this study. This data can be found here: https://journals.sagepub.com/doi/pdf/10.1177/1077546313513604; https://journals.sagepub.com/doi/pdf/10.1177/1077546313513604/a.

YX and YD carried out the properties tests of the viscoelastic damper. XH formulated the mathematical model. YL and SZ conducted the application and verification of the mathematical model with viscoelastic dampers and materials. YX and YL wrote and checked the manuscript.