Tire simulation method and tire simulation device

Abstract

Improve the accuracy of tire simulations. [Solution] A tire simulation method comprising: a viscoelastic model creation step of creating a viscoelastic model that shows the viscoelastic properties of the rubber material constituting the tire based on a master curve obtained by a dynamic viscoelastic test; an input step of inputting a modeled tire model with the viscoelastic model set; and a rolling step of bringing the tire model into contact with a road surface and rolling it on the road surface, characterized in that in the viscoelastic model creation step, the viscoelastic model is created using characteristic values ​​at frequencies of 10,000 Hz or less from the master curve obtained by the dynamic viscoelastic test.

Description

[Technical Field]

[0001] This invention relates to a tire simulation method and a tire simulation apparatus. [Background technology]

[0002] Previously, a tire simulation method was widely known that involved rolling a tire model on a virtual road surface and analyzing the tire's characteristics under rolling conditions. However, in tire models rolling on a virtual road surface, vibrations that would not occur in an actual tire rolling on a road surface may be calculated. Patent Document 1 discloses a method in which the frequency band in which such vibrations occur is analyzed in advance, and viscoelastic characteristic values ​​such as loss tangent are adjusted only in the frequency band in which such vibrations occur. [Prior art documents] [Patent Documents]

[0003] [Patent Document 1] Japanese Patent Publication No. 2024-71990 [Overview of the Initiative] [Problems that the invention aims to solve]

[0004] The method described in Patent Document 1 requires prior analysis of the frequency bands in which vibrations occur, resulting in a high workload for the simulation. The present invention aims to provide a tire simulation method that suppresses vibrations that cannot actually occur in tires rolling on a road surface, with less workload. As a result, the behavior of actual tires can be reproduced, and the accuracy of the simulation can be improved. [Means for solving the problem]

[0005] A tire simulation method according to one aspect of the present invention comprises: a viscoelastic model creation step of creating a viscoelastic model that shows the viscoelastic properties of the rubber members constituting the tire based on a master curve obtained by a dynamic viscoelastic test; an input step of inputting a tire model with the viscoelastic model set; and a rolling step of bringing the tire model into contact with a road surface and rolling it on the road surface, wherein in the viscoelastic model creation step, the viscoelastic model is created using characteristic values ​​at frequencies of 10,000 Hz or less from the master curve obtained by the dynamic viscoelastic test.

[0006] Furthermore, a tire simulation device according to one aspect of the present invention comprises: a viscoelastic model creation unit that creates a viscoelastic model showing the viscoelastic properties of the rubber members constituting the tire based on a master curve obtained by a dynamic viscoelastic test; a tire model input unit that inputs a tire model with the viscoelastic model set; and a rolling calculation unit that brings the tire model into contact with a road surface and rolls it on the road surface, wherein the viscoelastic model creation unit creates the viscoelastic model using characteristic values ​​at frequencies of 10,000 Hz or less from the master curve obtained by the dynamic viscoelastic test. [Effects of the Invention]

[0007] According to a tire simulation method in one aspect of the present invention, vibrations that cannot occur in actual tires rolling on a road surface can be suppressed with minimal effort. As a result, the behavior of actual tires can be reproduced, and the accuracy of the simulation can be improved. [Brief explanation of the drawing]

[0008] [Figure 1] This is a block diagram of a simulation device, which is an example of an embodiment. [Figure 2] This figure shows an example of a master curve measured by dynamic viscoelasticity testing. [Figure 3]This figure shows an example of a generalized Maxwell model constructed by the viscoelastic model creation unit based on the master curve shown in Figure 2. [Figure 4] This figure shows an example of the state during rolling in a simulation device, which is an example of an embodiment. [Figure 5] These figures show the behavior of vertical axial force when the master curve upper limit is set to 10,000 Hz and 1.0 × 10⁷ Hz, respectively. [Figure 6] This flowchart shows the processing procedure of a simulation method, which is an example of an embodiment. [Figure 7] This flowchart shows the processing procedure for the viscoelastic model creation step in a simulation method, which is an example of an embodiment. [Figure 8] This flowchart shows the processing procedure for the rolling step in a simulation method, which is one example of an embodiment. [Figure 9] This flowchart shows the processing procedure for the acoustic analysis step in a simulation method, which is one example of an embodiment. [Modes for carrying out the invention]

[0009] Previously, tire simulation methods were widely known that involved rolling a tire model on a road surface and analyzing the tire's characteristics under rolling conditions. For example, a simulation method is known that uses the tire's characteristics under rolling conditions (such as the shape of the tire surface) to analyze the sound emitted from a tire during driving. In such simulation methods, the behavior of the tire model under rolling conditions greatly affects the final analysis results of the emitted sound. In other words, reproducing the behavior of an actual tire under rolling conditions is extremely important for improving the accuracy of the simulation.

[0010] On the other hand, in a tire model that rolls on a road surface, vibrations that cannot occur in an actual tire rolling on the road surface may be calculated. In order to improve the accuracy of the simulation, it is necessary to set the tire model and the like so that such vibrations are not calculated.

[0011] In addition to the shape of the tire, a viscoelastic model showing the frequency dependence of the viscoelastic property values (storage elastic modulus, loss elastic modulus, etc.) of the constituent rubber members is set in the tire model. The viscoelastic model to be set preferably reflects the viscoelastic property values of the actual rubber members. For example, based on the master curve obtained by a dynamic viscoelasticity test using an actual rubber member, it is conceivable to construct and set a viscoelastic model such as a generalized Maxwell model.

[0012] Generally, after measuring the frequency dispersion of the viscoelastic property values of the rubber material at a plurality of temperatures with a viscoelasticity tester, the shift factor is obtained using the WLF equation (experimental equation (13) described in JIS K 6394), and the frequency dependence curve at each measurement temperature is shifted according to this shift factor to create the master curve. Therefore, usually, the master curve shows the frequency dependence of the viscoelastic property values in a wide frequency range such as 1.0×10

[0013] , , ~1.0×10 7 That is, it shows the frequency dependence of the viscoelastic property values in a wide frequency range.

[0013] As will be described in detail later, as a result of the study by the present inventors, it has been found that by constructing a viscoelastic model using only the viscoelastic property values at frequencies of 10,000 Hz or less among the master curves in the above wide frequency range, vibrations that cannot occur in an actual tire rolling on the road surface can be reduced. That is, according to this method, by simply changing the upper limit value of the frequency of the master curve when constructing the viscoelastic model, vibrations that cannot occur in an actual tire rolling on the road surface can be suppressed. Therefore, it can be said that the behavior of an actual tire can be reproduced with less man-hours and the accuracy of the simulation can be improved.

[0014] Hereinafter, an example of an embodiment of a tire simulation apparatus according to the present invention and a tire simulation method using the same will be described while referring to the drawings. The embodiment described below is merely an example, and the present invention is not limited to the following embodiment. Also, a configuration formed by selectively combining each component of the plurality of embodiments and modification examples described below is included in the present invention.

[0015] FIG. 1 is a block diagram of a simulation apparatus 1 which is an example of an embodiment. The simulation apparatus 1 includes a control device 10, an input device 20, and a display device 30. The input device 20 is an input interface for inputting information necessary for executing the simulation. An example of the input device 20 is a keyboard. The display device 30 is a display on which an input screen, an output screen such as a simulation result, etc. are generated. Examples of the display device 30 include a liquid crystal display and an organic EL display. The input device 20 and the display device 30 are connected to the control device 10.

[0016] The control device 10 includes a processor 11 and a memory 12. The processor 11 executes arithmetic processing for controlling the control device 10. The memory 12 stores a control program for controlling the processor 11 and is composed of, for example, a RAM, a ROM, a hard disk, etc. The control program includes a program for causing the control device 10 to execute a simulation method for creating a viscoelastic model and performing a rolling calculation. The control device 10 may be composed of one information processing device or may be composed of a plurality of information processing devices.

[0017] The control device 10 includes a viscoelastic model creation unit 13 that creates a viscoelastic model based on a master curve obtained by a dynamic viscoelastic test, a tire model input unit 14 that receives a tire model that models a tire, a rolling calculation unit 15 that brings the tire model into contact with the road surface and makes it roll on the road surface, and an acoustic analysis unit 16 that performs acoustic analysis based on the results calculated by the rolling calculation unit 15. In other words, the simulation device 1 of this embodiment is a simulation device that analyzes the sound radiated from a tire during driving using the tire characteristics in a rolling state.

[0018] The viscoelastic model creation unit 13 obtains a master curve previously determined by dynamic viscoelastic testing and creates a viscoelastic model. The calculation of the master curve can be performed using a known method. For example, as described above, the master curve is created by measuring the frequency dispersion of the viscoelastic properties of the rubber material at multiple temperatures using a viscoelastic testing machine, determining the shift factor using the WLF formula (empirical formula (13) described in JIS K 6394), and shifting the frequency-dependent curve at each measurement temperature according to this shift factor.

[0019] Figure 2 shows an example of a master curve measured by a dynamic viscoelasticity test. In Figure 2, the storage modulus and loss modulus are shown as viscoelastic property values. The master curve is created using the time-temperature superposition rule, so 1.0 × 10⁻⁶ -2 ~1.0×10 7 This shows the frequency dependence of viscoelastic properties over a wide frequency range.

[0020] The viscoelastic model creation unit 13 creates a viscoelastic model based on the acquired master curve. The viscoelastic model to be used can be appropriately set depending on the type of rubber material, etc., and examples include the generalized Maxwell model and the generalized Voigt model. Among these, the generalized Maxwell model is preferred. The viscoelastic model creation unit 13 creates a viscoelastic model for each rubber material that makes up the tire, such as the tread rubber, sidewall rubber, bead, belt, and carcass.

[0021] The generalized Maxwell model is a type of mathematical model used to describe the viscoelastic properties of materials, and it is a combination of multiple Maxwell models in parallel. The Maxwell model is a model that can simply describe both the viscous and elastic behavior of materials, and by generalizing it, it becomes possible to describe its behavior over a wide frequency range.

[0022] In the generalized Maxwell model, the storage modulus and loss modulus can be expressed as functions of angular frequency (ω) by the following equations (I) and (II), respectively. i θ represents the modulus of elasticity of each Maxwell unit, and τ i This shows the relaxation time for each Maxwell unit. Then, the parameters (modulus of elasticity and relaxation time) of the generalized Maxwell model are adjusted for the master curve obtained from dynamic viscoelastic testing to construct the generalized Maxwell model.

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[0023] In this case, it is preferable to determine the parameters of the generalized Maxwell model so as to minimize the value of the error function (ERR) in equation (III) below. In this case, the accuracy of the viscoelastic model is improved. Note that the method for determining the parameters of the generalized Maxwell model is not limited to this, and known methods can be used.

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[0024] Here, the viscoelastic model creation unit 13 constructs a generalized Maxwell model based on the storage modulus and loss modulus of the master curve obtained by dynamic viscoelastic testing at frequencies below 10,000 Hz. In other words, the viscoelastic model creation unit 13 constructs a generalized Maxwell model to fit the storage modulus and loss modulus of the master curve at frequencies below 10,000 Hz, and excludes the frequency range above 10,000 Hz from the curve fitting process.

[0025] This allows us to suppress vibrations that would not occur in a real tire rolling on a road surface when the tire model with the created generalized Maxwell model is rolled. As a result, the behavior of a real tire can be reproduced with high accuracy, improving the accuracy of the simulation.

[0026] Figure 3 shows an example of a generalized Maxwell model constructed by the viscoelastic model creation unit 13 based on the master curve shown in Figure 2. The example shown in Figure 3 shows a case where a generalized Maxwell model was constructed based on the storage modulus and loss modulus of the master curve obtained by dynamic viscoelastic testing at frequencies from 0.01 Hz to 10000 Hz or less.

[0027] As shown in Figure 3, the generalized Maxwell model constructed by the viscoelastic model creation unit 13 fits the storage modulus and loss modulus of the master curve accurately at frequencies up to 10,000 Hz. On the other hand, the generalized Maxwell model constructed by the viscoelastic model creation unit 13 does not fit the storage modulus and loss modulus of the master curve in the frequency range above 10,000 Hz.

[0028] The upper limit of the master curve frequency when the viscoelastic model creation unit 13 constructs a generalized Maxwell model (hereinafter sometimes referred to as the "master curve upper limit") may be 10,000 Hz or less, 6,000 Hz or less, or 4,000 Hz or less. In other words, the viscoelastic model creation unit 13 may construct a generalized Maxwell model to fit the storage modulus and loss modulus of the master curve at frequencies of 6,000 Hz or less, or to fit the storage modulus and loss modulus of the master curve at frequencies of 4,000 Hz or less. The master curve upper limit can be set appropriately depending on the characteristic value to be analyzed. For example, when analyzing radiated sound as in this embodiment, the master curve upper limit may be 4,000 Hz or less. The lower limit of the master curve frequency when the viscoelastic model creation unit 13 constructs a generalized Maxwell model (hereinafter sometimes referred to as the "master curve lower limit") is not particularly limited and may be, for example, 0.01 Hz or 1 Hz.

[0029] The tire model input unit 14 receives a tire model that has been modeled from a tire. The tire model is, for example, a finite element model in which the three-dimensional shape of the tire is divided into a finite number of elements. The elements of the tire model are preferably computer-compatible elements, such as solid elements like tetrahedral solid elements, pentahedral solid elements, and hexahedral solid elements, shell elements like triangular shell elements and quadrilateral shell elements, and surface elements. Each element of the tire model has an element number, node number, node coordinates, etc., as well as a viscoelastic model constructed by the viscoelastic model creation unit 13 set as a material property value. The tire model input unit 14 may also have a function to create a tire model divided into a finite number of elements based on the tire shape.

[0030] The rolling calculation unit 15 contacts the tire model with the road surface, rolls it on the road surface, and acquires information necessary for, for example, acoustic analysis. FIG. 4 shows an example of the state during rolling. As shown in FIG. 4, the rolling calculation unit 15, for example, performs an internal pressure filling process (inflation process) on the tire model 40, applies centrifugal force to the tire model 40, and rotates it at a target speed. Then, the drum 50 is approached to contact the tire model 40 with the road surface 51 and rolled for a certain period of time. After the tire reaches a steady state, information necessary for acoustic analysis (for example, the surface shape of the tire) is acquired.

[0031] As a result of the study by the present inventors, when the master curve upper limit value exceeds 10,000 Hz (for example, 1.0×10 7 Hz), when the tire reaches a steady state, vibrations that cannot occur in a tire actually rolling on the road surface occur, and it has been clarified that the fluctuation amount of the vertical axis force increases. On the other hand, when the master curve upper limit value is 10,000 Hz or less as in the present embodiment, the fluctuation amount of the vertical axis force can be reduced. The vertical axis force means a force acting in a direction perpendicular to the ground contact surface of the tire.

[0032] FIG. 5 shows the behavior of the vertical axis force when the master curve upper limit value is 10,000 Hz and 1.0×10 7 Hz when the tire reaches a steady state. The master curve lower limit value is 0.01 Hz in both cases. As shown in FIG. 5, when the master curve upper limit value is 10,000 Hz, when the tire reaches a steady state, the vertical axis force fluctuates within a range of ±1.0% with respect to the median value. On the other hand, when the master curve upper limit value is 1.0×10 7 Hz, when the tire reaches a steady state, the vertical axis force fluctuates within a range of ±1.8% with respect to the median value. By setting the master curve upper limit value to 10,000 Hz or less and reducing the fluctuation amount of the vertical axis force, the behavior of an actual tire can be reproduced more accurately, and the accuracy of the simulation can be improved.

[0033] The acoustic analysis unit 16 performs acoustic analysis based on the results calculated by the rolling calculation unit 15. For example, the acoustic analysis unit 16 acquires the surface shape of the tire in a rolling state and creates an acoustic spatial model using the acquired surface shape. The acoustic spatial model is, for example, an analysis model using the finite element method, and is a spatial model in which fluid (air) fills the space around the tire model.

[0034] The acoustic analysis unit 16 creates a sound pressure distribution in the created acoustic space model when the tire is rotated or vibrated. Specifically, it acquires information about the fluid (air) (flow velocity, pressure, etc.) when the tire is rotated or vibrated. Then, for example, it creates a sound pressure distribution based on the acquired information about the fluid pressure change. The calculation of fluid flow velocity and pressure, and the creation of the sound pressure distribution can be performed using commercially available finite element analysis application software such as Wave6 from Dassault Systems.

[0035] Next, the simulation method of the present invention will be described with reference to Figures 6 to 9. Figure 6 is a flowchart showing the processing procedure of the simulation method of the present invention, Figure 7 is a flowchart showing the processing procedure of the viscoelastic model creation step, Figure 8 is a flowchart showing the processing procedure of the rolling step, and Figure 9 is a flowchart showing the processing procedure of the acoustic analysis step.

[0036] As shown in Figure 6, the simulation method of the present invention comprises a viscoelastic model creation step S10 for creating a viscoelastic model, an input step S20 for inputting a tire model that models a tire, a rolling step S30 for rolling the tire model on a road surface, and an acoustic analysis step S40 for analyzing the noise generated during driving.

[0037] As shown in Figure 6, a viscoelastic model creation step S10 is performed to create a viscoelastic model based on a master curve obtained by a dynamic viscoelastic test. As shown in Figure 7, in the viscoelastic model creation step S10, a master curve obtained by a dynamic viscoelastic test of the rubber components constituting the tire (e.g., tread rubber) is first acquired (step S11). The method for calculating the master curve is not particularly limited, and known methods can be used. The frequency range of the master curve can be any range used when creating the viscoelastic model, and high frequency ranges (e.g., 1.0 × 10⁻⁶) are also acceptable. 7 It is not necessary to include Hz.

[0038] Next, the frequency range to be used when curve-fitting the generalized Maxwell model to the acquired master curve is set (step S12). At this time, the upper limit of the master curve is set to 10,000 Hz or less. This makes it possible to suppress vibrations that would not occur in a tire actually rolling on a road surface when the tire model with the created generalized Maxwell model set is rolled. As a result, the behavior of an actual tire can be reproduced with high accuracy, and the accuracy of the simulation can be improved. Note that the upper limit of the master curve may be 6,000 Hz or less, or 4,000 Hz or less. Also, the lower limit of the master curve is not particularly limited and may be, for example, 0.01 Hz or 1 Hz.

[0039] Then, based on the frequency range entered in step S12, the parameters (modulus and relaxation time) of the generalized Maxwell model are determined such that the error in the viscoelastic properties (storage modulus and loss modulus) is minimized (step S13). In this case, equation (III) below may be used as the error function. Note that the method for determining the parameters of the generalized Maxwell model is not limited to this.

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[0040] Next, as shown in Figure 6, the input step S20 is performed to input the tire model created by modeling the tire. As described above, the tire model is, for example, a finite element model in which the three-dimensional shape of the tire is divided into a finite number of elements. Each element of the tire model is set with an element number, node number, node coordinates, and the viscoelastic model constructed in the viscoelastic model creation step S10.

[0041] Next, as shown in Figure 6, a rolling step S30 is performed in which the tire model is rolled on the road surface. As shown in Figure 8, in the rolling step S30, first the tire model is subjected to an internal pressure filling process (inflation process), and the tire model after internal pressure filling is calculated (step S31). Then, a centrifugal force corresponding to the evaluation target speed is applied to the calculated tire model, and the tire model after centrifugal force is applied is calculated (step S32). By including a step to apply centrifugal force, it becomes possible to reach a steady state more quickly. Then, the tire model is rotated at a preset target speed (step S33).

[0042] Subsequently, the tire model is brought closer to the road surface until the predetermined displacement is achieved, a set load is applied, and the tire model at that time is calculated (step S34). Then, the calculated tire model is rolled on the road surface for a certain period of time (step S35). By rolling the tire model on the road surface after performing steps S31 to S34 described above, the behavior of a tire actually rolling on the road surface can be reproduced more accurately, making it possible to perform analysis with high precision.

[0043] Subsequently, it is determined whether the tire model has reached a steady state (step S36). The determination of whether a steady state has been reached may be made, for example, using the value of the vertical axial force applied to the tire model. Specifically, it may be determined that the tire model has reached a steady state if the value of the vertical axial force applied to the tire model has reached a predetermined target value, and the amount of fluctuation of the vertical axial force over a predetermined time is less than or equal to a predetermined value (for example, 1-3%). However, the determination of whether a steady state has been reached is not limited to this.

[0044] If it is determined that the tire model has reached a steady state (step S36: Yes), the necessary physical property data of the tire (for example, data related to the surface shape of the tire) is acquired in the acoustic analysis step S40 (step S37). On the other hand, if it is determined that a steady state has not been reached (step S36: No), the process returns to step S35, and the rolling motion on the road surface continues until the tire model reaches a steady state.

[0045] Next, as shown in Figure 6, an acoustic analysis step S40 is performed to analyze the noise generated during driving. As shown in Figure 9, in the acoustic analysis step S40, an acoustic spatial model is first created using the tire's physical property data (for example, data related to the tire's surface shape) acquired in the rolling step S30 (step S41). This acoustic spatial model may, for example, be a model in which sound sources are placed that assume pumping sounds originating from air blowouts, based on the acquired tire surface shape, obtained by fluid analysis of air blowouts from grooves such as main grooves. Note that the form of the acoustic spatial model is not limited to this and can be set appropriately according to the type of radiated sound to be desired.

[0046] Next, a sound radiation analysis is performed using the created acoustic space model (step S42). In the sound radiation analysis, for example, information about the fluid (air) (flow velocity, pressure, etc.) when the tire is rotated or vibrated is obtained. Then, based on the information obtained in step S42, physical quantities related to the radiation are obtained and the effect of the tire shape on the radiation is evaluated (step S43). Steps S41 to S43 above can be calculated using commercially available finite element analysis application software such as Wave6 from Dassault Systems.

[0047] As described above, the simulation method of the present invention constructs a viscoelastic model using only the viscoelastic characteristic values ​​at frequencies below 10,000 Hz from a master curve over a wide frequency range. This makes it possible to reduce vibrations that would not actually occur in a tire rolling on a road surface. In other words, with this method, vibrations that would not actually occur in a tire rolling on a road surface can be suppressed simply by changing the upper limit of the frequency of the master curve when constructing the viscoelastic model. Therefore, it can be said that the behavior of an actual tire can be reproduced with less effort, and the accuracy of the simulation can be improved.

[0048] In the above embodiment, a simulation method for analyzing the sound radiated from a tire during driving using the tire characteristics (such as the shape of the tire surface) in a rolling state was described, but the invention is not limited to this. For example, the braking performance and turning performance of a tire may be analyzed using the tire characteristics in a rolling state of a tire model to which the viscoelastic model of the present invention has been set. [Explanation of Symbols]

[0049] 1 Simulation device, 10 Control device, 11 Processor, 12 Memory, 13 Viscoelastic model creation unit, 14 Tire model input unit, 15 Rolling calculation unit, 16 Acoustic analysis unit, 20 Input device, 30 Display device, 40 Tire model, 50 Drum, 51 Road surface

Claims

1. A viscoelastic model creation step involves creating a viscoelastic model that shows the viscoelastic properties of the rubber components constituting the tire, based on a master curve obtained by dynamic viscoelastic testing. An input step in which a tire model with the aforementioned viscoelasticity model is set is input, A rolling step in which the tire model is brought into contact with the road surface and rolled on the road surface, A tire simulation method comprising: A tire simulation method comprising the step of creating a viscoelastic model, wherein the viscoelastic model is created using characteristic values ​​at frequencies of 10,000 Hz or less from the master curve obtained by dynamic viscoelastic testing.

2. The tire simulation method according to claim 1, wherein the viscoelastic model is a generalized Maxwell model.

3. The tire simulation method according to claim 1, further comprising an acoustic analysis step for analyzing noise generated during driving based on data acquired in the rolling step.

4. The tire simulation method according to claim 3, wherein in the viscoelastic model creation step, a viscoelastic model is created using characteristic values ​​at frequencies of 4000 Hz or less from the master curve obtained by dynamic viscoelastic testing.

5. A viscoelastic model creation unit creates a viscoelastic model that shows the viscoelastic properties of the rubber components constituting the tire, based on a master curve obtained by dynamic viscoelastic testing. A tire model input unit for inputting a tire model with the aforementioned viscoelastic model, A rolling calculation unit that brings the tire model into contact with the road surface and rolls it on the road surface, A tire simulation device comprising: The viscoelastic model creation unit is a tire simulation device that creates a viscoelastic model using characteristic values ​​at frequencies below 10,000 Hz from a master curve obtained by dynamic viscoelastic testing.

6. The tire simulation apparatus according to claim 5, wherein the viscoelastic model is a generalized Maxwell model.

7. The tire simulation device according to claim 5, further comprising an acoustic analysis unit that analyzes noise generated during driving based on data acquired in the rolling step.

8. The tire simulation apparatus according to claim 5, wherein the viscoelastic model creation unit creates a viscoelastic model using characteristic values ​​at frequencies of 4000 Hz or less from a master curve obtained by dynamic viscoelastic testing.

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