Theoretical and experimental method for determining the frequency-dependent dynamic modulus of elasticity and damping characteristics of a titanium alloy OT-4

A significant reduction in the dynamic modulus of elasticity of the titanium alloy OT-4 during cyclic deformation in the frequency range of 0 ÷ 25 Hz and its further stabilization at frequencies of 25 ÷ 80 Hz on the basis of studies of damped bending vibrations of a series of test samples is shown. The averaged amplitude dependence of the logarithmic vibration decrement of this alloy under tension-compression is obtained by minimizing the quadratic discrepancy between the experimental and calculated logarithmic decrements of the oscillations of the test samples.


Introduction
Titanium alloys are distinguished by high specific strength and rigidity in combination with high thermal and corrosion resistance, which opens up great prospects for their use in the aviation industry, rocket production and transport engineering [1,2]. This determines the need for an adequate determination of their elastic and damping properties that correspond to the operating conditions of the structure corresponding to the actual conditions.
The least studied mechanical characteristics of titanium and its alloys include the damping ability, which influences the dynamic strength of structural elements in cyclic deformation modes and, above all, at resonance, realized in the structure when the frequencies of its natural oscillations coincide with the frequency of external cyclic exposure. In this mode of loading, as is well known, the amplitude values of the parameters of the dynamic stress-strain state increase many times over. Correct and reliable their theoretical definition with the accuracy necessary for practical purposes requires proper consideration in the design ratios of the damping properties of materials of structures caused by internal friction.
The most reliable information about the characteristics of the elasticity of the material gives a dynamic method based on the experimental measurement of the frequency of bending vibrations of cantilever test specimens and comparing it with the calculated frequency of an ideally elastic sample [3,4]. During testing of test specimens from titanium alloy OT-4 in the mode of damped flexural vibrations, a significant decrease and stabilization of the elastic modulus of this alloy with increasing frequency of oscillations compared to its static nominal value was established, which gives grounds to  [5] (in contrast to the wellknown interpretation of the dynamic modulus of elasticity existing in the theory of energy dissipation [6,7]).
The damping properties of a material under tension-compression are determined by the logarithmic vibration decrement ) ( 0   , depending on the strain amplitude 0  [8,9]. The dependence can be determined by comparing the calculated and experimental logarithmic vibration decrement of a test sample with several amplitudes of oscillations of its free end [10,11].

Determination of the frequency dependence of the dynamic modulus of elasticity of titanium alloy OT-4
The tests were carried out in the mode of damped bending vibrations of 15
(1) that follows from the known relation for finding the lowest frequency of free bending vibrations of a fixed beam [12] ) . The values of d E obtained from (1) have some experimental scatter. To build a smoothing dependency, function was used. Parameters . The search for the minimum of the objective function was carried out by the Hook-Jeeves method [13,14]. Thus, we found the dependence giving values of d E in MPa. Figure 1 shows the dependences of ) ( f E d : the points by the formula (1), the line by the formula (2). There is a significant decrease in the dynamic modulus of elasticity d E relative to the static modulus E in the frequency range 25 0   f Hz (approximately 14%) and its further practical stabilization at frequencies 25  f Hz.

Determination of the logarithmic vibration decrement of titanium alloy OT-4
To represent the dependence ) ( 0   , which determines the damping properties of the OT-4 alloy, the function , it is proposed to use the finite element method [18][19][20][21][22][23][24][25][26][27] with modeling the test sample by beam elements. Matrices of internal and external aerodynamic damping of a beam finite element based on the Thomson-Kelvin-Voigt model [28], representing the viscous-elastic properties of the material, and Morison approximation [29,30] Figure 2, was obtained.

Conclusion
Using the theoretical-experimental method, we obtained the frequency dependence of the dynamic elastic modulus and the amplitude dependence of the logarithmic vibrations decrement of the OT-4 titanium alloy based on the study of damped bending vibrations of a series of test samples. The considered method can be applied to determine the dynamic characteristics of elasticity and damping properties of other structural materials.

Acknowledgments
The study was carried out by a grant from the Russian Science Foundation (project No. 14-19-00667).