AMPLITUDE-FREQUENCY CHARACTERISTICS OF A BEAM WITH HYSTERESIS-TYPE ELASTIC DISSIPATIVE PROPERTY AND A MOVING DYNAMIC ABSORBER

ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 608 AMPLITUDE-FREQUENCY CHARACTERISTICS OF A BEAM WITH HYSTERESISTYPE ELASTIC DISSIPATIVE PROPERTY AND A MOVING DYNAMIC ABSORBER Z.S.Yuldoshova Assistant Lecturer at the Department of Digital Economy and Information Technologies, Samarkand Branch of Tashkent State University of Economics yuldoshovazarnigor[email protected] Abstract. In this work, the amplitude-frequency characteristics of the transverse vibrations of a beam with a hysteresis-type elastic dissipative property, interacting with a moving dynamic absorber of the same type, has been analytically determined depending on the system parameters and variables, and analyzed on the basis of numerical calculations. Key words. Beam, dynamic absorber, hysteresis, vibrations, the amplitude-frequency characteristics. Introduction. Taking into account the nonlinearity of complex mechanical systems, a number of works have been devoted to the analysis of their vibrations and dynamics [1–7]. To analyze the dynamics of the beam protected from the considered vibrations, we write down its differential equation of motion. 𝑞󰇘𝑖+{(1+𝐶0(−𝜂1+𝑗𝜂2))𝑝𝑖2+3𝐸𝐼 𝜌𝐴𝑑2𝑖(−𝜂1+𝑗𝜂2)× ×∑𝐶𝑘𝑞𝑖𝑎 𝑘 𝑛 𝑘=1 ℎ𝑘 2𝑘(𝑘+3)∫𝑢𝑖 𝑙 0𝜕2 𝜕𝑥2(𝜕2𝑢𝑖 𝜕𝑥2|𝜕2𝑢𝑖 𝜕𝑥2|𝑘)𝑑𝑥}𝑞𝑖− −𝜇𝜇0𝑛0 2𝑢𝑖0(1+(−𝜃1+𝑗𝜃2)(𝐷0+𝑓(𝜁𝑜𝑡)))𝜁𝐻(1 𝑣− 𝑡)=− 𝑑𝑖𝜕2𝑤0 𝜕𝑡2; (1) 𝑢𝑖0𝑞󰇘𝑖+𝜁󰇘+𝑛0 2(1+(−𝜃1+𝑗𝜃2)(𝐷0+𝑓(𝜁𝑜𝑡)))𝜁=−𝜕2𝑤0 𝜕𝑡2, where 𝜇= 𝑚 𝜌𝐴𝑙;𝜇0=𝑙 𝑑2𝑖;𝑑𝑖=𝑑1𝑖 𝑑2𝑖; 𝑑1𝑖 =∫𝑢𝑖 𝑙 0𝑑𝑥; 𝑑2𝑖 =∫𝑢𝑖2 𝑙 0𝑑𝑥; 𝜌 , 𝐴 are, respectively, the density of the beam material and the cross-sectional area; 𝑤(𝑥0) is the displacement of the beam point where the dynamic absorber is located; 𝑥0=𝑣𝑡 is the point where the dynamic absorber is located; 𝑣 is the velocity of the dynamic absorber; 𝑡 is time; 𝛿(𝑥) is Dirac delta function; 𝐻(𝑙 𝑣) is Heaviside function; 𝑙 is the length of the beam; c, 𝑚 are the stiffness and the mass of the dynamic absorber, respectively; 𝜁 is the relative deformation of the dynamic absorber; 𝑤𝑎 is the absolute displacement of the beam 𝑤𝑎=𝑤0+𝑤, 𝑤0 is the displacement of the base; 𝑤 is the deflection of the beam; 𝜃1,𝜃2=𝜃22𝑠𝑖𝑔𝑛(𝜔) are constant coefficients depending on the elastic dissipative properties of the dynamic absorber material, determined from the hysteresis loop; 𝑗2= −1; 𝑓(𝜁𝑜𝑡) is the decrement of vibrations, 𝜁𝑜𝑡 is a function of the absolute value of the relative deformation, ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 609 𝑓(𝜁𝑜𝑡)=𝐷1𝜁𝑜𝑡 +𝐷2𝜁𝑜𝑡 2…+𝐷𝑛𝜁𝑜𝑡 𝑛, 𝐷0,𝐷1,…,𝐷𝑛 are the parameters of the hysteresis node determined experimentally, which depend on the damping properties of the dynamic absorber material [7]. 𝜂1,𝜂2=𝜂22𝑠𝑖𝑔𝑛(𝜔) are constant coefficients depending on the elastic dissipative properties of the beam material, determined from the hysteresis loop; 𝑓(𝜁𝑜) is the decrement of vibrations, which is a function of the absolute value of the relative deformation 𝜁𝑜. 𝑓(𝜁𝑜)=𝐶1𝜁𝑜+𝐶2𝜁𝑜 2…+𝐶𝑛𝜁𝑜 𝑛, 𝐶0,𝐶1,…,𝐶𝑛 are the parameters of the hysteresis node determined experimentally, which depend on the damping properties of the beam material [7]. Materials and methods. Using the system of differential equations (1), we determine the transfer function in order to analyze the dynamics of the beam protected from the considered vibrations. First, for this purpose, we transform the system of differential equations (1) 𝑆= 𝑑 𝑑𝑡 we reduce it to a system of algebraic equations using the differential operator. 𝑆2=−𝜔2 taking into account that 𝐻(1 𝑣−𝑡) of the Heaviside function 1 𝑣−𝑡>0 when 𝐻(1 𝑣−𝑡)=1 taking into account the property, 𝑞𝑖 and 𝜁 we solve with respect to the variables. 𝑞𝑖(𝑡)=−𝐴1+𝑗𝐴2 𝐵1+𝑗𝐵2𝑊0; (2) 𝜁(𝑡)=−𝐴3+𝑗𝐴4 𝐵1+𝑗𝐵2𝑊0, where 𝐴1=−𝜔2+(1+𝑑𝑖𝜇𝜇0𝑢𝑖0)𝑛0 2(1−𝜃1(𝐷0+𝑓(𝜁𝑜𝑡)); 𝐴2=(1+𝑑𝑖𝜇𝜇0𝑢𝑖0)𝑛0 2𝜃2(𝐷0+𝑓(𝜁𝑜𝑡)); 𝐴3=(−𝜔2+(1−𝜂1𝐶0)𝑝𝑖2−𝜂13𝐸𝐼 𝜌𝐴𝑑2𝑖 × ×∑𝐶𝑘𝑞𝑖𝑎 𝑘 𝑛 𝑘=1 ℎ𝑘 2𝑘(𝑘+3)∫𝑢𝑖 𝑙 0𝜕2 𝜕𝑥2(𝜕2𝑢𝑖 𝜕𝑥2|𝜕2𝑢𝑖 𝜕𝑥2|𝑘)𝑑𝑥)𝑑𝑖+𝑢𝑖0𝜔2; 𝐴4=𝑑𝑖𝜂2(𝐶0𝑝𝑖2+ +3𝐸𝐼 𝜌𝐴𝑑2𝑖∑𝐶𝑘𝑞𝑖𝑎 𝑘 𝑛 𝑘=1 ℎ𝑘 2𝑘(𝑘+3)∫𝑢𝑖 𝑙 0𝜕2 𝜕𝑥2(𝜕2𝑢𝑖 𝜕𝑥2|𝜕2𝑢𝑖 𝜕𝑥2|𝑘)𝑑𝑥); 𝐵1=[−𝜔2+(1−𝜂1𝐶0)𝑝𝑖2−𝜂13𝐸𝐼 𝜌𝐴𝑑2𝑖 × ×∑𝐶𝑘𝑞𝑖𝑎 𝑘 𝑛 𝑘=1 ℎ𝑘 2𝑘(𝑘+3)∫𝑢𝑖 𝑙 0𝜕2 𝜕𝑥2(𝜕2𝑢𝑖 𝜕𝑥2|𝜕2𝑢𝑖 𝜕𝑥2|𝑘)𝑑𝑥]×[−𝜔2+ +𝑛0 2(1−𝜃1(𝐷0+𝑓(𝜁𝑜𝑡)))]−𝜂2[𝐶0𝑝𝑖2+3𝐸𝐼 𝜌𝐴𝑑2𝑖 × ×∑𝐶𝑘𝑞𝑖𝑎 𝑘 𝑛 𝑘=1 ℎ𝑘 2𝑘(𝑘+3)∫𝑢𝑖 𝑙 0𝜕2 𝜕𝑥2(𝜕2𝑢𝑖 𝜕𝑥2|𝜕2𝑢𝑖 𝜕𝑥2|𝑘)𝑑𝑥]× ×𝑛0 2(1+𝜃2(𝐷0+𝑓(𝜁𝑜𝑡)))−𝜇𝜇0𝑛0 2𝑢𝑖0 2𝜔2(1−𝜃1(𝐷0+𝑓(𝜁𝑜𝑡))); ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 610 𝐵2=[−𝜔2+(1−𝜂1𝐶0)𝑝𝑖2−𝜂13𝐸𝐼 𝜌𝐴𝑑2𝑖 × ×∑𝐶𝑘𝑞𝑖𝑎 𝑘 𝑛 𝑘=1 ℎ𝑘 2𝑘(𝑘+3)∫𝑢𝑖 𝑙 0𝜕2 𝜕𝑥2(𝜕2𝑢𝑖 𝜕𝑥2|𝜕2𝑢𝑖 𝜕𝑥2|𝑘)𝑑𝑥]× ×𝑛0 2(1+𝜃2(𝐷0+𝑓(𝜁𝑜𝑡)))+𝜂2[𝐶0𝑝𝑖2+3𝐸𝐼 𝜌𝐴𝑑2𝑖 × ×∑𝐶𝑘𝑞𝑖𝑎 𝑘 𝑛 𝑘=1 ℎ𝑘 2𝑘(𝑘+3)∫𝑢𝑖 𝑙 0𝜕2 𝜕𝑥2(𝜕2𝑢𝑖 𝜕𝑥2|𝜕2𝑢𝑖 𝜕𝑥2|𝑘)𝑑𝑥]× ×[−𝜔2+𝑛0 2(1−𝜃1(𝐷0+𝑓(𝜁𝑜𝑡)))]−𝜇𝜇0𝑛0 2𝑢𝑖0 2𝜃2(𝐷0+𝑓(𝜁𝑜𝑡)))𝜔2. Expression (2) represents the amplitude-frequency characteristic of the transverse vibrations of a beam with a hysteresis-type elastic dissipative property together with a moving dynamic absorber of the same type. Results and discussion. Let us assume that the moving dynamic absorber is moving with a linear velocity of 𝑣=0.125 𝑚 𝑠 . In this case, when 𝑡=1;2 𝑠 we analyze the amplitude-frequency characteristics of the considered system. The experimentally determined parameters of the hysteresis node of the beam material are taken as follows [7]. 𝐶1=6.760624; 𝐶2=−8278.5937;𝐶3=5894761; 𝜂1=3 4;𝜂2=1 𝜋. For the elastic damping element of the moving dynamic absorber, we assume the following: 𝜃1=𝜃2=0,𝐷0=0,𝐷1=0,𝐷2=0. The remaining parameters are as follows: 𝑑1=0.3183098862;𝑑2=0.25;𝑑𝑖=1.273239545;𝜇=0.1;𝜇0=2.0 𝑢𝑖0 =𝑢10 =1;𝐼=1.066666667∙10−10𝑚4;𝜀𝑝0=10−5 2 𝑚. When 𝑡=1 𝑠 the moving dynamic absorber is located at the point 𝑥0=0.125 𝑚 of the beam. Based on the values of the parameters given above, we plot the amplitude–frequency characteristic of the system. ISSN: 2582-4686 SJIF 2021-3.261,SJIF 20222.889, 2024-6.875 ResearchBib IF: 9.948 / 2024 VOLUME-5, ISSUE-11 611 Figure 1. Amplitude–frequency characteristic. (𝒙𝟎=𝟎.𝟏𝟐𝟓 𝒎). In Figure 1, the variation of the amplitude–frequency characteristic at the point where the dynamic absorber is installed is shown for the transverse vibrations of a beam with a hysteresis-type elastic dissipative property together with a moving dynamic absorber located at 𝑥0=0.125 𝑚 . From these graphs, it can be observed that as the stiffness of the dynamic absorber increases, its frequency approaches the natural frequency of the beam, and the resonance curve shifts along the frequency axis from left to right (c=10∙102𝑁 𝑚 (black), c=12∙102𝑁 𝑚 (red), c=15∙102𝑁 𝑚 (blue)). At these stiffness values, the maximum amplitudes around the resonance frequency are 𝑞0= 0.0036;0.0035;0.0024 𝑚,respectively . From this, it can be concluded that for the point 𝑥0= 0.125 𝑚 with the above parameter values, c=15∙102𝑁 𝑚 represents the optimal value of the dynamic absorber stiffness. CONCLUSION The obtained expression is an analytical representation of the amplitude–frequency characteristic of the transverse vibrations of a beam with a hysteresis-type elastic dissipative property together with a moving dynamic absorber of the same type, which makes it possible to evaluate the efficiency of the moving hysteresis-type elastic dissipative dynamic absorber. REFERENCES 1. Wu J.J. Study on the inertia effect of helical spring of the absorber on suppressing the dynamic responses of a beam subjected to a moving load. Journal Sound and Vibration, 297, 2006, pp. 981999 2. Mohamed G., Sinan M. Transverse vibration of two axially moving beams connected by an elastic foundation. Proceedings of 2005 ASME International Mechanical Engineering Congress and Exposition, 2005, Florida. DOI:10.1115/IMECE2005-80377 3. 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