Technology and Management

Аннотация

This study examines the hybrid method for solving dynamic problems in elastic half- spaces under sinusoidal impulsive vertical surface loads. The method, which combines spatial discretization and modal superposition, is evaluated for stability and accuracy under varying time steps and discretizations. The relationship between stability and time step selection is analyzed, highlighting the importance of meeting Nyquist frequency conditions. Similarly, the accuracy of computed modes and natural frequencies is investigated, emphasizing the significance of fine vertical discretization. Numerical examples reveal critical stability thresholds and optimal discretization parameters for minimizing errors in surface displacement predictions. Recommendations for improving the method’s stability and accuracy are provided, offering insights into its practical application in dynamic modeling. Keywords: Hybrid method, dynamic modeling, half-space, sinusoidal impulsive loads, time step stability, vertical discretization, Nyquist frequency, accuracy, modal superposition, Rayleigh waves. Annotatsiya: Ushbu tadqiqot ishida sinusoidal impulsli vertikal sirt yuklari ta’sirida elastik yarim fazodagi dinamik masalalarni yechish uchun gibrid usulni o‘rganiladi. Fazoviy diskretlashtirish va modal superpozitsiyadan foydalanadigan ushbu usul turli vaqt bosqichlari va diskretizatsiya sharoitlarida barqarorlik va aniqlik jihatidan baholanadi. Barqarorlik va vaqt bosqichini tanlash o‘rtasidagi bog‘liqlik tahlil qilinib, Nyquist chastotasi shartlarini bajarish muhimligi ta’kidlanadi. Shuningdek, hisoblangan modellar va tabiiy chastotalarning aniqligi o‘rganilib, vertikal diskretlashtirishning nozik darajada bajarilishi katta ahamiyatga ega ekani ko‘rsatiladi. Sonli natijalar orqali sirt siljish xatolarini minimallashtirish uchun muhim barqarorlik chegaralari va optimal diskretizatsiya parametrlarini aniqlash mumkinligi ko‘rsatib beriladi. Ushbu usulning barqarorlik va aniqligini oshirish bo‘yicha tavsiyalar taqdim etilib, dinamik modellashtirishda amaliy qo‘llanilishi bo‘yicha tushunchalar beriladi. Kalit so‘zlar: Gibrid usul, dinamik modellashtirish, yarim fazo, sinusoidal impuls yuklar, vaqt bosqichi barqarorligi, vertikal diskretizatsiya, Nyquist chastotasi, aniqlik, modal superpozitsiya, Reley to‘lqinlari. Introduction. As Kausel [1] himself describes, the hybrid method is a semi- analytic method in the sense that it is based on a spatial discretization of the half-space in the heterogeneous direction. Moreover, the Green's function in the time domain is obtained by a modal superposition, which theoretically should be infinite, but in practice has to be truncated for a certain maximum value of the natural frequency. And on the 15 www.wordlyjournals.com AXBOROT TEXNALOGIYALARI VA MENEJMENT UNIVERSITETI INTERNATIONAL JOURNAL OF SCIENCE Technical sciences ISSN: 3060-4583 VOLUME 2, ISSUE 1, 2025 other hand, the coupling between the strata and the half-space is performed in a discrete way, which requires choosing a certain time step. In this section we study the stability and accuracy of the method. The stability, as can be intuited, depends almost exclusively on the time step chosen to solve the system of equations that couples the half-space with the layered plate. As for accuracy, the TLM is mainly responsible, and depends on the vertical discretization performed. This method is obviously more accurate the smaller the thickness of the elements taken. In fact, the modes and natural frequencies calculated by the method are not accurate, but present a dispersion that is accentuated for high frequencies, and depends precisely on the vertical discretization. In this sense Park introduced the parameter N which means the number of elements per wavelength, and informs about the accuracy of the chosen discretization in relation to the maximum natural frequency. Research methodology: We then proceed to discuss stability and accuracy based on a series of numerical examples in which we play mainly with the time step and the number of elements per wavelength. For this purpose a homogeneous half-space is considered whose dynamic properties are: 1500kg / m3, c 1000m/ s, c 2.08m/ s . For the purpose of testing the method a s p stratum of 1000m depth is virtually considered. In addition, a vertical surface loading is assumed in the form of a sine pulse both temporally and spatially, i.e., a sine pulse is assumed: Pr,,z,t,z f rhtze (1) z Where 1 f (r)  e(r/a)2 ; r 0 (2) a2 2 t h(r) sin2 ;  0t t (3) d t t p p where the periods t 0.1s and a 100m have been taken. p Performing the Fourier transform to the time function, and the Hankel transform to the spatial function we have: 1 (k a/2)2 f (k )  e r (4) r sint / 2 p h() (5) 2 2 t / 2( (t / 2) ) p p It can be observed that the amplitude of the function h() decreases with the cube of the frequency, such that for  (4/t ) it is practically negligible. The same max p 16 w