OBRABOTKAMETALLOV TECHNOLOGY Vol. 24 No. 2 2022 ( ) cos( ) / 2, x C ch kx kx ( ) sin( ) / 2. x D sh kx kx To fi nd the constants, the following equations for derivatives should be used: 1 2 3 4 ( ), m x x x x k C D C A C B C C 2 1 2 3 4 ( ), m x x x x k C C C D C A C B 3 1 2 3 4 ( ). m x x x x k C B C C C D C A . The factor k is the wave multiplier depending on the material properties and vibration frequency: 2 4 , m k EI (1) where E is the Young modulus of the waveguide material (E=71 GPa for AMg4), m is the waveguide weight per unit of length (m = b · h· l · ρ = 0.03×0.004×1×2,670 = 0.320 kg/m for the case under consideration), angular frequency 2 p f , where p f is the resonance frequency of the self-induced vibrations of the plate. To determine the nature of vibration propagation depending on the plate fi xing conditions, let’s use the algorithm described by Bulgakov (1954). Boundary conditions for this algorithm are written in expanded form, which results in heterogeneous equations relative to the constants. To avoid the constants being zero, the determinant made for the equation system coeffi cients should be equal to zero. The calculation scheme is given in Fig. 3. The ultrasonic vibration application spot x should be selected, so that in the welding zone lweld there is a maximum amplitude of vibrations. For these fi xing conditions (free plate ends on both sides): for lsc = 0 and for lsc = lp: 0 m and 0 m , constants C3 = 0 and C4 = 0. Substituting these values into the solution of the vibration equation, the following frequency equation is resulted: ï ï ( ) cos( ) 1 ch kl kl The roots of the equation are: ï / 2, kl n (2) where n =1, 2, 3... Let’s express the coeffi cient k from equation (2) and equate it to equation (1): 2 4 ï / 2 . n m l EI Taking into account that 2 p f one can obtain an equation to fi nd the plate length depending on the vibration frequency (3): ï 2 4 / 2 . (2 ) ï n l m f EI (3) Fig. 3. Scheme for calculating bending vibrations: lp is the length of the plate, x is the place of ultrasonic vibrations application, lweld is the place where the weld is applied
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