The Effect of Thickness Ratio in Cantilever Pipe for Rectangular Cross Section Conveying water at turbulent flow on Transverse Free Vibrations (Journal of University of Babylon for Engineering Sciences, 2018)
Raighly – Ritz is the approximate mathematical method which is used to study the  vibrations in engineering structures, which employed in this search to guess the natural frequency of the  of pipes conveying water at turbulent flow for rectangular cross section at tapered thickness this method have clamped –  free boundary conditions in the two cases. The first involves the pipe have a constant wall thickness (h1) at clamped end equal to (1mm & 2mm) while the thickness (h2) at free end changes according to the ratio (h2/h1=0.25, 0.5, 0.75, 1). In the second case the thickness at free end (h2) is constant (1mm & 2mm) whereas the thickness at clamped end (h1) changes at ratio ( h1/h2=0.25, 0.5, 0.75, 1). The pipe has a constant inner high of cross section (w2) is (5 cm & 10 cm) with different values of width (w1) vary at ratio of (w1//w2 = 0.5, 1, 1.5, 2 )  for different lengths of pipe are (1m & 2m). This study shows in the first case at any value of thickness (h1) and the height (w2), the natural frequency decreased with increasing the ratio (h2/h1) & the ratio of (w1/w2) at the same length. While  the frequency increase with increasing the thickness (h1) & the high (w2).  On the other hand the critical velocity increase with increasing thickness (h1), the high (w2) and the ratio (h2/h1) but decreased with increasing the length of pipe and the ratio (w1/w2). In the second case the natural & frequency critical velocity of the system increase with increasing the thickness at free end (h2), thickness ratio (h1/h2) & the high (w2) but decreasing with increasing the width (w1) and the length (L). At any formation of the pipe for uniform section the natural frequency decreased when the velocity of flow of water increased from zero to critical velocity. Because of the absence of studies about the turbulent flow induced vibrations in pipes with a rectangular section it has been to compare with the analytical method for different models for pipeline were obtained  excellent results.