By Ranjan Ganguli
This e-book addresses the answer of rotating beam free-vibration difficulties utilizing the finite point procedure. It offers an advent to the governing equation of a rotating beam, earlier than outlining the answer techniques utilizing Rayleigh-Ritz, Galerkin and finite aspect tools. the opportunity of enhancing the convergence of finite aspect equipment via a really apt collection of interpolation features, that are towards the matter physics, is additionally addressed. The booklet bargains a priceless advisor for college kids and researchers engaged on rotating beam difficulties – very important engineering constructions utilized in helicopter rotors, wind generators, gasoline generators, steam generators and propellers – and their functions. it might probably even be used as a textbook for specialised graduate classes on complicated purposes of finite aspect analysis.
Read Online or Download Finite Element Analysis of Rotating Beams: Physics Based Interpolation PDF
Best analysis books
Analysis of Reliability and Quality Control: Fracture Mechanics 1
This primary e-book of a 3-volume set on Fracture Mechanics is especially situated at the mammoth variety of the legislation of statistical distributions encountered in numerous clinical and technical fields. those legislation are necessary in knowing the chance habit of elements and mechanical constructions which are exploited within the different volumes of this sequence, that are devoted to reliability and qc.
- The Analysis of Thermally Stimulated Processes
- Hegel's Phenomenology, part I: analysis and commentary, Volume 1
- Water and Solute Permeability of Plant Cuticles: Measurement and Data Analysis
- Stability analysis in terms of two measures
- Foundations of analysis;: The arithmetic of whole, rational, irrational, and complex numbers. A supplement to textbooks on the differential and integral calculus
Extra resources for Finite Element Analysis of Rotating Beams: Physics Based Interpolation
Example text
Q1 . K18 ⎪ ⎪ ⎪ ⎪ . K28 ⎥ q ⎪ 2 ⎥⎪ ⎪ ⎪ . ⎥ . ⎪ ⎥⎪ ⎨ . ⎥ ⎥ . ⎥ . ⎥⎪ . ⎪ ⎪ ⎪ . ⎥ . ⎪ ⎥⎪ ⎪ . ⎦⎪ . ⎪ ⎪ ⎩ . K88 q8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ + ⎧ Q1 ⎪ ⎪ ⎪ ⎪ Q ⎪ 2 ⎪ ⎪ ⎪ . ⎪ ⎪ ⎨ . = . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ Q8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Or in matrix form, Here, KTOT is singular because rigid body modes are present. These are zero frequency modes and prevent the matrix from being inverted. However, this equation is valid for a free−free rotating beam.
14. The mass matrix is given by, ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ mij = m ⎢ ⎢ ⎢ ⎢ ⎣ 13 l 35 11 2 l 210 9 l 70 11 2 l 210 1 3 l 105 13 2 l 420 9 l 70 13 2 l 420 13 l 35 −13 2 l 420 −1 3 l 140 −11 2 l 210 −13 2 ⎤ l 420 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ −11 2 ⎥ ⎥ l 210 ⎥ ⎦ −1 3 l 140 l3 105 For example, l m13 = mH1 H3 dx 0 l = m 2 x l 3 −3 0 =m 9 l 70 Fig. 14 Element on the blade x l 2 +1 −2 x l 3 +3 x l 2 dx 44 1 Introduction The stiffness matrix is given by, ⎡ 12 l3 ⎢ ⎢ ⎢ 6 ⎢ l2 ⎢ kij = EI ⎢ ⎢ −12 ⎢ l3 ⎢ ⎣ 6 l2 ⎡ −mi 3 x 5 i + 6 l2 −12 l3 4 l −6 l2 −6 l2 12 l3 2 l −6 l2 ⎢ ⎢ ⎢ l l2 ⎢ 10 xi + 28 ⎢ 2⎢ ⎢ ⎢ −3 6l ⎢ 5 xi − 35 ⎢ ⎣ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥+ −6 ⎥ ⎥ l2 ⎥ ⎦ ⎢ ⎢ ⎢ 1 2 Ai ⎢ ⎢ 10 ⎢ 2 ⎢ −6 ⎢ 5l ⎢ ⎣ 4 l 6 5l 1 10 + l2 28 −3 x 5 i 3 − 6l 35 2 l x 30 i + l 105 −l x 10 i − l 28 −l x 10 i − l2 28 3 x 5 i + 6l 35 −l2 x 60 i − l3 140 2 −l2 70 ⎡ 2 l l x 10 i 6l 35 ⎤ 6 l2 1 10 −6 5l 2l 15 −1 10 −1 10 6 5l −l 30 −1 10 2l 15 ⎤ ⎥ ⎥ l ⎥ −l x − 140 ⎥ 60 i ⎥ ⎥ ⎥ ⎥ l2 ⎥ 70 ⎥ ⎦ l3 l2 x + 30 i 105 3 N Ai = 2 mj xj+1 − xj2 j=i To understand the origin of the Ai term, consider, (Fig.
K18 ⎪ ⎪ ⎪ ⎪ . K28 ⎥ q ⎪ 2 ⎥⎪ ⎪ ⎪ . ⎥ . ⎪ ⎥⎪ ⎨ . ⎥ ⎥ . ⎥ . ⎥⎪ . ⎪ ⎪ ⎪ . ⎥ . ⎪ ⎥⎪ ⎪ . ⎦⎪ . ⎪ ⎪ ⎩ . K88 q8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ + ⎧ Q1 ⎪ ⎪ ⎪ ⎪ Q ⎪ 2 ⎪ ⎪ ⎪ . ⎪ ⎪ ⎨ . = . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ Q8 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Or in matrix form, Here, KTOT is singular because rigid body modes are present. These are zero frequency modes and prevent the matrix from being inverted. However, this equation is valid for a free−free rotating beam.