ASEN5022 Dynamics of Aerospace Structures Lecture 8 Finite Element Modeling of Vibration of Bars In this lecture we will be introduced to finite element approximations of bar for vibration analysis Specifically we study 1 how to approximate the bar displacement for an element 2 how to discretize the kinetic and potential energy terms for an element 3 how to carry out variations to obtain elemental mass and stiffness matrices 4 how to assemble many elements to obtain the global discrete equations of motion for a bar 5 Use a Matlab based code to conduct vibration analysis of a bar string cable or shaft 1 Fig 1 A bar with attached masses and springs 8 1 Problem Statement for Vibration of Bars Consider a continuum bar with attached masses and spring at ends as shown in Fig 1 For finite lement discretization of the system we do not carry out variations of the energy functionals Instead we first discretize the energy functionals then carry out variations of nodal variables afterwards To this end we list the required energy functionals below 2 Kinetic energy of the continuum bar Tbar Potential energy of the continuum bar Vbar Z 1 L du x t m x ut x t 2dx ut x t 2 0 dt Z L 1 EA x ux x t 2 dx 2 0 du x t ux x t dx Z L External energy f x t u x t dx Wbar 0 Kinetic energy of the discrete masses Potential energy of four springs 1 1 Ts M0 u 20 t ML u 2L t 2 2 1 Vs k01 u20 t k02 u0 t u 0 t 2 2 kL1 u2L t kL2 uL t u L t 2 8 1 With the preceding energy functionals we state Hamilton s principle as follows Z t2 Ttotal Vtotal Wbar dt 0 Ttotal Tbar Ts Vtotal Vbar Vs t1 3 8 2 8 2 Key departure in FEM modeling of bars from continuum modeling 1 Instead of carrying out the variation the process of Hamilton s principle given in 8 2 in terms of the continuum variable u x t the energy expressions T V W etc are approximated on a completely free bar with an arbitrary length area A and mass density m x 2 Ideally the interpolation of the axial displacement u x t over the completely free bar segment 0 x is chosen to satisfy the homogeneous differential bar equation i e EA u x t xx 0 8 3 if possible If not it is chosen such that as the length of the bar gets smaller and smaller it approximately satisfies the homogeneous equation This is referred to as the patch test 8 2 1 Linear quadratic and cubic bar elements Consider general polynomial approximation of the displacement field of a bar given by u x t c0 t c1 t x c2 t x2 cn t xn We now specialize to several elements below 4 8 4 Fig 2 The concept of an element in a bar 1 Linear Element or Constant Strain Element Retain the first two terms in 8 4 viz c0 c1 to result in u x t c0 t c1 t x 5 8 5 2 Quadratic Element or Linear Strain Element Retain the first three terms in 8 4 viz c0 c1 c2 to result in u x t c0 t c1 t x c2 t x2 8 6 3 Cubic Element or Quadratic Strain Element Retain the first four terms in 8 4 viz c0 c1 c2 c3 to result in u x t c0 t c1 t x c2 t x2 c3 t x3 8 2 2 8 7 The shape function of a linear bar element 1 We specify the two discrete nodes viz nodes 1 and 2 in the previous figure Namely At node 1 At node 2 u 2 t u1 t u 2 t u2 t 6 8 8 Fig 3 Nodal variables in linear and quadratic elements so that we have the following two equations u1 t c0 t c1 t 2 u2 t c0 t c1 t 2 1 1 u x t x u1 t x u2 t 2 2 1 1 u t 1 u1 t 1 u2 t 2x 2 2 7 8 9 The preceding linear approximation can be expressed in a familiar standard form of two noded bar element u t N1 u1 t N2 u2 t 1 N1 1 2 1 N2 1 1 1 2 u t N q t N N1 N2 q t T u1 t u2 t T 8 8 10 8 2 3 Generation of a Linear Bar Element Elemental kinetic energy and elemental potential energy are obtained via el Tbar 1 2 Z 1 2 1 m x u 2 x t dx 2 12 Z 1 1 m u 2 t d 2 1 1 x 2 8 11 el Vbar 1 2 Z 2 2 1 EA x u2x x t dx 2 Z 1 1 2 1 EA 2u2 t d 2 When the linear interpolation 8 10 is used to evaluate them we have the following elemental mass and stiffness matrices 9 Z 1 Z 1 1 1 1 1 el Tbar A q t T NT N q t d m u 2 t d 2 1 2 2 1 2 Z 1 1 1 1 q T A NT N d q t q t T mel q t 2 2 1 2 A 2 1 el m where 6 1 2 Z 1 Z 1 2 1 2EA 1 1 el EA 2u2 t d q t T NT N q t d Vbar 2 1 2 2 1 Z 1 1 1 2EA T q t T N N d q t q t T kel q t 2 2 1 EA 1 1 where kel 1 1 10 8 12 8 13 Finally the elemental external energy is discretized by Z 1 Z 2 1 NT f x t d q t T f el f x t u x t dx q t T W el 2 1 2 Z 1 1 where f el NT f x t d 2 1 8 14 mel kel f el derived in 8 12 8 1 are called the elemental mass matrix elemental stiffness matrix and elemental applied force respectively They will be used extensively in constructing the discrete approximate equations of motion for a bar and summarized below Table 10 1 Summary of elemental energy expressions el Kinetic energy Tbar 1 el T 2 q t mel q t el el Potential energy Vbar 1 el T 2 q t kel q t el el External work Wbar Mass matrix Stiffness matrix 11 q t el T f el 2 1 A el m 6 1 2 1 1 kel EA 1 1 8 3 Question How do we model a long bar with bar elements Answer 1 Partition the long bar into small elements This needs to be explained 2 Generate the elemental energy expression This is done see Table …