Trunnion Hub Girder Bascule bridge Trunnion-Hub Assembly A civil engineer is ove
ID: 1711986 • Letter: T
Question
Trunnion Hub Girder Bascule bridge Trunnion-Hub Assembly A civil engineer is overseeing the replacement of a trunnion in a Bascule bridge. The process of shrink-fit is being adopted for the insertion of the trunnion into the hub. To achieve this the trunnion is first immersed in a -118° C dry ice/alcohol mixture, until the outer diameter of the trunnion has contracted enough for easy insertion into the hub. Once inserted into the hub the trunnion will heat up by absorbing heat from the surounding ambient air temperature and expand in order to be a tight fit. The thermal expansion co-efficient equation for the trunnion is defined as where the constants, ai, az and as, of the model are found by solving the following simultaneous linear equations 24 a1 2860 a2 + 726 000 as 0.0001057 a3 =0.0001057 726 000 a2 186 472 000 a3 -0.0104 162 = 726 000 a1 186 472 000 a2 52 435 700 000 a 2.56799 i) This is matrix that does not exhibit diagonal dominance and cannot be manipulated to become ly dominant. However the matrix is symmetrical and positive definite and so is able to be used in its original form to obtain convergence. Demonstrate that these aspects are correct; the matrix is symmetrical and positive definite. Explain why is it important for the matrix to be either diagonally dominant or symmetrical and positive definite ii) Solve the system of equations using Gauss-Seidel iteration. Perform five (5) iterations starting with the zero vector and retain four decimal places in your working. Comment upon the answers obtained in the final iteration.Explanation / Answer
Hi,
A i ]
WE can observe here that the a11 element is 24 is less than the element a31 and a13 which will cause the results to converge after a more number of iterations. That is the reason the matrix shall be always diagonally dominant.
The matrix if is symmetrical then their eigenvalues are real and each has a complete set of orthonormal eigenvectors. Positive definite matrices are even better as the eigen values obtained are positive and all real.