Question

Assicamenty 1 in Engineering Mathematis (MA MATH31S e each of the two mass-displacements to be denoted by x g Newton's 2nd and 3rd law of motion to Student Name Student No. Name: 1- Consider the two spring-mass system, and x,, and let us assume has the solutions can be written in the form x,-A, sin(ar-t). me each spring has the same spring constant .From vibration theory, develop a force-balance for each mass we have we have: 10A,@ -150-24,+A)- -204,o'-15(4-4) = 0 where A amplitude of the vibration of mass i, frequency of vibration, 0-phase shift. Find the frequency of the system (eigenvalues) and the amplitude of the vibration (eigenvector) Solution ANEON* NPS

Answer 1

The equations given are - 10 A_1 w^2 - 15 times (- 2 A_1 + A_2) = 0 - 20 A_2 w^2 - 15 (A_1 - A_2) = 0 Rewriting Them we have A_1 (30 - 10 w^2) - 15 A_2 = 0 - 15 A_1 + (15 - 20 w^2) A_2 = 0 Vertical-bar (30 - 10 w^2) - 15 - 15 (15 - 20 w^2)] [A_1 A_2] = [0 0] ..(1) A_1 notequalto 0 A_2 notequalto 0 because we are not interested in trivial solution so Vertical-bar 30 - 10 w^2 - 15 -15 15 - 20 w^2 Vertical-bar = 0 rightdoublearrow Vertical-bar (6 - 2 w^2) - 3 - 3 3 - 4 w^2 Vertical-bar = 0 rightdoublearrow (6 - 2 w^2) (3 - 4 w^2) - 9 = 0 8w^2 - 30 w^2 + 9 = 0 \r\n Solving we have w^2_1 = 15/8 + 3/8 squareroot 17 w^2_2 = 15/8 - 3/8 squareroot 17 w_1 = + 1.84 rad/s w_2 = 0.573 rad/s when w_1 = 1.84 rad/s w^2_1 = 3.42 so substituting in equation (1) [30 - 34.2 - 15 A_1 - 15 15 - 68.4]