Problem 7.We have a line of two equal masses m connected by three springswith spring constants c₁ = 1, c₂ = 1, and c3 = S. Spring 1is fixed at the top and spring 3 at the bottom, so xo = x3 =0.(a) Find the stiffness matrix K in the equation Kx = f for the massdisplacements.(b) Solve for the displacements x₁ and x2.(c) If the third spring constant S becomes very large or very small(S→∞ and S→ 0) what are the limiting values of thedisplacements x₁ and .x₂?(d) If the third spring constant S becomes very large or very small(S→∞ and S→ 0) what are the limiting values of the springforces y1, y2, and y3?91wwwm3-WWwwwC₁X₁"Xo = 0C2₂X2C3X3 = 0

Given: To find: Stiffness matrix Displacements Limiting values of displacementsa) Equation of motion for mass 1: mx¨1+c1x1+c2x1-x2=mg mx¨1+c1+c2x1-c2x2=mg Equation of motion for mass 2: mx¨2+c2x2-x1+c3x2=mg mx¨2-c2x1+c2+c3x2=mg Writing in matrix form: m00mx¨1x¨2+c1+c2-c2-c2c2+c3x1x2=mgmg Stiffness matrix: k=[c1+c2-c2-c2c2+c3]=[2-1-11+S]b) k-mω2X=F where ω is forcing frequency [2-1-11+S]X1X2=mgmg X1X2=2-1-11+S-1mgmg X1X2=11+2S1+S112mgmg x1=mg2+S1+2S x2=mg31+2S c) If S is very large: x1~∞ x2~0 If S is very small: x1=2mg x2=3mg d) If S is very large: y1=∞ y2=∞ y3=0 If s is very small: y1=2mg y2=mg y3=0

Engineering

Mechanical Engineering

We have a line of two equal masses m connected by three springs with spring constants c₁ = 1, c₂= 1, and c3 = S. Spring 1 is fixed at the top and spring 3 at the bottom, so xo = x3 = 0. (a) Find the stiffness matrix K in the equation Kx = f for the mass displacements. (b) Solve for the displacements x₁ and x₂. (c) If the third spring constant S becomes very large or very small (S→∞ and S→ 0) what are the limiting values of the displacements x₁ and x2? (d) If the third spring constant S becomes very large or very small (S→∞ and S→0) what are the limiting values of the spring forces y1, 32, and y3?

We have a line of two equal masses m connected by three springs with spring constants c₁ = 1, c₂= 1, and c3 = S. Spring 1 is fixed at the top and spring 3 at the bottom, so xo = x3 = 0. (a) Find the stiffness matrix K in the equation Kx = f for the mass displacements. (b) Solve for the displacements x₁ and x₂. (c) If the third spring constant S becomes very large or very small (S→∞ and S→ 0) what are the limiting values of the displacements x₁ and x2? (d) If the third spring constant S becomes very large or very small (S→∞ and S→0) what are the limiting values of the spring forces y1, 32, and y3?

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

Problem 7.
We have a line of two equal masses m connected by three springs
with spring constants c₁ = 1, c₂ = 1, and c3 = S. Spring 1
is fixed at the top and spring 3 at the bottom, so xo = x3 =
0.
(a) Find the stiffness matrix K in the equation Kx = f for the mass
displacements.
(b) Solve for the displacements x₁ and x2.
(c) If the third spring constant S becomes very large or very small
(S→∞ and S→ 0) what are the limiting values of the
displacements x₁ and .x₂?
(d) If the third spring constant S becomes very large or very small
(S→∞ and S→ 0) what are the limiting values of the spring
forces y1, y2, and y3?
91
www
m
3-WW
www
C₁
X₁
"Xo = 0
C2₂
X2
C3
X3 = 0

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