Mathematics, 26.12.2020 05:00, Snowball080717
Let y 00 + by0 + 2y = 0 be the equation of a damped vibrating spring with mass m = 1, damping coefficient b > 0, and spring constant k = 2. (a) Convert this second order equation into a system of two first order equations. (b) Express the eigenvalues for this system in terms of b. (c) Describe the stability of the equilibrium solution ~0 for b > 2 β 2. Justify your claim with information about the eigenvalues of the matrix for the system. (d) Connect the behavior of solutions near an equilibrium of this type with the spring mass system with damping coefficient b > 2 β 2 and explain why your answer for part (c) is (or is not) what one should expect.
Answers: 2
, then Β Ξ»β < 0 and Ξ»β < 0. Thus, the node is stable at equilibrium.
![v = \left[\begin{array}{ccc}+1&1\\-2&-2\\\end{array}\right] \left[\begin{array}{c}v_1\\v_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right]](/tpl/images/1007/7299/52ef7.png)
![\sim v = \left[\begin{array}{ccc}1&1\\0&0\\\end{array}\right] \left[\begin{array}{c}v_1\\v_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right]](/tpl/images/1007/7299/50e68.png)
![v = \left[\begin{array}{c}-1\\1\\\end{array}\right]](/tpl/images/1007/7299/c644d.png)
![m = \left[\begin{array}{ccc}2&1\\-2&-1\\\end{array}\right] \left[\begin{array}{c}m_1\\m_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right]](/tpl/images/1007/7299/675de.png)
![\sim v = \left[\begin{array}{ccc}2&1\\0&0\\\end{array}\right] \left[\begin{array}{c}m_1\\m_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right]](/tpl/images/1007/7299/c934f.png)
![m = \left[\begin{array}{c}-1/2 \\1\\\end{array}\right]](/tpl/images/1007/7299/7ae2d.png)
![\left[\begin{array}{c}y_1\\y_2\\\end{array}\right]= C_1 e^{-t} \left[\begin{array}{c}-1\\1\\\end{array}\right] + C_2e^{-2t} \left[\begin{array}{c}-1/2\\1\\\end{array}\right]](/tpl/images/1007/7299/4be49.png)
![\mathbf{ \left[\begin{array}{c}y_1\\y_2\\\end{array}\right]= \left[\begin{array}{c}0\\0\\\end{array}\right] \ \ so \ stable \ at \ node \ \infty }](/tpl/images/1007/7299/22a41.png)
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Let y 00 + by0 + 2y = 0 be the equation of a damped vibrating spring with mass m = 1, damping coeffi...
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