A mass with mass 4 is attached to a spring with spring constant 25.5625 and a dashpot giving a damping 20. The mass is set in motion with in

Question

A mass with mass 4 is attached to a spring with spring constant 25.5625 and a dashpot giving a damping 20. The mass is set in motion with initial position 1 and initial velocity 0. (All values are given in consistent units.) Find the position function x(t):

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Thiên Thanh 2 months 2021-07-30T03:05:02+00:00 1 Answers 4 views 0

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    2021-07-30T03:06:58+00:00

    m = 4

    k = 25.56

    b = 20

    x_o=1\\v_o=0

    The differential equation for damping motion is:

    x''+\gamma x'+\omega_o^2 x=0

    where,

    \gamma = \frac{b}{m} = \frac{20}{4}=5

    \omega_o^2=\frac{k}{m} = \frac{25.56}{4}=6.39

    Substitute the values in the differential equation and consider x” = r², x’ =r and solve:

    x''+5x'+6.39x=0\\r^2+5r+6.39=0\\r=-2.5 \pm0.37i

    Therefore, solution is given by:

    x(t) = e^{-2.5t}[C_1cos0.37t+C_2sin0.37t][tex]\\[/tex]

    at t = 0, x = 1

    C_1=1

    x'(t) =-2.5e^{-2.5t}[C_1cos0.37t+C_2sin0.37t]+e^{2.5t}[-0.37C_1sin0.37t+0.37C_2cos0.37t]

    at t = 0

    x’ =0

    \\[tex]C_2=6.76[/tex]

    x(t) = e^{-2.5t}[cos0.37t+6.76sin0.37t]

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