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5.8(C). The Dutch roll motion can be approximated using the following equations: egin{bmatrix} Delta dot{eta} \ Delta dot{r} end{bmatrix} = egin{bmatrix} frac{Y_eta}{u_0} & -left(1 - frac{Y_r}{u_0} ight) \ N_eta & N_r end{bmatrix} egin{bmatrix} Delta eta \ Delta_r end{bmatrix} + egin{bmatrix} frac{Y_{delta_r}}{u_0} \ N_{delta_r} end{bmatrix} Delta delta_r Assume the coefficients in the plant matrix have the following numerical values: $Y_eta = -7.8 ext{ ft/s}^2 quad N_r = -0.34 ext{ 1/s} quad Y_{delta_r} = -5.236 ext{ ft/s}^2$ $Y_r = 2.47 ext{ ft/s} quad u_0 = 154 ext{ ft/s} quad N_{delta_r} = 0.616 ext{ 1/s}^2$ $N_eta = 0.64 ext{ 1/s}^2$ (a) Determine the Dutch roll eigenvalues. (b) What is the damping ratio and undamped natural frequency? (c) What is the period and time to half amplitude of the motion? (d) Determine the response of the system if the initial conditions are as follows: egin{bmatrix} Delta eta \ Delta r end{bmatrix} = egin{bmatrix} 0.1 \ 0.0 end{bmatrix} From the time history plot, estimate the period and time to half amplitude. (e) Determine the response of the system to a step input. For this part assume that the initial conditions are both 0. The last two parts of this problem should be solved by computer.

          5.8(C). The Dutch roll motion can be approximated using the following equations:

egin{bmatrix} Delta dot{eta} \ Delta dot{r} end{bmatrix} = egin{bmatrix} frac{Y_eta}{u_0} & -left(1 - frac{Y_r}{u_0}
ight) \ N_eta & N_r end{bmatrix} egin{bmatrix} Delta eta \ Delta_r end{bmatrix} + egin{bmatrix} frac{Y_{delta_r}}{u_0} \ N_{delta_r} end{bmatrix} Delta delta_r

Assume the coefficients in the plant matrix have the following numerical values:
$Y_eta = -7.8 	ext{ ft/s}^2 quad N_r = -0.34 	ext{ 1/s} quad Y_{delta_r} = -5.236 	ext{ ft/s}^2$
$Y_r = 2.47 	ext{ ft/s} quad u_0 = 154 	ext{ ft/s} quad N_{delta_r} = 0.616 	ext{ 1/s}^2$
$N_eta = 0.64 	ext{ 1/s}^2$

(a) Determine the Dutch roll eigenvalues.
(b) What is the damping ratio and undamped natural frequency?
(c) What is the period and time to half amplitude of the motion?
(d) Determine the response of the system if the initial conditions are as follows:
egin{bmatrix} Delta eta \ Delta r end{bmatrix} = egin{bmatrix} 0.1 \ 0.0 end{bmatrix}
From the time history plot, estimate the period and time to half amplitude.
(e) Determine the response of the system to a step input. For this part assume that the initial conditions are both 0.

The last two parts of this problem should be solved by computer.

$5.8(C). The Dutch roll motion can be approximated using the following equations: eginbmatrix Delta doteta Delta dotr endbmatrix = eginbmatrix fracYetau0 -left(1 - fracYru0 ight) Neta Nr endbmatrix eginbmatrix Delta eta Deltar endbmatrix + eginbmatrix fracYdeltaru0 Ndeltar endbmatrix Delta deltar Assume the coefficients in the plant matrix have the following numerical values: Yeta = -7.8 ext ft/s^2 quad Nr = -0.34 ext 1/s quad Ydeltar = -5.236 ext ft/s^2 Yr = 2.47 ext ft/s quad u0 = 154 ext ft/s quad Ndeltar = 0.616 ext 1/s^2 Neta = 0.64 ext 1/s^2 (a) Determine the Dutch roll eigenvalues. (b) What is the damping ratio and undamped natural frequency? (c) What is the period and time to half amplitude of the motion? (d) Determine the response of the system if the initial conditions are as follows: eginbmatrix Delta eta Delta r endbmatrix = eginbmatrix 0.1 0.0 endbmatrix From the time history plot, estimate the period and time to half amplitude. (e) Determine the response of the system to a step input. For this part assume that the initial conditions are both 0. The last two parts of this problem should be solved by computer.$

Added by Nicole G.

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