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Consider the differential equation below.

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Answer #1

(a):

$m\ddot{y} +b\dot{y}+ky= k z(t)+b\dot{z}(t)$

Laplace transform of the above equations is:

$Y= \frac{bs+k}{ms^2+bs+k}Z$

(b):

For m = 1 and $\zeta$ = 0.05, b = 2*0.05*w_n and k = w_n². the lapalce transform is:

0.181 el

Substituing s = jw,

0.11 2

0.1212

Matlab code to plot the graph is given below:

wwn = 0:0.01:100; %w/wn
R = ((1^2+0.1^2*wwn.^2).^0.5)./(((1-wwn.^2).^2+wwn.^2).^0.5); % Amplitude
semilogx(wwn,R)
grid on
xlabel('Frequency');ylabel('Amplitude')

(c):

For frequencies above 100, amplitudes of the sginals are small. Therefore, all the frequencies above 100rad/s can be neglected

(d):

(0-2)-tan-1 = tan-1

Matlab code to plot the phase is given below:

wwn = 0:0.01:100; %w/wn
ang = atan2(0.1*wwn,1) - atan2(0.1*wwn,(1-wwn.^2)); % Phase
semilogx(wwn,ang)
grid on
xlabel('Frequency');ylabel('Phase')

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Answer 1

(a):

$m\ddot{y} +b\dot{y}+ky= k z(t)+b\dot{z}(t)$

Laplace transform of the above equations is:

$Y= \frac{bs+k}{ms^2+bs+k}Z$

(b):

For m = 1 and $\zeta$ = 0.05, b = 2*0.05*w_n and k = w_n². the lapalce transform is:

0.181 el

Substituing s = jw,

0.11 2

0.1212

Matlab code to plot the graph is given below:

wwn = 0:0.01:100; %w/wn
R = ((1^2+0.1^2*wwn.^2).^0.5)./(((1-wwn.^2).^2+wwn.^2).^0.5); % Amplitude
semilogx(wwn,R)
grid on
xlabel('Frequency');ylabel('Amplitude')

(c):

For frequencies above 100, amplitudes of the sginals are small. Therefore, all the frequencies above 100rad/s can be neglected

(d):

(0-2)-tan-1 = tan-1

Matlab code to plot the phase is given below:

wwn = 0:0.01:100; %w/wn
ang = atan2(0.1*wwn,1) - atan2(0.1*wwn,(1-wwn.^2)); % Phase
semilogx(wwn,ang)
grid on
xlabel('Frequency');ylabel('Phase')