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2  Consistence study for a one DOF system

The idea is here to run a numerical simulation of a system with known behavior in order to check that you understand how to used basic tools : time integration, differences between the continuous and discrete Fourier transform, properties of the basic oscillator.

2.1  Free response and FFT

Questions :

Programming suggestions

% Create your deriv function which will have (t,y,PA) as inputs
% Store all parameters in a structure
% Use t,y for time, F,Y for frequency response

% Damping study, store multiple results as a structure, then finalize plot
damp=[ ??? ];
C1=struct('X',[],'Y',[]); % prepare structure for result
for j1=1:length(damp)
 C1.X=f; C1.Y(:,j1)=Y(:,1); % store first column for each result in loop

% Finalize plot
plot(C1.X,C1.Y);xlabel('x');ylabel('y');set(gca,'xlim',[0 50])

2.2  Forced response

Questions :

  1. Apply a sinusoidal force (choose the amplitude and frequency) and illustrate how the corresponding frequency appears in the fft of the response.
  2. Illustrate windowing issues by keeping an integer/non-integer number of periods, in other words, a frequency that exists/does not exist in the spectrum.
  3. Illustrate how the transient decreases and depends on initial conditions. (Tip : compare fft for a full signal or a signal starting at T>0).
  4. Illustrate Shannon's theorem by using an excitation above the half sampling frequency (spectrum symmetry point).

Programming suggestions

% Forced response
PA.u=@(t)sin(10*t); % Define an "anonymous" function for response
% Truncate the first 10 s

2.3  Non-linear response

One now considers the classical cubic non-linearity (even if this rarely corresponds to something mechanical)

M  q + C q + K (q+α q3) = F(t)     (1)

α=10−6 is a decent starting value. Pay attention to the need to eliminate the transient associated with the initial condition.

2.4  Sweep excitation

One considers an excitation through a swepped sine

 PA=struct('A', ??? ,'B',[0;1],'u',[]); % Adjust damping here
 fmin=15;fmax=25;PA.u=@(t)cos((fmin+(fmax-fmin)/2*t/tspan(end))*t);  % Sweep

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