Contents Functions PDF Index |

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.

Questions :

- 1. Write the state space model associated with a spring mass system with M = 1, K=4e2
- 2. Choose C to have a damping ratio of 1 % (tip : you need to compute eigenvalues of the state matrix A at see the ration between the real part and the amplitude of poles)
- 3. Use a time integration function ode45 (in the engine cover application, we will also use lsim which is faster but only works for LTI systems), to compute the response to a non zero initial condition. Plot this response.
- 4. Chooze a time interval and compute the Fourier transform (using fft). Find the resonance frequency and in this FFT (compute the frequencies from the time vector).
- 5. Illustrate the influence of the damping ration by overlaying the fft for three distinct values of C.

Programming suggestions

% Create your deriv function which will have (t,y,PA) as inputs % Store all parameters in a structure PA=struct('A',[],'B',[],'u',[]); [t,y]=ode45(@deriv,tspan,y0,odeset,PA) % 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 end % Finalize plot plot(C1.X,C1.Y);xlabel('x');ylabel('y');set(gca,'xlim',[0 50])

Questions :

- Apply a sinusoidal force (choose the amplitude and frequency) and illustrate how the corresponding frequency appears in the fft of the response.
- 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.
- Illustrate how the transient decreases and depends on initial conditions. (Tip : compare fft for a full signal or a signal starting at T>0).
- 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 ind=(t>10);t=t(ind);y=y(ind,:);

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

M q + C q + K (q+α q^{3}) = F(t)
(1) |

- Apply a sinusoidal force. What frequencies are seen in the response fft ?
- How does the response at harmonics of the excitation frequency vary with amplitude (or the α coefficient).

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

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

- In the time response, how is the resonance visible ?
- How does the variance (var) change with damping ?

©1991-2020 by SDTools