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The objective of this session is to illustrate equations given in the modal basis (??) and (3). For this, one first analyzes the frequency response then simulation the test process using time transients and some signal processing steps.
To help through various steps, one will use different MATLAB toolboxes
To get the data
The objective of this part is to understand the influence of terms in equation (2) for the computation of transfers as a sum of modal contributions (see course notes section 2.2.4)
H(ω) = [c] [−Mω^{2}+K]^{−1} [b] ≈ 

 (2) 
and thus to analyzed equations of motion in the modal basis
[Is^{2}+[^{\} 2ζ_{j}ω_{j} _{\} ]s+[^{\} ω_{j}^{2} _{\} ]]{q_{r}}=[φ^{T} b]{u} et {y} = [cφ] {q_{r}} (3) 
A FEM mesh of the considered engine cover was done earlier and can be loaded using
load cc_model cf=feplot; cf.model=model; cf.def=def;fecom('ColorDataEvalY'); model def
The data contains two data structures
>> model % model contains the model model = Node: [2031x7 double] % sdtweb('node') Elt: [2021x18 double] % sdtweb('elt') >> def % contains the modes def = def: [10195x30 double] DOF: [10195x1 double] data: [30x1 double] fun: [1 2 0 8 0 0] label: 'Normal modes' lab: {30x1 cell}
model gives nodes, elements, properties ... def contains the first 30 modes.
The following commands allow the generation of transfer functions. Fill in the ??? to obtain an input at 426y (DOF 426.02) and an output at 1976y (DOF 1976.02) (one uses the low level calling format sys=nor2ss(def,damp,InDOF,OutDOF,'Hz acc')).
load cc_model cf=feplot; cf.model=model; cf.def=def;fecom('ColorDataEvalY'); damp=.01; f=linspace(0,5000,2048)'; % frequencies sys=nor2ss(def,damp, ??? , ??? ,'Hz acc'); qbode(sys,f*2*pi,'iiplot "simul1" po'); ci=iiplot;iicom('sub1 1'); % pointer to responses ci.Stack{'simul1'} % Data structure containing response
sys1=nor2ss(def, ??? , ??? , ??? ,'Hz acc'); qbode(sys1,f*2*pi,'iiplot "simul1" po'); sys2=nor2ss(def, ??? , ??? , ??? ,'Hz acc'); qbode(sys2,f*2*pi,'iiplot "simul2" po');
% commandability computation [(1:size(def.def,2))' def.def'*fe_c(def.DOF,[426;911]+.02)'] % Show node numbers fecom('ColorDataEvalY alpha.3');%fecom('showline'); fecom('textnode 426 911','FontSize',16,'Color','r')
From the FEM model, one builds a statespace model using SDT commands, then generate the transient and draws the response using cc_simul
load cc_model cf=feplot; cf.model=model; cf.def=def; uf.t=linspace(0,.5,2048)'; % time vector uf.u0=uf.t*0;uf.u0(find(uf.t<=1e4))=1; % ch1 (input =impact 0.1 ms); uf.window='None'; uf.noise=0; uf.filt=[]; damp=.001; uf.sys=nor2ss(def,damp,426.02,1976.02,'Hz acc'); % State space model uf=cc_simul('simul',uf);cc_simul('plot',uf);
Run the script above. Questions:
load cc_model; cf=feplot; cf.model=model; cf.def=def; uf.t=linspace(0,.5,2048)'; % time vector fmax=1500;[num,den]=butter(16,fmax*2*pi,'s');uf.filt=tf(num,den); uf.u0=uf.t*0;uf.u0(find(uf.t<=1e4))=1; % ch1 (input impact 0.1 ms); uf.window='None'; uf.noise=0;damp=.001; uf.sys=nor2ss(def,damp,426.02,1976.02,'Hz acc'); % statespace model uf=cc_simul('simul',uf);cc_simul('plot',uf)
Describe the usefulness of the antialiasing filter in this simulation.
If you have a little time
uf.window='exponential 0 10 3'; % 0 pts at 0, 10 pts at 1, end at exp(3) uf=cc_simul('simul',uf);cc_simul('plot',uf)
Indication : look at the low frequency response and at the resonances. If you really want to look, it is possible to estimate the damping induced by the exponential window.
uf.t=linspace(0,.5,8000)'; % time vector fmin=200;fmax=5000; u=@(t)cos(2*pi*(fmin+(fmaxfmin)/2*t/uf.t(end)).*t); uf.u0=u(uf.t);
End of lab work 2