acceleration amplitude increases from 5 mm.s−2 to 150 mm.s−2. In Figure3.9b, the transverse acceleration amplitude is fixed at 150mm.s−2 and the damping ratio increases from 0% to 1%.
(a) Frequency-response curves (0.3 % damping) at different acceleration amplitudes (solid=1000 rpm, dashed=5000 rpm).
(b) Frequency-response curves (150 mm.s−2acceleration amplitude) at different damping ratios (solid=1000 rpm, dashed=5000 rpm).
Figure 3.9: Frequency-response curves for increasing damping and amplitude of excitation.
We notice that the higher the damping the lower the response and that for high enough damping, the jump phenomenon and hysteresis disappear. In both cases we notice that the amplitude of the response for a given hub acceleration level decreases with the angular velocity and thus the amount of stretching in the structure.
the centrifugal force and the Coriolis force are added to the simulations. First, we find the equilibrium of a spinning membrane deflected by gravity is calculated. Then, the membrane is deformed according to the first axisymmetric mode of vibration. Finally, a nonlinear dynamic simulation of the free vibration of the membrane with little stiffness proportional damping is performed. This is similar to the technique described bySoares and Gon¸calves(2012).
We consider a linear elastic material. The membrane is modeled with S4R shell elements: 4-node doubly curved thin shell elements with reduced integration, hourglass control, finite membrane strains, and 5 thick- ness integration points elements. The mesh contains 2,151 nodes. We follow the steps summarized in Figure 3.10. We first apply a centrifugal and Coriolis loads (step 1) with angular velocityωto stiffen the membrane.
Then gravityg is added (step 2) to obtain the equilibrium deflected shape. For those two steps we use a linear “static” step (Newton’s method). From this equilibrium we can evaluate the mode shapes and natural frequencies in a linear perturbation step (step 2’) and introduce this as an initial perturbation in a “static”
step (step 3). To introduce this initial perturbation we impose the appropriate displacement as boundary conditions to each node of the membrane using a Python script. As Abaqus imposes boundary conditions relative to the initial shape (flat and unstressed) the imposed displacements are the equilibrium plus a chosen amplitudeAmultiplied by the mode shape. Finally we run a nonlinear dynamic implicit simulation (step 4) with small stiffness proportional damping. This damping is necessary to decrease the amplitude of oscillation and to be able to monitor the associated change in frequency. Too high damping damps the oscillations too fast to precisely estimate the backbone curve, whereas too little damping unnecessarily increases the time of the simulation.
Figure 3.10: Steps of FEM simulation.
We ran simulations at two angular velocities: 1200 rpm and 4000 rpm. 1200 rpm is close to the buckling angular velocity and 4000 rpm to the maximum speed of the motor in the experimental setup detailed in part 3.4. In this step we use the Hilber-Hughes-Taylor integration scheme with no numerical damping (α=0) and small time step (half increment residual). A small time step is necessary to obtain accurate results. Large time steps would for example change the apparent resonant frequency of oscillation of the membrane. A
convergence analysis was carried out to estimate a proper time step and a maximum time step was imposed.
The time step was tuned to obtain accurate natural frequency at low amplitude of oscillation (linear system).
We found that the parameters shown in Table 3.2 give accurate results. Figures 3.11 and3.11a show the time response at a point on the outer edge for the two simulations.
ω (rpm) 1200 4000
g (m.s−2) 9.81 9.81 β (s) 4×10−5 5×10−5
A(mm) 0.5 1
δtmax(s) 0.01 0.002
T (s) 22 35
Total CPU time (s) 4.3×105 1.4×106 Table 3.2: Dynamic simulations parameters.
Time (s)
0 5 10 15 20
EdgeDeflection(mm)
-0.5 0 0.5
(a) Free vibration at 1200 rpm.
Time (s)
0 10 20 30
EdgeDeflection(mm)
-1 -0.5 0 0.5 1
(b) Free vibration at 4000 rpm.
Figure 3.11: FEM results.
3.3.2 Post Processing
We use a method similar to the one introduced by Londo˜no et al. (2015). It consists of estimating the instantaneous frequencies and envelope amplitudes based on the decaying response. First, the zeros of the response are found. In order to minimize noise, due to the imprecise estimation of the zeros for example, we don’t estimate the frequency and amplitude between each of those zeros, but base on 10 periods. The instantaneous frequencies are defined as ten times the inverse of the time between 10 periods. As the damping ratio is very small in these decay curves, it doesn’t affect the instantaneous frequency. The instantaneous amplitude is the maximum amplitude within each ten periods.
Figure 3.12aand3.12bshow the instantaneous frequency and amplitude, respectively, at 1200 rpm and 4000rpm. First, we notice that the data are more sparse at 1200 rpm than 400 rpm. Even if the signal
runs for the same amount of time, as the frequency is higher at 4000 rpm, we have more periods to process.
One could run a longer simulation with less damping at 12000 rpm to obtain more results. We notice that the frequency increases with amplitude at 1200 rpm, characteristic of softening behavior. Conversely, the frequency decreases with amplitude at 4000 rpm, characteristic of hardening behavior.
5 10 15 20
21.7 21.8 21.9
5 10 15 20
0.2 0.4
(a) 1200 rpm.
5 10 15 20 25 30
46.5 47
5 10 15 20 25 30
0.5 1
(b) 4000 rpm.
Figure 3.12: Instantaneous frequencies and amplitudes.
3.3.3 Comparison Between Reduced Order Model and FEM Simulations
In Figures 3.13a and 3.13b we compare the reduced order model and FEM backbone curves at 1200 rpm and 4000 rpm, respectively. We notice an excellent agreement between the reduced order model and the nonlinear FEM simulations. This confirms the validity of the reduced order model.