Resultados obtenidos

Achieving uniform motion is crucial for the Floppy Wire experiments. Since the fluid damping is low, avoiding transient oscillations is not straightfor- ward. The method used to drive the oscillator was originally developed by Dmitry Zmeev (also independently by Roch Schanen) to ensure uniform

Figure5.3: Floppy Wire frequency sweep around the fundamental mode

of resonance at 173 µK in a magnetic field of 132 mT. The plot shows the in-phase and out-of-phase components of measured voltage. The full line shows fits of the Lorentzian curves to determine the characteristics of the

resonance.

motion of a grid to generate turbulence in4He [73]. The same method can be applied to a Floppy Wire and is described in the following section.

5.2.1

Equation of motion

The Floppy Wire motion can be described by the equation of motion

F(t) =mx¨(t) +λx˙(t) +mω_vac2 x(t) , (5.1)

where F is the time-dependent driving force, λ is the damping term and

ωvac is the resonant frequency in vacuum. The force relates to a drive cur-

rent I simply as

F= BIl , (5.2)

whereB is the intensity of the external magnetic field and l is the length of the Floppy Wire crossbar (9 mm).

A successful ramp is achieved when the Floppy Wire is moved from the initial equilibrium position to another equilibrium position at distance

d at a uniform velocity v without any overshoot or transient oscillations. Previous efforts have shown that simply increasing the driving current I

over distance d at constant velocity v. We choose the time dependence of position as follows x(t) =         

vtacc(1−_2tt_acc)(_tacct )3, 0 <t≤ tacc

vt, tacc <t≤ t3−tacc

d−vtacc(1−_2tt_acc)(_tacct )3, t3−tacc <t≤ t3 ,

(5.3)

where tacc is the acceleration time to velocity v as well as the deceleration

time to stop the Floppy Wire . The chosen x(t) results in a parabolic and continuous acceleration profile ¨x(t) ∼ t2. This profile shows no transient oscillations when driving a vibrating grid [73] and is just as effective for our case. The absence of oscillations is apparent on the emf signature of the Floppy Wire during a short burst pulse as shown in Figure5.4b.

5.2.2

Fork constant calibration

In order to derive the force from measurements of the displacement x(t), knowing the spring constant k of the Floppy Wire is necessary. First, we can calibrate the displacement by slowly increasing the DC drive current through the Floppy Wire and measuring the response of the pick-up coils as shown in Figure 5.4a. An emf is induced in the pickup coils due to the small high frequency signal added on top of the DC drive current. When the coil signal stops increasing the Floppy Wire has hit the cell wall and can move no further.

Alternatively, when an AC signal close to the resonant frequency is added to a DC offset, the Floppy Wire oscillates around its new equilibrium

position and its velocity can be estimated from the emf across the wire. On increasing the DC offset, once the cell wall is reached the emf vanishes as the Floppy Wire is no longer able to move, see Figure 5.4b. Both methods are consistent in determination of the touching voltage.

Figure5.4: Calibration of the Floppy Wire wire position within the cell. a)

The emf of the pickup coils as the wire is moved across the cell. When the emf stops increasing the cell wall was reached. b) Alternatively, the emf of the Floppy Wire wire directly related to its velocity can be measured. The plot shows the induced voltage at two positions near the wall and touching

the the wall.

The process is repeated on both sides of the cell. Since the displacement of the top of the wire is small compared to the Floppy Wire’s leg length, we assume it to be linear with applied current. Knowing the cell dimensions, the total distance between the both wall positions is d = 12.1 mm. The spring constant is then

k(B) = Vt1+Vt2

d , (5.4)

whereVt1 and Vt2are touching voltages shown in Figure 5.4a.

The calibration was performed at 80 mT (1 A in the main magnet, cor- responding to HWD0 = 5.025×104nV Hz/mV for the Floppy Wire reso-

nance) leading to k0 = 0.29 mm/V. The spring constant linearly depends

5.2.3

Resulting driving force

We have introduced the desired profile of displacement as well as the esti- mation of the spring constantk. We can now rewrite the driving force from (5.1) in the form F =k x¨+2π∆f2x˙ + (2πf0) 2 x (2πf0)2 . (5.6)

Before every set of ramps it is necessary to perform a Floppy Wire frequency sweep to determine the three temperature and magnetic field dependent parameters that influence the driving force: width of resonance ∆f2 and

resonant frequency f0 for the force equation (5.6) and the HWD parameter

to determine spring constant in (5.5). An example of a calculated drive voltage together with the expected Floppy Wire position and velocity is shown in Figure5.5. -10 0 10 20 30 40 50 0.0 0.5 1.0 1.5 2.0 Time (ms) v = 50 mm/s d = 1 mm t_acc= 4 ms 0.00 0.25 0.50 0.75 1.00 Velocity (mm/s) Drive (A) Position (mm) 0 10 20 30 40 50

Figure5.5: Calculated driving voltage to move the Floppy Wire by 1 mm at

constant velocity of 50 mm s−1. The expected time dependence of position and velocity are also shown.