Hello, my name is Emil Garnell, and I am a PhD Student in the lab for mechanics at the ENSTA Paristech. I will present you quickly my research about a new type of loudspeakers based on an active elastomeric material. You will see in this presentation how modal analysis can be used on non-linear structures. In standard loudspeakers, the membrane is moved by the magnetic field, created by a magnet and a voice coil. This design have not changed for years so good perfomances have been reached. However they have two main drawbacks: they are heavy and quite thick. I am looking at a completely new type of loudspeakers. They are thin, lightweight and should be more efficient than the standard ones. For my loudspeakers, I use a new type of materials, that have been discovered about 20 years ago - they are called dielectric elastomers. They have first been used as artificial muscles or micro-pumps, but their possible range of applications is huge. They consist of an elastomeric material sandwiched between flexible electrodes. The elastomer can typically be silicone, and carbon grease is a good choice for the electrodes. The working principle is that when an electric voltage is applied, electric charges are moved to the electrodes. Positive charges on one side of the membrane and negative ones on the other side. As you well know, positive and negative charges are attracted to each other. This results is an applied pressure on both sides of the membrane. As the elastomer is incompressible, the change in thickness of the membrane creates a change in area. We will use this effect to create sound later. Now, if the membrane is put on a tube, and inflated with a static pressure delta P, what is going to happen when you apply the electric field? As we said before, the membrane is going to expand in area. But because of the pressure, the membrane will move outwards, and this globally results in an increased volume. Imagine now that we do that periodically, we obtain a pulsating volume source, that is to say a loudspeaker! I use an electro-mechanical model to analyse the behaviour of my loudspeakers. It is highly non linear because of the large deformations, and because of the non linear behaviour of the material. We apply this model to compute the deformed shape of the membrane when it is inflated and when a voltage is applied. It is performed using finite elements. On this plot in axisymmetric coordinates, you can see in blue the deformed shape of the membrane when it is inflated. No voltage is applied yet. In red, you have the deformation that is obtained when a DC voltage is applied. The yellow zone is where the electrode is located. You see that the voltage increases the deformation. The shape of the deformed membrane depends on: - what voltage is applied - where the electrode is located. The question now is what happens when the voltage is the sum of a DC component and an alternative component. Indeed, this is how we intend to make the membrane vibrate. The idea is that the membrane will oscillate around a state of equilibrium, that is determined by the static inflation pressure, and the DC component of the applied voltage V_c. We are mostly interested in small vibrations around this equilibrium, so we will use the framework of linear modal analysis. We linearise the non linear equations around the equilibrium. This allows us to compute the modes. We will use those modes to describe the vibration behaviour of the membrane. We used finite elements to compute the structural modes. On this plot (again in axisymmetric coordinates), you can see : - in black the equilibrium shape, that the membrane reaches when a static inflation pressure and a DC voltage is applied. - in colors the first mode-shapes of the structure around the equilibrium. As the computation has been carried out in axisymmetric coordinates, only the axisymmetrical modes are obtained. You can notice that these modes have an increasing number of nodal circles . faire apparaitre et lancer video mode.mp4 en boucle I also measured the modes, to check that my model is valid. On this video you can see the third measured mode, at 250 Hz. It has two nodal circles, as the computed one. A good loudspeaker has a flat frequency response, meaning that the same level of sound is emitted at all frequencies. I measured the frequency response of my prototype, by sending a sweep signal and measuring the velocity of one point of the membrane. lancer la video chirp.mp4 The measured transfer function is plotted here. Obsiously it is not flat, you can see many peaks, which are caused by the membrane modes. The idea in my PhD is that we should be able to improve this frequency response if we control the membrane modes. If we have a closer look at mode 3 for example, it has 2 nodal circles. Now, let us try to understand how we can excite this mode. If for example we apply a force where you see the red arrows, is this mode going to respond? Obviously not, as the modal force is the integral of the modeshape multiplied by the force. As the modeshape is zero where the force is applied, the modal force is zero and this mode will not move. On the other hand, if we now apply a force at the green arrows, the modeshape is large at those points, so the projection of this force on this mode will be non zero. This means that this mode is excited by this force. Here we applied exterior forces on the membrane to illustrate that we can select which modes are going to be excited depending on where the force is applied. In the case of our elastomeric membrane, they are no exterior forces, but the force is created by the charges on the electrodes. The idea of my PhD is that by changing the shape of the electrodes, we can select which modes are excited. We can then optimize the shape of the electrodes to control the excited modes. The goal is to radiate sound. The different mode-shapes have different radiation properties: different directivity, different radiation efficiency. So by selecting the right ones, we should be able to improve the quality of the radiated sound. Thanks to the innovative material that we use, we can do that just by changing the shape of the electrode. For example, the electrode could be just a circular patch in the middle of the membrane. We have then 1 control parameter, the outer radius. This should allow us to control maximum 1 mode. But if the electrode is a circular annulus, there is a second control parameter: the inner radius of the electrode. It should therefore be possible to control up to 2 modes. Using axi-symmetric designs of electrodes only axi-symmetric modes can be controlled. This is our primary goal. But what if we use non axi-symmetric designs of electrodes, are we going to be able to control the other modes too? Maybe there will be an answer in a year or two! Thank you for your attention.
