Supplementary information
Acoustic levitation of a large solid sphere
Marco A. B. Andrade1, Anne L. Bernassau2, Julio C. Adamowski3
1
Institute of Physics, University of São Paulo, São Paulo 05508-090, Brazil
2
School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
3
Department of Mechatronics and Mechanical Systems Engineering, Escola Politécnica, University of
São Paulo, São Paulo 05508-030, Brazil
1. Numerical simulation of the axial acoustic radiation force on the sphere as a function
of the sphere radius
Numerical simulations were carried out in COMSOL to obtain the axial acoustic
radiation force on the sphere as a function of R and H, where R is the sphere radius and H is
the distance between the sphere surface and the transducer. As illustrated in Fig. 1, a 2D
axisymmetric linear acoustic model was implemented. The axial acoustic radiation force
produced by only one transducer on the sphere is presented in Fig. 2. In the results of Fig. 2,
the transducer operates at 25230 Hz with displacement amplitude of 15 m.
FIG. 1. Numerical model geometry.
FIG. 2. Simulated axial acoustic radiation force on a sphere as a function of R and H.
In the conditions of the experiment, two transducers were inclined by an angle of 50o
and a third transducer was inclined by 60o. In this configuration, the vertical acoustic radiation
force Ftotal produced by the three transducers on the sphere is given by:

 
 
Ftotal  2 sin 50o  sin 60o F  2.4F ,
(1)
where F is the axial acoustic radiation force produced by one transducer. By replacing the
peak values of Fig. 2 in Eq. (1) we obtain the maximum acoustic radiation force on the sphere
as a function of R (Fig. 3). As shown in Fig. 3, the acoustic radiation force increases almost
linearly from R ranging from 5 to 35 mm. In contrast, the gravitational force on the sphere
increases with R3. By finding the point where the two curves intersect (“x” symbol in Fig. 3)
we can estimate the maximum sphere size that can be levitated. For the transducers operating
at 25230 Hz with displacement amplitude of 15 m, the maximum sphere radius is
approximately 32.5 mm for an expanded polystyrene sphere. It is important to note that this is
the maximum sphere size in the conditions of our experiment and a larger object can be
levitated by increasing the transducer displacement amplitude. It is also worthy to mention
that the acoustic radiation force is proportional to square of the transducer displacement
amplitude.
FIG. 3. Gravitational force and the maximum acoustic radiation force on an expanded
polystyrene sphere as a function of R.
2. Extending the proposed levitation concept
The levitation concept proposed in the paper “Acoustic Levitation of a large solid
sphere” is not restricted to spheres and it can be extended to objects of different shapes and
sizes. Figure 4 illustrates how a large solid object could be acoustically levitated in Earth’s
environment. In Fig. 4, the object is suspended in air by using three ultrasonic transducers at
the bottom of the solid object and lateral stability is achieved by employing side transducers.
FIG. 4. Proposed configuration for levitating large solid objects in Earth’s environment.
We also believe that the levitation concept can be applied to trap large liquid drops in
microgravity. Figure 5 shows how the proposed levitation concept could be modified to allow
the contactless handling of large liquid samples in space. In this figure, a large liquid sample
is confined by the standing waves produced by transducers located around the sample.
FIG. 5. Proposed configuration for levitating large liquid drops in microgravity.