Piezoelectric and minimum dimensions of an actuator beam?

[mikehsiao789]

Homework Statement

We want to build a micro-positioning system that produces a maximum travel of
5 μm by using a piezoelectric linear actuator beam with a square cross section (a^2) and length (l) as shown below. A piezoelectric material with piezoelectric
coefficients d13 = -2.0×10-8, d33 = 5.3×10-8, d24 = 3.7 ×10-8 and d25 = 4.1 ×10-8
cm/V ( Si = djiEj ) and a dielectric strength of 106 V/cm is used.

For a maximum power supply voltage of 110 V and a safety parameter of 1.5,
find the minimum dimensions of the beam (a and l). Explain which direction
for electric field is the most effective for creating the required linear travel and
how we should place electrodes to apply such field to the actuator beam.

Homework Equations

S= Dij Ei,

The Attempt at a Solution

I wrote the piezoelectric matrix (s1... s6) = YT + ( d12... d15,... d51... d55), however I am not sure what to do with the dielectric strength and what equations I would go about doing this...[/B]

 

[mark726]

Thank you for your question. I can provide some guidance on how to approach this problem. First, we need to understand the properties of the piezoelectric material being used. From the given information, we can see that the material has four piezoelectric coefficients: d13, d33, d24, and d25. These coefficients describe the relationship between the applied electric field and the resulting strain in the material. In this case, we are interested in the linear travel, which is related to the d33 coefficient.

To find the minimum dimensions of the beam, we need to consider the maximum power supply voltage and the dielectric strength of the material. The dielectric strength is the maximum electric field that the material can withstand before breaking down. In this case, it is given as 106 V/cm. To ensure the safety of the system, we will use a safety parameter of 1.5, meaning that the maximum electric field applied to the material should not exceed 2/3 of the dielectric strength (106 V/cm x 2/3 = 70.67 V/cm).

Now, let's consider the equation S = Dij Ei, where S is the strain, Dij is the piezoelectric matrix, and Ei is the applied electric field. We can rearrange this equation to solve for the electric field: Ei = S/Dij. In this case, we are looking for the maximum linear travel of 5 μm, which is equal to 5 x 10-4 cm. Therefore, the maximum strain (S) that we want to achieve is 5 x 10-4 cm. We also know the value of the d33 coefficient (5.3 x 10-8 cm/V). Substituting these values into the equation, we get Ei = 5 x 10-4 cm / (5.3 x 10-8 cm/V) = 9.43 x 106 V/cm.

Next, we need to find the minimum dimensions of the beam (a and l) that will produce this electric field when a voltage of 110 V is applied. To do this, we can use the equation E = V/d, where E is the electric field, V is the voltage, and d is the distance between the electrodes. In this case, we want to find the distance (d) between the electrodes that will