Example23 Air Damping of Cantilever

General

• In this example, a cantilever is analyzed with damping by the air resistance taken into account, and the damping ratio (1/(2*Q)) of the fundamental mode is obtained. The damping is given by the acoustic impedance boundary condition. Its value is presumed from the drag applied to the sphere in the fluid.
• The obtained damping ratio matches with the experiment value by the difference of about 20%.
• Piezoelectricity is ignored in the calculation.
   
• Unless specified in the list below, the default conditions will be applied.

Analysis Condition

Resonant Analysis Tab

Model

The model is a cantilever of SiO2 (300ｘ40ｘ1[um]) having an Au layer (thickness: 0.1 m) around it. The length of the cantilever (L) is changed to 300, 250, 200, and 150[um] for analysis.

Body Attribute and Material Property

Boundary Condition

(*)Zre and zim are defined by variables for easy input.

To perform analysis taking air damping into account, it is required to set a proper value for the acoustic impedance (Z).

As a method to determine the acoustic impedance value having the equivalent effect of air resistance, it is possible to determine the acoustic impedance value so that the analysis results match the experimental results. In this example, however, a method is devised to determine the acoustic impedance by calculation so that a reasonable analysis is possible even if there is no experimental results.

Since the method to calculate the acoustic impedance directly is not known, an approximate method is used. In the air, an force (F) applied to a sphere which vibrates harmonically in a small amplitude can be acquired theoretically. An equation is shown as in (1).

(1)

 where R: radius of sphere, ω: angular frequency, u: velocity, η: air viscosity=1.81e-5[Pa･s] ρ: air density=1.18[kg/m3]

Using this equation, the acoustic impedance is determined so that, when the disc with the same radius (R) as the sphere rotates at the same velocity as the sphere, receives the same force from the air as the sphere. (Figure 1) The input value (Z) for the boundary condition of acoustic impedance is Z=pressure/velocity. The pressure is acquired by dividing the force in equation (1) by the surface area of the disc (2πR2). The front and back sides of the disc are taken into account, and the influence of the side face is ignored. Based on this idea, the acoustic impedance can be given by the equation (2). The acceleration can be written as du/dt=jωu as the vibration is assumed to be harmonic.

Figure 1 (A) Sphere that receives force from air by vibration (B) Disc that receives force from air by vibration

Acoustic impedance of the disc is determined so that the disc receives the same force as the sphere.

(2)

Assuming the sphere has a diameter equivalent to the depth of the cantilever (b), the acoustic impedance is obtained by the equation (2).

As the angular frequency is necessary to calculate the equation (2), it is acquired by the resonant analysis without setting the acoustic impedance (Z=0).

The table below shows the acquired resonant frequencies and acoustic impedances by the equation (2). Since the depth of the cantilever is 40μm, the calculation is performed with R=20μm.

Table 1 The resonant frequencies without setting the resonant impedances (Z=0) and the resonant impedances acquired by the equation (2)

Go to Analysis Condition > Resonant Analysis tab > Approximated Frequency, and enter f0 [Hz] of the Table 1. By setting the acoustic impedance value, that is given with the frequency, to the outer boundary condition for analysis, the complex resonant frequency will be acquired.

Damping ratio and Q factor are acquired from the complex resonant frequency.

• Analysis 1 Run the solver for the project file as is. The length (l) of the cantilever is 300[um].
• Result 1 Resonant frequency (real part only) is acquired. (f0 of the Table 1 is acquired.)
• Analysis 2 Go to Analysis Condition > Resonant Analysis tab > Approximated Frequency, and enter f0. Enter the acoustic impedance Z in the Analysis Condition.
• Result 2 Complex resonant frequency is acquired. Damping ratio and Q factor are acquired from the complex resonant frequency.
• In the project file in this example, the length (l) of the cantilever is set to 300[um], and the acoustic impedance (Z) is set to０[Pa*s/m]. As these are defined as variables, modification is easy.

Results

The result diagram is shown as below.

The fundamental vibration mode of the cantilever can be observed. Color of the contour correspond to the magnitude of displacement.

The list below shows the acquired complex resonant frequencies (f0). The damping ratio is acquired as a ratio of real part and imaginary part of f0. The values are compared with the experimental values described in the source below.

Please take a look at the right column. It shows the ratio of calculated value and experimental value of the damping ratio. They match within error of 20%.

For your reference, the relation of damping ratio (ζ), Q , and complex resonant frequency (f0=fre+j fim) is shown below.

[Source]

Christian Bergaud, Liviu Nicu and Augustin Martinez ： Multi-Mode air damping analysis of composite cantilever beam ,Jpn.J.Appl.Phys., 38,(1999)6521