Semi-analytical modeling of anchor loss in plate-mounted resonators.

TLDR: It is shown how the semi‐analytical approach of stitching together analytical Lamb wave expressions to the finite element model can be utilized, and the approach is demonstrated for single and double cantilever configurations on a plate.

About: This article is published in Ultrasonics.The article was published on 2018-01-01 and is currently open access. It has received 1 citations till now. The article focuses on the topics: Lamb waves & Finite element method.

Figures

Figure 2: Response graphs for the discrete mass m0 coupled to a semi-infinite beam

Figure 5: Complex roots (unit [rad/mm]) of the dispersion equations for an aluminum substrate plate with 2H = 1.78 mm at f = 100 kHz. Considered modes in semi-analytical approach are highlighted in red

Figure 6: FRFs at reference node for single cantilever problem. Solid lines correspond to the l = 20 mm model and dashed lines denote results for the l = 10 mm model. Magnitudes are normalized to maximum value

Table 2: Results for the double cantilever problem

Figure 10: Transient FE results for double cantilever problem with symmetric (anti-phase) and anti-symmetric loading (inphase) through initial velocity field are shown as red solid lines. Signals are captured at the reference node on the top left corner of the left cantilever and are normalized to maximum value of envelope functions. The impulse responses of fitted SDOF systems are shown as gray dashed lines

Figure 3: Frequency response approach: the resonator and a small segment of the substrate plate are modeled with finite elements. The resonator is harmonically loaded, and the response is determined for a given frequency range

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Semi-analytical modeling of anchor loss in plate-mounted resonators" ?

A semi-analytical technique for estimating the energy loss in a resonator mounted to an infinite plate substrate is proposed in this paper. 

Q2. What have the authors stated for future works in "Semi-analytical modeling of anchor loss in plate-mounted resonators" ?

Improving the method ’ s computational efficiency will be part of future research, as well its application to more realistic resonator models including three dimensional problems, such as hemispherical resonator gyroscopes. 

Q3. What is the FE model for a left-propagating wave?

Since waves through the left interface must propagate to the left (towards −∞), the wavenumber k is replaced by −k to transform the expression for right-propagating waves into the appropriate counterparts for left-propagating waves. 

Q4. What other methods have been developed to extend the scope of such analyses?

Other semi-analytical methods, such as the distributed point source method (DPSM) [26], have recently been developed extend the scope of such analyses. 

Q5. What is the simplest way to determine the harmonic response of a wave?

In this paper, the wave and finite element (WFE) approach [27, 38, 28] is used to determine the real, imaginary and complex roots of the aforementioned dispersion equations due to the method’s versatilityA semi-analytical method is used to determine the harmonic response of a harmonically forced resonator mounted to a plate. 

Q6. How many real, imaginary and complex roots are retained for the analysis?

For the considered plate segment, the interfaces between the FE and analytical domain are about six plate thicknesses away from the cantilever “source,” and only nine roots (real, imaginary and complex) are retained for the analysis. 

Q7. What is the scalar unknown modal amplitudes ai?

T̂ are composed of the corresponding mode shape vectors, i.e.Û = [ û1 û2 . . . ûnm ] and T̂ = [ t̂1 t̂2 . . . t̂nm ] . (16)The scalar unknown modal amplitudes ai are dependent on any wave sources in the FEA model and are combined into the column matrix a. 

Q8. How is the first eigenfrequency of the damped cantilever identified?

The first eigenfrequency of the damped cantilever is identified by fitting an SDOF system to the data, resulting in f = 84.4 kHz and σ = 14.0 1/ms. 

Q9. What is the common practice for creating a model of the resonator?

In any case, it is common practice to create a model of the resonator that includes a small segment of the plate with a finite element (FE) software in conjunction with absorbing boundary elements [11, 5]. 

Q10. What is the simplest way to determine the displacements and stresses of a wave?

The expressions for the displacements and stresses are typically given for right-propagating waves, i.e. wavesthat propagate towards +∞. 

Q11. What is the modal frequency and the harmonic response of the resonator?

The semi-analytical approach requires the specification of a harmonic load and the determination of the subsequent harmonic response at a point on the resonator. 

Q12. How many modes are considered for the least square problem?

Note that for the convergence graphs in Fig. 7, a maximum of nm = 9 modes has been considered since the least-square problem does not yield accurate results for a larger number of modes. 

Q13. What is the simplest way to eliminate the inner degrees of freedom?

Although the inner degrees of freedom uI can be eliminated from Eq. (11) by applyinguI = D−1II (f The author−DIBuB) , (13)this would involve a numerically costly inversion of the matrix DII. 

Q14. How many modes are considered in a second step?

In a second step, the influence of the number of considered modes is investigated, while keeping the length l = 20 mm of the plate segment and the mesh size δ = 100 µm constant. 

Q15. What is the challenge for the semi-analytical steady state analysis?

A related challenge for the semi-analytical steady state analysis is the selection of the forcing term(s) to preferentially excite a given mode. 

Q16. What is the modal frequency and damping of a gyroscope?

The modal frequency and damping can be estimated from the frequency response function (FRF) that has been computed on a frequency grid. 

Q17. What are the parameters of the equations for Lamb waves in multi-layered plates?

A number of so-called semi-analytical methods have recently been developed to determine the roots of the dispersion equations corresponding to propagating, non-propagating and evanescent Lamb waves in multi-layered waveguides [30, 31, 27].