WO2024149731A1

WO2024149731A1 - Method and device for calculating frequency shifts of angular frequencies occurring in a mems system, and mems system

Info

Publication number: WO2024149731A1
Application number: PCT/EP2024/050347
Authority: WO
Inventors: Peter Degenfeld-Schonburg
Original assignee: Robert Bosch Gmbh
Priority date: 2023-01-11
Filing date: 2024-01-09
Publication date: 2024-07-18

Abstract

In a computer implemented method for calculating frequency shifts of angular frequencies occurring in a micro-electromechanical system, MEMS, the MEMS system is modelled by a finite element (FE) model. Modal angular frequencies and form factors are calculated for a plurality of major modes of the FE model. The entirety of the modes of the FE model are thereby divided into a plurality of main modes and into a plurality of minor modes. The major modes comprise a drive mode of the MEMS system. For each node of the FE model, a first force on the node dependent on the minor modes is calculated using the form factors, and a second force on the node dependent on the minor modes and the drive mode is calculated. First auxiliary modes are calculated, which correspond to the first force. Second auxiliary modes are calculated, which correspond to the second force. A first energy contribution and a second energy contribution are calculated using the first and second auxiliary modes. Using the modal angular frequencies, a frequency shift of one of the angular frequencies is calculated from the first energy contribution and the second energy contribution, which is dependent on an amplitude of the drive mode.

Description

Title

Method and apparatus for calculating frequency shifts of angular frequencies occurring in a MEMS system, and MEMS system

The present invention relates to a method and a device for calculating frequency shifts of angular frequencies occurring in a microelectromechanical system (MEMS system). The invention further relates to a MEMS system.

State of the art

Increasing demands for performance and functionality at low cost require miniaturized and more complex MEMS designs. As a result of miniaturization, challenges in the development phases arise from unexpected nonlinear behavior in MEMS systems, such as gyroscopes and micromirrors, which are due to geometric nonlinearities.

An important feature of geometric nonlinearities are frequency shifts of angular frequencies that affect all modes due to large changes in the oscillation amplitudes of the driven mode. These frequency shifts can generate unexpected resonances that lead to undesirable nonlinear effects and affect the reliability of the MEMS system.

When developing MEMS systems, numerous simulations and evaluations of performance parameters are carried out to support the decision-making processes in product development. The simulation of geometrically nonlinear phenomena is particularly challenging.

An approximate approach to reducing the model order is known from Touze et al. “Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques”, Nonlinear Dyn. 105, 1141-1190 (2021). However, there is a need for efficient approaches to determine the frequency shifts of angular frequencies caused by geometric nonlinearities in a real MEMS system with high mode density.

Disclosure of the invention

The invention provides a method and a device for calculating frequency shifts at angular frequencies occurring in a MEMS system, as well as a MEMS system having the features of the independent patent claims.

Preferred embodiments are the subject of the respective subclaims.

According to a first aspect, the invention therefore relates to a computer-implemented method for calculating frequency shifts of angular frequencies occurring in a microelectromechanical MEMS system. The MEMS system is modeled by a finite element (FE) model. Modal angular frequencies and form factors are calculated for a plurality of main modes of the FE model. The totality of the modes of the FE model is divided into the plurality of main modes and a plurality of secondary modes. The main modes comprise a drive mode of the MEMS system. For each node of the FE model, a first force on the node that depends on the secondary modes is calculated using the form factors, and a second force on the node that depends on the secondary modes and the drive mode is calculated. First auxiliary modes are calculated that correspond to the first force. Second auxiliary modes are calculated that correspond to the second force. A first energy contribution and a second energy contribution are calculated using the first and second auxiliary modes. Using the modal angular frequencies, the first energy contribution and the second energy contribution, a frequency shift of one of the angular frequencies is calculated, which depends on an amplitude of the drive mode.

According to a second aspect, the invention relates to a device for calculating frequency shifts in angular frequencies occurring in a MEMS system, with a computing device. The computing device is designed to model the MEMS system using an FE model and to calculate modal angular frequencies and to calculate form factors for a plurality of main modes of the FE model, wherein a totality of modes of the FE model is divided into the plurality of main modes and secondary modes, wherein the main modes comprise a drive mode of the MEMS system. The computing device is further designed to calculate, for each node of the FE model, a first force on the node that depends on the secondary modes and a second force on the node that depends on the secondary modes and the drive mode using the form factors. The computing device is further designed to calculate first auxiliary modes that correspond to the first force and second auxiliary modes that correspond to the second force, and to calculate a first energy contribution and a second energy contribution using the first and second auxiliary modes. The computing device is further designed to calculate a frequency shift of the angular frequency that depends on an amplitude of the drive mode using the modal angular frequencies, the first energy contribution and the second energy contribution.

According to a third aspect, the invention relates to a MEMS system with a device for calculating frequency shifts of angular frequencies occurring in the MEMS system. The MEMS system further comprises a control device which is designed to control at least one component of the MEMS system using a frequency shift of the angular frequency calculated by the device.

Advantages of the invention

The invention makes it possible to take into account nonlinear frequency shifts caused by geometric nonlinearities of the MEMS system. The drive mode (driven mode) can have a changing large amplitude, which leads to geometric nonlinearities, so that in turn frequency shifts arise, which can excite other modes in the event of resonance.

The efficient and accurate evaluation of these frequency shifts is of great importance for the MEMS design and the product reliability of MEMS systems. The invention in particular provides a method and a device for the efficient and precise calculation of such frequency shifts caused by geometric nonlinearities.

The method can only be carried out by using design methods and linear solution methods and is therefore not dependent on convergence criteria of numerical solvers or the like. The method is particularly applicable to oscillating mechanical sensor and actuator systems as MEMS systems with a high degree of quality in the limiting case of large mechanical strains but small mechanical stresses within the framework of elasticity theory.

In the context of this invention, the calculation of the frequency shifts of the angular frequencies can be understood to mean that the angular frequency is calculated as a function of the amplitude of the drive mode. The frequency shift of the angular frequency is thus calculated as a function of the amplitude of the drive mode, relative to the angular frequency when the amplitude of the drive mode disappears.

According to a further embodiment of the method for calculating frequency shifts of angular frequencies occurring in the MEMS system, the FE model comprises a mass matrix and a stiffness matrix. The modal angular frequencies and the form factors for the main modes of the FE model are calculated using the mass matrix and the stiffness matrix. The first auxiliary modes and the second auxiliary modes are calculated using the stiffness matrix of the FE model. The FE model can be generated using known methods.

According to a further embodiment of the method for calculating frequency shifts of angular frequencies occurring in the MEMS system, mode coupling coefficients are calculated using the form factors, the first force and the second force. The first energy contribution and the second energy contribution are further calculated using the mode coupling coefficients. The frequency shift of the angular frequency is further calculated using the mode coupling coefficients. The mode coupling coefficients correspond to effective interactions of modes. According to a further embodiment of the method for calculating frequency shifts of angular frequencies occurring in the MEMS system, a nonlinear mechanical mode stress is calculated using the form factors. A modal Duffing coefficient and modal cross-Duffing coefficients are calculated using the nonlinear mechanical mode stress. The frequency shift of the angular frequency is further calculated using the modal Duffing coefficient and the modal cross-Duffing coefficients. The Duffing coefficient ß denotes the coefficient of the cubic term in the so-called Duffing equation, where the same mode occurs cubically. With the cross-Duffing coefficient, different modes occur cubically.

According to a further embodiment of the method for calculating frequency shifts of angular frequencies occurring in the MEMS system, the mode coupling coefficients, the modal Duffing coefficient and the modal cross-Duffing coefficients depend on the frequency shift of the angular frequency. The frequency shift of the angular frequency depends on the amplitude of the drive mode. The frequency shift of the angular frequencies is calculated iteratively. The iterative solution makes the calculation self-consistent in order to take into account a highly nonlinear dependence of the angular frequencies on the amplitude of the drive mode.

According to a further embodiment of the method for calculating frequency shifts of angular frequencies occurring in the MEMS system, in order to calculate the frequency shift of the angular frequency, a contribution of the secondary modes is calculated using the modal Duffing coefficient and modal cross-Duffing coefficient, the first energy contribution and the second energy contribution. The secondary modes are thus described by the effective energy contributions, i.e. the summation over the secondary modes is carried out effectively.

According to a further embodiment of the method for calculating frequency shifts of angular frequencies occurring in the MEMS system, a contribution of the main modes is calculated to calculate the frequency shift of the angular frequency, wherein contributions of all main modes are summed. The Contributions of the main modes are thus explicitly taken into account.

According to a further embodiment of the method for calculating frequency shifts of angular frequencies occurring in the MEMS system, 3-mode coupling coefficients are calculated using the form factors, the first force and the second force. The frequency shift of the angular frequency is further calculated using the 3-mode coupling coefficients. The 3-mode coupling coefficients (or 3-wave coupling coefficients) denote effective interactions of three modes.

According to a further embodiment of the method for calculating frequency shifts of angular frequencies occurring in the MEMS system, the MEMS system is further controlled using the calculated frequency shift of the angular frequency. For example, the control of micromirrors or rotation rate sensors can be dependent on the calculated frequency shift of the angular frequency.

Further advantages, features and details of the invention emerge from the following description, in which various embodiments are described in detail with reference to the drawing.

Short description of the drawings

Show it:

Figure 1 is a schematic block diagram of a MEMS system with a device for calculating frequency shifts of angular frequencies occurring in a MEMS system according to an embodiment of the invention; and

Figure 2 is a flow chart of a method for calculating frequency shifts of angular frequencies occurring in a MEMS system according to an embodiment of the invention. The numbering of process steps is for clarity and should not generally imply a specific chronological order. In particular, several process steps can be carried out simultaneously.

Description of the embodiments

Figure 1 shows a schematic block diagram of a MEMS system 1 with a device 2 for calculating frequency shifts of angular frequencies that occur in the MEMS system 1. The MEMS system 1 can be, for example, a sensor device or an actuator device. The MEMS system 1 comprises, for example, a micromirror and/or a rotation rate sensor, such as a MEMS gyroscope.

The device 2 comprises a computing device 4 and a storage device 5. The computing device 4 can comprise software and/or hardware components, such as processors, microprocessors, microcontrollers, integrated circuits, application-oriented integrated circuits or the like. The storage device 5 can comprise a volatile or non-volatile memory, such as a hard disk, memory card or the like.

The computing device 4 is designed to model the MEMS system using an FE model. The FE model enables a mode analysis and the computing device 4 determines two global FE matrices, the mass matrix M and the stiffness matrix K.

The computing device 4 further performs a mode analysis, where the generalized eigenvalue problem:

is solved to determine the modal angular frequencies a> _n and the modal shape functions S _n . The terms "modal" and "mode-..." indicate that the quantities thus specified refer to the modes occurring in the FE model. In principle, the number of modes corresponds to the number of degrees of freedom (DOF) in the FE model. During the mode analysis, A main modes are determined so that n E {1,2, ... A} with N « DOF. The number of A main modes can be in the order of 10 to 100, the total number of nodes can be several million. The total modes of the FE model are thus divided into main modes and secondary modes. The secondary modes correspond to the modes with n > N.

The main modes include in particular at least one drive mode of the MEMS system. The drive mode is a mode that corresponds to the drive of the MEMS system 1, such as a primarily occurring oscillation of a micromirror or rotation rate sensor.

Knowledge of the angular frequencies is very important for the design process. They are constant quantities in the linear region of device operation, but they change due to geometric nonlinearities as the oscillation amplitude of the drive mode increases. Such a drive mode is always present in MEMS angular rate sensors and resonant MEMS micromirrors as part of their functionality. The angular frequencies therefore depend on the amplitude A of the drive mode.

For each node n of the FE model, the computing device calculates a first force F _n on the node that depends on the secondary modes and a second force F _dn on the node that depends on the secondary modes and the drive mode using the form factors.

The first force and the second force can be called renovation forces because they include the contributions of all secondary modes and thus allow an effective description of the forces on the modes.

Using the modal shape functions S _{n ,} the n-th renormalization force F _n (ie the first force with respect to mode n) is determined at each node n of the FE model using standard discretization methods for finite elements. The n-th renormalization force at the node // in the -tcn spatial direction is given by:

Here, the Einstein summation convention is used, in which a sum is implicitly formed over duplicate indices. The calculation can be carried out on the basis of standard FE methods, whereby the element shape function /V is calculated and the integration of the volume is carried out over the element integration points. St also denotes the Kronecker symbol,

denotes the linear mechanical mode stress,

denotes the nonlinear mechanical mode stress and D _{i ] k} i the material tensor of the linear-elastic material described by the FE model.

In addition to the n-th renormalization force (first force), the n-th renormalization pair force F _dn (second force with respect to mode n) is calculated, which acts on the node p in the -tcn spatial direction, given by

Equations (1) and (2) can be computed numerically, requiring only construction steps and vector and matrix multiplications for given modal shape functions S _n . Numerically computationally intensive steps such as matrix inversion are not required and the number of construction steps scales linearly with the number of principal modes N.

Furthermore, the computing device 4 is designed to calculate first auxiliary modes, which correspond to the first force, and second auxiliary modes, which correspond to the second force, for each node n. The first auxiliary modes can also be referred to as renormalization modes p _n and the second auxiliary modes can be referred to as renormalization mode pairs p _dn , where the index d refers to the drive mode. For this purpose, the following linear algebraic equations are solved, which depend on the stiffness matrix K and the first and second forces:

Kp _n = F _n (3) and Pdn ⁼ ^dn- (4)

The calculation can be performed with conventional FE solvers to obtain the first auxiliary modes (renormalization modes) p _n and the second auxiliary modes (renormalization mode pairs) p _dn .

The computing device 4 further calculates mode coupling coefficients a _ncc and a _dnc using the form factors S _n , the first force F _n and the second force F _dn . The mode coupling coefficients a _ncc and a _dnc are mode coupling coefficients (three-mode coupling coefficients or three-wave coupling coefficients corresponding to an effective point interaction of three modes) and are given by the following formulas:

and dnc = -F _än fiirn * d * c * n (6)

The computing device 4 further calculates for each node n a first energy contribution R _d (with respect to the drive mode) and a second energy contribution R _n (with respect to the other main modes) using the first auxiliary modes p _n , the second auxiliary modes p _dn , the mode coupling coefficients a _ddn , a _dnm , the stiffness matrix K and the mass matrix M according to the following formulas:

and )

Here, a> _d denotes the angular frequency of the drive mode and o) _c the angular frequency of the main modes, where o) _d (0), a> _c (0) indicates that the dependence on the oscillation amplitude A _d of the drive mode d is neglected or the oscillation amplitude A _d of the drive mode d is set equal to zero.

The computing device 4 further calculates a nonlinear mechanical

Mode voltage

using the form factors S _n according to the following formula:

The computing device 4 calculates modal cross-Duffing coefficients and the modal Duffing coefficient ß using the mechanical mode stress ?7 ' ^m according to the following formulas:

and

The quantities required for the coefficients in equations (10a) and (10b) were already used to calculate the renormalization forces in equations (1) and (2). Therefore, the calculation does not cause any significant numerical effort and scales linearly with the number N of the main modes.

The computing device 4 is further designed to use the modal angular frequencies, the first energy contribution, the second energy contribution, the modal Duffing coefficient and the modal cross-Duffing coefficients and the mode Coupling coefficients are used to calculate a frequency shift of the angular frequency (ie a dependence of the angular frequency on the amplitude of the drive mode) which depends on an amplitude of the drive mode. This is described in more detail below.

The angular frequency dependent on the amplitude of the drive mode on a main mode n for n

d. where d is the index of the drive mode, is initially given by the following equation: for n

For the drive mode d, the angular frequency is given by the following equation:

The oscillation amplitude A _d of the drive mode, the modal angular frequencies m _n (A _d ) dependent on the oscillation amplitude A _d of the drive mode, the cross-Duffing coefficients

the modal coupling coefficient ß, the 3-mode coupling coefficients a _nmi and the dynamic transfer functions ^dnc and c _dnc .

The dynamic transfer functions a _ddc and b _ddc depend on the angular frequencies a> _n and the modal quality factor Q _c of mode c, which are given in the FE model for all main modes.

The dynamic transfer functions are defined by

The dynamic transfer functions depend on the oscillation amplitude A _d of the drive mode, since the angular frequencies depend on the drive amplitude.

Each angular frequency a> _n of the n-th mode depends on the angular frequencies a> _m of all other modes in the system, which is evident from the sum Xc=i ■■■ i ⁿ in the above equations (11a), (11b).

By approximating the dynamic transfer functions for a> _c » | a> _d + a> _n | up to the second order, we obtain:

In the last approximation, the dependence of the frequencies on the oscillation amplitude A _d of the drive mode is neglected. The approximation is valid in the limit a> _c » 2oj _d , since the dynamic transfer functions have a small slope in this limit and the changes in the angular frequencies over the oscillation amplitude A _d of the drive mode are small compared to their absolute values.

Within these approximations, the closed expression for the n-th angular frequency dependent on the oscillation amplitude A _d of the drive mode is given by

for n G {1,2, ... N \nd}, with the n-th renormalization energy independent of the drive amplitude (first renormalization energy for mode n):

The same applies to the angular frequency of the drive mode:

with the renormalization energy of the driving mode dependent on the amplitude of the driving mode (second renormalization energy):

The first and second renormalization energies have already been calculated according to the above formulas (7) and (8). The calculation device 4 has also already calculated the other quantities required for equation (12) and can thus calculate the frequency shifts of the angular frequencies j _n (A _d ), j _d (A _d ).

In particular, the modal coefficients ß, V _n and a _dnm for modes within the main mode set, i.e. n, m G {1,2, ... , Ai} were efficiently calculated and the summation over quantities not within the main mode set, i.e. C>N ■■■ was also carried out. Finally, the solution was determined for the highly nonlinear case where both the angular frequencies and the dynamic transfer functions depend on the amplitude of the driving mode.

Furthermore, a spectrally self-consistent renormalization procedure can be used to take into account the highly nonlinear dependence of the angular frequencies on the amplitude of the driving mode. The dynamic transfer functions a _ddc , ^C ddc ^a dnc ^C dnc depend on the angular frequencies, which in turn depend on the amplitude of the driving mode. By introducing the renormalized Duffing coefficient /? ^Är 4 _d ) and the renormalized cross-Duffing coefficients l^ ^r (A _d ) depending on the amplitude of the driving mode, given by:

an incremental, iterative solution procedure is implemented, according to

for nd (ie for main modes which are not the driving mode) and

for n = d (i.e. for the drive mode). The incremental drive amplitude steps A _d j for j G {1,2, ... ,/} with A _dl = 0, A _dJ = A _d are used, where A _d is the target amplitude of the calculation, and the increment AA = A _dj+1 — A _d . The solution procedure of equations (14) and (15) is iterative. For the starting value A _{d 0} ⁼ 0, the angular frequencies a) _n (0) for all main modes n E {1,2, ... , A} are known, from which the dynamic correction factors and finally the renormalized quantities in equations (12) and (13) can be determined. From this, a) _n (A _dx ) etc. can then be calculated.

The MEMS system 10 further comprises a control device 3, which controls a component 6 of the MEMS system 1 using the frequency shifts calculated by the device 2, such as a micromirror or a rotation rate sensor.

Figure 2 shows a flow chart of a method for calculating frequency shifts of angular frequencies occurring in the MEMS system 1. The method is particularly applicable to the MEMS system 1 described above and can be carried out by the device 2 described above. Conversely, the device 2 described above can be designed to carry out the method described below. In a method step S1, the MEMS system 1 is modeled by a finite element (FE) model, wherein the FE model comprises a mass matrix M and a stiffness matrix K.

In a method step S2, modal angular frequencies and form factors for a large number of main modes of the FE model are calculated using the mass matrix M and the stiffness matrix K. The totality of the modes of the FE model is divided into the large number of main modes and a large number of secondary modes. The main modes include a drive mode of the MEMS system 1.

In a method step S3, for each node of the FE model, a first force on the node that depends on the secondary modes is calculated using the form factors, and a second force on the node that depends on the secondary modes and the drive mode is calculated.

In a method step S4, mode coupling coefficients are calculated using the form factors, the first force and the second force

In a method step S5, first auxiliary modes, which correspond to the first force, are calculated using the stiffness matrix K of the FE model. Second auxiliary modes, which correspond to the second force, are calculated using the stiffness matrix K of the FE model.

In a method step S6, a first energy contribution and a second energy contribution are calculated using the first and second auxiliary modes and the mode coupling coefficients.

In a process step S7, a nonlinear mechanical mode stress is calculated using the form factors.

In a process step S8, the modal Duffing coefficient and the modal cross-Duffing coefficients are calculated using the mechanical mode stress. In a method step S9, 3-mode coupling coefficients are calculated using the form factors, the first force and the second force.

In a method step S10, a frequency shift of an angular frequency is calculated using the modal Duffing coefficient and the modal cross-Duffing coefficients, the 3-mode coupling coefficients, the mode coupling coefficients, the modal angular frequencies, the first energy contribution and the second energy contribution, which frequency shift depends on an amplitude of the drive mode. The calculation can be carried out for all main modes.

To calculate the force, a contribution from the secondary modes is calculated using the modal Duffing coefficient, the modal cross-Duffing coefficients, the first energy contribution, and the second energy contribution. A contribution from the main modes is calculated by summing over contributions from all main modes.

Furthermore, the calculation can be done iteratively, as described above.

Optionally, the MEMS system 1 can be controlled using the calculated force on the main modes.

Claims

Expectations

1. Computer-implemented method for calculating frequency shifts of angular frequencies occurring in a microelectromechanical, MEMS, system (1), comprising:

Modelling (Sl) of the MEMS system (1) by a finite element, FE, model;

Calculating (S2) modal angular frequencies and form factors for a plurality of main modes of the FE model, wherein a total of modes of the FE model is divided into the plurality of main modes and a plurality of secondary modes, wherein the main modes comprise a drive mode of the MEMS system (1);

Calculating (S3), for each node of the FE model, a first force on the node dependent on the secondary modes and a second force on the node dependent on the secondary modes and the driving mode, using the form factors;

Calculating (S4) first auxiliary modes corresponding to the first force and second auxiliary modes corresponding to the second force;

Calculating (S5) a first energy contribution and a second energy contribution using the first and second auxiliary modes; and

Calculating (S6) a frequency shift of one of the angular frequencies, the frequency shift depending on an amplitude of the drive mode, using the modal angular frequencies, the first energy contribution and the second energy contribution.

2. The method of claim 1, wherein the FE model comprises a mass matrix and a stiffness matrix; wherein calculating the modal angular frequencies and the form factors for the main modes of the FE model using the mass matrix and the stiffness matrix; and wherein the calculation of the first auxiliary modes and the second auxiliary modes is carried out using the stiffness matrix of the FE model.

3. The method according to claim 1 or 2, further comprising the step:

Calculating mode coupling coefficients using the form factors, the first force and the second force; wherein calculating the first energy contribution and the second energy contribution is further performed using the mode coupling coefficients; and wherein calculating the frequency shift of the angular frequency is further performed using the mode coupling coefficients.

4. Method according to one of the preceding claims, further comprising the step:

Calculating a nonlinear mechanical mode stress using the form factors; and

Calculating a modal Duffing coefficient and modal cross-Duffing coefficients using the mechanical mode stress; wherein calculating the frequency shift of the angular frequency is further performed using the modal Duffing coefficient and the modal cross-Duffing coefficients.

5. Method according to claim 3 and 4, wherein the mode coupling coefficients, the modal Duffing coefficient and the modal cross Duffing coefficients depend on the frequency shift of the angular frequency, wherein the frequency shift of the angular frequency depends on the amplitude of the drive mode, and wherein the calculation of the frequency shift of the angular frequencies is carried out iteratively.

6. The method of claim 4 or 5, wherein for calculating the frequency shift of the angular frequency, a contribution of the secondary modes is calculated using the modal Duffing coefficients, the modal cross-Duffing coefficients, the first energy contribution and the second energy contribution.

7. The method of claim 6, wherein the frequency shift is calculated by summing the angular frequency over the main modes.

8. Method according to one of the preceding claims, further comprising the step:

Calculating mode coupling coefficients using the form factors, the first force and the second force; wherein calculating the frequency shift of the angular frequency is further performed using the mode coupling coefficients.

9. Device (2) for calculating frequency shifts of angular frequencies occurring in a microelectromechanical MEMS system (1), with a computing device (4) which is designed to: model the MEMS system (1) using a finite element FE model; calculate modal angular frequencies and form factors for a plurality of main modes of the FE model, wherein a total of modes of the FE model is divided into the plurality of main modes and a plurality of secondary modes, wherein the main modes comprise a drive mode of the MEMS system (1); calculate for each node of the FE model a first force on the node dependent on the secondary modes and a second force on the node dependent on the secondary modes and the drive mode using the form factors; first auxiliary modes which correspond to the first force and second auxiliary modes, corresponding to the second force; calculating a first energy contribution and a second energy contribution using the first and second auxiliary modes; and calculating a frequency shift of one of the angular frequencies that depends on an amplitude of the drive mode using the modal angular frequencies, the first energy contribution, and the second energy contribution.

10. Microelectromechanical MEMS system (1), comprising: a device (2) for calculating frequency shifts of angular frequencies occurring in the MEMS system (1); and a control device (3) which is designed to control at least one component of the MEMS system (1) using a frequency shift of the angular frequency calculated by the device (2).