Understanding the behaviour of mechanical systems can be facilitated and improved by employing electro-mechanical analogies. These analogies enable the use of network analysis tools as well as purely analytical treatment of the mechanical system translated into an electric circuit. Recently, we developed a novel kind of sensor set-up based on two coupled cantilever beams with matched resonance frequencies (co-resonant coupling) and possible applications in magnetic force microscopy and cantilever magnetometry. In order to analyse the sensor's behaviour in detail, we describe it as an electric circuit model. Starting from a simplified coupled harmonic oscillator model with neglected damping, we gradually increase the complexity of the system by adding damping and interaction elements. For each stage, various features of the coupled system are discussed and compared to measured data obtained with a co-resonant sensor. Furthermore, we show that the circuit model can be used to derive sensor parameters which are essential for the evaluation of measured data. Finally, the much more complex circuit representation of a bending beam is discussed, revealing that the simplified circuit model of a coupled harmonic oscillator is a very good representation of the sensor system.

Electro-mechanical analogies are a suitable tool to describe, understand and
analyse the behaviour of mechanical systems

In the following, we will start with a short description of the mechanical system which is translated into an electric circuit model. From there we analyse the system's behaviour for different cases, i.e. without and with damping, and then show a possible way to derive useful information to characterize the sensor.

Cantilever-based measurement techniques are widely spread, for example for
the determination of magnetic sample properties as in magnetic force
microscopy or cantilever magnetometry

The cantilever's motion is usually detected by laser interferometry or laser
deflection

For our experiments we used a commercially available atomic force microscopy (AFM) microcantilever and
a carbon nanotube (CNT) as nanocantilever. Frequency matching between the two
subsystems has been achieved by adding some mass to the free end of the
nanocantilever. The frequency matching introduces a very strong interplay
between the two subsystems, resulting in two main effects. First, a
significant amplitude amplification between micro- and nanocantilever is
observed, leading to very high oscillation amplitudes of the nanocantilever

The behaviour of the co-resonant sensor concept has been studied thoroughly
in terms of analysing the mechanical model and the system's differential
equations

In the following, we will derive an electric circuit model for the sensor and
discuss its behaviour for various simplifications. Analytical expressions
will be given, which are evaluated with the software

Example of a coupled sensor consisting of a micro- and a nanocantilever. The sensor is driven with a piezoelectric actuator, and the oscillation is detected with laser deflection.

The sensor employs two cantilevers which are coupled in succession as
depicted in Fig.

Mechanical model for the sensor depicted in Fig.

Electric circuit model for a co-resonant cantilever sensor whose
mechanical representation is shown in Fig.

Given the mechanical representation of the sensor, we can derive the electric
circuit model by employing the analogies in Table

Analogies used for the conversion of mechanical systems into an electric circuit.

With these analogies, the structure of the mechanical representation is
preserved with only the masses being treated specifically as one of their
connection points always has to be the frame, hence ground in the electric
case

The sensor concept itself is of a general nature, but in order to show some
quantitative evaluation of the following calculations, the mechanical
properties of one fabricated co-resonant sensor are given in Table

The damping constants

Since the eigenfrequencies

At this point it is useful to review the terms

Values for the mechanical and electrical elements,
the latter being calculated according to Table

The simplest case we will consider is the circuit from Fig.

Simplified circuit model from Fig.

The quantities of interest are the velocities

Underscores will be used throughout the text to indicate variables with complex values, and variables without underscores denote the magnitude.

The expressions forThese expressions are evaluated for the numerical values given in
Table

Amplitude curves of both cantilevers of the circuit from Fig.

Please note that the complete solution for

Comparison of calculated and
measured resonance frequencies

The electric circuit given in Fig.

Bode plots for both subsystems of Fig.

With this simplified circuit model it is possible to obtain the resonance frequencies as well as find typical features of the coupled system like frequency shifting compared to the eigenfrequencies of the subsystems and antiresonance behaviour.

The model so far does not represent the finite amplitudes of the resonance
peaks; therefore, the expected amplitude amplification between the nano-
and the microcantilever cannot be studied. Hence, we will include the damping
in this section and derive an analytical expression for this case. We again
consider a coupled system without external interaction, i.e.

Circuit model with damping and without external interaction,

With that we can again apply the rule for voltage division to derive
expressions for

This time the expressions are complex, but by separating the real and
imaginary parts of numerator and denominator and applying the rules for
complex numbers

In this work we focus on the amplitude expressions since the amplitude
response is used as the main measurement signal in our experiments. In case
the phase of the oscillation becomes important, for example if a phase-locked
loop is used to track the change of the resonance frequencies of the coupled
system, the corresponding phase relations are given for completeness. With
these phase relations, one is able to conduct a similar discussion as follows
for the amplitude expressions. Plotting Eqs. (

Amplitude curves obtained with Eqs. (

First, the two resonance frequencies

Calculated resonance frequencies

Furthermore, it is also possible to find the frequency

In addition, it is possible to study the
amplitude ratio between the amplitudes of nano- and microcantilever, which we
will term amplitude amplification in the following. The amplitude of the
nanocantilever is increased compared to that of the microcantilever due to
the co-resonant coupling; the amplitude amplification factor

A general approach to find the amplitude amplification factor is to determine
the resonance frequencies

Bode plot of amplitude curves for both cantilevers of the coupled
system from Fig.

When looking at Fig.

Amplification factor for the higher resonance peak of the coupled
system for varying eigenfrequency

In order to describe the coupled systems' behaviour with the circuit model,
the properties of the subsystems have to be known. They can be determined
from their geometric and material properties as well as from measurements.
However, it is particularly difficult to determine the eigenfrequency

Applying these values in Eq. (

The most general description of the coupled sensor is the one depicted in
Fig.

First, parallel branches are combined into complex impedances to simplify the
circuit. As depicted in Fig.

The same is done for the parallel branches of

With these transformations we get the circuit depicted in
Fig.

Simplified circuit model for Fig.

In order to express all elements of the circuit in Euler form so that the
following calculations will be facilitated, the capacitor

With the above definitions and combinations it is now possible to give
expressions for

These expressions can easily be separated into real and imaginary part to
obtain magnitude and phase. Figure

In the following discussion we will only consider the additional spring

Amplitude curves for both cantilevers for the circuit model in
Fig.

In the last section, the frequency shift for both resonance peaks of the
coupled system caused by an additional spring

Transformation of the circuit model without external interaction

The approach of determining the effective sensor properties is based on a
series/parallel conversion of the circuit model in such a way that the
coupled system is expressed as a complex impedance

The transformation steps are depicted in Fig.

Further transformation of the resulting circuit leads to a total complex
impedance

This total impedance consists of a real and an imaginary part which can be
compared to their respective counterparts of the series resonance circuit
depicted in Fig.

The comparison yields

This formula can also be applied for the coupled system when each resonance
peak is described as a single cantilever with effective properties. Hence,

As the discussion in Sect.

With these considerations, all elements of the series resonance circuit are
defined. Hence, another series/parallel transformation of the resistor and
inductor is carried out to obtain a circuit which is structurally equivalent
to that of a cantilever, i.e. with inductor and resistor in parallel as shown
in Fig.

As the numerical values in the next section indicate, it is a valid approach
to represent each of the coupled system's resonance peaks by an equivalent
single cantilever, hence a single harmonic oscillator. With this in mind,
there is an alternative to calculate

Calculated values for the equivalent
circuit model for both resonance peaks of the coupled system in Fig.

In the experiment, the nanocantilever was made of an iron-filled carbon
nanotube (FeCNT). Therefore, the external interaction was given by a
magnetostatic interaction between the iron nanowire and a magnetic field
introduced by a permanent magnet

Comparison of analytically derived
and measured frequency shift values for an external interaction of

The value for

As already mentioned above, we can also test the equivalent circuit model for
each of the two resonance peaks

Amplitude response curve of the coupled system measured with laser
deflection at the microcantilever under high vacuum (

In the previous sections we have seen that a simple coupled harmonic oscillator model describes the behaviour of a co-resonantly coupled sensor very well. However, one has to keep in mind that this model is only valid for one resonance frequency. This is not a limitation for the case given, as only one flexural vibration mode of both beams is of interest.

However, the rather simple model does not take any shear deformation into
account

Each element contains an inductor in the rotatory and a resistor and
capacitor in the translational part. These two parts are connected via a
current-controlled voltage source. For the set-up of two coupled cantilever
beams, the corresponding circuit is given in Fig.

Circuit model of two coupled cantilever beams with translational and rotatory degrees of freedom of the movement.

Bode plot for both cantilevers of the coupled system of
Fig.

Amplitude curves for both cantilevers for the first bending mode
obtained with the discrete beam model in Fig.

Figure

Application of electro-mechanical analogies is a commonly used tool to describe and analyse mechanical systems. We have applied the electric circuit model to our recently developed concept of a co-resonantly coupled sensor consisting of a micro- and a nanocantilever. The mechanical representation of the sensor set-up can be considered as a simple coupled harmonic oscillator, which then has been transferred into an electric circuit. Analysis of the circuit can be performed completely analytically and with the software LTspice in order to gain profound understanding of the system's behaviour and to examine various features of the coupled system. This knowledge can be used to predict the behaviour of the sensor and help interpret and understand measurement results. Furthermore, the circuit model can also be employed for sensor design, allowing for determination of expected sensor characteristics such as the effective spring constant, which determines the sensitivity of the system.

Additionally, the more complex model of a beam taking bending and shear deformation into account has been applied for the coupled sensor system. Generally, it has been shown that the simple coupled harmonic oscillator model describes the sensor's behaviour sufficiently well.

The data presented in this paper is available on request from the corresponding author.

The authors acknowledge U. Marschner for helpful discussions regarding electro-mechanical networks. Furthermore, the authors acknowledge funding by the DFG (grant no. MU 1794/3-2). Edited by: S. Schmid Reviewed by: two anonymous referees