Eigenfrequency of micromechanical resonators for mass sensing applications

I will show the explicit calculation of the eigenfrequency of micromechanical resonators in the typical form of cantilever and bridge. The eigenfrequency is the frequency corresponding to the natural mode of vibration of the resonators and it is one of the fundamental physical characteristics for mass sensing applications.

Micromechanical resonators Euler – Bernoulli equation

Euler – Bernoulli equation describes the motion of an elastic structure in vacuum under the assumption of small deflections and neglecting shear strain and rotational inertia:

\rho \Gamma \dfrac{\partial^2 }{\partial t^2}u(z, t) + EI_z \dfrac{\partial^4 }{\partial z^4} u(z, t) = 0, \qquad (1)

where:

  • u(z, t) is the displacement field in x direction,
  • \rho is the mass density,
  • \Gamma = wh is the cross-sectional area,
  • E is the Young modulus,
  • I_z is the geometric moment of inertia.

The solution of the partial differential equation are harmonics which can be separated in a spatial and temporal part:

u(z, t) = u_n(z) \cdot exp(-i \omega_n t) \qquad(2)

where n is the modal number \omega_n is the angular frequency. By inserting the solution (2) in equation (1) the partial differential equation will separate. The spatial part can be written as follows:

\dfrac{d^4}{dz^4} u_n(z) = \kappa_n^4 u_n(z) \qquad \kappa_n^4 = \dfrac{\omega_n^2 \rho \Gamma}{EI_z} \qquad(3)

Solutions u_n(z) of equation (3) are called eigenfunctions and the parameter \kappa_n are known as modal constants. Modal constants are determined by considering the boundary conditions.

Microcantilever

A microcantilever is a cantilever-like micromechanical resonator with one fixed end, whose boundary conditions are the following:

u_n(z = 0) = 0, \qquad \dfrac{d}{dz}u_n(z)\big|_{z=0}=0,

\dfrac{d^2}{dz^2}u_n(z)\big|_{z=l} = 0, \qquad \dfrac{d^3}{dz^3}u_n(z)\big|_{z=l}=0\qquad(4)

By inserting boundary conditions (4) into the eq. (3) and some after math, the spatial part of the solutions of eq. (2) will take the form:

u_n(z) = A_n \cdot (\cos {\kappa_n z} - \cosh{\kappa_n z})+ B_n \cdot (\sin{\kappa_n z} - \sinh{\kappa_n z})\qquad (5)

In which the amplitude A_n and B_n verify the following relations:

A_n/B_n = -1.362; \ 0.982; \ -1.001; \ -1.000; \ \ldots \qquad (6)

The boundary conditions (4) also define the characteristic equation of modal constant:

1 + \cos{\kappa_n l} \cdot \cosh{\kappa_n l} = 0 \qquad (7)

The solutions of the transcendental equation (7) are:

\lambda_n = \kappa_n l = 1.875; \ 4.694; \ 7.855; \ 10.996; \ \ldots \pi /2 (2n - 1) \qquad (8)

Eigenfunctions corresponding to the first 4 vibrations mode of a cantilever are depicted in Fig.1, where the vibration amplitude are measured in unity of A_n and the position in unity of length l.

Microcantilever natural vibrations mode corresponding to the 4 eigenfunctions

Figure 1 Microcantilever natural vibrations mode, corresponding to the fundamental mode of vibration and the first 3 higher mode.

Microbridge

A microbridge is bridge-like micromechanical resonator with both fixed end, whose boundary conditions are the following:

u_n(z = 0) = 0, \qquad \dfrac{d}{dz}u_n(z)\big|_{z=0}=0,

u_n(z = l) = 0, \qquad \dfrac{d}{dz}u_n(z)\big|_{z=l}=0\qquad(9)

By inserting boundary conditions (9) into the eq. (3) and after some math, the spatial part of the solutions of eq. (2) has the same form of eq. (5), but the amplitude verify these relations:

A_n/B_n = -1.01178; \ 0.9992; \ -1.0000; \ -0.999; \qquad (10)

The boundary conditions (9) also determine the characteristic equation of modal constants \kappa_n 

1 - \cos {\kappa_n l} \cdot \cosh {\kappa_n l} = 0 \qquad (11)

The solution of the transcendental eq. (11) are:

\lambda_n = \kappa \cdot l = 4.7300;\ 7.8532:\ 10.9956;\ \ldots \pi/2 (2n+1) \qquad (12)

Eigenfunctions corresponding to the first 4 vibrations mode of a bridge are depicted in Fig.2 where the vibration amplitude are measured in unity of  A_n and the position z in unity of the length l.

Microbridge eigenfunctions natural vibration mode

Figure 2 Microbridge natural vibrations mode, corresponding to the fundamental mode of vibration and the first 3 higher mode.

The solutions of the Euler – Bernoulli equation show that micromechanical resonators naturally vibrate in a particular modes, each of them are characterized by a specific spatial shape. The first mode corresponding to n = 0 is known as the fundamental mode of vibration and the others n \ge 1 generally as the higher mode of vibrations. By looking at Figure 2 e 3, it is clear that each mode of vibration has section with great amplitude and other section, known as nodal points, where the vibration amplitude is null. The number of nodal number increases with the modal number.

The vibration mode described here are also known as bending mode, since we have neglected shear strain and rotational inertia in the Euler – Bernoulli eq. 1. Otherwise the micromechanical resonator shows also shear and rotational mode of vibration.

Micromechanical Resonators Eigenfrequency

The eigenfrequency corresponds to the frequency of the natural mode of vibration of a resonator. By solving the characteristic equation relative to the modal constants \kappa_n it is possible to calculate the resonator eigenfrequencies f_n by using eq. (3):

f_n = \dfrac {1}{2\pi} \dfrac{\lambda_n^2}{l^2}\sqrt{ \dfrac{EI_z}{\rho \Gamma} }\qquad (13)

 Example 1 eigenfrequency

Let consider a Silicon monocrystalline micromechanical resonators with rectangular cross-section with the following characteristics:

Table 1 Geometrical and physical spec of micromechanical resonators
Length l = 500 \mu m
Width w = 100 \mu m
Height h = 10 \mu m
 Young modulus  E_{Si} = GPa
 Density  \rho_{Si} = 2330 kg
 Mass  m = \rho l w h = 1.16 \mu g
Geometric moment of inertia  I_z = h^3w/12 = 33.3 \times 10^{3} \mu m^4

The eigenfrequency of the fundamental bending mode can be calculated by using eq. 13 and the modal constant eq. (8) for microcantilever and (12) for microbridge with n = 0.

Table 2 Eigenfrequency of fundamental mode of vibration of micromechanical resonator
microcantilever f_{0C} = 164.000 \ kHz
micorbridge f_{0B} = 1043.300 \ kHz

It is important to note that by fixing the physical and geometrical configuration of the micromechanical resonators, the eigenfrequency of the fundamental bending mode of a microbridge is roughly 6 times of that of the microcantilever. The microbridge naturally vibrates at higher frequency respect to microcantilever.

Principle of Mass Sensing with Micromechanical Resonators

The eigenfrequency of micromechanical resonators is a crucial quantity in mass sensing applications. In this field it is common use to define an equivalent  one-dimensional harmonic oscillator model for each mode of vibration of th micromechanical resonators. The eigenfrequency f_n corresponding to n-th mode of vibration is given by:

f_n = \dfrac{1}{2\pi} \sqrt{ \dfrac{k_{eff}}{m_{eff}} } \qquad (14)

where k_{eff} is the effective spring constant and m_{eff} is the effective mass of the micromechanical resonator in the n-th mode of natural vibration. The effective physical quantity depends basically on:

  1. Mode of vibration of the micromechanical resonator
  2. Type of mass load (distributed, point-like, multiple point-like…)

In general,  the fundamental principle in mass sensing with micromechanical resonators is that the addition of a mass causes the variation in the eigenfrequency. By differentiating the eq. (14) respect to \Delta m one can derive the first order variation of the eigenfrequency: \dfrac{\partial}{\partial m} f_n = - \dfrac{f_n}{2m_{eff}} \Rightarrow

\Delta f_n \simeq -\dfrac{f_n}{2m_{eff}} \cdot \Delta m \qquad (15)

Note that the variation in eigenfrequency caused by the deposition of an extra mass is directly related to the natural eigenfrequency of the micromechanical resonators and it is inversely proportional to its effective mass.

Let us consider the simplest case, that is the micromechanical resonators vibrate at the fundamental mode (n = 0) and mass load is point like.

For a microcantilever with a point mass located at the free apex one has:

k_{effC} = \dfrac{3EI_z}{l^3}\qquad m_{effC} = \dfrac{3m_0}{\lambda_{0C}^4}=0.241\cdot m_0\qquad (16)

where m_0 = \rho \Gamma l is the microcantilever mass and \lambda_{0C} = 1.875 is given by relations (8)

For a microbridge with a point mass located at the centre one has:

k_{effB} = \dfrac{192EI_z}{l^3}\qquad m_{effB} = \dfrac{192m_0}{\lambda_{0B}^4}= 0.383 \cdot m_0\qquad (17)

where m_0 = \rho \Gamma l is the microbridge mass and \lambda_{0B} = 4.73 are given by relations (12).

The mass sensitivity S of a micromechanical resonators is defined as the ratio between is defined as the ratio between the variation in eigenfrequency and the added mass, by considering eq.15 one has:

S = \dfrac{\Delta f}{\Delta m} = \dfrac{f_n}{2m_{eff}}

Example 2 mass sensitivity

Consider the same micromechanical resonators of example 1, let calculate the mass sensitivity of the sensors for a point-like mass located at the free apex of the cantilever and at the centre of the microbridge respectively. By using eq. 15 one has:

Table 3 Mass sensitivity of micromechanical resonators
Microcantilever S_C = 0.3 Hz/pg
Microbridge S_B = 1.2 Hz/pg

 

 

 

It is important to note that the microbridge configuration is roughly 4 times more sensitive than microcantilever.

References

Detail on micromechanical resonators in sensor applications, with solution of Euler – Bernoulli equation and calculation of eigenfrequency, can be found in Boisen Anja 2011 Rep. Prog. Physics “Cantilever like micromechanical sensors”

Experimental work presenting a microbalance based on microcantilever can be found in Mauro Marco 2014 arXiv “Single microparticles mass measurement using an AFM cantilever resonator”.

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