# A Bioinspired Tilt Sensor Model with Adaptive Gain and Enhanced Sensitivity

Authors: Lijun Liu; Ying Lei
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Although various types of tilt sensors have been proposed in the past decade, it is still essential to develop rugged, cheap, simple-structured tilt sensors with wide measuring range and high sensitivities for efficient monitoring of infrastructures and early warning of natural disasters. It has been investigated that stereocilia in some fishes’ inner ear organs are the basic sensory units of nature’s inertial sensors and are highly sensitive over broad dynamic range because of a combination of adaptation and negative stiffness mechanisms. In this paper, a bioinspired tilt sensor model is proposed that mimics the mechanism of stereocilia in adaptive signal amplification to mechanical stimuli, leading to high sensitivity to weak input and low sensitivity to high input, thus expanding the dynamic range through adaptive amplification. The negative stiffness mechanism is implemented by magnet forces. The tilt motion is measured by the strain gauge at the end of the flexible cantilever beam element in the model. Measurements of both static and dynamics tilt motion are investigated. Numerical simulation results are used to demonstrate the capability of the proposed model for the measurements of tilt motions with adaptive amplification and enhanced sensitivity.


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## Body

## 1. Introduction

Tilt sensors, also known as inclinometers, are used to measure the angular deflection of an object against a reference plane or line. Tilt angle measurements have wide applications in many fields [1, 2]. In the past decade, various types of tilt sensors were reported based on the principles and implementations such as resistive, capacitive, inductive, magnetic, optical, mechanics, thermodynamics, fiber-optic, and electrolyte [2–8]. However, with the measurement tilt increases, the sensitivity of a tilt sensor usually decreases seriously; that is, the larger tilt it measures, the lower sensitivity it becomes. Therefore, how to improve the sensitivity of tilt sensor is the technical difficulty to achieve an accurate and wide range measurement in tilt [9]. It is still essential to develop rugged, cheap, simple-structured tilt sensors with wide measuring range and high sensitivities for early warning of natural disasters; for example, it is still changing to develop tilt with high-sensitive and wide range measurement for monitoring geological disasters, such as landslides [10, 11]. Also, it is important to measure the dynamic rotational angle responses of buildings, bridges, and other civil infrastructures, but it is still difficult to accurately measure these small dynamic rotational responses in practice. Commonly, tilt or inclination has been mathematically derived from another measurement response; however, there is inherent error in any indirect measurements [12–14]. Therefore, there is also a need to develop tilt sensors with high-precision to small dynamic rotational angles for efficient health monitoring of infrastructures [15–19].In marine biology, some experiments have shown that some fishes can detect very weak motion with their inner ear organs [20, 21]. Stereocilia in these inner ear organs are the basic sensory units of nature’s inertial sensors and are highly sensitive over broad dynamic range because they display adaptive signal amplification to mechanical stimuli, leading to high sensitivity to weak input and low sensitivity to high input, thus expanding the dynamic range through adaptive amplification [21, 22]. Some researchers have explored that the high sensitivity that is maintained by stereocilia is hypothesized to exist due to a combination of adaptation and negative stiffness mechanisms, which shift the region of highest sensitivity toward the active operation range of the stereocilia [23, 24]. Based on such adaptation and negative stiffness mechanisms, some bioinspired sensor model has been developed [25, 26]. The authors also have investigated the mechanism of a hair cell bioinspired sensor with ultrasensitivity to weak and low frequency vibration signals [27].In this paper, based on the combination of adaptation and negative stiffness mechanisms of the stereocilia in some fishes’ inner ear organs, a bioinspired mechanical model of tilt sensor with adaptive gain and enhanced sensitivity is proposed. The negative stiffness mechanism is implemented by a magnet pair attached to the top of a fixed cantilever beam element and a rigid bar, respectively, which emulates the negative resistance of the tip-link due to the transient stiffness softening by the gating ion channel [24]. The tilt motion is measured by the strain gauge at the end of the cantilever beam element. Measurement of the static tilt motion is first studied. Then, the measurement of the dynamic tilt motion is investigated. Both numerical simulation results are used to demonstrate the capability of the proposed model for the measurements of tilt motions with adaptive amplification and high sensitivity.


## 2. The Bioinspired Mechanical Model of the Tilt Sensor

As shown in Figure1(a), the proposed model consists of a fix light weight flexible cantilever beam (right) with a concentrated mass at the tip, which mimics the stereocilia bundle and the otolith in some fish’s inner ear organs, and a fix rigid bar (left). The bending stiffness of the beam in the x-z plan is much larger than that in the x-y plan, so the beam is assumed to be deflected in the x-y plan. To generate the stiffness softening by the gating spring [27], a magnet pair facing the same pole is attached to the top of the beam element and rigid bar, respectively, to generate a repulsive force against each other. When the base is not tilted in the x-y plan, the magnet pair is perfectly aligned such that there is no net force to bend the beam element in the x-y plan; this represents a closed ion channel without gating. However, when the base is tilted in the x-y plan as shown in Figure 1(b), there is a force at the top mass due to the fact that gravity increases in horizontal direction of the x-y plan; the beam element deflects in the x-y plan accordingly. The repulsive magnetic force enhances the bending movement of the beam element in the plan until the repulsive force is in equilibrium with the elastic restoring force. This mimics the negative resistance of the stereocilia bundle in response to the miniscule stimuli during the channel opening by the gating.The bioinspired model of the tilt sensor.
(a)
(b)The magnet pair generates repulsive forces that are inversely proportional to the distance squared [28]; that is,
(1)F=J×J′4×π×μ0×∑i=01∑j=01∑k=01∑l=01∑p=01∑q=01(-1)i+j+k+l+p+qφ(Uij,Vkl,Wpq,r),
where F is the vector of magnet force, μ0 is the permeability of vacuum,
(2)μ0=4π×10-7N/A2,J and J′ are the magnetizations of the two magnets on the top of the rigid bar and the flexible beam, respectively. φ is a vector with the component of φy and φx as
(3)φy(U,V,W,r)=(U2-W2)2ln⁡(r-V)+UVln⁡(r-U)+UWtg-1(UVWr)+12Vrφx(U,V,W,r)=(V2-W2)2ln⁡(r-U)+UVln⁡(r-V)+VWtg-1(UVWr)+12Ur,Uij=x0+(-1)jA-(-1)ia;Vij=y0+(-1)lB-(-1)kb;Wpq=z0+(-1)qC-(-1)pcr=Uij2+Vkl2+Wpq2
in which (x0,y0,z0) is the relative coordinates between the two centers of the magnet pair. A, B, C and a, b, c are the size of the two magnets in the x, y and directions, respectively.In the model, the sizes of the two magnets are selected as2A=2a=0.006m, 2B=2b=0.006 m, 2C=2c=0.006m, and the two magnetizations are J=J′=0.38. Other parameters are shown in Table 1.Table 1
Parameters of mechanical model.
Item
Value
Definition
l
0.1 m
Length of the beam element
E
9
×
10
5 N/m2
Young’s modulus of beam element
I
z
6.36
×
10
-
11 m4
Moment of inertia inz direction of beam element
I
y
inf
Moment of inertia iny direction of beam element
m
0.002 kg
Tip mass
ζ
0.865
Damping ratio of model, the otolith and hair cells are surrounded by endolymph, the damping is simplified as a large value viscous damping
z
0
0.02 m
The relative coordinate of the two magnets inz direction


## 3. Measurements of Tilt Motions

The proposed model in Figure1(a) is used to measure the tile motion θ of the base in Figure 1(b). The tilt motion is measured by the strain gauge at the end of the flexible cantilever beam element in the model as shown in Figure 1(b).


### 3.1. Measurements of Static Tilt Motions

A static tilt angle is applied at the base shown in Figure1(b). The equations of motion for the two models with magnet and without magnet can be obtained as (4) and (5), respectively,
(4)kδ=(mg+Fx)sin(θ+δl)+Fycos⁡(θ+δl)(5)kδ=mgsin(θ+δl)
in which δ is the vertical deflection at the tip of the beam, k is its stiffness, and Fx and Fy are the components of the magnetic force in the x and y direction, respectively, which are determined by (1)–(3).To examine the adaptive amplification of the model with magnet, the base is tilted angleθ from 0 to 60 degree with an interval of 0.01 degree. Figure 2(a) shows the comparisons of strain at the bottom of beam element subject to tilt angle  θ with and without the magnetic force. It is shown that the measured strains in model with magnet are more sensitive to small tilt angles than those in the model without magnet. The amplification sensitivity is defined as the ratio of strain of model with magnet and that of model without magnet. As shown in Figure 2(b), the amplification sensitivity is high for small tilt angles and the sensitivities decrease for larger tilt angles. Therefore, the model has adaptive amplification with high sensitivity to slight tilt motion and low sensitivity to large tilt motion due to the negative stiffness contributed by magnet force.Measurement results of the model under static tilt angle.
(a)
Comparisons of strain under static tilt angle
(b)
Sensitivity under static tilt


### 3.2. Measurements of Dynamic Rotational Motions

The two mechanical models are subject to the dynamic rotational motionθ(t) of the base. Then, the equations of motion for two models with magnet and without magnet can be obtained as (6)
(6)kδ(t)=-mδ¨(t)-cδ˙(t)-mlθ¨(t)+(mg+Fx)sin⁡[θ(t)+δ(t)l]+Fy[cos⁡θ(t)+δ(t)l]kδ(t)=-mδ¨(t)-cδ˙(t)-mlθ¨(t)+mgsin⁡[θ(t)+δ(t)l],
where θ¨(t), θ˙(t), and θ(t) are the angular acceleration, angular velocity, and angular displacement of tilt motion, respectively, and c is the viscous damping as shown in Table 1.


#### 3.2.1. Measurements of Dynamic Rotational Motions with Varying Amplitudes

To examine the adaptive amplification of the model with magnet to the rotational motion with varying amplitudes, it is assumed that the base has the sinusoidal rotation motion; that is,θ(t)=θ0sin⁡ωt. The amplitude of rotational angle θ0 ranges from 0 to 60 degrees.Figures3(a)–3(d) show the time history of the strain at the bottom of flexible beam under base rotation motion with frequency equal to 0.1 Hz and with different amplitude θ0.Time histories of strains under dynamic base rotations with varying amplitudes.
(a)
θ
0
=
0.5
7
°
(b)
θ
0
=
5
.
7
°
(c)
θ
0
=
28
.
5
°
(d)
θ
0
=
5
7
°Figure4(a) shows the comparisons of the amplitudes of strain at the bottom of the flexible beam in two models with and without magnets under dynamic base tilt motions with varying amplitudes  θ0.Measurement results of the model under dynamic tilt motion with varying amplitudes.
(a)
Comparisons of strain amplitudes under dynamic tilt motion
(b)
Sensitivity under dynamic tilt motionIt is shown that the measured strains in model with magnet are more sensitive to slight tilt motion than those in the model without magnet. The amplification sensitivity is defined as the ratio of amplitude of strain of model with magnet and that of model without magnet. As shown in Figure4(b), the amplification sensitivity is in highly rising tendency for small amplitude of base rotation angles and is in descent tendency for large amplitude of base rotation angles. This confirms the adaptive amplification capability of the proposed model due to the negative stiffness effect by the magnet forces.


#### 3.2.2. Measurements of Dynamic Rotational Motions with Varying Frequencies

The two models with magnet and without magnet are subject to dynamic base sinusoidal rotational motion with varying frequencies. The amplitude of rotational displacementθ0 is assumed as 0.1 rad (5.7°). Figures 5(a)-5(b) compare the time histories of the strain responses at the bottom of the flexible beam in the two models subject to dynamic base sinusoidal rotational motions with varying frequency ratios ω/ωn, in which ωn is the natural frequency of the model.Time histories of strains under dynamic base rotations with varying frequencies.
(a)
ω
/
ω
n
=
0.002
(b)
ω
/
ω
n
=
0.2The amplification sensitivities defined under dynamic base rotational motions with varying frequencies are shown in Figure6. It is clear that the amplification sensitivities are high for base rotational motions with low frequency (0≤ω/ωn≤0.6) than those for base rotational motions with high frequency (ω/ωn>0.6). Therefore, it is demonstrated that proposed model is more sensitive to low frequency tilt motion due to the negative stiffness effect by the magnet forces.Figure 6
The amplification sensitivities for base rotational motion with varying frequencies.


## 3.1. Measurements of Static Tilt Motions

A static tilt angle is applied at the base shown in Figure1(b). The equations of motion for the two models with magnet and without magnet can be obtained as (4) and (5), respectively,
(4)kδ=(mg+Fx)sin(θ+δl)+Fycos⁡(θ+δl)(5)kδ=mgsin(θ+δl)
in which δ is the vertical deflection at the tip of the beam, k is its stiffness, and Fx and Fy are the components of the magnetic force in the x and y direction, respectively, which are determined by (1)–(3).To examine the adaptive amplification of the model with magnet, the base is tilted angleθ from 0 to 60 degree with an interval of 0.01 degree. Figure 2(a) shows the comparisons of strain at the bottom of beam element subject to tilt angle  θ with and without the magnetic force. It is shown that the measured strains in model with magnet are more sensitive to small tilt angles than those in the model without magnet. The amplification sensitivity is defined as the ratio of strain of model with magnet and that of model without magnet. As shown in Figure 2(b), the amplification sensitivity is high for small tilt angles and the sensitivities decrease for larger tilt angles. Therefore, the model has adaptive amplification with high sensitivity to slight tilt motion and low sensitivity to large tilt motion due to the negative stiffness contributed by magnet force.Measurement results of the model under static tilt angle.
(a)
Comparisons of strain under static tilt angle
(b)
Sensitivity under static tilt


## 3.2. Measurements of Dynamic Rotational Motions

The two mechanical models are subject to the dynamic rotational motionθ(t) of the base. Then, the equations of motion for two models with magnet and without magnet can be obtained as (6)
(6)kδ(t)=-mδ¨(t)-cδ˙(t)-mlθ¨(t)+(mg+Fx)sin⁡[θ(t)+δ(t)l]+Fy[cos⁡θ(t)+δ(t)l]kδ(t)=-mδ¨(t)-cδ˙(t)-mlθ¨(t)+mgsin⁡[θ(t)+δ(t)l],
where θ¨(t), θ˙(t), and θ(t) are the angular acceleration, angular velocity, and angular displacement of tilt motion, respectively, and c is the viscous damping as shown in Table 1.


### 3.2.1. Measurements of Dynamic Rotational Motions with Varying Amplitudes

To examine the adaptive amplification of the model with magnet to the rotational motion with varying amplitudes, it is assumed that the base has the sinusoidal rotation motion; that is,θ(t)=θ0sin⁡ωt. The amplitude of rotational angle θ0 ranges from 0 to 60 degrees.Figures3(a)–3(d) show the time history of the strain at the bottom of flexible beam under base rotation motion with frequency equal to 0.1 Hz and with different amplitude θ0.Time histories of strains under dynamic base rotations with varying amplitudes.
(a)
θ
0
=
0.5
7
°
(b)
θ
0
=
5
.
7
°
(c)
θ
0
=
28
.
5
°
(d)
θ
0
=
5
7
°Figure4(a) shows the comparisons of the amplitudes of strain at the bottom of the flexible beam in two models with and without magnets under dynamic base tilt motions with varying amplitudes  θ0.Measurement results of the model under dynamic tilt motion with varying amplitudes.
(a)
Comparisons of strain amplitudes under dynamic tilt motion
(b)
Sensitivity under dynamic tilt motionIt is shown that the measured strains in model with magnet are more sensitive to slight tilt motion than those in the model without magnet. The amplification sensitivity is defined as the ratio of amplitude of strain of model with magnet and that of model without magnet. As shown in Figure4(b), the amplification sensitivity is in highly rising tendency for small amplitude of base rotation angles and is in descent tendency for large amplitude of base rotation angles. This confirms the adaptive amplification capability of the proposed model due to the negative stiffness effect by the magnet forces.


### 3.2.2. Measurements of Dynamic Rotational Motions with Varying Frequencies

The two models with magnet and without magnet are subject to dynamic base sinusoidal rotational motion with varying frequencies. The amplitude of rotational displacementθ0 is assumed as 0.1 rad (5.7°). Figures 5(a)-5(b) compare the time histories of the strain responses at the bottom of the flexible beam in the two models subject to dynamic base sinusoidal rotational motions with varying frequency ratios ω/ωn, in which ωn is the natural frequency of the model.Time histories of strains under dynamic base rotations with varying frequencies.
(a)
ω
/
ω
n
=
0.002
(b)
ω
/
ω
n
=
0.2The amplification sensitivities defined under dynamic base rotational motions with varying frequencies are shown in Figure6. It is clear that the amplification sensitivities are high for base rotational motions with low frequency (0≤ω/ωn≤0.6) than those for base rotational motions with high frequency (ω/ωn>0.6). Therefore, it is demonstrated that proposed model is more sensitive to low frequency tilt motion due to the negative stiffness effect by the magnet forces.Figure 6
The amplification sensitivities for base rotational motion with varying frequencies.


## 3.2.1. Measurements of Dynamic Rotational Motions with Varying Amplitudes

To examine the adaptive amplification of the model with magnet to the rotational motion with varying amplitudes, it is assumed that the base has the sinusoidal rotation motion; that is,θ(t)=θ0sin⁡ωt. The amplitude of rotational angle θ0 ranges from 0 to 60 degrees.Figures3(a)–3(d) show the time history of the strain at the bottom of flexible beam under base rotation motion with frequency equal to 0.1 Hz and with different amplitude θ0.Time histories of strains under dynamic base rotations with varying amplitudes.
(a)
θ
0
=
0.5
7
°
(b)
θ
0
=
5
.
7
°
(c)
θ
0
=
28
.
5
°
(d)
θ
0
=
5
7
°Figure4(a) shows the comparisons of the amplitudes of strain at the bottom of the flexible beam in two models with and without magnets under dynamic base tilt motions with varying amplitudes  θ0.Measurement results of the model under dynamic tilt motion with varying amplitudes.
(a)
Comparisons of strain amplitudes under dynamic tilt motion
(b)
Sensitivity under dynamic tilt motionIt is shown that the measured strains in model with magnet are more sensitive to slight tilt motion than those in the model without magnet. The amplification sensitivity is defined as the ratio of amplitude of strain of model with magnet and that of model without magnet. As shown in Figure4(b), the amplification sensitivity is in highly rising tendency for small amplitude of base rotation angles and is in descent tendency for large amplitude of base rotation angles. This confirms the adaptive amplification capability of the proposed model due to the negative stiffness effect by the magnet forces.


## 3.2.2. Measurements of Dynamic Rotational Motions with Varying Frequencies

The two models with magnet and without magnet are subject to dynamic base sinusoidal rotational motion with varying frequencies. The amplitude of rotational displacementθ0 is assumed as 0.1 rad (5.7°). Figures 5(a)-5(b) compare the time histories of the strain responses at the bottom of the flexible beam in the two models subject to dynamic base sinusoidal rotational motions with varying frequency ratios ω/ωn, in which ωn is the natural frequency of the model.Time histories of strains under dynamic base rotations with varying frequencies.
(a)
ω
/
ω
n
=
0.002
(b)
ω
/
ω
n
=
0.2The amplification sensitivities defined under dynamic base rotational motions with varying frequencies are shown in Figure6. It is clear that the amplification sensitivities are high for base rotational motions with low frequency (0≤ω/ωn≤0.6) than those for base rotational motions with high frequency (ω/ωn>0.6). Therefore, it is demonstrated that proposed model is more sensitive to low frequency tilt motion due to the negative stiffness effect by the magnet forces.Figure 6
The amplification sensitivities for base rotational motion with varying frequencies.


## 4. Conclusions

In this paper, based on the mechanisms of adaptation and negative stiffness of the stereocilia in some fishes’ inner ear organs, a bioinspired mechanical model of tilt sensor with adaptive gain and enhanced sensitivity is proposed. The negative stiffness effect is implemented by magnet forces and tilt motion can be measured by the strain at the end of flexible cantilever beam element in the model. Numerical simulation results of the measurements of the static and dynamic tilt motions have demonstrated that the proposed tilt model is more sensitive to slight and low frequency tilt motion. Therefore, the proposed tilt model has the capability of adaptive gain and enhanced sensitivity.The proposed model can be used for the design of bioinspired tilt sensors with adaptive amplification and high sensitivity. More investigations on practical implementation issues of the design of such tilt sensors with small sizes by Micro-Electro-Mechanical System (MEMS) are necessary.



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*Source: 957850-2014-04-27.xml*