Keywords

1 Introduction

In nature, all mammals except humans possess specialised tactile hairs, called vibrissae or whiskers. These thin long hairs grow usually in groups in different locations on an animal’s body. Among the most familiar to us are mystacial vibrissae orderly distributed over the whiskerpad on a muzzle, supraorbital vibrissae above the eyes, and carpal vibrissae that are located on the downside aspect of the forelimbs (Fig. 1). For terrestrial and marine mammals, vibrissae are significantly important tactile sensors used for the exploration of the surrounding area, texture discrimination, or even detection of water vibrations (Dehnhardt and Kaminski 1995; Niederschuh et al. 2014; Vincent 1912).

Fig. 1
figure 1

a Mystacial and supraorbital vibrissae of a house cat; b Triple of carpal vibrissae at a forelimb of Rattus norvegicus, photo taken by Voges D (TU Ilmenau)

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The structure and characteristics of vibrissae differ considerably from normal body hairs of the same mammal individual. They are thicker and stiffer than other types of hairs and have a conical shape with the linearly decreasing diameter (Voges et al. 2012). Another significant difference lies in the more complicated base of the vibrissa. The vibrissal hair grows from a special follicle, called the follicle-sinus complex (FSC). It incorporates lots of sensory nerve endings and a capsule of blood (Ebara et al. 2002). For some groups of vibrissae, the FSC is surrounded by two types of muscles that allow an animal to move its vibrissae back and forth (Dörfl 1982). These sniffing active movements of vibrissae, called whisking, play an important role during the exploratory behaviour of mammals.

The fascinating biological paradigm of the vibrissal tactile sensing has inspired a large number of research investigations. Scientists, biologists and engineers would like to understand the functional principles of the vibrissal sensing system from different aspects. Neuroscientists are trying to figure out how nerve impulses are processed from vibrissae to the brain and how an animal encodes the information. They perform laboratory experiments with living creatures, e.g. (Jadhav and Feldman 2010; Wolfe et al. 2008). Biologists usually describe the anatomy and morphology of vibrissae from species to species. They believe that the vibrissal sensing system played an important role during evolutionary development of mammals (Mitchinson et al. 2011).

For the engineers, the biological vibrissa gives an inspiration to design various artificial tactile sensors and find possible applications for them, as it promises to be competitive with artificial vision. Most often, the research focus lies on the development of single sensors. Some interesting examples are the rotary actuated whisker for obstacle avoidance and the whisker-like sensor for underwater applications (Scholz and Rahn 2008; Valdivia y Alvarado et al. 2012). Furthermore, a variety of developed prototypes of mobile robots with a vibrissa-like sensory array can be found, e.g., in (Fend et al. 2004; Solomon and Hartmann 2006; Zimmer 1995). One of the latest platforms is the SCRATCHbot—a rat-like robot, which can move and position its whiskers toward specific objects with three degrees of freedom (Pearson et al. 2011).

Recent experimental investigations performed at the Department of Mechanical Engineering (TU Ilmenau) show that it is possible to reconstruct a profile of an object by one single quasi-static sweep of the straight technical vibrissa made of steel (Fig. 2). By measuring the clamping forces and the bending moment at the fixed base of the beam, the contact point with an object may be quantified. The theoretical analysis of the problem is based on the non-linear Euler–Bernoulli theory of the static deformable beam (Will et al. 2016).

Fig. 2
figure 2

The experimental setup at the Department of Mechanical Engineering (TU Ilmenau) for the quasi-static profile scanning

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Summarising the research on mathematical-mechanical modelling of vibrissae, two approaches may be marked out: rigid rod models, e.g. (Behn 2013; Berg and Kleinfeld 2003; Volkova et al. 2016a), and continuum elastic models, e.g. (Neimark et al. 2003; Scholz and Rahn 2008; Volkova et al. 2016b; Yan et al. 2013). The continuum models are in turn closer to the biological paradigm, as they take into account the inherent dynamical behaviour and the bending stiffness of the vibrissal hair. As for the modelling of the viscoelastic support of the hair inside the FSC, several spring and damping elements may be used, e.g. (Behn 2013). In (Volkova et al. 2016a), the FSC incorporating the blood capsule is modelled as a continuum volume filled with a viscous magnetic fluid. There, an approach to realise and control three-dimensional oscillations of the rigid rod is presented using an applied uniform magnetic field.

The present paper focuses on the non-linear process of the vibrissa sweeping across a surface of an object during texture discrimination. Frictional interactions between the tip of the vibrissal hair and a rough surface generate vibrations of the hair (Fig. 3). Since it has no receptors along the length, the tactile signals are transmitted by these mechanical vibrations to sensory receptors inside the FSC.

Fig. 3
figure 3

Schematic drawing of a vibrissa sweeping past a rough surface

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In this paper, the vibrissal hair is modelled as a straight elastic beam with linearly decreasing diameter along its length. The model describing in-plane vibrations of the beam stimulated by a periodic following force is developed. Based on asymptotic methods of mechanics, the parametric resonance of the beam is analysed theoretically. According to the so-called resonance hypothesis proposed in (Andermann and Moore 2008; Neimark et al. 2003), the vibrissa resonance may be a potentially useful mechanism during texture discrimination. It could amplify small tactile signals arising from the contact with an object and enchase the sensitivity of these biological tactile sensors. In the second part of the work, the numerical analysis of the problem using a finite element model is presented. The simulations are performed for various values of the excitation frequency of the applied force. The comparison of the theoretical and numerical results is made.

2 Theoretical Analysis

2.1 Geometry and Assumptions of the Model

Consider a truncated Euler–Bernoulli beam whose undeformed neutral axis is a straight line of a length L (Fig. 4). The radius \( r(x) \) of the beam’s circular cross-section evolves linearly along the axial direction:

Fig. 4
figure 4

Deflection of the straight truncated beam under the following force applied toward the cross-section at the tip

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$$ r(x) = r_{b} - \frac{{r_{b} - r_{t} }}{L}x, $$
(1)

where \( r_{b} \) and \( r_{t} \) are, respectively, the radii at the base and the tip of the beam. The Cartesian coordinate system \( (x,y,z) \) is placed such that the x axis lies along the neutral axis of the beam, and the origin is in the middle of the base’s cross-section. The Cartesian basis vectors are \( {\text{e}}_{x} ,{\text{e}}_{y} ,{\text{e}}_{z} \). The material properties of the beam are constant, i.e., the density ρ = const and the Young’s modulus E = const. The support condition of the beam at the base is assumed to be pinned. It allows the beam to rotate around the z axis, but not to translate in any direction. At the end of the beam, the roller support is considered, which provides free rotation around the z axis and the horizontal deflection (Fig. 4). The beam is stimulated by a periodic force applied directly toward the cross-section of the beam at the tip:

$$ {\text{F}}(t) = - F_{0} \cos (2\uppi\,\Omega \,t){\text{n}}_{t} , $$
(2)

where \( F_{0} \) is the amplitude, Ω is the excitation frequency, t is the time, and \( {\text{n}}_{t} \) is the normal to the cross-section at the tip. The assumed force \( {\text{F}}(t) \) corresponds to the frictional force, which arises from interactions between the vibrissal tip and the complex roughness profile of a surface (Fig. 3). Thus, it may be considered as a first harmonic component of the Fourier transform spectrum of the friction force.

It is supposed throughout that the transverse vibrations of the beam caused by the force \( {\text{F}}(t) \) about its straight natural configurations are in the x-y plane and small. The displacement vector of the axis points has the transverse component: \( {\text{u}}(x,t) = v(x,t){\text{e}}_{y} . \)

2.2 Equation of Motion and Its Approximation

The small in-plain vibrational motion of the straight beam can be described with a single partial differential equation of the fourth order. With regard to the rotary inertia and in the absence of the force of viscous damping, the equation of motion in terms of \( v(x,t) \) has the following form (Svetlitsky 2005; Zentner 2014):

$$ m_{0} (x)\frac{{\partial^{2} v(x,t)}}{{\partial t^{2} }} + \frac{{\partial^{2} }}{{\partial x^{2} }}\left( {E\,I_{z} (x)\frac{{\partial^{2} v(x,t)}}{{\partial x^{2} }}} \right) + F_{0} \cos \left( {2\uppi\,\Omega \,t} \right)\frac{{\partial^{2} v(x,t)}}{{\partial x^{2} }} = 0. $$
(3)

Here, \( m_{0} (x) =\uppi \uprho r^{2} (x) \) is the mass of the beam’s element, and \( I_{z} (x) =\uppi\,r^{4} (x)/4 \) is the moment of inertial of the cross-section.

The bending moment \( {\text{M}}(x,t) = M(x,t){\text{e}}_{z} \) at any cross-section of the beam can be expressed by the transverse displacement as:

$$ M(x,t) = E\,I_{z} (x)\frac{{\partial^{2} v(x,t)}}{{\partial x^{2} }}. $$
(4)

According to the chosen support combination, and since no external bending moment is applied, we have the following boundary conditions for the displacement function itself as well as for the second derivative of it:

$$ v(0,t) = v(L,t) = 0,\quad \left. {\frac{{\partial^{2} v(x,t)}}{{\partial x^{2} }}} \right|_{(0,t)} = \left. {\frac{{\partial^{2} v(x,t)}}{{\partial x^{2} }}} \right|_{(L,t)} = 0. $$
(5)

In order to find the solution of the partial differential Eq. 3, let us first reduce it to an ordinary one using the Galerkin method (Kantorovich and Krylov 1958). We shall seek one-term approximate solution of Eq. 3 in the form:

$$ v(x,t) = \sin \left( {\frac{{\uppi\,x}}{L}} \right)\;f(t), $$
(6)

which satisfies the boundary conditions of the beam (Eq. 5). Substituting this expression in Eq. 3 with the requirement of the orthogonality of Eq. 3 to the function \( \sin \left( {{{\uppi\,x} \mathord{\left/ {\vphantom {{\uppi\,x} L}} \right. \kern-0pt} L}} \right) \), we obtain an ordinary second-order differential equation for the function \( f(t) \). It can be written in the dimensionless form as:

$$ {\ddot f}(\uptau) + \left({1 -\upvarepsilon\cos (\upgamma\,\uptau)} \right)f(\uptau) = 0. $$
(7)

Here, the dot notation is used to represent a time derivative of a function, and the dimensionless variables are introduced as follows:

$$ \upvarepsilon = \frac{{2\uppi{\kern 1pt} L^{2} F_{0} }}{{b_{1} E{\kern 1pt} r_{b}^{4} }},\;\upgamma = \frac{\Omega }{{\upomega_{1} }},\;\uptau = 2\uppi\,\upomega_{1} {\kern 1pt} t,\;b_{0} = \frac{1}{6}\left( {1 +\updelta +\updelta^{2} } \right) - \frac{1}{{4\pi^{2} }}\left( {1 -\updelta} \right)^{2} > 0, $$
(8)
$$ b_{1} = \frac{{\uppi^{4} }}{10}\left( {1 +\updelta +\updelta^{2} +\updelta^{3} +\updelta^{4} } \right) - \frac{{\uppi^{2} }}{2}\left( {1 -\updelta} \right)\left( {1 -\updelta^{3} } \right) + \frac{3}{4}\left( {1 -\updelta} \right)^{4} > 0,\;\updelta = \frac{{r_{t} }}{{r_{b} }}{ \in }[0,1]. $$

The natural frequencies of transverse vibrations of the beam are (in Hz):

$$ \upomega_{n} = \frac{{n^{2} r_{b} }}{{4\uppi{\kern 1pt} L^{2} }}\sqrt {\frac{{b_{1} E{\kern 1pt} }}{{b_{0}\uprho}}} ,\quad n = 1,2,3, \ldots $$
(9)

In particular, for the cylindrical beam with constant radius \( r(x) = r_{b} \), the assumed solution (Eq. 6) is an exact one. In this case, Eq. 3 reduces to the same Eq. 7 with the parameters \( b_{0} = {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2} \) and \( b_{1} = {{\uppi^{4} } \mathord{\left/ {\vphantom {{\uppi^{4} } 2}} \right. \kern-0pt} 2} \) as \( \updelta = 1 \).

3 Parametric Resonance

Equation 7, known as the Mathieu equation, describes periodically excited vibrations of a system (McLachlan 1947). It has a cosinusoidal time-dependent coefficient of the term \( f(\tau ) \). The dimensionless parameters ε and γ, defined by Eq. 8, depend on parameters of the beam and the periodic applied force. It appears that for different values of ε and γ solutions of Eq. 7 may be periodic and bounded for all time, or the amplitude of the vibrations in solutions may grow progressively in time (parametrical resonance). Let us determine analytically and numerically ranges of the parameters, when the parametric resonance of the beam takes place.

3.1 Procedure of Averaging

For the approximate analysis of the oscillating process described by Eq. 7, the method of averaging is used (Bogolyubov and Mitropoliskii 1961). First, in order to reduce Eq. 7 to the standard form, it is assumed that the function \( f(\uptau) \) and its derivative have the form: \( f(\uptau) = a(\uptau)\cos (\uppsi(\uptau)) \) and \( \dot{f}(\uptau) = - a(\uptau)\sin (\uppsi(\uptau)). \)

Then, the differential equation of the second order (Eq. 7) converts to the system of two equations of the first order for the new variables \( a(\tau ) \) and \( \uppsi(\uptau) \):

$$ \dot{a}(\uptau) = - \frac{\upvarepsilon}{2}a(\uptau)\sin (2\uppsi(\uptau))\cos (\upgamma \uptau ),\quad {\dot{\uppsi }}(\uptau) = 1 -\upvarepsilon\cos^{2} (\uppsi(\uptau))\cos (\upgamma \uptau ). $$
(10)

As it will be shown, the most intense parametric resonance occurs for values of the excitation frequency Ω close to \( 2\upomega_{1} \) of free vibrations of the beam. Therefore, we can set \( \upgamma = 2 +\upvarepsilon{\kern 1pt}\Delta \), where Δ corresponds to the amplification factor of the parametric excitation. Consider the amplitude \( F_{0} \) to be small in comparison with the elastic forces of the beam element, so that ε can be treated as a small positive parameter: \( 0 <\upvarepsilon \ll 1{\kern 1pt} \). If \( a(\uptau) \) and \( \xi (\uptau) =\upgamma{\kern 1pt}\uptau - 2\uppsi(\uptau) \) are smooth functions of the time such that their derivatives are small terms of order ε, the values of these functions can be seen as the superposition of slowly varying part and small rapidly oscillating terms. Thus, they can be averaged on \( \uppsi \) over one period:

$$ \dot{a}_{0} (\uptau) = \frac{1}{{2\uppi}}\int\limits_{0}^{{2\uppi}} {\dot{a}(\uptau)d\uppsi} = \frac{\upvarepsilon}{4}a_{0} (\uptau)\sin (\upxi_{0} (\uptau)), $$
(11)
$$ {\dot{\xi }}_{0} (\uptau) = \frac{1}{{2\uppi}}\int\limits_{0}^{{2\uppi}} {{\dot{\xi }}(\uptau)\,d\uppsi} =\upvarepsilon{\kern 1pt}\Delta + \frac{\upvarepsilon}{2}\cos (\upxi_{0} (\uptau)). $$

3.2 Region of the Principal Parametric Resonance

The non-linear system (Eq. 11) may be simplified to a linear one with constant coefficients by defining new variables \( \upeta(\uptau) = a_{0} (\uptau)\cos ({{\upxi_{0} (\uptau)} \mathord{\left/ {\vphantom {{\upxi_{0} (\uptau)} {2 + {\pi /4}}}} \right. \kern-0pt} {2 + {\pi /4}}}) \) and \( {\varsigma }(\uptau) = - a_{0} (\tau )\sin ({{\upxi_{0} (\uptau)} \mathord{\left/ {\vphantom {{\upxi_{0} (\uptau)} {2 +\uppi/4}}} \right. \kern-0pt} {2 +\uppi/4}}) \):

$$ {\dot{\eta }}(\uptau) = - \frac{\upvarepsilon}{4}\upeta(\uptau) + \frac{{\upvarepsilon{\kern 1pt}\Delta }}{2}{\varsigma }(\uptau).\quad {\dot{\varsigma }}(\uptau) = - \frac{{\upvarepsilon{\kern 1pt}\Delta }}{2}\upeta(\uptau) + \frac{\upvarepsilon}{4}{\varsigma }(\uptau). $$
(12)

The matrix corresponding to the system (Eq. 12) has the following eigenvalues:

$$ \uplambda^{2} = \frac{{\varepsilon^{2} }}{4}\left( {\frac{1}{4} -\Delta ^{2} } \right). $$
(13)

Thus, the solution of Eq. 12 is aperiodic and unstable, when there is an eigenvalue with positive real part, i.e., for \( \left|\Delta \right| < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2} \) around the dimensionless frequency value \( {\Omega \mathord{\left/ {\vphantom {\Omega {\upomega_{1} }}} \right. \kern-0pt} {\upomega_{1} }} = 2 \). This means that the resonance takes place within the interval

$$ 0 <\upvarepsilon \ll 1{\kern 1pt} \quad \text{and}\quad 2 - \frac{\upvarepsilon}{2} < \frac{\Omega }{{\upomega_{1} }} < 2 + \frac{\upvarepsilon}{2}. $$
(14)

It is called the region of the principal parametric resonance. The width of this range is proportional to the parameter ε.

Other frequency ranges of the parametric resonance are close to the values \( \Omega = {{2{\kern 1pt}\upomega_{1} } \mathord{\left/ {\vphantom {{2{\kern 1pt}\upomega_{1} } n}} \right. \kern-0pt} n} \) for any natural number n (Landau and Lifshitz 1969). However, the width of them gets narrow proportionally to the value \( \upvarepsilon^{n} \) as n increases. In practice, the cases for n = 1, 2, and 3 (rarely) are usually observed.

4 Numerical Analysis and Results

The small transverse vibrations of the parametrically excited beam are described by the partial differential Eq. 3. The solution of it with respect to the boundary conditions (Eq. 5) may be obtained by means of the finite element method. The numerical simulations are performed using software ANSYS Workbench 16.2.

Two particular models of the beam are considered:

  1. 1.

    A straight cylindrical beam (CB) with a constant radius of the cross-section.

  2. 2.

    A straight truncated beam (TB), when the radius at the tip is half of that at the base: \( r_{t} = {{r_{b} } \mathord{\left/ {\vphantom {{r_{b} } 2}} \right. \kern-0pt} 2} \).

4.1 Remarks on the Simulations

For both beam models, the vibrational characteristics (natural frequencies and mode shapes) are evaluated using an analysis system ANSYS “Modal”. The dynamic response of a structure under the action of a periodic following load is calculated within the ANSYS module “Transient structural”.

The CB is modelled as a line body with a thin circular solid cylinder attached at the right end. The latter one is needed to apply the periodic following force perpendicular to the cross-section. The cylinder may be considered massless and rigid. The transition between beam elements (beam 188, 50 elements) of the line body and solid elements (solid 186, 16 elements) of the cylinder is defined with the multi-point constraint bonded contact. Both translational and rotational degrees of freedom are accounted for. The TB is modelled as a whole using solid elements (solid 186, 504 elements). The values of the parameters for both models are given in Table 1. The boundary conditions are applied according to the theoretical formulation of the problem. Within the simulations, no damping is considered. The numerical damping is defined manually to be zero.

Table 1 Values of the parameters used in numerical simulations
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For the chosen parameter sets, the natural frequencies are listed in Table 2. It is shown that the values of the CB obtained numerically for three different meshing elements coincide with the theoretical eigenfrequencies given by Eq. 9. It has to be noted that the values of the parameters assigned to the beam (Table 1) do not correspond to vibrissae values reported in literature, e.g. (Neimark et al. 2003; Voges et al. 2012). They are chosen in such a way that \( \upomega_{1} \) ≈ 1 Hz of the CB. This in turn further allows using a reasonable adequate time step size (10−2–10−3 s) in transient simulations. The theoretical natural frequencies of the TB give upper boundary values, i.e., they are slightly higher than the numerical results (Table 2).

Table 2 Natural frequencies of transverse vibrations of the CB and TB
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The transient analysis of parametric vibrations requires an initial deflection of the beam from the equilibrium position. Thus, the load process is developed in three steps: (1) [0, 0.25] s, (2) [0.25, 0.5] s, and (3) [0.5, 30] s. Within the first two steps, the impulse force is applied and discharged linearly at the midst of the beam in the transverse direction. Its maximum magnitude is 10–7 N. Then, within a time range [0.25, 30] s, the main load is applied as a pressure \( P(t) = - P_{0} \cos (2\pi \,{\kern 1pt}\Omega \,t) \), \( P_{0} = 3 \) Pa, normal to the cross-section at the right end.

4.2 Regions of the Parametric Resonance

The dynamic response of the system under the parametric excitation is simulated for various frequency values. For the CB, the solution is found to be unstable for the frequencies close to the doubled eigenfrequency of the beam (region of the principal parametric resonance), as well as for the frequencies around this value (second range of the parametric resonance). For the TB, the region of the principal parametric resonance is obtained. Within these frequency intervals, the vibrations of the beam are unstable, i.e., their amplitude increases exponentially with time.

The comparison of the theoretical (Eq. 14) and numerical regions of the parametric resonance is given in Table 3. It may be seen that the results of the CB correspond to each other. Moreover, it is shown that the response of the system on the parametric excitation is qualitatively stronger within the region of the principal parametric resonance than within the second range. For the TB, a slight discrepancy between the theoretical and numerical results is observed. It is associated with the accounting of the variable radius of the cross-section.

Table 3 Regions of the parametric resonance of the CB and TB (in Hz)
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In Fig. 5, some simulation results of the CB and TB are presented. The averaged amplitude of the theoretical solution is plotted for comparison as a dash line.

Fig. 5
figure 5

Transverse displacement of the CB (a, b) and the TB (c, d): a, c unstable solution of the principle parametric resonance; b, d steady-state solution; — numerical simulations; ‒ ‒ the theoretical averaged amplitude a 0(t) for: a a 0(0) = 0.024 mm, ψ(0) = 0; b a 0(0) = 0.0265 mm and ψ(0) = 1.4573; c a 0(0) = 0.18 mm, ψ(0) = 0, Ω = 2ω1 = 1.5669 Hz; d a 0(0) = 0.078 mm, ψ(0) = 1.7453, Ω = 1.461 Hz

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5 Conclusions

In this work, a model of a straight truncated beam is developed to study the vibrational motion of mammal’s vibrissae. The transverse oscillations of the beam stimulated by a periodic following force are investigated theoretically and numerically. It is shown that the parametric resonance of the beam occurs at the specific ranges of the excitation frequency, within which the vibrations of the beam are unstable with exponentially increasing amplitude. Together with the realisation of the viscoelastic support of an artificial tactile sensor, it may be a potentially useful method for amplifying small signals arising from the contact with an object. Future investigations will also focus on the design of sensor arrays consisting of several vibrissa elements.