Beat Phenomenon Analysis of Concrete Beam with Piezoelectric Sensors

Abstract

The focus of this paper is to give a better understanding of beat phenomenon in the free vibration test of a concrete beam with piezoelectric ceramic sensors from the view of mathematics. The cause of beat phenomenon from piezoelectric ceramic sensors embedded in the concrete beam is illustrated and the influence factors of beat phenomenon are discussed. The results show that the beat phenomenon from piezoelectric ceramic sensors in the concrete beam is caused by the coupled responses with similar model frequencies in different directions. The influence factors of beat phenomenon due to damping effect, impact direction, sensor position and sectional dimension are discussed. As the damping ratios increased, the amplitude of beat signal will die out in an exponential decay. Meanwhile, the damping has a tiny influence on the beat frequency of system response, the amplitudes of beat signal both in the time and frequency domain are changed with the variation of impact direction. In addition, the amplitude of beat signal will be also changed with the position of sensors altered. The beat frequency will get more with the greater difference of sectional dimension.

1. Introduction

Piezoelectric materials have a broadly applicative prospect in structure health monitoring due to their characters of electromechanical coupling, simple structure, low cost, good reliability, and wide frequency response range. The piezoelectric transducers based on positive and negative piezoelectric effect have been applied as actuator and dynamic measurement in damage identification [1–3], impact force location [4, 5], and fatigue crack detection [6, 7] in composite structures. Besides, the piezoelectric transducers have been successfully utilized in civil engineering structures. Song et al. [8] have developed an overheight collision detection and evaluation system for concrete bridge girders using piezoelectric transducers. Li et al. [9] applied a new type of cement-based piezoelectric sensor to monitor the traffic flows and concluded that there is a good potential for the cement-based piezoelectric sensor in the engineering application for monitoring traffic flows in the field of transportation. Gu et al. [10] provided a method of piezoelectric-based strength monitoring, in which an innovative experimental approach is proposed to conduct the concrete strength monitoring at early ages. Song et al. [11] exploited smart aggregate (SA), an innovative multifunctional transducer, fabricated by embedding a wired, waterproof piezoelectric ceramic patch into a small concrete block. The proposed SA has been successfully utilized in the structural health monitoring of a full-size bridge girder [12], reinforced concrete shear walls [13], a two-story reinforced concrete frame subjected to progressive collapse [14], circular RC columns under cyclic combined loading [15], and circular reinforced concrete columns after seismic excitations [16].

The free vibration of undercritically damped system would be decayed exponentially, as shown in Figure 1 [17]. However, when conducting the free vibration test with piezoelectric ceramic sensors, instead of an exponential decay, the vibration tends to decrease or increase periodically as shown in Figure 2, which is characterized as the classical beat phenomenon. It will be difficult to evaluate the frequency or damage of structures from the beat signals. However, there is no definite conclusion about the cause and influence factors of beat phenomenon in piezoelectric transducers currently. In this paper, the cause of beat phenomenon in piezoelectric ceramic sensors of a simply supported beam is illustrated from the view of mathematics. In addition, the influence factors of beat phenomenon due to damping effect, impact direction, sensor position, and sectional dimension are discussed.

Figure 1

Free-vibration response of undercritically damped system.

Figure 2

The classical beat phenomenon from piezoelectric ceramic sensors.

2. The Analysis of Beat Phenomenon from Piezoelectric Ceramic Sensor Embedded in the Simply Supported Beam

A simply supported beam embedded with piezoelectric ceramic sensors is taken as an example to analyze the cause of beat phenomenon. The hypothesis is put up that the calculative model is the ideal simply supported beam, and the material of structure is in elastic phase. The size of the beam is $B\times W\times L$ , and the piezoelectric ceramic sensor is embedded with the electrode parallel to the beam axis at the length of $z_0$ to the end of the beam. The detailed illustration of calculative model is shown in Figure 3. When subject to an impact with initial velocity of v on the middle of the beam from θ-direction, the beam would oscillate as free vibration.

Figure 3

The calculative model of the simply supported beam.

2.1. The Stress Condition on the Surface of Sensor Dipole

The electrode surface of piezoelectric ceramic sensor has only axial stress when subjected to the curvature movement. The initial velocity can be resolved into orthometric components $v_x$ and $v_y$ , and the axis stress ${\sigma }_3$ is the composition of stress component ${\sigma }_{3x}$ and ${\sigma }_{3y}$ caused by $v_x$ and $v_y$ , respectively. Consequently, the stress on the electrode surface is expressed by (1)

$$\begin{array}{cc}& {\sigma }_3={\sigma }_{3x}+{\sigma }_{3y},\end{array}$$

(1)

where ${\sigma }_{3x}$ and ${\sigma }_{3y}$ can be analyzed, respectively. When subjected to impact force by $v_x$ , the dynamic equation in undercritically damped system is given by [17]

$$\begin{array}{l}{E}_b{I}_y\frac{\partial {u_x}^{\left(4\right)}\left(z,t\right)}{\partial z^4}+m\frac{{\partial }^2{u}_x\left(z,t\right)}{\partial t^2}\\ \hspace{1em}\hspace{1em}+a_1{E}_b{I}_y\frac{\partial {u_x}^{\left(5\right)}\left(z,t\right)}{\partial z^4\partial t}+a_{0}m\frac{\partial u_x\left(z,t\right)}{\partial t}=0,\end{array}$$

(2)

where $u_x(z,t)$ is the displacement response of the beam in x-direction, $E_b$ is the modulus of elasticity, $I_y$ is the moment of inertia of the beam in y-direction, m is the mass of the beam in unit length, and $a_0$ and $a_1$ represent the Rayleigh damping coefficient related to the mass and stiffness, respectively. Solving (2) with the method of separating variables, the solution is expressed by $Z_x(z)Y_x(t)$ , which indicates that the free vibration motion is of a specific mode shape $Z_x(z)$ having a time-dependent amplitude $Y_x(t)$ . Then (2) is transformed by

$$\begin{array}{cc}& {Z_x}^{\left(4\right)}\left(z\right)-a^4{Z}_x\left(z\right)=\mathrm{0},\end{array}$$

(3)

$$\begin{array}{cc}& {\ddot{Y}}_x\left(t\right)+\mathrm{2}{\xi }_x{\omega }_x\dot{Y}\left(t\right)+{\omega }_x^2{Y}_x\left(t\right)=\mathrm{0},\end{array}$$

(4)

where a is a parameter related to the frequency of the beam, in which the natural frequency in x-direction is conveyed as

$$\begin{array}{cc}& {\omega }_x=a^{\mathrm{2}}\sqrt{\frac{E_b{I}_y}{m}},\end{array}$$

(5)

where ${\xi }_x$ is the damping ratio of the beam in x direction given by

$$\begin{array}{cc}& {\xi }_x=\frac{a_{\mathrm{0}}}{\mathrm{2}{\omega }_x}+\frac{a_{\mathrm{1}}{\omega }_x}{\mathrm{2}}.\end{array}$$

(6)

The general solution of (3) is

$$\begin{array}{cc}& Z_x\left(z\right)=A_{\mathrm{1}}\cos \hspace{0.17em}\left(az\right)+A_{\mathrm{2}}\sin \hspace{0.17em}\left(az\right)+A_{\mathrm{3}}\cosh \hspace{0.17em}\left(az\right)+A_{\mathrm{4}}\sinh \hspace{0.17em}\left(az\right).\end{array}$$

(7)

The boundary conditions of simply supported beam are shown in (8), where $M(z)$ is the bending moment at the length of z:

$$\begin{array}{cc}& Z_x\left(\mathrm{0}\right)=Z_x^{\prime }\left(\mathrm{0}\right)=M\left(\mathrm{0}\right)=M\left(L\right)=\mathrm{0}.\end{array}$$

(8)

Substituting (8) into (7), then the value a is acquired by the vibration $Z_x(z)$ , and natural frequency of each mode in x direction can be calculated in each order. The first mode provides the greatest contribution to the vibration so as the $u_x(z,t)$ in 1st mode can be considered as the total displacement response approximately. The value of a in 1st mode is displayed in (9), and the natural frequency is given by (10)

$$\begin{array}{cc}& a=\frac{\pi }{L},\end{array}$$

(9)

$$\begin{array}{cc}& {\omega }_x={\pi }^{\mathrm{2}}\sqrt{\frac{E_b{I}_y}{m{L}^{\mathrm{4}}}}.\end{array}$$

(10)

The initial condition 1st mode is calculated as (11) based on the mode shape orthogonality:

$$\begin{array}{cc}& {\dot{Y}}_x\left(\mathrm{0}\right)=\frac{{\int }_{\mathrm{0}}^{L}m{Z}_x\left(z\right)v\left(z,\mathrm{0}\right)dz}{{\int }_{\mathrm{0}}^{L}m{Z}_x^2\left(z\right)dz}\cos \hspace{0.17em}\theta .\end{array}$$

(11)

By solving (4) $Y_x(t)$ is obtained as (12), which is a harmonic signal decayed exponentially:

$$\begin{array}{cc}& Y_x\left(t\right)=\frac{{\dot{Y}}_x\left(\mathrm{0}\right)}{{\omega }_{Dx}}\sin \hspace{0.17em}\left({\omega }_{Dx}t\right)\exp \hspace{0.17em}\left(-{\xi }_x{\omega }_{x}t\right),\end{array}$$

(12)

where ${\omega }_{Dx}$ is the free vibration frequency of the damped system which is equal to ${\omega }_x\sqrt{\mathrm{1}-{{\xi }_x}^{\mathrm{2}}}$ . The stress on the surface of sensor dipole caused by $v_x$ can be described as

$$\begin{array}{l}{s}_{\mathrm{3}x}=\frac{x_{\mathrm{0}}{E}_b{I}_y{Z}_x^{\prime \prime }\left(z_{\mathrm{0}}\right)Y_x\left(t\right)}{I_y}\\ \hspace{1em}\hspace{0.17em}=x_{\mathrm{0}}{E}_b{Z}_x^{\prime \prime }\left(z_{\mathrm{0}}\right)Y_x\left(t\right),\end{array}$$

(13)

where $z_0$ depicts the position of the piezoceramic sensor. $x_0$ is the distance between x-axis and the centroid of piezoelectric ceramic sensor axial section. ${\sigma }_{3x}$ is also a harmonic signal with natural frequency of ${\omega }_x$ . Likewise, the stress on the surface of sensor dipole caused by $v_y$ is a harmonic signal with natural frequency of ${\omega }_y$ , which can be conveyed as

$$\begin{array}{cc}& {\sigma }_{\mathrm{3}y}=y_{\mathrm{0}}{E}_b{Z}_y^{\prime \prime }\left(z_{\mathrm{0}}\right)Y_y\left(t\right),\end{array}$$

(14)

where $y_0$ is the distance between x-axis and the centroid of piezoelectric ceramic sensor axial section. The stress on the electrode surface is acquired by submitting (13) and (14) into (1).

2.2. The Output Voltage of Piezoelectric Ceramic Sensor

By means of the piezoelectric sensors and signal acquisition system, the changes of stress are transformed into voltage signal. Following assumptions are made to the computational model of piezoelectric ceramic sensors to facilitate the analysis according to the actual background and the need of the analysis. First, the piezoelectric ceramic can be considered as the ideal elastic material without free charge. In addition, the electrode is isopotential, which means that the electric field between both electrodes is uniform, and there is no electric field in any other directions. The piezoelectric equation can be formulated as (15) [18]:

$$\begin{array}{cc}& D=d^d\sigma +e^{\sigma }E,\\ \epsilon =s^E\sigma +d^{c}E,\end{array}$$

(15)

where D is the electric displacement vector, which means the charge on unit area. ε denotes the strain vector. σ expresses the stress vector, and E is the electric intensity. $s^E$ is the elastic compliance matrix, and $e^{\sigma }$ is the dielectric permittivity matrix. The superscripts σ and E indicate that the quantity is measured at constant stress and constant electric field, respectively. $d^d$ and $d^c$ are piezoelectric coefficient matrix, where c and d have been added to differentiate the converse and direct piezoelectric effects. In practice, $d^d$ is equal to $d^c$ . The positive direction of computation model is shown in Figure 4. Equations (16) and (17) are the matrix form of piezoelectric equation:

$$\begin{array}{l}\left[\begin{array}{l}{D}_{\mathrm{1}}\\ D_{\mathrm{2}}\\ D_{\mathrm{3}}\end{array}\right]=\left[\begin{array}{ll}\begin{array}{lll}& & \\ & & \\ d_{\mathrm{31}}& d_{\mathrm{32}}& d_{\mathrm{33}}\end{array}& \begin{array}{ll}& d_{\mathrm{15}}\\ d_{\mathrm{24}}& \\ & \end{array}\end{array}\right]\left[\begin{array}{l}\begin{array}{l}{\sigma }_{\mathrm{1}}\\ {\sigma }_{\mathrm{2}}\\ {\sigma }_{\mathrm{3}}\end{array}\\ \begin{array}{l}{\sigma }_{\mathrm{4}}\\ {\sigma }_{\mathrm{5}}\\ {\sigma }_{\mathrm{6}}\end{array}\end{array}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}+\left[\begin{array}{lll}{e}_{\mathrm{11}}& & \\ & e_{\mathrm{22}}& \\ & & e_{\mathrm{33}}\end{array}\right]\left[\begin{array}{l}{E}_{\mathrm{1}}\\ E_{\mathrm{2}}\\ E_{\mathrm{3}}\end{array}\right],\end{array}$$

(16)

$$\begin{array}{ll}& \begin{array}{l}\left[\begin{array}{l}\begin{array}{l}{\epsilon }_{\mathrm{1}}\\ {\epsilon }_{\mathrm{2}}\\ {\epsilon }_{\mathrm{3}}\end{array}\\ \begin{array}{l}{\epsilon }_{\mathrm{4}}\\ {\epsilon }_{\mathrm{5}}\\ {\epsilon }_{\mathrm{6}}\end{array}\end{array}\right]=\left[\begin{array}{llllll}{s}_{\mathrm{11}}& s_{\mathrm{12}}& s_{\mathrm{13}}& & & \\ s_{\mathrm{21}}& s_{\mathrm{22}}& s_{\mathrm{23}}& & & \\ s_{\mathrm{31}}& s_{\mathrm{32}}& s_{\mathrm{33}}& & & \\ & & & s_{\mathrm{44}}& & \\ & & & & s_{\mathrm{44}}& \\ & & & & & s_{\mathrm{66}}\end{array}\right]\left[\begin{array}{l}\begin{array}{l}{\sigma }_{\mathrm{1}}\\ {\sigma }_{\mathrm{2}}\\ {\sigma }_{\mathrm{3}}\end{array}\\ \begin{array}{l}{\sigma }_{\mathrm{4}}\\ {\sigma }_{\mathrm{5}}\\ {\sigma }_{\mathrm{6}}\end{array}\end{array}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}+\left[\begin{array}{lll}& & d_{\mathrm{31}}\\ & & d_{\mathrm{32}}\\ & & d_{\mathrm{33}}\\ & d_{\mathrm{24}}& \\ d_{\mathrm{15}}& & \end{array}\right]\left[\begin{array}{l}{E}_{\mathrm{1}}\\ E_{\mathrm{2}}\\ E_{\mathrm{3}}\end{array}\right].\end{array}\end{array}$$

(17)

Figure 4

The positive direction of computation model.

When the poling of sensor is in 3-direction, the eternal electric field is zero, and (16) is expressed by

$$\begin{array}{cc}& D_{\mathrm{3}}=d_{\mathrm{31}}{\sigma }_{\mathrm{1}}+d_{\mathrm{32}}{\sigma }_{\mathrm{2}}+d_{\mathrm{33}}{\sigma }_{\mathrm{3}}.\end{array}$$

(18)

And the charge Q is

$$\begin{array}{cc}& Q=\iint D_{3}d{A}_3,\end{array}$$

(19)

where $A_3$ is the area of electrode.

Piezoelectric ceramic has very high impedance, in which the output current is weak. The output signal needs to be collected through charge amplifier which can provide low impedance to signal. Figure 5 is the operating principle of charge amplifier, where $C_P$ is the capacity of piezoelectric ceramic sensor, $C_C$ is the lead wire capacitance, $C_F$ is the feedback capacitance, and $K_F$ is the gain of charge amplifier. The total charge $(C)$ can be written as

$$\begin{array}{cc}& C=C_C+C_P+\left(1+K_F\right)C_F,\end{array}$$

(20)

$C_F$ is far larger than $C_C$ and $C_P$ , and the influence of accuracy by $C_C$ and $C_P$ is less than 0.1% [19]. Therefore, it can be considered that the output voltage V only depends on Q and $C_F$ , just as

$$\begin{array}{cc}& V=\frac{Q}{C_F}.\end{array}$$

(21)

Figure 5

The operating principle of charge amplifier.

For typical piezoelectric sheet, the axial thickness is so thin that the shear stress in 1 and 2 directions can be neglected. Substituting (18) and (19) into (21) leads to

$$\begin{array}{cc}& V=\frac{d_{33}{\sigma }_3{A}_3}{C_F}.\end{array}$$

(22)

The sensitivity of piezoelectric ceramic sensor can be defined as (23), which represents the relation between output voltage and stress on the surface of electrode, and V can be depicted as (24):

$$\begin{array}{cc}& S_q=\frac{d_{33}{A}_3}{C_F},\end{array}$$

(23)

$$\begin{array}{cc}& V=S_q{\sigma }_3.\end{array}$$

(24)

Substitute (1), (13), and (14) into (24), and the coupled response from piezoelectric ceramic sensors can be acquired as (25). Note that for the simply supported beam subject to eccentric impact, the rotating component has no influence on the output voltage of piezoelectric ceramic sensors, because the piezoelectric coefficient ( $d_{36}$ ) in the direction of rotation about PZT axis is zero. Then, the output voltage V can be expressed by

$$\begin{array}{cc}& V=A_x\sin \hspace{0.17em}\left({\omega }_{Dx}t\right)+A_y\sin \hspace{0.17em}\left({\omega }_{Dy}t\right),\end{array}$$

(25)

where

$$\begin{array}{l}{A}_x=S_q\frac{x_0}{{\omega }_{Dx}}{E}_b{Z}_x^{\prime \prime }\left(z_0\right){\dot{Y}}_x\left(0\right)\exp \hspace{0.17em}\left(-{\xi }_x{\omega }_{x}t\right)\\ \hspace{1em}\hspace{0.17em}=A_{cx}\exp \hspace{0.17em}\left(-{\xi }_x{\omega }_{x}t\right),\\ A_y\hspace{0.17em}=S_q\frac{y_0}{{\omega }_{Dy}}{E}_b{Z}_y^{\prime \prime }\left(z_0\right){\dot{Y}}_y\left(0\right)\exp \hspace{0.17em}\left(-{\xi }_y{\omega }_{y}t\right)\\ \hspace{1em}\hspace{0.17em}=A_{cy}\exp \hspace{0.17em}\left(-{\xi }_y{\omega }_{y}t\right),\end{array}$$

(26)

where $A_{cx}$ and $A_{cy}$ are constants unrelated to time. To illustrate the cause of beat phenomenon, it is assumed the amplitude of voltages from the piezoceramic sensor has the same value, which means that $A_x$ equals $A_y$ . Then, (25) can be derived as [20]

$$\begin{array}{l}V=2A_{cx}\exp \hspace{0.17em}\left(-{\xi }_x{\omega }_{x}t\right)\cos \hspace{0.17em}\left(\frac{{\omega }_{Dy}-{\omega }_{Dx}}{2}t\right)\sin \hspace{0.17em}\left(\frac{{\omega }_{Dx}+{\omega }_{Dy}}{2}t\right)\\ \hspace{1em}=2A_{cx}\exp \hspace{0.17em}\left(-{\xi }_x{\omega }_{x}t\right)\cos \hspace{0.17em}\left({\omega }_{B}t\right)\sin \hspace{0.17em}\left({\omega }_{A}t\right),\end{array}$$

(27)

where ${\omega }_B=({\omega }_{Dx}-{\omega }_{Dy})/2$ and ${\omega }_A=({\omega }_{Dx}+{\omega }_{Dy})/2$ . Equation (27) shows the coupled response is an amplitude-modulated harmonic function with the frequency equal to ${\omega }_B$ and amplitude varying function with the frequency equal to ${\omega }_A$ , as illustrated in Figure 6. ${\omega }_B$ is defined as the beat frequency and ${\omega }_A$ is the average frequency.

Figure 6

Illustration of the beat phenomenon with equal $A_x$ and $A_y$ .

To give a general understanding of beat phenomenon, one can consider the solution of system response by the way of rotate vector method. As shown in Figure 7, ${\stackrel{⃑}{A}}_x$ can be seen as a vector rotating around X-axis with the length of $A_x$ , where ${\omega }_{Dx}$ is the angular velocity of ${\stackrel{⃑}{A}}_x$ . Similarly, ${\stackrel{⃑}{A}}_y$ is a vector rotating around X-axis with the length of $A_y$ and the angular velocity of ${\omega }_{Dy}$ . $\stackrel{⃑}{A}$ is the summation of ${\stackrel{⃑}{A}}_x$ and ${\stackrel{⃑}{A}}_y$ , and the mathematical meaning of V in (25) is the projective length of $\stackrel{⃑}{A}$ in Y-axis and can be expressed by (28), in which ${\omega }^{\prime }t$ is the phase given by (29), and A is the length of $\stackrel{⃑}{A}$ shown in (30). In this stage, the cause of beat phenomenon can be further examined in Figure 8. $A_x$ and $A_y$ are illustrated as the amplitudes of system responses caused by $v_x$ and $v_y$ , respectively; V can be described as a simple harmonic oscillation with the amplitude of A and the phase of ${\omega }^{\prime }t$ . A is the upper envelope of output signal, where the amplitude appears in a way of time-varying with the beat frequency of $|({\omega }_{Dx}-{\omega }_{Dy})/\mathrm{2}|$ . Meanwhile, the phase is altered with time and $A_y/A_x$ . The beat phenomenon will become easily observable if the beat frequency is in an appropriate scope:

$$\begin{array}{cc}& V=A\hspace{0.17em}\sin \hspace{0.17em}\left({\omega }^{\prime }t\right),\end{array}$$

(28)

$$\begin{array}{cc}& \omega \text{'}t=\{\begin{array}{cc}\text{arccos}\left(\frac{1+\left(A_y/A_x\right)\cos \hspace{0.17em}\hspace{0.17em}\left[\left({\omega }_{Dy}-{\omega }_{Dx}\right)t\right]}{\sqrt{1+{\left(A_y/A_x\right)}^2+2\left(A_y/A_x\right)\cos \hspace{0.17em}\hspace{0.17em}\left[\left({\omega }_{Dy}-{\omega }_{Dx}\right)t\right]}}\right)+{\omega }_{Dx}t\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{if}\mathrm{\hspace{0.17em}\hspace{0.17em}}\left({\omega }_{Dy}-{\omega }_{Dx}\right)t\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{is}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{in}\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{1}\text{st}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{2}\text{nd}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{quadrant}\hfill & \hfill \\ -\text{arccos}\left(\frac{1+\left(A_y/A_x\right)\cos \hspace{0.17em}\hspace{0.17em}\left[\left({\omega }_{Dy}-{\omega }_{Dx}\right)t\right]}{\sqrt{1+{\left(A_y/A_x\right)}^2+2\left(A_y/A_x\right)\cos \hspace{0.17em}\hspace{0.17em}\left[\left({\omega }_{Dy}-{\omega }_{Dx}\right)t\right]}}\right)+{\omega }_{Dx}t\hfill & \hfill \\ \hspace{1em}\hspace{1em}\text{if}\mathrm{\hspace{0.17em}\hspace{0.17em}}\left({\omega }_{Dy}-{\omega }_{Dx}\right)t\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{is}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{in}\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{3}\mathrm{rd}\hspace{0.17em}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathrm{4}\text{th}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{quadrant},\hfill & \hfill \end{array}\end{array}$$

(29)

$$\begin{array}{cc}& A=\sqrt{{A_x}^2+{A_y}^2+2A_x{A}_y\cos \hspace{0.17em}\left[\left({\omega }_{Dy}-{\omega }_{Dx}\right)t\right]}.\end{array}$$

(30)

Figure 7

Illustration of V by rotation vector method.

Figure 8

Anatomy of the beat phenomenon with unequal $A_x$ and $A_y$ .

From previous discussion, the beat frequency ${\omega }_B$ is the major parameter related to beat phenomenon. The zero value of ${\omega }_B$ means that beat phenomenon could not be observed. The beat phenomenon is characterized as the amplitude modulated periodically. The exponentially decayed amplitude implies that there is be phenomenon in the system response. Hence, the beat frequency ${\omega }_B$ and upper envelope A are researched further in the following.

3. The Influence Factor of Beat Phenomenon

In this section, the influences of beat phenomenon including damping effect, impact direction, the position of sensor and anisotropic properties are discussed. Unless mentioned otherwise, the damping ratio of calculative model is 0.01 in both orthometric directions, and other parameters are listed in Table 1.

Table 1

The parameters of the simply supported beam.

$B\times W\times L\hspace{0.17em}\hspace{0.17em}({\text{m}}^{\mathrm{3}})$	$m\hspace{0.17em}\hspace{0.17em}(\text{kg/m})$	$E_b\hspace{0.17em}\hspace{0.17em}({\mathrm{10}}^{\mathrm{10}}\text{N/}{\text{m}}^{\text{2}})$	$z_{\mathrm{0}}\hspace{0.17em}\hspace{0.17em}(\text{m})$	θ  (°)	$v\hspace{0.17em}\hspace{0.17em}(\text{m/s})$	$S_q\hspace{0.17em}\hspace{0.17em}(\text{V/MPa})$
$0.1\times 0.11\times 1$	23.57	$2.5$	0.25	45	1	0.813

3.1. Damping Effect

Figure 9 shows the upper envelope of output signal with equal ${\xi }_x$ and ${\xi }_y$ . As the damping ratios increase, the output signal dies out in an exponential decay. In practice, the damping ratios in different modes are diverse more or less. Figure 10 shows the upper envelope of output signal in which ${\xi }_x$ is 0.01 and ${\xi }_y$ is unequal to ${\xi }_x$ . With the increasing of ${\xi }_y$ , the single response with modal frequency of ${\omega }_y$ decays rapidly, and the coupled response of sensor tends to the one with the modal frequency of ${\omega }_x$ .

Figure 9

The upper envelope of output signal with equal ${\xi }_x$ and ${\xi }_y$ ( ${\xi }_x={\xi }_y=\xi$ ).

Figure 10

The upper envelope of output signal with unequal ${\xi }_x$ and ${\xi }_y$ .

The damping not only leads to the attenuation of amplitude but also reacts on the beat frequency. Figure 11 depicts the change of beat frequency in which ${\xi }_y$ / ${\xi }_x$ is in the range of 0.9 to 1.1, which shows that the difference of damper makes tiny influence on beat frequency.

Figure 11

The change of beat frequency with different ${\xi }_y$ / ${\xi }_x$ ( ${\omega }_B^{\prime }$ is the beat frequency when ${\xi }_x$ equals ${\xi }_y$ ).

3.2. Impact Direction

With the variation of impact direction, the initial conditions in orthorhombic modes would be different, and the amplitude of coupled response of sensor and the magnitudes in each modal frequency are also changed. Figure 12 is the time history and frequency domain chart of output signals with the impact direction of 0°, 45°, 60°, and 90°. From the figures of time history, the beat phenomenon is observed obviously with the eccentric impact force. If the impact direction is parallel to the symmetric axis of the section, beat phenomenon would not happen. Also, from the figures of frequency domain, it is difficult to evaluate the structural frequency if beat phenomenon happened.

Figure 12

The time history and frequency domain chart of output signals.

Inversely, the maximum and minimum of the upper envelope expressed as $A_{\max \hspace{0.17em}}$ and $A_{\min \hspace{0.17em}}$ can be observed from beat signal. The relation between $A_{\max \hspace{0.17em}}$ / $A_{\min \hspace{0.17em}}$ and $\tan \hspace{0.17em}\theta$ is shown as (31), where ${\beta }_0$ is the $A_y$ / $A_x$ when θ equals 45°. Then the impact direction can be determined, which can be a practical approach to determine impact direction, such as the collision monitoring of the ocean platform or bridge in a traffic accident:

$$\begin{array}{cc}& \frac{{\text{A}}_{\max \hspace{0.17em}}}{{\text{A}}_{\min \hspace{0.17em}}}=\left|\frac{\mathrm{1}+{\beta }_{\mathrm{0}}\tan \theta }{\mathrm{1}-{\beta }_{\mathrm{0}}\tan \theta }\right|.\end{array}$$

(31)

3.3. The Sensor Position

The stress on the surface of sensor dipole rather than the system frequency will be changed with the alteration of sensor position. Therefore, the position of piezoelectric ceramic sensor has no influence on beat frequency but has influence on the amplitude of sensor response. Figure 13 represents the upper envelope of beat signal with various $z_0$ . Figure 14 shows the ratio of single amplitudes ( $A_y$ / $A_x$ ) in orthometric directions with various $z_0$ . Note that the change of $z_0$ has no influence on the proportion of response in each single mode. It can be concluded that the beat phenomenon can be observed more easily if the piezoelectric ceramic sensors are placed near to the middle of the beam.

Figure 13

The upper envelope of beat signal with different $z_0$ .

Figure 14

The $A_y/A_x$ with different $z_0$ .

Figure 15 is the upper envelope of beat signal with different $y_0$ , and the $A_y$ / $A_x$ with different $y_0$ is charted in Figure 16. For the system with similar orthometric modes, the dipole surface stress caused by each single mode is different with the variation of position in the same section, so the amplitude in single mode will be also changed, which can cause the alteration of the amplitude of coupled response and the proportion of response in each single mode.

Figure 15

The upper envelope of beat signal with different $y_0$ ( $x_0=0.025$  m).

Figure 16

The $A_y$ / $A_x$ with different $y_0$ ( $x_0=0.025$  m).

3.4. Sectional Dimension

Sectional dimension can be regarded as a significant factor accounting for the beat phenomenon, for the natural frequencies in orthogonal directions of the beam are related to material properties. Figure 17 is the beat frequency in which the $B/W$ is from 0.9 to 1.1, and Figure 18 is the upper envelope of output signal with different $B/W$ . It can be concluded that the beat frequency will get more with the greater difference of sectional dimension.

Figure 17

The beat frequency with different $B/W$ .

Figure 18

The upper envelope of output signal with different sectional dimension.

4. Conclusion

The beat phenomenon in the free vibration test of a simply supported beam is analyzed from the view of mathematics. The results show that the beat phenomenon of piezoelectric ceramic sensors is caused by coupled response with similar frequency in different directions. As the damping ratios increase, the amplitude of beat phenomenon will die out in an exponential decay. However, the damping effect has a tiny influence on beat frequency. With the variation of impact direction, the proportion of response in each single mode will be changed. Hence, the amplitude of coupled response of sensor and the magnitude in frequency domain are changed. The amplitude of system response can also be changed with the variation of sensor position in axial direction, which has no influence on the proportion of response in single mode. The location shift of sensors in the same section will change not only the coupled response but also the proportion of amplitude in both single modes. The sectional dimension is a vital factor for the beat frequency, which will get more with the greater difference of sectional dimension.