Identification of inertia force between cantilever beam and moving mass based on recursive inverse method

2.1. Differential equations of modal responses and inertia force of cantilever

The differential equation of the deflection of an Euler-Bernoulli beam is given as below [1]:

where $\rho$ is a constant of mass per unit length of cantilever, $v(x,t)$ is the beam deflection at point $x$ and time $t$, $EI$ is a constant of flexural stiffness of cantilever, $\delta$ is the Dirac delta function, $f\left(t\right)$ is the time-varying inertia force. Because the interaction surface between the cantilever and moving mass is very smooth and is lubricated by lubricating oil, the damping of the motion process can be neglected.

Based on the modal superposition principle, the solution of Eq. (1) can be written as:

where $q_i\left(t\right)(i=1,2,3,\dots )$ are the modal displacements of the beam structure.

Substitute Eq. (2) to Eq. (1) and each side of Eq. (1) is multiply by $sin(i\pi x/L)$, which is the mode shape function. Then integrate the equation with respect to $x$ between 0 and $L$, use the properties of Dirac delta function $\delta$ and the boundary conditions of simple supported beam:

the resultant equation can be obtained as:

where:

are the modal frequency of the simple support beam at the $j$-th modal.

However, the boundary conditions of cantilever beam are different from simple support beam and Eq. (3) should be modified as [5]:

What’s more, the modal frequency ${\omega }_{\left(j\right)}$ and vibration modal function of the cantilever beam can only be accessed as approximate value by numerical calculation methods because it can’t be obtained as any analytical form. The frequency equation of cantilever beam is given as [10]:

and the modal function of cantilever beam can be obtained as below:

where $x$ is the distance from the sampling points to the start point.

By combining Eq. (4) and Eq. (6), the differential equation of the modal response and time-varying inertia force can be obtained as:

where $q_{i,k}$ is the $i$-th modal displacement and ${\ddot{q}}_{i,k}$ is the $i$-th accelerate of the modal vibration of the cantilever, $n$ is the maximum order of modal, $l\left(k\right)$ is the distance between the sampling point and the start point of the cantilever at $k$-th sampling moment.

2.2. Discretized state function of the identification system

Eq. (7) can be rewriten as follow:

where $\ddot{\mathbf{q}}$ and $\mathbf{q}$ are the $n\times 1$ vector of the acceleration and displacement of modal vibration respectively, $\mathbf{\omega }$ is the $n\times n$ matrix of the modal frequency, $\mathbf{\phi }$ is the $n\times 1$ matrix of modal function and $p$ is the inertia force between cantilever beam and moving mass.

The differential equation Eq. (8) can be transform to the continuous-time state equations as below:

and the measurement equation can also be given as:

where $\mathbf{A}=\left[\begin{array}{c}{0}_{n\times n}{\mathbf{I}}_{n\times n}\\ -\omega 0_{n\times n}\end{array}\right]$, $\mathbf{B}=\frac{1}{\rho L}\left[\begin{array}{c}{0}_{n\times 1}\\ \phi \end{array}\right]$, $\mathbf{X}\left(t\right)=\left[\begin{array}{c}q\left(t\right)\\ \dot{\mathbf{q}}\left(t\right)\end{array}\right]$, $\mathbf{H}={\mathbf{I}}_{2n\times 2n}$ is the measurement matrix and $\mathbf{Z}\left(t\right)$ is the observation vector.

Eq. (9) can be discretized over time intervals of length $\Delta t$ with process noise input as follow:

where $\mathbf{X}\left(k\right)$ is the state vector of the system, $\varphi$ is the state transition matrix, $\Gamma$ is the input matrix, $p\left(k\right)$ is the inertia force between cantilever and moving mass which is going to be estimate and $w\left(k\right)$is the input system noise which is assumed to be zero mean and white with variance $E\left\{w\right(k_1\left)w\right(k_2\left)\right\}=Q{\delta }_{k_1{k}_2}$, where $Q$ is the process noise covariance and ${\delta }_{k_1{k}_2}$ is the Dirac delta function.

The measurement noise also should be considered and therefore Eq. (10) can be discretized as follow:

where the observation vector is:

and the measurement noise vector:

is assumed to be zero mean and white. The variance of $\mathbf{v}\left(k\right)$ is:

where the elements ${\sigma }_i$ is the standard deviation of measurement noise of $v_i\left(k\right)$.

2.3. Recursive algorithm of inertia force identification

The recursive identification algorithm consists of two parts: Kalman filter part, which is adapted to calculate the residual innovation sequence, and recursive least-square estimation part, which estimates the inertia force between cantilever and moving mass.

The Kalman filter part is given as:

The recursive least-square estimation part is given as:

where $\overline{\mathbf{X}}(k/k-1)$ is the state prediction without considering the system input $p\left(k\right)$, $\overline{\mathbf{X}}(k/k)$ is the state updated estimation, $\mathbf{P}(k/k-1)$ is the covariance of state prediction, $\mathbf{P}(k/k)$ is the updated state covariance, $\mathbf{K}\left(k\right)$, $\mathbf{S}\left(k\right)$, and $\overline{\mathbf{Z}}\left(k\right)$ are the Kalman gain, covariance and innovation, respectively, ${\mathbf{K}}_b\left(k\right)$ is the gain of recursive least-square algorithm, ${\mathbf{P}}_b\left(k\right)$ is the error covariance of the estimation of inertia force, $\overline{\mathbf{B}}\left(k\right)$ and $\mathbf{M}\left(k\right)$ are the sensitivity matrices. What’s more, there is a scalar parameter $\gamma$, which is between 0 and 1, can adjust the performance of the algorithm [8]. When $\gamma =$1, the algorithm becomes to be the regular sequential least-square algorithm. For $0<\gamma <1$, it can affect the ability of noise elimination and fast adaption to the signal of the algorithm. If $\gamma$ is close to 1, it has a very strong ability to estimate the inertia force from the data affected by the noise, however it will lose its fast adaptive capability. If $\gamma$ is close to 0, it can track the fast-varying signal but the identification results will strongly affected by the noise. Therefore, we need a compromising between the fast adaptive capability and the noise elimination capability. The value of $\gamma$ should be decided carefully based on the noise the signals have and adaptive capability we need.

The detailed derivation of this estimation algorithm can be found in Tuan’s work [8].

The identification procedure of inertia force between cantilever and moving mass by using the recursive inverse method are illustrated as bellow:

(i) Measure the vibration data of the cantilever, transform it to the modal response and form the observation vector $\mathbf{Z}\left(k\right)$.

(ii) Obtain the innovation covariance $\mathbf{S}\left(k\right)$, Kalman gain $\mathbf{K}\left(k\right)$, innovation $\overline{\mathbf{Z}}\left(k\right)$ by using the Kalman filter.

(iii) Calculate the inertia force $p\left(k\right)$ by substituting $\mathbf{K}\left(k\right)$, $\mathbf{S}\left(k\right)$, and $\overline{\mathbf{Z}}\left(k\right)$ into the recursive least-square algorithm.

3.1. None-noise data simulation

As it shown in the previous study [11], the recursive inverse method can estimate the inertia force between the cantilever and moving mass very accurately in the none-noise simulation environment. The effectiveness of the recursive inverse method has been tested in three situations with different velocities of moving mass: $c=$ 5 m/s, 10 m/s and 20 m/s. The RPE value of three tests is shown in Table 1 and the identification results are shown in Fig. 2 4.

As it shown in the Table 1, even though there is a very short delay and some jitters at the beginning of the identification process, the algorithm can estimate the inertia force based on the none-noise deflection data very accurately.

Fig. 2The identification result (c= 5 m/s)

Fig. 3The identification result (c= 10 m/s)

Fig. 4The identification result (c= 20 m/s)

3.2. Noise data simulation

As it mentioned above, the recursive inverse method has a good performance in dealing with the none-noise deflection data of the cantilever beam. However, the deflection data obtained from the field experiment will be affected by the measurement noise and therefore, the capability of reducing the effect of noise is very important for the algorithm.

In order to test the recursive inverse method’s capability of dealing with the noise data, Gaussian noise is added to the none-noise simulation deflection data to imitate the data obtained from the sensors in field experiments. Modified IMII method proposed by Minzhuo Wang [5] is also introduced here to provide contrast.

The recursive inverse method is adapted in two different situations with different measurement noise intensity: the standard deviations of measurement noise are ${\sigma }_1=\mathrm{}$10^-7 and ${\sigma }_2=\mathrm{}$10^-6, which match the measurement noise of two kinds of sensors in our laboratory.

The identification results are shown in Fig. 5-6 and RPE are listed in Table 2.

As it shown in the Fig. 5-6, the recursive inverse method has a very strong ability to identify the inertia force between the cantilever and moving mass in the strong noise environment, where the IMII method fails. At the initial part of the identification process, the results is very unstable because the deflection is very small when the mass has just moved onto the cantilever and the noise data will “submerge” the true deflection data. What’s more, the algorithm itself has not converged yet and some parameters are not stable, either. As the mass moves forward, the deflection of cantilever becomes more severe and the algorithm soon converge, which make the recursive inverse method can estimate the inertia force very accuracy.

Fig. 5The identification result (c= 10 m/s, σ= 10-7)

Fig. 6The identification result (c= 10 m/s, σ= 10-6)

4.1. Experiment setup

As it shown in Fig. 7, a set of experimental apparatus consists of cantilever and moving mass, whose parameters are the same as mentioned in section 3, are build up. A nitrogen propulsion unit is designed to provide the initial impact to the moving mass. Ten laser displacement sensors (Keyence, LK-G400) are evenly placed under the cantilever beam that the length between each of two sensors is 150 mm. The measurement noise of the laser displacement sensor is tested before the experiment and the standard deviation of its measurement noise is approximately equal to 10^-7. Data acquisition instrument (Dewetron 1201) is used to provide the sync signals to the laser displacement sensors and record the deflection data of cantilever.

The contact surfaces between the cantilever beam and moving mass are manufactured very smooth and the lubricating oil is added between the surfaces, therefore, the moving damp can be neglected and the velocity of the moving mass can be assumed to be constant. In fact, in the prep experiments, a high speed camera (IDT-Y3) is set up to record the motion process of the mass and a motion analysis software (ProAnalyst) is used to calculate the speed of the moving mass. The results show that the decrease amount of the speed is very small and therefore the model assumption that the damp can be neglected is appropriate.

As soon as the mass moves on the cantilever, the laser displacement sensors begin to record the displacement of the cantilever and a high speed camera (not be shown in Fig. 8), which is synchronous with the data acquisition device, is set at the fixed end of the cantilever to record the exact moment the mass move out the fixed end of the cantilever beam which can be seen as the start signal of the data recording.

Fig. 7The experiment schematic diagram

4.2. Experiment results

Three experiments with different mass velocity, $c=$10 m/s, 15 m/s and 20 m/s, are conducted and the identification result of the inertia force of each experiment is shown in Fig. 8-10.

Fig. 8The experiment identification result (c= 10 m/s)

Fig. 9The experiment identification result (c= 15 m/s)

Fig. 10The experiment identification result (c= 20 m/s)

As it can be seen in the Fig. 8-10, the initial part of the identification process is very unstable, even much more unstable than the noise-data simulation results and it is obvious not caused by the signal noise or instability of the algorithm. After checking the deflection data, it is found that when mass has just moved on the cantilever, it will bring an impact to the structure and even when it pass the fixed end, the impact wave can’t vanish in such short time and it will interfere the deflection of cantilever beam. After the impact wave vanish in a short period, the recursive inverse method can identify the inertia force between cantilever and moving mass very accurately. The RPE of the stable part of each identification result is listed in Table 3.