Since lock erection, no works were performed that could have impacted on its structure or performance. In 2005, works related to renovation were delivered, including repair of concrete surfaces, sealing of expansion joints, delivery of head crown top surfaces and chamber walls, delivery of curbstones, and application of the concrete protective coating. During the following few years after the renovation, some irregularities were evident during gates’ operation, namely, sudden vibrations during water level change in the lock. The so-called sudden movement occurred only sometimes and was an accidental and occasional phenomenon.

#### 2.2. Measurements and Methods

Firstly, static analysis was performed with the numerical model. For this purpose, the 3D model was created with FEM SOFiSTiK. Since the structural geometry of the gates’ leaves and the hydrostatic loading are symmetric about the leaf center line, only one of the gates’ leaves was modeled. Two-dimensional, four-nodal shell elements and one-dimensional beam elements were used for the construction [

39]. The gates’ structural elements were elaborated pursuant to design documentation. However, the following assumptions and simplifications were applied in the model. Supports were taken into account by point constraints or elastic supports (springs). The effect of contact of the side and the bottom gaskets was not considered. Due to the different boundary conditions of the gates during opening (separated gates’ leaves) and after closing (connected gates’ leaves), two different models were established. The first one, named “opened model” (

Figure 5a), provided simulation for open gates and allowed the identification of modal parameters. In this model, the rotational bearings were articulated supports. Because other bearings were not involved in the load transfer, they were omitted and only non-structural masses of those bearings and the upper deck were added to the model. The additional spring was mounted in place of the hydraulic cylinder. Its stiffness was established in the calibration process so that the natural frequencies of the modal analysis were consistent with those obtained from field tests. The second model, named “closed model” (

Figure 5b), provided simulation for the closed gates and was purposed for static analyses. In this model, the rotational bearings were vertical, articulated, sliding supports. In place of the miter and the support bearings, rigid springs were mounted. Loadings to the “closed model” were provided with dead weight and the following water static pressure:

A, maximum top water level (57.30 m above sea level) and bottom water zero level (41.80 m above sea level);

B.1, top water level 1/3 and bottom water level + 1 m;

B.2, top water level increase from 1/3 to 2/3 and bottom water level + 1 m;

B.3, top water level increase from 2/3 to 3/3 and bottom water level + 1 m.

In both models, mesh convergence analysis was established. For the applied load with maximum magnitude, displacements in representative nodes were checked. The maximum dimension of the element was reduced iteratively until the displacements in subsequent steps did not change by more than 3%. In each subsequent step, the element size was reduced twice.

Surveying measurements were based on measurements performed with Leica TS30 (Leica Geosystems AG, Heerbrugg, Switzerland ) (with a precise mode about 0.6 mm + 1 ppm / typ. 7 s) at data recording precision of 0.1 mm.

Figure 6 specifies the locations of control points (from PP3 to PP8), measurement positions (OS1 and OS2), and measuring points (from R1 to R10).

For comprehensive survey and assessment regarding the entire gate deformation, measurements were performed with laser scanning. All measurements were performed using the Faro branded device, Focus 3D X130 (FARO Swiss Holding GmbH, Beringen, Switzerland) (with an accuracy of about 2 mm and a measurement speed up to 1 million points per second). Survey results allowed assessment regarding gates’ element deformation during lock filling. Before measurement, stabilization followed 6 flat marks (

Figure 6a) to ensure proper coordination of the system, and 6 positioning balls (

Figure 7) providing data records at the required accuracy. Due to the gates’ dimensions (12.50 × 16.00 m) and difficultly in accessing the gates, there was a need to take measurements from two different positions (from the left side “OS1” and the right side “OS2”,

Figure 6) when performing the scanning.

Measurements with a tachymeter and a laser scanner were performed at the following 4 different water levels (

Figure 4): (1) level 0/3, (2) level 1/3, (3) level 2/3, and (4) level 3/3. These were static measurements correlated with a constant water level in the lock. Each measuring series was taken for each of the two measuring positions, OS1 and OS2.

Figure 8 includes the laser scanner operation and a cloud of points, along with the RGB (red-green-blue) color code.

In addition, acceleration measurements were performed on the gates’ steel structure. As mentioned, the dynamic measurements had two main purposes, i.e., validation of the numerical FEM model and detecting of the unexpected movements of bearings. The measuring system consisted of the HBM PMX Amplifier (Hottinger Baldwin Messtechnik GmbH, Darmstadt, Germany) system, 4 MEMS (Micro Electro Mechanical System) Direct Current (DC) response accelerometers, and wires. The amplifier was connected by an Ethernet wire to a PC computer. HBM CatmanEasy (Hottinger Baldwin Messtechnik GmbH, Darmstadt, Germany) software installed on the PC was used in the configuration of the system and data acquisition. The used amplifier was designed for, and typically utilized in, industry control systems. However, after some modifications, its parameters allowed researchers to use it in laboratory and field tests. Improvements were carried out by the crew of the Field Test Laboratory of the Technical University of Gdansk. The amplifier was covered with a stiff box to protect it from mechanical damage and adverse weather conditions. Additionally, a battery was inserted into the box. Therefore, after connection with the PC computer (notebook type), it created a stand-alone, portable, 16-channels measuring system, which could be used for up to 8 hours without access to electricity. Another advantage was the elimination of redundant, additional vibrations coming from the electric generator’s work. It was especially significant to use low-noise sensors when low-magnitude vibrations are measured. Thus, the created acquisition system is very universal and convenient for field tests on bridges and other infrastructure constructions. In the system, following MEMS DC response, 4 to 20 mA signal output sensors were used as follows: 2 tri-axial accelerometers TE 4332M3-002 (TE Connectivity Ltd., Schaffhausen, Switzerland) and 2 one-axial accelerometers TE 4312M3-002. The following characteristic parameters of these accelerometers were: range ± 2 g, sensitivity 4.0 mA/g, frequency response 0 to 200 Hz, and natural frequency 700 Hz. The noise measured on the gates’ leaf, if no intentional load was applied, was in the range of ± 0.005 m/s

^{2}. The acquisition sampling rate was established to be 300 Hz. If predicted, interesting natural frequencies of the gates were below 30 Hz, it was evaluated as high enough to ensure the obtainment of all information about vibrations.

Figure 9 shows the instrumentation for the acceleration measurements.

In each of the 6 positions, the gates’ leaf was excited three times to natural vibrations, and then was left until vibrations damped. The measurements were carried out separately for the left leaf and separately for the right, as shown in

Figure 4.

Figure 10 specifies the location of the accelerometers placed on the gates’ steel structure. The same sensors were used for both leaves. Therefore, the letter "L" (left leaf) or "R" (right leaf) was added, as shown in

Figure 10, for channel markings. The gates were excited to vibrations by shaking the bridge railing, by one of the researchers standing on the top of the gates’ leaf. The direction of the shakes was perpendicular to the gates’ surface. The starting frequency of shaking was approximately 4 Hz, and during excitation, it was adopted to the natural frequency. After obtaining acceleration amplitudes close to 1 m/s

^{2}, the shaking stopped. For analysis, when the shaking stopped and vibrations were damped to about 0.5 m/s

^{2} was used as a signal.

During dynamic tests, time-domain free-response data were obtained. There exist numerous techniques for identifying the modal parameters from free vibration signals [

49,

50]. In this case, the ERA (eigensystem realization algorithm) was adopted [

51,

52,

53,

54]. It is an efficient method of modal parameter identification, confirmed in many real-world problems. A basic description of the method is presented below.

We consider the state-space representation of a discrete-time, linear time-invariant system in the form of:

where

$x\left(k\right)$ is the vector of states,

$u\left(k\right)$ is the vector of system inputs, and

$y\left(k\right)$ is the vector of system outputs at the

$k$-th step.

$A,B,C,D$ are the discrete-time state-space matrices. The order of the model is the number of components of the state vector

$x$. For

$x\left(0\right)=0$ and for the unit impulse excitation in all input elements:

the results can be described as follows:

These matrices in the sequence are known as the Markov parameters and describe the pulse response of the system. A system realization is the estimation of the set of matrices $A,B,C$ from the Markov parameters for which the discrete-time model is satisfied. The minimum realization is the solution, where the order of the model is minimal.

In ERA, the first step is the formulation of the block Hankel matrix, composed from the Markov parameters in the form:

The shape of the Hankel matrix is equal $\left(\alpha m\times \beta \right)$, where $m$ is the number of output points (i.e., the number of sensors available for an identification).

The second step is the factorization of

${H}_{0}$ using singular value decomposition (SVD):

where

$R$ and

$S$ are orthonormal matrices and

$\Sigma $is the rectangular matrix (Equation (6)).

where

${\Sigma}_{n}$ is a diagonal matrix with shape

$\left(n\times n\right)$, and

$n$ is the model order. Due to the existence of noise in the acquired data, a minimal realization is obtained by eliminating relatively small singular values along the diagonal of

${\Sigma}_{n}$. Corresponding to eliminated values, rows and columns are removed, and from

$R$ and

$S$ arise

${R}_{n}$ and

${S}_{n}$, respectively.

System matrices, which are minimum realization, can be calculated as:

where

${E}_{m}^{T}=\left[{I}_{m},{0}_{m},\cdots ,{0}_{m}\right]$,

${I}_{m}$ is an identity matrix of order

$m$, and

${0}_{m}$ is a null matrix of order

$m$. The accent

$\left(\widehat{}\right)$ means estimated values as opposed to true values. The modal parameters can be found by solving the eigenvalue problem:

where

$\Psi $ is the matrix of eigenvectors and

$\widehat{\Lambda}$ is the diagonal matrix of corresponding eigenvalues. Eigenvalues of

$\widehat{A}$ are complex conjugates and each pair is associated with a mode of vibration. Before calculation of frequencies, a discrete-time system must be transformed to the corresponding continuous-time system using the relation:

Modal damping rates and damped natural frequencies can be obtained from the real and imaginary parts of the

${\Lambda}_{c}.$ In turn, the mode shapes can be obtained from the column vectors of the matrix

$\widehat{C}\Psi $. The ERA algorithm was implemented in Python. To assess the assumed model order, a stabilization diagram was created. To determine the diagram, the identification process was repeated with different (increasing) model order. If the modes are stable, they should remain constant in most iterations. Additionally, a filtered version of the stabilization diagram was generated [

55]. It can be created using the criteria of evaluation of the solution modes. In our case, MAC (modal assurance criterion [

56]) and MPC (modal phase collinearity [

57]) were employed.