## 1 Introduction
### Directions of Research Work  
Despite difficulties with precise analytical determination of the relationship between the energy of the source of vibrations and the vibrations of the tested object, work is underway to create mathematical models for vibration analysis and the design of new types of sensors. Long-term measurements in real conditions allow for a more exact assessment of the impact of vibrations transmitted through the ground on buildings. The analysis of online values that were collected or monitored earlier is primarily conducted for the dependable prediction and detection of future problems and failures. In the exemplary vibration monitoring system of a single coal mill, researchers rely on the kernel extreme learning machine model based on feature extraction. The model reduces information redundancy, reduces computing costs, and provides a new idea for the fault diagnosis of a coal mill based on feature extraction [6]. Another approach to vibration monitoring is predicting object vibrations based on the vibration properties of its selected elements. In [30], only the vibration data at certain points of the structure were selected for analysis. A random forest regression model for predicting any location of the structure, amplitude, and frequency of the force was used [30]. During construction works, it seems necessary to obtain reliable values of the predicted ground vibrations. Research shows that, in special conditions, forecasting is possible using the Lagrangian-Eulerian (ALE) adaptive mesh method [28; 31]. These simulations were performed to simulate the pile penetration in the continuous piling process. The term continuous piling describes the piling process from the ground surface to the final penetration depth. The simulations showed that the numerical model could efficiently calculate the velocity of the front of waves of various types in a homogeneous soil [31]. Some scientists go further and try to implement heterogeneous soil in the model. In [32], the finite element method in time domain (FEM-TD) is employed for predicting pile-driving-induced vibrations. A soil model with four layers of different thickness was used in the simulations. Layer thickness and properties were determined using actual measurements. The obtained values of the soil layer properties were used in the model. The results of the simulation were compared with the results of the low-strain test performed with a handheld impact hammer. The results of the simulation and practical test confirmed that the developed numerical model is highly consistent with in situ measurements [32]. Another interesting research direction is developing vibration sensors without an external power source. Papers on mechanical vibration sensors with an integrated energy collection function with a piezoelectric transducer are published. Typical generators for generating mechanical energy are equipped with a movable lever (cantilever) made of piezoelectric material. The piezoelectric lever generator may be a self-powered sensor that detects the state of vibration of the support. With a known resonant frequency of the lever, it is possible to determine the vibrations based on the damping of the signal coming from the lever [33]. The physical feasibility of building such a sensor has been proven, and the current work concerns estimating potential errors [34]. This makes it possible to build self-powered vibration sensors based on a unique phenomenon. Another type of generator is a nano-generator that uses a combination of the triboelectric and electrostatic effects. The source of energy is flexible nanofibers, fabricated by electrospinning with a strong dielectric compound and the relatively negative triboelectric material polyvinylidene fluoride. This type of sensor can be an efficient, self-powered nanofiber vibration sensor (SNVS) for the safety detection of rail fastener tightness [35]. The lack of an external power supply allows for mobile work of the sensor.