# Vibration Analysis and Empirical Law Definition for Different Equipment in a Civil Construction
## 1 Introduction
### Literature Review  
Vibration issues can be generated by the operation of construction equipment, blasting and traffic traveling. The potential damage to constructions includes superficial and structural damage [15], caused by excavation and demolition activities or earth movement. Besides, it can also affect sensitive equipment, i.e., microelectronic manufacturing equipment, which is very sensitive to ground vibrations and individuals close to the vibration source [16; 17; 18]. Usually, the vibration amplitude of traffic is not high enough to cause damage to a construction due to suspension systems and pneumatic tires [19]. On the other hand, railways and trains can also have a significant impact on vibrations, making it necessary to apply countermeasures [20]. The concepts of particle displacement, velocity and acceleration are used to describe ground vibration, being the peak particle velocity the most appropriate variable to assess the potential damage to a construction [14]. Vibration amplitude is described by three components: two horizontal components, transverse and longitudinal, and one vertical component, which generally has the highest amplitude and it is easy to measure [21]. The duration and amplitude of the vibration change depends on the type of operation and equipment, ranging from a high amplitude and short duration to lower and longer characteristics, respectively. The equipment or activities can be classified as continuous, excavation and vibratory compaction equipment, among others, and single impact with or without a high-rate repeated impact vibration, like a jackhammer or other similar breakers. The maximum velocity component for construction vibration, peak particle velocity (PPV), is used as a descriptor of the wave effect. This preference results from the close association of construction vibration with blast vibration monitoring, where particle velocity correlates with the appearance of cracking [2]. The vibration source creates a disturbance that propagates away, being the R-wave the primary concern for foundations close to the surface [2]. According to [PERSON] [22], R-waves account for 67% of the total energy, S- waves for 26% and P-waves for 7% when the exciting force is applied vertically to the propagation direction. The vibratory excitation propagates radially outward, causing a spreading loss as the wave finds an increasing volume, reducing the amplitude of the displacement. The general expression to model the spreading loss is defined in Section 2.3. When the rock mass is highly fractured and deteriorated, or it is a soil, its behavior is not perfectly elastic, having a damping effect, influenced by multiple variables such as the type of material, moisture or frequency of the vibration source. This behavior was defined by [PERSON] et al. [23], also explained in Section 2.3, having two parameters, \(\gamma\) and \(\alpha\), that represent the geometric attenuation coefficient, depending on the wave type, and the attenuation coefficient of the material. [PERSON] [2] and Jones and Stokes Associates [24] gathered the typical values of the attenuation coefficient, however, there is a large possible range in each type of material. On the other hand, [PERSON] [25] proposed an interesting approach to forecast the maximum particle velocity on the basis of blasting design and rock parameters, detailing an analytical approach to support the empirical expressions used. The dependence on frequency caused by construction equipment is commonly considered as weak. It is often assumed independent from frequency [26]. Therefore, the greatest concern is related to the distance from the vibration source. [PERSON] [27] and [PERSON] [21] gathered many of the damping coefficients depending on the type of material. Besides, [PERSON] [27] mentions that it is a proper methodology to determine the impact to structures and people. However, most of the information is for soils and not for rocks.