Oscillations in mechanical systems reflect the stress-strain states (SSS) resulting from external and internal factors that may result in displacement or deformation of a critical part.

We are able to assess the degree to which such factors influence the critical object’s performance through vibration and oscillation parameters.

**However, widely-spread amplitude analysis is a simple approximation of an average SSS assessment and therefore does not produce full information about the object.**

In reality, each point of the observed object oscillates along a hodograph (elliptic trajectory) at each frequency of the observed spectrum (superposition) of oscillations, in full accordance with the *classical mechanics laws for elastic continuums*. This also refers to stress (Lamé theorem) and strain (Cauchy stress tensor).

Stress reflects the *kinetic energy* of impacts whereas strain – the *potential energy* of the resistance of the elastic continuum to these impacts. This holds true for the full spectrum of observed frequencies.

The sum total of kinetic (**Е**** _{kin}**) and potential (

**Е**

**) energy determines the perturbed level of the object (**

_{pot}**Е**

**) which must be evaluated (to be subject to diagnostics from the standpoint of**

_{Σ}*operational danger*) and eliminated. This is why vibration diagnostics and vibration regulation (dynamic balancing) are performed.

By measuring the time-space (vector-phase) diagnostic 3D parameters of oscillations in the form of a 4D-hodograph spectrum, one may quantitatively **evaluate the perturbed level through total energy Е**_{Σ} **which is proportionate to the area of the hodograph at each diagnostic frequency**.

A mechanical system which functions properly in compliance with the design has minimum **Е**_{Σ} and any levels in excess of the relative design (proper) level of oscillations are evidence of potential failure, which must be subject to diagnostics. Then statistical levels of critical diagnostic parameters can be known which, if exceeded, pose the risk of failures and accidents.

Download paper 'Wave Phase-Sensitive Transformation of 3d-Straining of Mechanical Fields''