Relative stress

Stress is often normalized to create a relative stress so that the stress-performance of two different resonators can be easily compared. Two methods of normalization are common:

1. Normalizing the stress in a resonator at location \( X \) to the maximum stress in a thin wire when both have the same amplitude u at a specified location \( Y \)  —
   
   

   \begin{align} \label{eq:13701a} \textsf{Relative stress} = \frac{\textsf{Stress in resonator at X (with specified amplitude u at Y)}}{\textsf{Max stress in thin wire (with specified amplitude u at antinode)}} \end{align}

   This relative stress has no units.

2. Normalizing the stress in a resonator at location \( X \) to the amplitude in the same resonator at a specified location \( Y \) —
   
   

   \begin{align} \label{eq:13702a} \textsf{Relative stress} = \frac{\textsf{Stress in resonator at X }}{\textsf{Amplitude u at Y}} \end{align}

   For this normalization, the amplitude u is usually chosen either as —

   • A convenient value that can be easily scaled (e.g., 1 micron peak or 100 microns peak).
   • The actual or expected amplitude of the resonator.

   Typical units are "MPa/micron" [psi/micron].

\( X \) is often the location where the resonator stress is a maximum. \( Y \) is often at a pertinent location on output face (typically the center) since this is where the ultrasonic energy is delivered to the load (although sometimes at the input stud if this is where the amplitude is known).