Different tools for different analysis

With Linear Analysis, the dynamic behavior of the system is studied through:

• the Eigenvalues of the system, representing the system natural modes. They allow the prediction of the possible resonances of the system occurring at specific frequencies [Hz] (the natural modes), as well as the associated damping ratio [%] of the oscillations: Figure 5: Eigenvalues with frequencies [Hz] and damping ratios [%] - Hydraulic line in closed/closed conditions

• the Modal Shapes, representing the space distribution for a natural mode (i.e. frequency and damping ratio) all along the system. They show the parts of the system that are impacted corresponding to a selected natural mode, and they highlight how these parts oscillate (in phase or out-of-phase). The parts involved are clearly identified and it is then easy to damp the oscillations with appropriate dampers: best design, at best location: Figure 6: Modal Shapes: First two modes of Hydraulic line in closed/closed conditions

• the Transfer Functions, representing the frequency response in gain G [dB] and phase [degrees] of an observer variable (output) linked to the excitation of a control variable (input). They allow predicting the amplitude of the oscillations for various frequencies of excitations. It shows how the natural modes are excited in relation with a selected excitation: Figure 7: Bode Plot - Transfer functions: P_left (output #1) / Q_left (input #1) & P_right (output #2) / Q_left (input #1)

• the Root Locus is a plot (real part/imaginary part), representing the trajectories in frequency [Hz] and damping ratio [%] of the natural modes due to some parameters changes. It is very efficient to reach the best design (for example, calibrating accurately the diameter of an orifice), or to study the stability of the closed-loop controlled systems (for example, stability of a pressure regulator). Figure 8: Root locus & Typical response vs. location of the Eigenvalues

Each of these approaches highlights one aspect of the structural dynamic properties of your system. By combining all these methods of analysis, the engineer gets the full understanding of the dynamic system behavior whatever the time-domain inputs are.