Accurate Determination of the Frequency Response Function of Submerged and Confined Structures by Using PZT-Patches†

Abstract

To accurately determine the dynamic response of a structure is of relevant interest in many engineering applications. Particularly, it is of paramount importance to determine the Frequency Response Function (FRF) for structures subjected to dynamic loads in order to avoid resonance and fatigue problems that can drastically reduce their useful life. One challenging case is the experimental determination of the FRF of submerged and confined structures, such as hydraulic turbines, which are greatly affected by dynamic problems as reported in many cases in the past. The utilization of classical and calibrated exciters such as instrumented hammers or shakers to determine the FRF in such structures can be very complex due to the confinement of the structure and because their use can disturb the boundary conditions affecting the experimental results. For such cases, Piezoelectric Patches (PZTs), which are very light, thin and small, could be a very good option. Nevertheless, the main drawback of these exciters is that the calibration as dynamic force transducers (relationship voltage/force) has not been successfully obtained in the past. Therefore, in this paper, a method to accurately determine the FRF of submerged and confined structures by using PZTs is developed and validated. The method consists of experimentally determining some characteristic parameters that define the FRF, with an uncalibrated PZT exciting the structure. These parameters, which have been experimentally determined, are then introduced in a validated numerical model of the tested structure. In this way, the FRF of the structure can be estimated with good accuracy. With respect to previous studies, where only the natural frequencies and mode shapes were considered, this paper discuss and experimentally proves the best excitation characteristic to obtain also the damping ratios and proposes a procedure to fully determine the FRF. The method proposed here has been validated for the structure vibrating in air comparing the FRF experimentally obtained with a calibrated exciter (impact Hammer) and the FRF obtained with the described method. Finally, the same methodology has been applied for the structure submerged and close to a rigid wall, where it is extremely important to not modify the boundary conditions for an accurate determination of the FRF. As experimentally shown in this paper, in such cases, the use of PZTs combined with the proposed methodology gives much more accurate estimations of the FRF than other calibrated exciters typically used for the same purpose. Therefore, the validated methodology proposed in this paper can be used to obtain the FRF of a generic submerged and confined structure, without a previous calibration of the PZT.

Keywords: PZT actuators; exciters; modal analysis; submerged structures.

Figure 1

(a) Tested structure with installed PZT; and (b) installed Accelerometer (back side of the disk).

Figure 2

Equipment used.

Figure 3

Time signals: (a) chirp excitation (PZT) and response; and (b) hammer excitation and response.

Figure 4

Sweep excitation (PZT): excitation and response.

Figure 5

Tested Structure (disk) submerged in water with a nearby rigid wall.

Figure 6

Obtaining the FRF: (a) One average of the frequency signals of Accelerometer and Hammer after applying the FFT; and (b) FRF estimated with 5 averages (impacts) and coherence.

Figure 7

Comparison of the FRF obtained with the Hammer and the transfer function ((m/s²)/V) obtained with the PZT (chirp excitation).

Figure 8

(a) First Mode Shape of the analyzed structure with measured points; (b) Representation of $H\left(j{\omega }_r\right)$ for the points with maximal deformation (Hammer and PZT).

Figure 9

Damping Ratio estimation with the half power method for the FRF and for the transfer function ((m/s²)/V) obtained with the PZT (chirp excitation).

Figure 10

(a) Comparison PZT-Transfer function within Sweep and Chirp; and (b) detailed Time–Frequency representation of the peak hold method analysis for the sweep excitation.

Figure 11

Flowchart of the proposed method to estimate the FRF using PZTs and a numerical simulation model.

Figure 12

(a) Simulation model including the position of the force and the measurement point; and (b) application of the harmonic response compared to experimental result for the 3rd mode.

Figure 13

Comparison FRF obtained with Hammer and FRF estimated with the PZT combined with the proposed method. Structure suspended in air.

Figure 14

Simulation model used: (a) Disk with “infinite” water medium; and (b) disk close to rigid wall (25 mm).

Figure 15

(a) Comparison PZT transfer function with FRF; and (b) comparison FRF estimated with the PZT and FRF obtained with the Hammer. Structure with infinite water medium.

Figure 16

Natural Frequency and Damping estimation of the fourth mode of the disk submerged in water close to a rigid wall.

Figure 17

(a) Comparison PZT transfer function with FRF; and (b) comparison FRF estimated with the PZT and FRF obtained with the Hammer. The Structure is close to a rigid wall.

Figure 18

Comparison modal parameters for the four analyzed modes. Ratio modal parameter estimated with the Hammer against modal parameter estimated with the proposed method (using PZTs: (a) Natural Frequency; (b) Damping Factor; and (c) Amplitude FRF.

Figure 19

Example of a submerged-confined structure (reversible Francis turbine).