Smart Active Vibration Control System of a Rotary Structure Using Piezoelectric Materials

Abstract

A smart active vibration control (AVC) system containing piezoelectric (PZT) actuators, jointly with a linear quadratic regulator (LQR) controller, is proposed in this article to control transverse deflections of a wind turbine (WT) blade. In order to apply controlling rules to the WT blade, a state-of-the-art semi-analytical solution is developed to obtain WT blade lateral displacement under external loadings. The proposed method maps the WT blade to a Euler-Bernoulli beam under the same conditions to find the blade's vibration and dynamic responses by solving analytical vibration solutions of the Euler-Bernoulli beam. The governing equations of the beam with PZT patches are derived by integrating the PZT transducer vibration equations into the vibration equations of the Euler-Bernoulli beam structure. A finite element model of the WT blade with PZT patches is developed. Next, a unique transfer function matrix is derived by exciting the structures and achieving responses. The beam structure is projected to the blade using the transfer function matrix. The results obtained from the mapping method are compared with the counter of the blade's finite element model. A satisfying agreement is observed between the results. The results showed that the method's accuracy decreased as the sensors' distance from the base of the wind turbine increased. In the designing process of the LQR controller, various weighting factors are used to tune control actions of the AVC system. LQR optimal control gain is obtained by using the state-feedback control law. The PZT actuators are located at the same distance from each other an this effort to prevent neutralizing their actuating effects. The LQR shows significant performance by diminishing the weights on the control input in the cost function. The obtained results indicate that the proposed smart control system efficiently suppresses the vibration peaks along the WT blade and the maximum flap-wise displacement belonging to the tip of the structure is successfully controlled.

Keywords: active vibration control; analytical vibration analysis; piezoelectric sensor-actuator; smart structure; transfer function method.

Figure 1

Representative shape of the beam including piezoelectric patches.

Figure 2

Section’s estimated diagram from the blade to the beam.

Figure 3

Flow chart of the total transfer function deriving process.

Figure 4

Block diagram for control.

Figure 5

The outcomes of the analytical results and FEM results for the twentieth sensor under loading number 1 (15 degrees attack angle).

Figure 6

The outcomes of the analytical results and FEM results for the thirty-fifth sensor under loading number 2 (30 degrees attack angle).

Figure 7

Error percentage for each sensor under loading number 1 (15-degree attack angle).

Figure 8

Error percentage for each sensor under loading number 2 (30-degree attack angle).

Figure 9

Error percentage for each sensor under loading number 3 (45-degree attack angle).

Figure 10

The outcomes of applying controlling rule for the fifteenth sensor under loading number 2 (30-degree attack angle).

Figure 11

The outcomes of applying controlling rule for the thirtieth sensor under loading number 2 (30-degree attack angle).

Figure 12

The outcomes of applying controlling rule for the forty-fifth sensor under loading number 2 (30-degree attack angle).

Figure 13

The lateral vibration domain with 2 different Gamma amounts (${\gamma }_1$ = ${10}^{-11}$ and ${\gamma }_2$ = ${10}^{-13}$) for the fifteenth sensor under loading number 2 (30-degree attack angle).

Figure 14

The lateral vibration domain with 2 different Gamma amounts (${\gamma }_1$ = ${10}^{-11}$ and ${\gamma }_2$ = ${10}^{-13}$) for the thirtieth sensor under loading number 2 (30-degree attack angle).

Figure 15

The lateral vibration domain with 2 different Gamma amounts (${\gamma }_1$ = ${10}^{-11}$ and ${\gamma }_2$ = ${10}^{-13}$) for the forty-fifth sensor under loading number 2 (30-degree attack angle).