#### Presentation Title

Vibration Control of Euler-Bernoulli Beams

#### Faculty Mentor

Xiaosong Li

#### Start Date

23-11-2019 8:45 AM

#### End Date

23-11-2019 9:30 AM

#### Location

222

#### Session

poster 2

#### Type of Presentation

Poster

#### Subject Area

physical_mathematical_sciences

#### Abstract

In this project, we study an Euler-Bernoulli beam with one end pinned and the other end fixed, which can be used to model the transverse vibrations of conveyor belts, satellite antennas, bridges, trackbeds, etc. caused by outside forces or high translation speeds. Our goal is to seek effective controls to suppress the deleterious vibration. For this goal, we compare two control methods. One is a pointwise control force modeled by a spring-mass-dashpot subsystem that is located at any position within the span of the beam. The other method puts linear feedback controls on the second and third derivatives of deflection at one end.

We solved both systems analytically using the Fourier Transformation and expanded the eigenvalues asymptotically. For the first method, we found that if the ratio of the distance from the location of the pointwise control force to the left end of the beam over the length of the whole beam is a rational number of the form (4m+2)/(4n+1) for some positive integers m and n, then the model achieves the maximum stability margin. For the second method, the parameter for the linear control of the second derivative of deflection has to be negative and the parameter for the linear control of the third derivative of deflection has to be positive. The larger the magnitudes of the parameters, the better the results. We also provided numerical results to support our conclusion.

The method developed can help to predict stability margins and optimize parameters for controlled beams with other types of boundary conditions and controller structures.

Vibration Control of Euler-Bernoulli Beams

222

In this project, we study an Euler-Bernoulli beam with one end pinned and the other end fixed, which can be used to model the transverse vibrations of conveyor belts, satellite antennas, bridges, trackbeds, etc. caused by outside forces or high translation speeds. Our goal is to seek effective controls to suppress the deleterious vibration. For this goal, we compare two control methods. One is a pointwise control force modeled by a spring-mass-dashpot subsystem that is located at any position within the span of the beam. The other method puts linear feedback controls on the second and third derivatives of deflection at one end.

We solved both systems analytically using the Fourier Transformation and expanded the eigenvalues asymptotically. For the first method, we found that if the ratio of the distance from the location of the pointwise control force to the left end of the beam over the length of the whole beam is a rational number of the form (4m+2)/(4n+1) for some positive integers m and n, then the model achieves the maximum stability margin. For the second method, the parameter for the linear control of the second derivative of deflection has to be negative and the parameter for the linear control of the third derivative of deflection has to be positive. The larger the magnitudes of the parameters, the better the results. We also provided numerical results to support our conclusion.

The method developed can help to predict stability margins and optimize parameters for controlled beams with other types of boundary conditions and controller structures.