Free Vibration and Sound Radiation of Bistable Clamped-Clamped Beams: An Experimental Study

A bistable buckled beam has two symmetric equilibrium positions which can be switched from one to the other under an external actuation. The switching process, known as snap-through, induces elastic energy exchange as well as post-buckling vibrations. These attributes have made bistable buckled beams useful in a wide range of applications, such as actuators and switches, resonators, energy harvesters, and vibration reduction. All these applications rely on a solid understanding of the buckling process and the post-buckling behavior. This study aims to investigate the free vibration of bistable clamped-clamped beams, focusing on the snap-through process and the induced free vibration.

First, an analytical model we previously developed is reformulated in a dimensionless form to predict the natural frequencies and mode shapes of post-buckling beams. We find from the first six modes that there are two groups of modes with different characteristics: symmetric modes and anti-symmetric modes. As the central static deflection increases, the natural frequencies of the anti-symmetric vibrational modes stay constant. The corresponding vibrational mode shapes stay unchanged, too. As for the symmetric modes, the natural frequencies increase monotonically to asymptotic values. The corresponding vibrational mode shapes also change gradually showing more nodes and anti-nodes.

We then carry out an experimental investigation into the snap-through process, with a special focus on the free vibration after the jump phenomenon. The experimental setup consists of four main components: (1) a clamped-clamped beam, (2) an actuator to switch the beam, (3) two laser displacement sensors: one as a reference sensor pointed at a fixed location (the center), and the other as a scanning sensor moving across the beam, and (4) a microphone to measure the generated sound. The idea is to consistently initiate the snap-through while using the scanning sensor to measure the deflection of one point on the beam at a time, and then align the measurements using the signal from the reference sensor.

We study the snap-through process and the post-buckling vibration from the measurement the laser sensors. We first get the 3D chronology of the beam through the whole snap-through process. The beam changes quasi-statically under the thrust of the actuator, but automatically jumps to the inverted stable buckling configuration once it passes a critical location. Afterwards, the beam vibrates around the buckling configuration until settling down. Then we investigate the post-buckling vibrations. It is found that the snap-through of the buckling beam provides an impulse excitation that causes the beam to vibrate. Multiple modes of the beam are excited and different modes contribute differently. In addition, the experimental vibrational mode shape is constructed by combining the amplitudes of the frequency response at the interested natural frequency of all the points on the beam.

Finally, we investigate the sound generated from the buckled beam as an indirect method to investigate the post-buckling vibrations. The multiple peak frequencies identified in the sound spectrum agree well with the analytical model and laser displacement sensor measurements. Due to the dipole nature of the radiation, we also find that the sound amplitude depends on the azimuth angle. As far as we know, this is the first study to study the sound radiation of buckled beam in the context of snap-through and relate to the free vibration.

In summary, we conduct our study to investigate the snap-through process of a bistable clamp-clamp beam. It shines a new light on a classical problem by directly measuring the whole beam's response in the whole process and verifying its response via the generated sound.