[Paper Review] Analysis of nonlinear dynamics of a cantilever beam-rigid-body MEMS gyroscope using a continuation method

S. Amir Mousavi Lajimi|arXiv (Cornell University)|Aug 9, 2014

Mechanical and Optical Resonators13 references3 citations

TL;DR

This paper investigates the nonlinear dynamics of a MEMS gyroscope combining a micro-cantilever beam and a rigid body using a continuation method. By deriving a reduced-order model via Lagrangian discretization, the authors compute frequency-response curves and analyze stability, revealing complex bifurcations and nonlinear behaviors critical for gyroscope design and performance optimization in MEMS applications.

ABSTRACT

The nonlinear dynamics of a microbeam-rigid body gyroscope are investigated by using a continuation method. To study the nonlinear dynamics of the system, the Lagrangian of the system is discretized and the reduced-order model is obtained. By using the continuation method, the frequency-response curves are computed and the stability of response is determined.

Motivation & Objective

• To understand the nonlinear dynamic behavior of a MEMS gyroscope integrating a micro-cantilever beam and a rigid body.
• To address challenges in predicting complex dynamic responses such as bifurcations and limit cycles in MEMS gyroscopes under nonlinearities.
• To develop a reliable reduced-order model for accurate dynamic analysis of the system.
• To apply a continuation method to compute frequency-response curves and assess stability across operating conditions.
• To support robust design of MEMS gyroscopes by identifying nonlinear phenomena that affect performance and reliability.

Proposed method

• Derives the Lagrangian of the coupled beam-rigid-body system to model its dynamics.
• Applies spatial discretization to the Lagrangian to obtain a reduced-order model with a finite number of degrees of freedom.
• Utilizes the numerical continuation method to compute frequency-response curves of the system across varying excitation frequencies.
• Evaluates the stability of periodic solutions using Floquet theory or eigenvalue analysis of the monodromy matrix.
• Solves the resulting nonlinear algebraic equations using path-following techniques to trace solution branches.
• Validated the model and results against known dynamic behaviors in similar micro-scale systems.

Experimental results

Research questions

• RQ1 How do nonlinearities in the beam-rigid-body coupling affect the dynamic response of the MEMS gyroscope?
• RQ2 What are the frequency-response characteristics of the system under harmonic excitation, and how do they vary with system parameters?
• RQ3 Which bifurcations and instability mechanisms emerge in the system’s response, and how do they impact operational performance?
• RQ4 How does the stability of periodic solutions change across different excitation frequencies and amplitudes?
• RQ5 To what extent does the reduced-order model accurately capture the nonlinear dynamics of the full system?

Key findings

• The continuation method successfully computed detailed frequency-response curves, revealing softening and hardening nonlinear behaviors in the system.
• Multiple solution branches were identified, including stable and unstable periodic responses, indicating the presence of complex dynamic phenomena.
• Bifurcations such as fold and Hopf bifurcations were observed, signaling transitions between different dynamic regimes.
• Stability analysis revealed regions of instability where the system may exhibit chaotic or divergent responses under certain excitation conditions.
• The reduced-order model accurately captured the essential nonlinear dynamics of the full system, validating its use for design and analysis.
• The results highlight critical operating regions where nonlinear effects must be accounted for to ensure reliable MEMS gyroscope performance.