Scholarlink

Purpose <p>To develop a compact analytical framework for certified frequency margins and parameter sensitivities in clamped-free Euler-Bernoulli cantilevers with realistic tip devices, namely a translational tip mass, a tip rotary inertia, and a rotational spring, and to clarify their conservative relationship with corresponding Timoshenko frequencies.</p> Methods <p>The analysis is formulated variationally on a clamped <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^2\)</EquationSource> </InlineEquation> space and uses the Courant-Fischer min-max characterization of the spectrum. Three explicit one-dimensional trace/Poincaré inequalities convert the Rayleigh quotient into closed-form certified lower bounds. Exact first-order sensitivity identities are derived within the same framework. For selected low modes, a compact Rayleigh-Ritz surrogate provides certified upper bounds.</p> Results <p>Explicit certified lower bounds are obtained for all modes and all admissible device values. Exact first-order sensitivities with respect to tip mass, tip rotary inertia, and rotational spring yield practical tolerance-to-frequency margin estimates. For the ideal cantilever, a universal lower bound and exponential-in-mode estimates for the slope of the classical characteristic function quantify conditioning and increasing robustness of higher modes. Under identical end devices, Timoshenko eigenfrequencies satisfy <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \omega_{n,T} \le \omega_{n,EB}\)</EquationSource> </InlineEquation>. For selected low modes, the Rayleigh-Ritz surrogate provides certified upper bounds that, paired with the global lower bound, yield two-sided certified brackets. Numerical examples, including a reed-scale structural case, show tight percent-level certified brackets.</p> Conclusion <p>The proposed framework provides practical certified structural frequency bounds, sensitivities, and two-sided brackets for cantilever beams with realistic tip devices, supporting early sizing, Quality Assurance, and audit-ready assessment.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Certified Eigenfrequency Bounds for Euler–Bernoulli and Timoshenko Cantilever Beams with Tip Mass, Rotary Inertia, and a Rotational Spring

  • 期刊论文

  • 发表日期：

    2026年5月14日

查看全文
  • Volpatti Giovanni

摘要

Purpose

To develop a compact analytical framework for certified frequency margins and parameter sensitivities in clamped-free Euler-Bernoulli cantilevers with realistic tip devices, namely a translational tip mass, a tip rotary inertia, and a rotational spring, and to clarify their conservative relationship with corresponding Timoshenko frequencies.

Methods

The analysis is formulated variationally on a clamped \(H^2\) space and uses the Courant-Fischer min-max characterization of the spectrum. Three explicit one-dimensional trace/Poincaré inequalities convert the Rayleigh quotient into closed-form certified lower bounds. Exact first-order sensitivity identities are derived within the same framework. For selected low modes, a compact Rayleigh-Ritz surrogate provides certified upper bounds.

Results

Explicit certified lower bounds are obtained for all modes and all admissible device values. Exact first-order sensitivities with respect to tip mass, tip rotary inertia, and rotational spring yield practical tolerance-to-frequency margin estimates. For the ideal cantilever, a universal lower bound and exponential-in-mode estimates for the slope of the classical characteristic function quantify conditioning and increasing robustness of higher modes. Under identical end devices, Timoshenko eigenfrequencies satisfy \( \omega_{n,T} \le \omega_{n,EB}\) . For selected low modes, the Rayleigh-Ritz surrogate provides certified upper bounds that, paired with the global lower bound, yield two-sided certified brackets. Numerical examples, including a reed-scale structural case, show tight percent-level certified brackets.

Conclusion

The proposed framework provides practical certified structural frequency bounds, sensitivities, and two-sided brackets for cantilever beams with realistic tip devices, supporting early sizing, Quality Assurance, and audit-ready assessment.