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Free in-plane vibration analysis of a rectangular plate with an edge V-notch

This paper presents an enriched Ritz method utilizing corner functions derived from Williams's asymptotic solution to accurately analyze the free in-plane vibration of rectangular plates with edge V-notches, demonstrating that notch depth is the dominant geometric parameter influencing modal characteristics.

Original authors: Xutao Sun

Published 2026-08-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xutao Sun

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Flat metal sheets form the backbone of countless structures, from the wings of airplanes to the hulls of ships and the floors of buildings. When these sheets vibrate, they do not just bend up and down; they also stretch and shear within their own flat plane. These in-plane vibrations, though often occurring at higher pitches than the familiar bending sounds, are critical for understanding how energy travels through a structure and how noise radiates into the environment. Engineers have long known that sharp cuts or notches along the edges of these plates—created by machining, assembly, or accidental damage—can drastically alter how the metal behaves. At the very tip of such a cut, the material experiences a severe concentration of stress, a phenomenon that makes the physics near the corner behave very differently from the rest of the sheet. While scientists have developed sophisticated ways to predict how plates vibrate when they are perfectly smooth, the specific challenge of calculating the vibrations of a plate with a sharp, V-shaped notch has remained largely unsolved, particularly for these in-plane movements.

A researcher at the University of Waikato has now filled this gap by developing a new mathematical tool to predict exactly how a rectangular plate with an edge V-notch will vibrate. The study focuses on a method that combines a broad, global view of the plate's movement with a highly specialized, local view of the sharp corner. Imagine the plate as a whole as a large, flexible sheet that can be described by a standard set of wave patterns. However, near the sharp V-cut, the material behaves in a unique, singular way that standard patterns cannot capture accurately. To solve this, the researcher re-derived a specific set of mathematical functions, originally developed for different types of stress problems, to describe the intense, singular motion right at the tip of the notch. By weaving these special corner functions into the standard description of the plate, the new method creates a hybrid model that is both efficient and incredibly precise.

The team tested this approach by simulating a square plate with a vertical V-notch cut into the center of one edge. They compared their results against complex computer simulations used in industry, known as finite element analysis, which are considered the gold standard but require massive computing power to get right. The new method proved to be highly accurate, matching the computer simulations to within three significant figures while converging much faster. The study revealed that the sharpness of the notch dictates how much this special corner treatment is needed. For a wide, gentle cut, standard calculations work reasonably well, but as the notch becomes sharper, the standard methods fail to capture the true physics, and the special corner functions become essential. Without them, the predicted vibration frequencies can be significantly wrong, especially for the most sensitive modes of vibration.

One of the most striking findings concerns the geometry of the notch itself. The researchers discovered that the depth of the cut is the dominant factor controlling the plate's vibration, far outweighing the influence of the angle of the cut. When the notch depth was increased from a shallow cut to one that reached halfway across the plate, the lowest vibration frequency dropped by approximately 70 to 73 percent. This massive reduction indicates that a deep notch makes the structure significantly more flexible in its in-plane movements. In contrast, changing the angle of the V-cut from a very sharp five degrees to a wider thirty degrees had a much smaller effect, altering the frequencies by less than one percent for shallow cuts and only becoming noticeable for the deepest cuts. Furthermore, the presence of the notch broke the natural symmetry of the square plate. In a perfect square, certain vibration modes occur at the exact same frequency, but the notch caused these paired frequencies to split, with one dropping significantly while the other remained relatively stable, effectively changing the order in which the plate vibrates.

The study also provided a detailed look at how the shape of the vibration changes. While the overall patterns of movement remained recognizable, the deep notch introduced significant relative motion along the two faces of the cut, creating a distinct local behavior that the rest of the plate did not share. The research confirms that for sharp notches, ignoring the singular stress field at the tip leads to poor predictions, whereas including the specialized corner functions allows for a rapid and accurate solution. The paper concludes by offering a set of benchmark results for a free square plate with a vertical V-notch, providing a reliable reference for future engineers and scientists. These findings establish that while the depth of a cut is the primary driver of change in a plate's vibrational character, the specific angle of that cut plays a secondary, yet measurable, role, and that accurate prediction requires a mathematical approach that respects the unique, singular physics of the sharp corner.

Technical Summary: Free In-Plane Vibration Analysis of a Rectangular Plate with an Edge V-Notch

Problem Statement
While the transverse vibration of plates is well-studied, the free in-plane vibration of rectangular plates containing edge V-notches remains largely unaddressed. In-plane vibrations are critical for high-frequency energy transmission in built-up structures and influence noise radiation and dynamic responses at structural junctions. Edge V-notches, resulting from machining, cutouts, or accidental damage, introduce strong stress singularities at their vertices. These singularities significantly alter the membrane load path and, consequently, the natural frequencies and mode shapes of the plate. The primary challenge in solving this problem lies in the coupled nature of the two in-plane displacement components and the difficulty of incorporating the asymptotic displacement field near the notch vertex into global solution methods.

Methodology
The paper proposes an enriched Ritz method specifically tailored for the coupled in-plane displacement field of a rectangular plate with an edge V-notch. The formulation involves the following key components:

Displacement Approximation: The in-plane displacement components ( $u$ and $v$ ) are approximated using a global trigonometric-polynomial basis. This regular basis is augmented with symmetric and antisymmetric corner functions. These corner functions are re-derived from Williams's asymptotic solution for a traction-free angular corner in extension, specifically adapted for the coupled in-plane field.
Boundary Conditions: Essential boundary conditions (clamped, free, and two types of simply supported) are enforced strongly through separate boundary multipliers for each displacement component.
Energy Formulation: The membrane strain energy and kinetic energy are integrated over the physical notched domain (excluding the triangular notch region). High-order triangular quadrature (degree-19 Gaussian quadrature) is employed to evaluate these integrals, with local mesh refinement near the notch vertex to accurately resolve the singular behavior of the corner functions.
Eigenvalue Problem: The stationarity condition of the total energy functional leads to a generalized eigenvalue problem $(\mathbf{K} - \omega^2\mathbf{M})\mathbf{g} = 0$ , where the stiffness and mass matrices are partitioned to separate regular and corner-enriched contributions.

Key Contributions

Novel Formulation: This work fills a gap in the literature by developing the first enriched Ritz method for the free in-plane vibration of V-notched plates. Unlike previous applications of Williams-type enrichment to transverse vibration, the corner functions here are explicitly re-derived for the coupled in-plane displacement field.
Convergence and Accuracy: Extensive convergence studies demonstrate that the method reproduces published solutions for intact plates with monotonic upper-bound convergence to at least four significant figures. For V-notched plates, where no benchmark solutions exist, the results converge from above to at least three significant figures and show close agreement with finite element analysis (ANSYS).
Role of Enrichment: The study quantifies the necessity of corner enrichment. It finds that while regular functions can represent mild geometric discontinuities, the inclusion of singular corner functions is crucial for sharp notches. The leading roots of the Williams eigenvalue problem (where $0 < \lambda < 1$ ) provide the most significant corrections to frequency parameters, with higher-order roots offering diminishing returns.
Benchmark Data: The paper presents new benchmark frequency parameters and mode shapes for a free square plate with a centrally located vertical V-notch, covering various notch depths and opening angles.

Results and Discussion
The numerical results for a free square plate with a vertical V-notch reveal several key behaviors:

Notch Depth vs. Angle: Notch depth is the dominant geometric parameter affecting natural frequencies. Increasing the depth ratio ( $d/b$ ) from 0.1 to 0.5 reduces the first frequency parameter by approximately 70–73%. The notch opening angle ( $\gamma$ ) has a secondary effect, becoming more pronounced only as the notch deepens.
Mode Splitting: The introduction of a V-notch breaks the rotational symmetry of the intact square plate, causing repeated eigenvalues to split. This alters the ordering of modes; for instance, a mode that was degenerate in the intact plate may become the fundamental mode of the notched plate.
Mode Sensitivity: The influence of the notch is highly mode-dependent. Some modes are significantly affected by the notch compliance, while others remain relatively insensitive.
Convergence Behavior: For sharp notches (e.g., $\gamma = 5^\circ$ ), the unenriched regular basis converges extremely slowly or fails to capture the correct frequency without corner functions. The enriched method achieves rapid convergence, often requiring only the first singular root from each symmetry family to reach stable results.

Significance
The paper establishes that the enriched Ritz method is an accurate and computationally efficient tool for predicting the in-plane dynamic characteristics of V-notched plates. By providing benchmark results and demonstrating the necessity of singular field enrichment for sharp notches, the work offers a validated framework for future analytical and numerical studies. The author notes that the method preserves the upper-bound characteristic of the Ritz method and can be extended to more complex configurations, such as multiple notches, asymmetric locations, or laminated composite plates, in future research.

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