Non-Uniform Euler–Bernoulli Beam Analysis

• Non-Uniform Euler–Bernoulli beams are flexural models with spatially varying properties such as bending stiffness and mass density, capturing discontinuities and singularities.
• Advanced analytical frameworks, including very weak and intrinsic distributional methods, extend classical theories to ensure well-posedness under impulsive and irregular loads.
• Spectral and control techniques, alongside numerical regularization, enable precise structural identification, stabilization, and multi-physics integration in engineering applications.

A non-uniform Euler–Bernoulli beam refers to a flexural beam model in which key physical properties—flexural rigidity (bending stiffness), mass per unit length, and possibly other structural coefficients—are functions of the spatial coordinate, often exhibiting discontinuities or singularities. This class covers both smoothly inhomogeneous beams and beams with distributional property variations such as jumps, cracks, or concentrated devices. Recent research rigorously extends the foundational theory and analytical techniques to include cases with distributional coefficients, singular loads, and boundary control, ensuring well-posedness, stability, and practical applicability across advanced engineering and physical contexts.

1. Governing Equations and Model Generalization

The canonical non-uniform Euler–Bernoulli beam equation on a finite interval $[0, L]$ takes the general form

$$m(x)\,\partial_{t}^2u(x,t) + \partial_x^2\left(A(x)\,\partial_x^2 u(x,t)\right) = q(x,t),$$

where $A(x) = E(x) I(x)$ is the bending stiffness, $m(x) = \rho(x)A(x)$ the line mass density, and $q(x,t)$ the external load, each possibly irregular or distributional in $x$ or $t$ (Blommaert et al., 2021). Boundary conditions may be clamped, simply-supported, free, or mixed. The corresponding operator is fourth order in $x$, with the coefficients encoding non-uniformity and possible singular features.

For static problems with an additional axial force $P(x)$: $(E(x)I(x) w''(x))'' + P(x) w''(x) = q(x).$ Applications include beams with step-changes in section properties, point masses, tip masses, cracks, or devices at the ends (Dias et al., 2020, Chatterjee et al., 23 Jul 2024).

2. Analytical Frameworks: Distributional and Very Weak Solutions

Classical weak-solution theory assumes $A(x), m(x) \in L^\infty$ and smooth enough data. For beams with distributional coefficients (e.g., step changes, delta functions), standard frameworks become ill-posed. Two primary approaches generalize solution theory:

A. Very Weak (Regularized) Solution Framework (Blommaert et al., 2021):

• Coefficients and loads are regularized via convolution with mollifiers.
• A net of regularized problems is solved in suitable spaces (e.g., $C([0,T]; H_0^2)$).
• The "very weak solution" is the net limit, provided it exhibits moderate growth (as quantified by energy-type norms).
• Guarantees of existence, uniqueness (up to negligible nets), and consistency with weak solutions when classical assumptions recover.

B. Intrinsic Distributional Product and ODE Formulation (Dias et al., 2020):

• The intrinsic product (*) extends the pointwise product and Hörmander's product to distributions with coincident singular supports.
• The static beam ODE is reformulated so that interface and singular terms are handled in a mathematically consistent way.
• Well-posedness includes explicit interface jump conditions at singularities, with solutions characterized as piecewise-polynomials plus possible delta contributions.

If the "singularity order" in coefficients exceeds the operator's order, solutions may have delta ("concentrated") components physically corresponding to concentrated curvatures or moments—critical for modeling cracked or prestressed beams.

3. Advanced Spectral and Inverse Methods

For beams with inhomogeneities or discrete features (the "discrete beam" case), spectral and inverse analyses uncover deep algebraic structure:

• The matrix-string formalism (Beals et al., 2020) rewrites the fourth-order beam problem as a coupled second-order system:

$$D^2 \begin{pmatrix} y \ y'' \end{pmatrix} = \lambda M(x) \begin{pmatrix} y \ y'' \end{pmatrix}$$

with $M(x)$ encoding non-uniform density and rigidity.

• The associated operator $T = D_0^{-1} M D_0^{-1}$ is compact and its eigenstructure gives the beam spectrum.
• For "discrete" beams (e.g., beams segmented by mass or stiffness point defects), the spectral data is recovered via non-commutative generalizations of continued fractions (a Stieltjes-type expansion), with segment lengths and masses determined from Hankel-like determinant formulas.
• Isospectral deformations, related to Camassa–Holm-type flows, are constructed so the spectral properties are invariant under certain nonlinear evolutions of property measures.

These operator-theoretic and inverse techniques are essential in structural identification, damage detection, and non-destructive evaluation.

4. Control, Motion Planning, and Stabilization in Non-Uniform Beams

Non-uniform beams with complex boundary features or dynamic control inputs (such as moving supports, tip masses, or delayed feedback) demand robust state-space and control-theoretic frameworks:

• Coupled PDE-ODE models accurately represent configurations with tip-masses and mobile constraints. State-space embeddings are constructed in weighted Sobolev spaces accounting for spatial inhomogeneity (Chatterjee et al., 23 Jul 2024).
• Well-posedness is established by verifying the operator generates a $C_0$-semigroup under compatibility conditions for the input and initial data.
• Flatness-based motion planning, extended via generating-function expansions, gives an explicit, spectrally-free construction of trajectories and controls between arbitrary initial and final states in a reachable submanifold. These series converge rapidly, as verified in experimental setups for non-uniform beams with tip-masses (Chatterjee et al., 23 Jul 2024).
• For beams with degenerate rigidity or delayed boundary feedback, energy multiplier and Lyapunov functionals adapted to the non-uniform configuration confirm exponential stabilization, with explicit decay rates tied to property bounds and control gains (Siriki et al., 20 Dec 2025).

5. Thermal, Damping, and Coupled Multi-Physics Extensions

Thermoelasticity and distributed damping in non-uniform beams require multi-physical PDE sets:

$$\rho(x)\,w_{tt}(x,t) + b(x)\,w_{t}(x,t) + \left( E\,I(x)\,w_{xx}(x,t) \right)_{xx} + \sigma\,\theta_{xx}(x,t) = 0,$$

$$\theta_t(x,t) - \kappa\,\theta_{xx}(x,t) - \sigma\,w_{xxt}(x,t) = 0,$$

where damping $b(x)$, coupling $\sigma$, and diffusivity $\kappa$ may be spatially variable (V. et al., 2015). The state-space and semigroup formulations in non-uniform weighted spaces guarantee well-posedness and lead to uniform energy decay via Lyapunov methods.

These models extend the reach of the non-uniform beam paradigm into fields requiring structural-thermal analysis or active damping, with numerical evidence supporting theoretical predictions of exponential decay.

6. Physical and Numerical Implications

The mathematically rigorous extension to distributional and discontinuous property beams has several practical implications:

• Classical weak-solution frameworks are insufficient in the presence of concentrated, discontinuous, or rapidly varying beam features; the very weak and intrinsic distributional approaches remain robust under these conditions (Blommaert et al., 2021, Dias et al., 2020).
• Interface conditions at jumps, cracks, or concentrated devices naturally arise from the mathematical formulation, capturing physically observable phenomena such as moment jumps, slope discontinuities, or concentrated deflections.
• Numerical schemes, including finite element and hybrid methods, can accommodate regularized coefficients and efficiently approximate solutions even in the presence of strong singularities or impulsive loading, with convergence rates reflecting underlying regularity (Blommaert et al., 2021).
• The practical modeling guidance is to regularize singular coefficients via mollifiers or intrinsic products, solve numerically or analytically, and apply the appropriate limiting procedures to recover physically admissible solutions.

7. Summary Table: Major Analytical Frameworks

Framework/Result	Handles Distributions	Interface/Jump Conditions	Consistency with Classical
Very Weak Solution (Blommaert et al., 2021)	Yes	Implicit, by regularization	Yes, when data regular
Intrinsic Product (Dias et al., 2020)	Yes	Explicit, algebraic	Yes, under smooth data
Matrix String / Spectral (Beals et al., 2020)	Yes	Via operator spectral data	Yes
State-Space/Control (Chatterjee et al., 23 Jul 2024, Siriki et al., 20 Dec 2025)	Yes	Encoded in boundary/input	Yes, for regularization

This structure underpins current rigorous treatment of non-uniform Euler–Bernoulli beams, encompassing highly singular configurations and supporting advanced control, identification, and physical modeling tasks across structural mechanics and engineering.