Model Order Reduction of a Sliding Beam using a Global Basis: Formulation and Evaluation

Technical Summary: Model Order Reduction of a Sliding Beam using a Global Basis

Problem Statement
Telescopic structures, ubiquitous in heavy machinery such as cranes and telehandlers, rely on sliding beams where effective lengths and boundary conditions (BCs) change continuously. Modeling these systems presents a fundamental challenge: classical Model Order Reduction (MOR) techniques assume fixed topologies, computing mode shapes once for a static configuration. In sliding beams, as the structure extends or retracts, the effective length and BCs evolve, rendering fixed modal coordinates invalid. Updating the modal basis during simulation to capture these changes typically requires recomputing coordinates, which negates the computational efficiency gains of reduction. While Parametric Model Order Reduction (PMOR) offers solutions, existing local approaches (interpolating between discrete parameter values) often result in reduced models that change with parameters, complicating the physical interpretation of coordinates. Furthermore, global PMOR approaches, which project the system onto a fixed subspace, have received limited attention within the context of constraint multibody dynamics (MBD) for sliding systems.

Methodology
The authors propose a snapshot-based global PMOR approach integrated within a constraint multibody formalism. The methodology proceeds in two main stages:

1. Construction of a Global Reduction Basis:

   • Snapshots: The system is discretized using planar Euler–Bernoulli (EB) finite elements. Snapshots are defined as modal matrices ($\Psi_i$) containing a set of eigenmodes ($\phi$) computed for specific slider positions ($s$).
   • Parameterization: Two parameters govern the snapshot generation: $n_{snap}$ (number of distinct slider positions sampled) and $r_{snap}$ (number of eigenmodes retained per snapshot). The sliding BC is enforced algebraically at specific nodes during the computation of these eigenmodes.
   • Compression: All snapshots are stacked to form a "pool" matrix ($\Psi_{all}$). Proper Orthogonal Decomposition (POD) via Singular Value Decomposition (SVD) is applied to this pool to extract a compact global reduction basis ($\Psi_{glob}$) consisting of the most significant left-singular vectors.

2. Integration into Multibody Formalism:

   • Constraint Enforcement: The sliding beam is modeled using a Lagrangian Type 1 framework with algebraic constraints. The constraint vector $g$ enforces zero displacement at the fixed end and zero transverse displacement at the moving slider position. The slider position is time-dependent (rheonomic) and holonomic.
   • Reduction: The full system coordinates $q$ are expressed as $q(t) = \Psi_{glob} p(t)$, where $p$ are the reduced modal coordinates. This projection is applied to the Differential Algebraic Equations (DAEs).
   • Implementation: The reduced mass ($M_{red}$) and stiffness ($K_{red}$) matrices are constant. However, the reduced constraint Jacobian ($C_{q,red}$) and force vector ($f_{red}$) are recomputed at every time step to account for the moving slider and external loads. The constraint Jacobian is reduced via $C_{q,red} = \Psi_{glob}^T C_q$.

Key Contributions

• Global Basis in Constraint MBD: The work demonstrates the integration of a global reduction basis into a constraint multibody formalism for sliding beams, a combination not systematically evaluated in prior literature.
• Algebraic Constraint Formulation: The authors formulate algebraic sliding constraints directly using Hermite shape functions within the EB framework, ensuring $C^1$-continuity as the slider moves between elements.
• Systematic Evaluation: A comprehensive study investigates the impact of snapshot quantity ($n_{snap}$) and eigenmode count per snapshot ($r_{snap}$) on accuracy, particularly under a fixed budget of global modes ($r_{glob}$).
• Benchmarking: The proposed method is validated against a high-fidelity reference solution using the Absolute Nodal Coordinate Formulation (ANCF) with a sliding joint implementation, comparing both kinematic responses and energy conservation.

Results

• Accuracy: In a challenging test case involving a highly flexible beam with time-dependent loading and rapid slider movement, the global reduction basis (using $n_{snap}=5, r_{snap}=6, r_{glob}=20$) maintained a normalized Root-Mean-Square (RMS) displacement error below 2% compared to the non-reduced EB solution. Specifically, the error was 1.82% for beam displacement and 1.57% for tip displacement in the "evolved" test case.
• Energy Conservation: The energy evolution of the reduced EB model matched the ANCF benchmark, confirming that the algebraic constraint enforcement introduces neither artificial energy dissipation nor injection.
• Computational Efficiency: The reduction of the coordinate space from 153 degrees of freedom to 20 resulted in a computation time reduction of approximately 90% compared to the non-reduced EB approach. Compared to the ANCF benchmark, the reduced approach was roughly 50% faster.
• Parameter Sensitivity: The study revealed a trade-off in snapshot selection. Under a fixed budget of global modes, a balance between the number of snapshots and modes per snapshot is critical. Too few snapshots yield an uninformed basis, while too many snapshots (with few modes each) can lead to inefficient truncation and higher errors. An optimal product of $n_{snap} \times r_{snap} \approx 26$ was identified for the specific test case.

Significance and Claims
The paper claims that the proposed global reduction basis successfully addresses the challenge of evolving topology in sliding beams without sacrificing the physical meaning of the modal coordinates. By projecting the system onto a fixed subspace derived from a snapshot pool, the method enables efficient simulation of flexible multibody systems with moving boundaries. The authors assert that the method is suitable for real-time simulation of telescopic structures, offering a significant speedup while maintaining acceptable accuracy (errors < 2%). The work establishes a foundation for applying global PMOR within constraint multibody dynamics, bridging the gap between classical reduction techniques and the complexities of parameter-dependent mechanical systems. Future work is noted to extend this approach to floating frame of reference formulations to handle large rigid-body motions.