## 1 Introduction  
In vibration analysis, ordinary differential equations are a powerful tool for modeling dynamic systems. However, accurate models often involve a large number of equations, which primarily aid in understanding complex physical phenomena. When used for simulation, control design, or vibration optimization, these models can be computationally inefficient and demand significant execution time and memory. To overcome this challenge, model order reduction techniques have been developed, creating reduced-order models that approximate the input-output behavior of large-scale systems with far fewer equations. Additionally, in engineering applications, system dynamics often depend on parameters such as material properties, geometry, or configuration. In such cases, parametric model order reduction (pMOR) [1] is necessary to account for these variations efficiently. Generating and reducing large-scale models can be highly resource-intensive. For instance, a complex computational fluid dynamics (CFD) simulation [2, 3] may take days to evaluate at a single value of the parameter. Similarly, optimizing a mechanical structure for vibration using the finite element method (FEM) often requires model evaluations at every step, even for systems with thousands of states. The primary objective of pMOR is to generate a reduced model that remains accurate for any parameter value, thus eliminating the need to repeat the modeling and reduction process. Typically, high-dimensional models that depend on parameters are only available at specific parameter values. As a result, most pMOR techniques leverage interpolation. The general approach of interpolation-based methods follows a two-step process: first, the high-dimensional models at selected parameter values are reduced independently. Then, reduced models for other parameter values are generated through interpolation using the existing reduced models ([4; 5]); see Figure 1.  
The figure represents a cantilever beam with varying lengths, which is treated as a parametric system. In this setup, reduced-order models are generated for different beam lengths. Once these reduced models are obtained, interpolation is performed between them to efficiently approximate the system's behavior for new, unseen lengths, without needing to run a full simulation each time. Because the behavior of the complex system at the desired parameter value is not directly known, additional prior information must be integrated into the interpolation algorithm. This ensures that the interpolated models closely capture the original behavior of the system beyond the initially available models. For instance, if the original system is known to be stable across the entire valid parameter domain, the interpolated system should also maintain stability. Various pMOR techniques have been developed and applied to different classes of systems. Balanced truncation for parameter-dependent systems was explored in [6], which extends classical balanced truncation to parameter-dependent systems (including differential-algebraic equations) by computing reduced models on a set of parameter samples and constructing a single projection subspace valid for all parameters. This approach has a computationally expensive offline part and a much more efficient online part. Matrix interpolation techniques for local reduced order model (ROM) interpolation appeared in [7]. The study introduces a framework for parametric MOR via matrix interpolation. High-fidelity models are first reduced at various fixed parameter values, producing stable local ROMs. Then, the system matrices of these ROMs are interpolated across the parameter domain. This method ensures the interpolated ROM remains stable (using a matrix-measure criterion). Stability-preserving interpolation was comprehensively tackled in [1; 8]. The papers propose an enhanced matrix-interpolation method that guarantees asymptotic stability of the parametric ROM.