Data-driven reduced-order modeling is an essential tool in constructing high-fidelity compact models to approximate physical phenomena when explicit models, such as state-space formulations with access to internal variables, are not available yet abundant input/output data are. When deriving models from theoretical principles, the resulting models typically contain differential structures that lead to certain system properties. A common example are dynamical systems with second-order time derivatives arising in the modeling of mechanical or electro-mechanical processes. In this case, data are often available in the frequency domain, where the systems' input-to-output behavior is described by rational functions rather than differential equations. Classical frequency domain approaches like the Loewner framework, vector fitting and AAA are available and can be used to learn unstructured (first-order) models from data. However, these models in their original formulation do not reflect the structure-inherited properties. The key element in the derivation of frequency domain approaches is the barycentric form of rational functions. In this work, we present structured extensions of the barycentric form for the case of mechanical systems. Building on these structured barycentric forms, we develop new algorithms for learning mechanical phenomena in the frequency domain, while enforcing the mechanical system structure in the model description.