# Vibrational Analysis of Building Structures with Irregularities
## 1 Introduction  
This paper presents a predictive mathematical model for vibrations in lightweight, timber-framed structures (LTFSs), which captures the random uncertainties inherent in both the materials and construction techniques. We are particularly concerned with sound transmission through the double leaf construction typically used in timber-based floor/ceilings or walls. As timber-based buildings gain popularity, the challenge of addressing their typically poorer sound insulation--compared to concrete structures--has led to complex, multi-component LTFS designs [1]. These structures, incorporating elements like wooden panels, ribs, joists, and fiberglass wool, require more sophisticated modeling approaches as traditional deterministic models struggle to predict vibration accurately, especially in the low- to mid-frequency ranges. Our focus here is on incorporating random component irregularities directly into the model. Our modeling approach shares the same theoretical foundation as the finite element method (FEM), relying on stiffness matrices derived from the Lagrangian of structural deformation. However, traditional FEM becomes computationally impractical at higher frequencies, where structural detail and irregularities are often unmeasurable, resulting in expensive solutions that deviate significantly from experimental measurements [2]. [PERSON] and [PERSON] [3; 4; 5] provided foundational insights into how cavity space and joists impact LTFS vibrations, while [PERSON] [6; 7] explored vibration propagation across junctions between flexible plates and beams. Building on these, our model introduces the following two key extensions: It incorporates plate-joist coupling and accounts for irregularities in joist properties. To make the model adaptable to real-world structures without needing a complete re-derivation each time, we employ a variational formulation, following [PERSON] and [PERSON] [8]. Here, deflection is determined as the structure's totalenergy minimum, allowing us to capture complex interactions in typical floor/ceiling configurations; see Figure 1.  
Recent studies have advanced the efforts to improve sound insulation in lightweight timber structures, with many focusing on mitigating impact sound--a challenge due to their lightness and flexible connections [1; 9; 10]. Traditional methods like statistical energy analysis (SEA) filter at low frequencies, where modal densities are lower, and energy transmission becomes more directional [11; 12; 13]. [PERSON] and colleagues showed how material choice significantly affects performance; for example, variations in wood species and strength classes in cross-laminated timber (CLT) panels influence their vibroacoustic behavior [14]. Optimized floor designs and construction techniques have shown promise in enhancing sound insulation [10; 11; 15]. Effective strategies include using resilient underlays and toppings, sand and sawdust damping layers, decoupling ceilings with resilient mounts or independent joists, and adding stiffeners to increase resonance spacing [10; 15]. Composite floor systems, combining timber with concrete layers, also improve low-frequency insulation by utilizing concrete's mass and stiffness while retaining timber's advantages [10; 11; 13]. We are particularly interested in exploring whether similar results can be achieved without a substantial increase in mass and stiffness. Research, such as that by [PERSON] and colleagues [16], shows that timber panels with varying levels of heterogeneity exhibit distinct sound transmission losses, particularly at mid frequencies. This finding prompts the following intriguing question: can material irregularities be leveraged to enhance design? Answering this requires a robust theoretical and computational model capable of characterizing such irregularities and accurately predicting the resulting vibrations and associated sound transmission loss [9]. This paper introduces such a model.