Certain materials exhibit different elastic behavior in tension and compression. These materials are typically modeled as bimodular, with different Young’s moduli in tension and compression, as a simplified engineering approximation. Due to this material property, the neutral axis of a beam composed of bimodular material shifts away from the geometric centroid of the cross-section, with its position either above or below the geometric centroid depending on the sign of the curvature. Consequently, the equation governing flexural oscillations is formulated using a model based on effective two-layer laminates, incorporating a discontinuous neutral axis. However, the position of the neutral axis for each of the two bending configurations is influenced not only by the material’s elastic properties but also by the characteristics of a cross-section, where the cross-section, in this case, varies along the beam axis. Depending on the cross-sectional properties, the dynamic response of a bimodular beam may exhibit either linear or nonlinear vibrational behavior. When a difference in the effective bending stiffness exists between upward and downward bending, additional varying internal forces arise, leading to a nonlinear dynamic response. For this purpose, the harmonic balance method is specifically applied to determine the frequency-response function to periodic excitation. A numerical investigation of a bimodular tapered beam with different cross-sectional properties, followed by a comparative analysis of the responses, reveals the substantial effect of bimodular materials, particularly concerning their cross-sectional properties.