Abstarct: Abstract Studying nonlinear systems demands analysis methods that can accommodate their complex behaviors. The fundamental feature of modal analysis—the ability to decouple equations of motion—has been the primary motivation for extending its application to nonlinear systems and developing the concept of nonlinear normal modes (NNMs). As an active topic of research interest, the search for more efficient analysis methods for accurately designing microelectromechanical systems (MEMS) is driven by their high reliability, low cost, and low power consumption. This dissertation employs a NNM approach baxsed on the concept of invariant manifolds to develop reduced-order models for studying various nonlinear microbeams with different applications. Similar to a linear mode, a NNM is an inherent characteristic of the system and refers to a specific state of vibrational motion where all the state variables of the system are constrained to a two-dimensional invariant manifold in the system's state space. Part of the present research uses NNMs to examine the effect of thermoelastic damping on the free vibrations of an Euler-Bernoulli microbeam with mid-plane stretching and subjected to electrostatic loading. Initially, the vibrational equation and the thermoelastic heat equation are reduced to a single equation baxsed on physical reasoning. After Galerkin discretization, the first NNM is computed from the first three linear modes baxsed on an asymptotic method using a polynomial expansion approximation around the equilibrium point. Using this approach, the pull-in voltage and free vibrations of the system were analyzed and compared, both with and without the effects of thermoelastic damping. Anlysis results indicated that increasing voltage intensifies thermoelastic damping. It was also found that frequency increases over time to reach a voltage-dependent limit, after which it remains constant. Additionally, the calculated static and dynamic pull-in voltages were found to be consistent with previous results from the literature, without being significantly influenced by thermoelastic damping. Another part of the research investigated the free vibrations of hyperelastic and dielectric microbeams in the presence of a control parameter. The Euler-Bernoulli beam model and Von Kármán strain were used, and the invariant manifold-baxsed NNM was computed similarly using the asymptotic method. This part included three case studies: a simple hyperelastic microbeam under axial force as a control parameter, a hyperelastic dielectric sandwich microbeam with conductive hydrogel electrodes, and a similar microbeam to the second case but with aluminum electrodes. The control Galerkin discretization, the first NNM is computed from the first three linear modes baxsed on an asymptotic method using a polynomial expansion approximation around the equilibrium point. Using this approach, the pull-in voltage and free vibrations of the system were analyzed and compared, both with and without the effects of thermoelastic damping. Anlysis results indicated that increasing voltage intensifies thermoelastic damping. It was also found that frequency increases over time to reach a voltage-dependent limit, after which it remains constant. Additionally, the calculated static and dynamic pull-in voltages were found to be consistent with previous results from the literature, without being significantly influenced by thermoelastic damping. Another part of the research investigated the free vibrations of hyperelastic and dielectric microbeams in the presence of a control parameter. The Euler-Bernoulli beam model and Von Kármán strain were used, and the invariant manifold-baxsed NNM was computed similarly using the asymptotic method. This part included three case studies: a simple hyperelastic microbeam under axial force as a control parameter, a hyperelastic dielectric sandwich microbeam with conductive hydrogel electrodes, and a similar microbeam to the second case but with aluminum electrodes. The control parameter in the second and third cases was voltage. In addition to analyzing static bifurcation and the effect of the control parameter on the response, the accuracy of the NNM method was compared with the Galerkin discretization method. Results showed that the discrepancy between the response from the NNM and the response from the three-mode Galerkin discretization is less than the discrepancy between the response from the single-mode Galerkin discretization and the three-mode Galerkin discretization. The capability of the NNM method to develop reduced-order models while retaining significant system information was demonstrated, with the detection of internal resonance serving as a notable example. In the next part, the harmonic forced vibrations of the first and third case study microbeams were analyzed using the NNM method. To obtain the frequency response, the invariant manifold was determined numerically, utilizing a Galerkin-baxsed approach to set the residue equal to zero. The excitation frequency was added as a new state variable to the two state variables describing the invariant manifold, and the geometry of the manifold was extracted numerically baxsed on Galerkin projection. Transient and steady-state time response, phase diagrams of the steady-state response, and frequency response diagrams indicated that the results Galerkin discretization, the first NNM is computed from the first three linear modes baxsed on an asymptotic method using a polynomial expansion approximation around the equilibrium point. Using this approach, the pull-in voltage and free vibrations of the system were analyzed and compared, both with and without the effects of thermoelastic damping. Anlysis results indicated that increasing voltage intensifies thermoelastic damping. It was also found that frequency increases over time to reach a voltage-dependent limit, after which it remains constant. Additionally, the calculated static and dynamic pull-in voltages were found to be consistent with previous results from the literature, without being significantly influenced by thermoelastic damping. Another part of the research investigated the free vibrations of hyperelastic and dielectric microbeams in the presence of a control parameter. The Euler-Bernoulli beam model and Von Kármán strain were used, and the invariant manifold-baxsed NNM was computed similarly using the asymptotic method. This part included three case studies: a simple hyperelastic microbeam under axial force as a control parameter, a hyperelastic dielectric sandwich microbeam with conductive hydrogel electrodes, and a similar microbeam to the second case but with aluminum electrodes. The control parameter in the second and third cases was voltage. In addition to analyzing static bifurcation and the effect of the control parameter on the response, the accuracy of the NNM method was compared with the Galerkin discretization method. Results showed that the discrepancy between the response from the NNM and the response from the three-mode Galerkin discretization is less than the discrepancy between the response from the single-mode Galerkin discretization and the three-mode Galerkin discretization. The capability of the NNM method to develop reduced-order models while retaining significant system information was demonstrated, with the detection of internal resonance serving as a notable example. In the next part, the harmonic forced vibrations of the first and third case study microbeams were analyzed using the NNM method. To obtain the frequency response, the invariant manifold was determined numerically, utilizing a Galerkin-baxsed approach to set the residue equal to zero. The excitation frequency was added as a new state variable to the two state variables describing the invariant manifold, and the geometry of the manifold was extracted numerically baxsed on Galerkin projection. Transient and steady-state time response, phase diagrams of the steady-state response, and frequency response diagrams indicated that the results obtained from NNMs align with the three-mode Galerkin results. Additionally, the softening behavior of the first microbeam and the hardening behavior of the third microbeam were examined. The next part analyzed the free vibrations of a hyperelastic micropipe conveying fluid, using a similar state-augmented method to determine the invariant manifold. The results confirmed the consistency of responses obtained from the NNM method and the Galerkin approach. It was also found that higher flow rates increases the nonlinear natural frequency. Overall, this dissertation contributes to expanding the scope of research for studying nonlinear vibrations of MEMS through the lense of NNMs.