The field of nanotechnology has been quickly growing over the last few decades and many different functional Nano Electro Mechanical Systems(NEMS) can now be made.
To further increase the possibilities and functionality of NEMS, new materials have to be characterized. Of the new materials which become available at the nanoscale, graphene is one of the most promising. This is mainly due to the combination of its extraordinary strength, electric and thermal conductivity, and low weight.
In order to be able to use graphene's full potential in future applications, the elastic properties, such as the Young's modulus and the bending rigidity, have to be known.

These elastic properties have been obtained following different approaches. However, the obtained values are scattered. This scattering has a few reasons.
Firstly, experimental research into graphene is difficult, due to the small scale, big influence of the environment, and the difficulty of fabricating well defined graphene membranes.
Secondly, graphene is a purely two-dimensional material. Therefore, continuum theory is not easily applied. The bending rigidity for example is not related to thickness as it is in a continuum plate.
Finally, graphene exhibits strong mode coupling and is always vibrating due to Brownian motion, the motion which results from the stochastic excitation due to temperature. Parameters extracted from static measures may thus not match the reality or experiments. In conclusion, the elastic properties have to be obtained from the dynamic response, following a multi-modal approach.

As graphene, only one atom thick, is close to the atomic scale, Molecular Dynamics simulations are used to investigate its behavior. To extract parameters, these simulations are compared to an analytical continuum model. In this way, the advantages of continuum mechanics and Molecular Dynamics simulations are combined with a dynamic, multi-modal approach to extract the bending rigidity and Young's modulus of graphene.

In the continuum model, the equations of motion of a circular graphene plate and membrane are derived from a Lagrangian approach. The governing equations are discretized using admissible functions that satisfy boundary conditions. Furthermore, the geometric nonlinearity is included, as graphene is so thin, and is thus easily driven into the nonlinear regime.

The basic principle of Molecular Dynamics simulations is to solve Newton's equation of motion for every single atom of a system. The force acting on the atoms is described by a potential field. The equations of motion are then integrated over time.

The mode coupling in graphene is shown to be so strong that the energy in all modes is equal after some time. Therefore, the only steady state attainable is the state in which the energy in all modes is equal, which corresponds to the Brownian motion. The eigenfrequencies are obtained from the time response of the atoms in the graphene membrane, excited by Brownian motion. The obtained eigenfrequencies are compared to the values obtained from the continuum model. From this comparison the bending rigidity of graphene is extracted. This is done following an optimization approach, which minimizes the difference between the eigenfrequencies obtained from the continuum and the Molecular Dynamics model. Including multiple modes is shown to be a necessity for reaching convergence.
Furthermore, the Young's modulus is obtained. This is done by comparing the geometric nonlinear behavior of graphene obtained in Molecular Dynamics with a continuum prediction of this behavior.

The bending rigidity and the Young's modulus thus have been obtained, independently, from the dynamic response of graphene obtained in Molecular Dynamics.

Biomechanics of cells have been identified as an important factor relating to their functionalities. Consequently, the need for sensitive measurement methods for mechanical analysis on the nano- and microscale is high. Micro-cantilever resonators can have a large impact in material determination at the nanoscale. When adding a material, the resulting shift in resonance frequency is associated with its mass and stiffness properties. The goal of this project is to exploit the contribution of higher flexural modes on the determination of both the density and Young's modulus of an added polymer on a micro-cantilever.
The identification of mass and stiffness is, in this thesis, restricted to the deposition of a two-photon polymerized layer on micro-cantilevers. A polymeric material is used, since cells are complex systems with viscoelastic properties, where polymers are less complex and mimic biological matter very well. Two configurations are investigated: an added polymer near the tip of the cantilever and a polymeric layer near the base of the beam. Experimentally and theoretically derived multi-modal analysis are linked, to decouple the mass and stiffness properties.
The added polymeric layer affects the resonance frequency of the system, which is found to be location and mode related. Decoupling of the mass and stiffness properties based on the modes with the largest frequency shift, lead to different outcomes compared to considering modes with the largest deflection and curvature at a specific location. The use of higher modes does affect the outcomes of the decoupling of mass and stiffness. Whether it leads to an increase on accuracy is yet elusive.

A new concept was developed to improve current airpot designs for vibration isolation solutions. The concept is based on an additional piston, or add-on, with certain characteristics, that is to be incorporated into an existing airpot-design. The effect of this add-on will passively improve the vibration isolation performance of the airpot. This research into this add-on concept consists of the modelling of this concept , the design of a demonstrator and a comparison between modelled and measurements.

Since its invention, the Atomic Force Microscope has emerged into one of the most useful tools in
nanotechnology due to its acclaimed abilities in exploring surface topography, micro- and nanoscale manipulation
and characterization. The nonlinear interaction between the cantilever tip and the sample surface
has been studied in great detail and a thorough understanding of the cantilever dynamics can improve
measurements and lead to new methods for identification, manipulation and characterization. One of the
biggest challenges in the field of AFM is related to the identification of the cantilever tip size. The accuracy of
measurements in AFM is directly related to the size and geometry of this tip. It is desired that the tip condition
can be continuously monitored in between or even throughout measurements in order to provide high
quality measurements. In this thesis project, a methodology for tip assessment is introduced based on the
nonlinear dynamic response of the cantilever. The method consists of the acquisition of frequency response
curves in the attractive regime where the influence of the tip radius is predominant. Experimental frequency
response curves are fitted to the tip-sample interaction model that includes the van der Waals forces. It is
found that exploiting the nonlinear dynamic response of the cantilever is a safe and accurate method for tip
assessment.

To facilitate research on the dynamics of micro-cantilevers immersed in liquid, a modal test setup was designed. Modal testing is a powerful form of dynamic testing whereby the system's natural frequencies, modal masses, modal damping ratios, mode shapes, and weight of each of these modes at a certain frequency are determined. Micro-cantilever resonators are often used for sensing applications, where information is extrapolated from the change in dynamical behavior. Such systems are used in liquid environments to characterize the fluid density and viscosity, to measure the mass and stiffness of particles suspended in this liquid, and to image biological samples using atomic force microscopy. The designed setup employs piezoelectric base actuation to excite the cantilevers, of which the response is measured in vacuum, air, and water. A clean and spurious free response is achieved in a bandwidth ranging from 20 kHz to 500 kHz in vacuum and air, and a bandwidth ranging from 50 kHz to 500 kHz in water.

Ever since its inception, graphene has been the subject of research in many parts of the
world. This is due to its exceptional mechanical and electrical properties, which makes it
ideal for NanoElectroMechanical (NEMS) devices. The inherent nature of NEMS devices,
includes low damping, large amplitudes of oscillation, resonant operating conditions, and
the presence of nonlinear force fields. This sets an ideal stage for the appearance of nonlinear
behavior. In this thesis, appearance of such nonlinear behavior in optothermally actuated
graphene nanodrum resonators is studied. Frequency response arising from parametric
excitation is explained based on, time modulated stiffness due to temperature variation in
the membrane. Also, the response arising from direct excitation is discussed based on initial
geometric imperfection present in the membrane. In order to explain the nonlinear response
seen in graphene resonators, novel analytical models are developed and its corresponding
limitations are discussed. A single differential equation is used to simulate the behavior
of both directly and parametrically excited graphene nanoresonator. This equation is
used to study the influence of nonlinear damping on response of the system. Then, an
illustration is provided on characterization of graphene properties from the parametric
response of the system. Finally, it is concluded that, alternative damping mechanism and
other physical phenomena could be influencing the system dynamics. Therefore, modeling
of these phenomena would lead to better matching of the experimental results.

A common structural design verification is to conduct modal analysis and to ensure that the vibration modes of the structure do not get harmonically excited during operation. Since the modal frequencies are dependent on the mass and stiffness properties of the structure, they are an influencing factor in the design optimization. Modal frequencies are obtainable using eigenvalue analysis after linearization approximations in the restoring force terms of the governing equation of motion. While this approximation is valid for lower loads, it is not acceptable for strong nonlinear vibrations. A measurable shift in the modal frequency is observable when the amplitude of vibrations is in the order of the thickness of the vibrating structure. This phenomenon is more pronounced in thin walled and flexible structures where the vibration amplitudes can exceed the linear threshold relatively more easily.

A credible approach in computation of nonlinear modes is the utilization of parametric continuation schemes for generating nonlinear frequency-amplitude response. However, utilization of this scheme with a large degree of freedom model is a computationally intensive approach which necessitate the development of reduced order models. A model reduction method in the finite element framework, termed as Hamiltonian Reduced Order Model (ROM), has been recently developed at Delft University of Technology. The present study is conducted for the experimental validation of the ROM. Governing equations of motion of a stiffened plate have been derived using the Hamiltonian ROM and the nonlinear frequency response has been generated using a continuation scheme. For validation, experiments have been conducted using the Laser Doppler Vibrometer to obtain similar frequency response curves. A comparison of the numerical and the experimental results shows excellent agreement. Furthermore, accurate numerical response is obtainable using only a single degree of freedom in the reduced order model which proves the effectiveness of the Hamiltonian ROM. The results also demonstrate the necessity of a nonlinear damping model for obtaining comparable results.

Nonlinear analysis of dynamic problems has become important for modern industrial design applications. The increasing pressure on airlines to decrease fuel costs demands the design of more efficient aircraft. This requires aircraft manufacturing companies to design lighter structural components. The result is the need for more realistic and accurate modelling of critical structural components. Over the years, more powerful finite element discretization methods and improved numerical methods and programming techniques for dynamic analyses of structures have been introduced. Despite these advances and the increase in available computer power, the analysis of nonlinear dynamic problems is yet a computationally demanding task, implying it is very expensive. To reduce the computational time of nonlinear finite element analyses, reduction methods have been developed. These methods have as aim to reduce the number of degrees of freedom, while retaining sufficient accuracy of the solution.

Recently, a new reduction method, applicable to nonlinear static stability problems, has been developed at Delft University of Technology. The aim of this thesis is to extend the reduction method for statics to nonlinear dynamics. This is achieved by using the Hamiltonian formulation to describe the motion of a system. A reduced order model (ROM) is constructed for free vibrations, forced vibrations and damped vibrations, using Hamilton’s equations of motion. These are integrated to obtain the response of the ROM, in terms of displacements and momenta. The displacements of the full finite element model are computed by back-substituting the reduced response into the displacement expansion. The ROM is implemented in a finite element framework.

The ROM is applied to beams, to plates as well as to shells. Overall, good agreement is found between the ROM and Abaqus. The big advantage of the ROM is found when the computational times for beams and plates are compared to that of Abaqus. A drastic reduction in time is observed for the ROM, while still maintaining accurate results. The ROM thus saves valuable computational time.