We study the free transverse vibrations of a uniform, simply supported Euler-Bernoulli beam carrying a two-degree-of-freedom mass-spring-damper(MSD) system at an arbitrary point along its longitudinal axis: a mass mounted directly on the beam is connected, through a linear spring and a viscous damper, to a second mass suspended below it. Such configurations are relevant both to vibration reduction by tuned mass dampers and to the vibration-based structural health monitoring of bridges, both relying on accurate knowledge of a beam's damping rates and frequencies. In contrast to the classical tuned-mass-damper model, the additional mass resting directly on the beam at the mounting point is considered. The beam is partitioned at the attachment point and coupled to the MSD-system through the shear forces acting at that point. After non-dimensionalizing the model and eliminating the degree of freedom of the suspended mass, separation of variables is used to obtain a spatial eigenvalue problem, a temporal problem and a characteristic equation whose complex roots give the damping rate and frequency of each oscillation mode. As this equation has no closed-form solution, it is split into its real and imaginary parts and solved numerically for a exemplary parameter set. The computed roots occur in complex-conjugate pairs with negative real parts, confirming that every computed mode is damped. This procedure provides a way to determine the modal damping rates and frequencies. ... Read more

Energy can be harvested from vibrations by using a damped harmonic oscillator with base excitation, providing a sustainable way of yielding energy. By solving the equations of motion for this oscillator and studying the steady state solution, an expression for the time-averaged power is obtained. Different damping values of the oscillator influence how much power is yielded. In this thesis, it is analytically shown that a constant damping value equal to $c_{v} = \frac{\sqrt{c_{m}^{2}\psi^{2}f_{s}^{2}+(f_{s}^{2}-1)^{2}}}{\psi f_{s}}$ yields the most time-averaged power for the case where there is no switch in damping value and when there is a singular arbitrary switch in damping value. It is numerically shown that this damping value also yields the most time-averaged power for multiple switches in the damping value. ... Read more

Vibrations in engineering structures can lead to severe instabilities, especially under low-frequency excitations that traditional linear isolators cannot effectively suppress. To address this, quasi-zero stiffness (QZS) vibration isolators, known for their high-static-low-dynamic stiffness properties, have gained increasing attention. This report investigates the reflection and absorption characteristics of a nonlinear string with a QZS mechanism applied as a boundary condition. The model is considered, and the governing equations are derived and nondimensionalized. Using regular perturbation methods and the method of multiple time scales, analytical solutions are obtained and evaluated. The analysis distinguishes between cases where the oblique springs are extended or compressed. It is found that with compressed springs, when the vertical damping coefficient is below unity, the system is counterintuitively stable. Furthermore, the inclusion of oblique dampers leads to unphysical energy growth. These phenomena are attributed to the singular nature of the system’s dynamics and the limitations of the chosen multiple time scale method. The results indicate that the current model does not fully capture the effects of the oblique springs and dampers, underscoring the need for further investigation into the system’s asymptotic expansions. Moreover, exploring second-order dynamics and external forcing could provide a deeper understanding of the system’s complex behaviour. ... Read more

Cables are fundamental components in numerous technical implementations, such as cable-stayed bridges. As cables are prone to vibration due to e.g. wind, it is necessary to find ways to reduce these oscillations.
This thesis aims to build upon the work conducted by Su et al. [1]; their paper studies the vibration of an inclined cable with an attached Tuned Mass Damper (TMD). In particular, as Su et al. assume that the cable takes the shape of a parabola in equilibrium, the goal is to find a better estimate of the equilibrium configuration of the cable. To this end, this thesis will utilise a modified version of the method used by Caswita [2]; Caswita derives the equations of motion of a cable without any attached mass by applying Lagrangian mechanics.
The results show that the equilibrium position differs meaningfully from a parabola. The ordinary differential equations that Su et al. obtain by using Galerkin’s method are also considerably different when using the alternative equilibrium position. These differences are mainly caused by the fact that the cable hangs on an incline, rather than by the addition of the TMD. ... Read more

In this Masters thesis, the dynamics of pipes conveying pulsating flow are investigated. The initial-boundary value problem associated with the linear beam equations of motions governing the pipe system is derived using the principles of Lagrangian mechanics. In this thesis, the fluid flow is assumed to have a small velocity with harmonic time dependence $V(t)=\varepsilon(V_0+ V_1 \sin(\Omega t))$, which allows us to investigate the effects of different pulsation frequencies on the pipe system. For certain $\Omega$ frequencies, the pipe system is observed to be exposed to more complex dynamical behaviours. By using the multiple time scale perturbation method, comprehensive insights into the stability and the dynamic behaviour of pipe systems are achieved.

The study focuses on investigating the primary resonance frequencies and understanding how pulsation frequencies near those resonance frequencies impact the stability of the system. Furthermore, we elaborate on special resonance cases where multiple oscillatory modes interact leading to even more complicated dynamics.

By building upon existing literature this research enhances our understanding of stability and dynamic behaviors under various flow pulsation frequencies. This study makes an important contribution to the present literature by exploring scenarios where multiple resonant modes interact, due to coinciding primary resonance frequencies, which has not been extensively discussed in the literature. Our findings suggests scepticism on the relevance of the existing solution methods and results in the literature for certain parameter values. ... Read more

The main aim of the research presented in this report is investigating analytical methods to model fluid-structure interaction in large-scale offshore floating photovoltaics. The model that was attempted to be solved analytically is based on a model presented by Pengpeng Xu (2022).
The dimensions in the equations were removed. Applying a perturbation method yielded hierarchic partial differential equations by introducing the wave amplitude divided by the depth of the ocean as a small perturbation parameter. The analytical solution of the first order problem was found by applying separation of variables and by using a Fourier transform. For certain classes of problems it is shown in this report that it is possible to analytically solve a model for fluid-structure interaction in offshore solar farms for various initial conditions. ... Read more

Varying-length cable systems are widely applied in a vast class of engineering problems which arise in industrial, civil, aerospatial, mechanical, and automotive applications. Due to external excitations, large oscillations can occur when cables are lifted up or down. This phenomenon is caused by resonance. In general, resonance is harmful, and can cause significient deformations and dynamic stresses in machinery and structures, and even can lead to accidents. Therefore, this doctoral dissertation is devoted to the study of transverse and longitudinal resonance phenomena and output feedback stabilization of varying-length cables.... ... Read more

The Ekman spiral is described by a coupled system of differential equations originally discussed by Walfrid Ekman (Ekman, 1905). This system is a simplified version of the Navier­Stokes equations. The differential equations, as discussed in Ekman’s paper, concern the currents of the ocean. However, it is also possible to interpret these equations so as to describe and predict the flow of wind. The research as presented is not only inspired by Walfrid Ekman’s original paper, but also by the master thesis from de Jong (2021). The main contribution of this thesis is to include the influence of a constant vertical wind speed on the classical Ekman spiral. After stuyding the classical Ekman spiral, the inclusion of a constant vertical wind speed is done step­wise. First, the vertical wind speed is discussed without having any vertical Coriolis forces. The classical Ekman spiral and the Ekman spiral with vertical wind, but no vertical Coriolis force, were solved exactly. Then, the vertical wind speed is included fully, giving rise to a non­linear coupled system of differential equations. For the non­linear system, an algorithm for solving it analytically using a general perturbation method is proposed. Next, the hodograph of the non­linear equations of motion including a constant vertical wind speed, is made using Euler’s Explicit numerical method and a shooting problem is solved. ... Read more

Vibrations or oscillations can be caused in overhead cable lines or bridge cables due to strong rain and winds, making the structure unstable.These vibrations can be mathematically described as a string like initial boundary value problem with non-classical boundary conditions. In this thesis, we consider a nonlinear attachment at the boundary which consists of a mass, nonlinear spring and a damper attached to a semi infinite string. In particular, we consider a weak nonlinearity and damping. In this study we used the D'Alembert solution and the multiple time scales perturbation method to obtain bounded solutions of the initial boundary value problem. We assumed travelling wave initial conditions, and obtained special cases and conducted detuning around these special cases to further study the reflected waves at the boundary and the stability of our solutions. Our main objective is to study the reflection of the incident wave on the boundary and compute how much energy is dissipated at the boundary due to the weak dissipative forces present at the boundary ... Read more

In this research a study of the response of a simply supported microbeam subject to an electric actuation is presented. A perturbation method called the method of multiple scales is explained and used to solve our problem. A model concerning the mid-plane stretching and an electric force with a direct and alternating current component is formulated. The method of multiple scales is used to construct a solution that is valid for a long time after the initial conditions. The effect of the frequency of the alternating current was studied by performing a stability analysis. The results show that for frequencies close to the eigenfrequency of the homogeneous problem, there is no stable equilibrium and resonance occurs. Furthermore, a start was made to study the effect of the damping coefficient. The results show that a smaller damping will always lead to resonance on a very small time scale. The results also indicate larger oscillations and an small increase of the importance of the non-linear terms for smaller damping. All results are validated by comparing them with a numeric solution and show excellent agreement. ... Read more

In this thesis the nonlinear spring system is considered. This system contains a semi-infinite string that is modelled by the wave equation with a pair of inititial conditions and a nonlinear boundary condition. The goal of this thesis is to find a good approximation of this system. Furthermore, the behaviour of the string is studied by plotting the reflected waves. Two methods are considered for estimating the solution. These are the Multiple Scales Perturbations method and the Fourth Order Runge Kutta method. The approximations of the two methods are compared to each other. From this, conclusions have been drawn on the accuracy of these approximations. ... Read more

In this paper, it is analysed how an infinite cable behaves when a wave from the right hits a TMD system attached to it. The TMD system consists of a mass M1, a spring with stiffness K, a damper with coefficient C and a second mass M2. The cable has a tension T. Then it is investigated what happens when a wave goes through the cable and hits this TMD system. The goal of this paper is to calculate
how effective this TMD system damps. Using the multiple time scale perturbation method, the displacements of the cable and TMD system are calculated. This results in the reflected wave being damped by the TMD system, but the transmitted wave having a larger is placement than expected. The displacement of the TMD system also had unrealistic outcomes. To calculate the effectiveness of the TMD system, the energy which has been lost has to be calculated. Since the displacement of the TMD system had unrealistic outcomes, the calculation of
the energy is not realistic either. ... Read more

In this thesis the stability type of y=0 is being considered for the delay differential equation y''(t) + ay(t) + by(t-1) = 0 with a and b real numbers. It is already known that y=0 is stable when b=0 and a>0 and unstable when b=0 and a<0. The aim of this project is to determine the stability of y=0 for all values of a and b. First, the general stability theory for delay differential equations was highlighted before giving an in-depth stability analysis of the equation y''(t) + ay(t) + by(t-1) = 0. It turns out that a theorem of Pontryagin (1908 -1988) is really helpful for answering these stability questions. Due to this theorem all values for a and b are determined such that y=0 is asymptotically stable for y''(t) + ay(t) + by(t-1) = 0. However, this does not cover the stability type of y=0 for all values of a and b. So more analysis was done in order to give a full answer of the stability problem. The full answer was not achieved as there are still values for a and b where the stability is unknown. Finally, numerical solutions of y''(t) + ay(t) + by(t-1) = 0 are shown to confirm the results that are obtained. ... Read more

In this thesis we construct a perturbation method for delay differential equations (DDEs) based on the method of multiple scales for ordinary differential equations (ODEs) and ordinary difference equations (O$\Delta$Es). The method works for nonlinear DDEs, which are linear DDEs in the unperturbed case. The validity of the method is proven under certain conditions, such as a Lipschitz condition on the perturbation, and we illustrate how the method can be applied by working out several examples. We consider a delayed version of Mathieu's equation, which is especially useful, because it can be used when one linearizes a nonlinear oscillator around a period soluction. We also consider a quadratic perturbation. For these examples we have to analyse the relationship between the solutions of the characteristic equation. There already exists a perturbation method for DDEs, for which one solves a corresponding ODE, and uses this solution as an approximation. This method is only applicable when the influence of the delay is small, and is not always accurate due to the different natures of DDEs and ODEs. We study an example for which this method can be used, and show when it fails to give an accurate approximation. We then show how to use our perturbation method for this example, to obtain an accurate approximation. ... Read more

Mass-spring systems are commonly used in structural components. Understanding their characteristics and pitfalls is an important issue, combining physics and mathematics to prevent such systems from resonating and causing structures to weaken or collapse. This way, suitable solutions can be found such as the use of dampers with the right characteristics. In this paper we focus on such systems, in particular with a snap-through mechanism. Their dynamical behaviour and the in uence of a small disturbance on the mechanism, such as a wind force, are analyzed. A behavioral model is set up to which Melnikov's Method is applied, in order to analyze the behaviour numerically. The latter is done using the Trapezoidal method. ... Read more

The numerical simulation of brittle failure with nonlinear finite element analysis (NLFEA) remains a challenge due to robustness issues. These problems are attributed to the softening material behaviour and the iterative nature of the Newton-Raphson type methods used in NLFEA. However, robust numerical simulations become increasingly important, for example due to recent developments in Groningen. 
To address these issues, sequentially linear analysis (SLA) was developed which exploits the fact that a linear analysis is inherently stable. By assuming a stepwise material degradation the nonlinear response of a structure can be approximated with a sequence of linear analyses. Although this approach has been proven to be effective for several case studies, the numerical performance is still a problem that has to be solved. After every linear analysis, a single element is damaged resulting in incremental damage. As a result, the system of equations only changes locally between these linear analyses. Traditional solution techniques do not exploit this property and calculate a matrix factorisation every linear analysis, resulting in high computational times per analysis step. Since SLA typically requires many linear analyses to obtain the desired structural response, this leads to unacceptable analysis times. The aim of this thesis is to improve the computational performance of SLA by developing numerical solution techniques which exploit the incremental approach of SLA. To this extend, the following methods have been developed.
A direct solution technique has been developed which is based on the Woodbury matrix identity. This identity allows for the numerically cheap computation of the inverse of a low-rank corrected matrix. In this approach, the expensive matrix factorisation does not have to be calculated every linear analysis step. Instead, the old factorisation can be reused along with some additional matrix- and vector multiplications and solving a significantly smaller linear system of equations. An optimal strategy is derived to determine at which point a new factorisation should be calculated.
An improved preconditioner for the conjugate gradient (CG) method has been developed. Instead of an incomplete factorisation, the complete factorisation is used as a preconditioner which reduces the number of required CG iterations significantly. The point at which too many CG iterations are required and a new factorisation is necessary, is determined using the same strategy as the first method. From numerical experiments it follows that both methods perform significantly better than the direct solution method, especially for large 3-dimensional problems. The best performance is achieved using the Woodbury matrix identity resulting in the solver no longer being the dominant factor in SLA. Furthermore, significantly larger problems are not solvable in time frames in which previously only smaller problems were solved. ... Read more

In this project the transversal vibrations of an accelerating elevator cable system are studied, with the aim to find the resonance times, the resonance duration and the resonance amplitude. The elevator cable is modelled as an axially moving string, with length given by l(t) = l0 + 1/2 at2, with a the acceleration and t the time. The cable is sinusoidally excited at the top and fixed at the bottom. It is assumed that the axial acceleration is small compared to the transversal acceleration, that the cable mass is small compared to the car mass, and that the excitation amplitude is small compared to the length of the cable. Using these estimations, the solution for the transversal displacement u is approximated up to O(ε) with ε a small parameter. The elevator cable goes through a cascade of autoresonances: the eigenfrequencies of the cable are varying because the cable length is varying, and at several times an eigenfrequency matches the excitation frequency. These are the resonance times, and they have been found as t+ = (2/εa1l0)1/2arccos((Ωl0/χk)1/2), with t+ a measure of oscillation of t, Ω the angular excitation frequency, l0 the initial length, χk the eigenfrequency of mode k and εa1 = a. The duration of the resonances (the timescale) is shown to be O(ε-1/4) if χk≠Ωl0 and O(ε-1/6) if χk=Ωl0 (a bifurcation of the problem). The amplitude scale is thus O(ε3/4) or O(ε5/6), respectively, and solutions for the amplitude are calculated both outside and inside the resonance zone.    ... Read more

In this report, the rain-wind induced vibrations of cables are studied. This is done by modeling the cable cross-section as a mass-spring system with two time-varying masses. Thereafter, the solution of this model is approximated using a multiple timescale perturbation method. Lastly, for some choices
of the time-varying masses the eigenfrequencies are analyzed, stability properties are derived, and approximations of the solutions are given. ... Read more