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State of the art
Modal and stability analysis is a well-established tool in structural engineering. It is a method to reveal intricate vibrations in structural dynamics. Modes give vibrational and stability information on the oscillatory patterns of a structure in equilibrium. Further-more, modal frequencies show when to expect resonance. This in turn can be utilized for assessment or optimization of structures.
An overview of stability analysis of time-periodic systems and classic modal analysis are treated in this chapter. These key concepts are meant as background for readers that are not specialized in time-periodic dynamics. What is introduced are relations between periodic and constant systems, modal and stability analysis.
There have been several methods developed to analyze time-periodic systems for sta-bility. Floquet theory shows these systems have linearly independent almost-periodic solutions, equivalent to modes, called Floquet forms (FFs). However we believe there is a current gap in literature, which consists mostly in a comprehensive guide for com-puting and using these almost-periodic modes for engineering, namely for helping in the understanding of the oscillations of structures in periodic elastic states.
Starting by specifying the place of time-periodic systems within nonlinear dynam-ics, classic linear time-invariant system are introduced as a special case of time-periodic systems. Stability of time-periodic systems and Floquet theory is then introduced and Floquet theory is split into time and frequency domain approaches.
Time-periodic systems
In structural dynamics time-periodicity arises from certain processes directly. Rotating machinery with imbalances such as damaged helicopter wings [17] or windturbines with varying atmospheric conditions [63], are a principal domain as a source of interest for time-periodic problems. In maritime engineering, parametric roll is considered as a time-periodic stability problem [64]. Another application is to enhance damping by periodically varying a parameter [30]. Classic textbooks on this subject have been written by Nayfeh
Relation between nonlinear and time periodic systems
The link between nonlinear, periodic and linear systems is described in this section.
The nonlinear dynamic of structures can usually be described in the N -dimensional discrete formalism y˙(t) = f (t, y(t)), y(t) ∈ ℜN (2.1) where the state of the system is described by the N -dimensional vector y(t). The state velocity y˙(t) = dy(t)/dt is a function of time t and the state of the system itself through the evolution function f . By looking at stationary solutions of dynamical systems as in (2.1), predictions can be made about their long-term behaviour. Possible stationary solutions to nonlinear systems include fixed points, limit cycles, limit tori and strange attractors [10]. The current discussion is limited to fixed points and limit cycles. These are referred to as constant and periodic stationary states respectively.
If the solution is a fixed point of a dynamical system, the structure described by this solution is often said to be in a constant equilibrium state. In structural dynamics, linearizing around an equilibrium state leads to the canonical linear equation of motion with mass, damping and stiffness matrices. Mathematically speaking, this linear system is described by linear ordinary differential equations with constant coefficients.
In case of a limit cycle, the structure is said to be in periodic state. Linearizing the system around the limit cycle leads to a time-periodic linear system. The equation of motion now has one or more time-periodic coefficients. These systems are sometimes re-ferred to as parametric systems since the parameter variation can be seen as an excitation of the system [11]. Computing stability and the characteristic response of these systems is a non-trivial task.
Other solutions such as tori and strange attractors, which lead to chaotic behavior [10], are beyond the scope of this work. Limit tori lead to almost-periodic systems, these can be approximated as periodic systems [65] to compute stability.
Almost-periodic functions [66] do not have a closed limit cycle, which would be the case for periodic functions. However the system almost repeats itself within an arbitrarily close distance ǫ given sufficient time τ0 . So that if we take a continuous function g(t), the system almost repeats itself: |g(t) − g(t + τ0)| < ǫ. In the work of Strogatz [10] this is referred to as a quasi-periodic function.
The state of a system y(t) is what determines the change in state y˙(t). If the system is at a fixed point y0 (t) = y0 , then it will stay in that state so that: f (t, y(t)) = 0. For periodic stationary state, the state is periodically repeating. When the limit cycle has a period T , the periodic equilibrium state and its derivative are periodic: y0 (t) = y0 (t + T ), y˙0 (t) = y˙0 (t + T ). In state space this results in a closed trajectory [67].
By looking at linearized systems around stationary states, properties of the stationary state such as stability and oscillatory modes can be analyzed. Liinearizing a dynamical system around a stationary solution consists in writing out a first order approximation of the nonlinear system at this stationary state. This reveals the properties of the latter such as stability and the linear vibratory response. Mathematically, this linearization process consists in a first order perturbation of the stationary solution
y(t) = y0 (t) + ǫy1 (t) + H.O.T . (2.2)
y(t) ≈ y0 (t) + ǫy1 (t).
by expanding the response into a power series with a sufficiently small parameter ǫ. Since the parameter ǫ is small, the higher order terms (H.O.T.) ǫ2 , ǫ3 , . . . , ǫn can be neglected. Since we are linearizing at a limit cycle, the zeroth order response y0 (t) is the limit cycle itself and is supposed to be known.
Analyzing first order response y1 (t) determines stability and shows the vibratory modes around the stationary state. By expanding the nonlinear function (2.1) into a Taylor serie, one obtains
y˙0 (t) + ǫy˙1 (t) = f (t, y0 (t)) + ǫ ∂f (t, y0 (t))y1(t). (2.3)
Separating the variables with and without ǫ gives: y˙0 (t) = f (t, y0 (t)) and ǫy˙1 (t) = ǫ ∂f (t, y (t))y (t).
The first order derivative with respect to vector y0 (t) is also referred to as the Jacobian matrix:
J (t) = J (t, y0 (t)) = ∂f (t, y0 (t)). (2.4) ∂y
By isolating the equation in ǫ, we get the first order dynamical equation around the studied stationary state y˙1 (t) = J (t)y1 (t). (2.5)
It is from this linear equation that the linear vibratory response around a stationary state, y 0 (t), is determined. Equation (2.5) is the linear ordinary differential equation that will be studied throughout the report. The difference compared to (2.4) is that in the rest of this work, the stationary state will be trivial and will not need to be computed in order to only focus on linear systems. A final note on notation, the first order response y1 (t) is simplified to y(t) throughout this thesis.
Time-periodicity in Ordinary Differential Equations
Linear time invariant (LTI) systems can be seen as a special case of linear time periodic (LTP) systems. The difference between LTP and LTI systems is that they have time-periodic and time-invariant coefficients, respectively. They are governed by a set of linear ordinary differential equations (ODEs) that for second order systems are in the form: a0 (t)x(t) + a1 (t)x˙(t) + a2 (t)x¨(t) = F (t). (2.6)
Note that the dot notation is used for derivatives with respect to t so that the derivative dx(t)/dt is noted as x˙. The second derivative d2 x(t)/dt2 is represented by double dots x¨(t). The coefficients ai(t), i = 0, 1, 2 depend periodically on time. Periodic coefficients repeat themselves after a certain period by definition. If the period is noted as T , the periodic coefficient ai(t) verifies ai(t) = ai(t+T ), where T is the period. The link between time periodic and invariant systems is that as the period approaches zero, the coefficients become constant. In other words, if a coefficient is constant, it could be considered as a special class of periodicity with T = 0.
Time-periodic vibrations in engineering
In this section, equations of motion of LTP systems are given in the context of time-periodic vibrations in engineering. The link between physical and state space equations is explained. When analyzing LTP systems based on Floquet theory, state space equations will be used which follows the notation from [21, 37, 39, 62].
When studying the vibrations around a structure in equilibrium state, the governing equations of LTI systems have the canonical form: M x¨(t) + Cx˙(t) + K x(t) = F (t). (2.7)
In the classic set of linear discrete N -dimensional equation (2.7), the stiffness K, damping C and mass M matrices are constant over time. This case corresponds to equations of motion with constant coefficients. In the following, we will consider no damping in our system so that C = 0.
If one wants to study the oscillations around a structure in periodic stationary state, the coefficients of the ODEs (2.7) can vary periodically. Any system parameter, such as mass, damping or stiffness, can be a source of periodicity. In the current work only elasticity is assumed to be periodic, which is the most usual case in engineering.
In structural vibrations, periodic elasticity translates as a periodic stiffness matrix: M x¨(t) + K(t)x(t) = F (t). (2.8)
Equation (2.8) can be the result of nonlinear processes or because of an imperfection in rotating machinery for example [59]. Note that limiting our framework to periodic stiffness terms does not mean other sources of periodicities can not be analyzed. Eventual periodicities in damping or mass can always be rewritten in the form of constant mass with periodic stiffness (as in equation (2.8)) by a change of coordinates [11].
In this work, the periodic stiffness matrix consists of a constant part K0 and a periodic part Kσ (t) with period T : K(t) = K0 + Kσ (t), Kσ (t) = Kσ (t + T ). (2.9)
Not only this is often the case in of practical problems (any slender structure under periodic compressive or tensile prestress for example) but it can be useful to separate the constant and periodic terms. This allows to compare the classic response of the system without periodicity to the system including periodicity. One can define the periodicity using the period T or the frequency β = 2π/T .
In structural vibration, it is computationally useful to rewrite equations of motion (2.7) in state space where they become first order differential equations. The state space formalism is adopted for time-periodic systems (2.8), leading to:
0 Iz˙(t) = I 0 z(t) + 0 (2.10)
M 0x˙(t) 0 −Kσ (t) − K0 x(t) F (t) .
In equation (2.10), the state variable z is introduced that reads z(t) = x˙(t).
The state matrices A(t) and B of sizes 2N × 2N are introduced along with state vector y(t) of size 2N × 1 to simplify the notation
A(t) = I 0 B = 0 I y(t) = z(t) (2.11)
0 −Kσ (t) − K0 , M 0 , x(t) .
Furthermore the Jacobian matrix of the system is written as J (t) = B−1 A(t) so that the canonical form of time-periodic dynamical systems can be obtain y˙ = J (t)y,J (t) = J (t + T ). (2.12)
This state space formulation with periodic Jacobian of size 2N × 2N will be used exten-sively in Floquet theory. It is interesting to note that equation (2.12) is identical to (2.5), but the difference is that the expression of the periodic Jacobian matrix here is trivial, when in (2.5), J (t) was depending on the nonlinear stationary state we had to compute.
Examples: vibrations of beams in periodic elastic states
In figure 2.1 the cases analyzed throughout this work is shown. The first system is a finite element model of the linear transverse vibrations of a cantilever beam under periodic prestress (Fig.2.1(a,b)). The second case concerns the oscillatory motion of a straight double pendulum under periodic compression load (Fig.2.1(c)).
The 2-DoF beam is an archetypical example of a time-periodic system. It is modeled as a lumped mass and stiffness system with a periodic compressive load. This load can be either constant in direction throughout deformation (η = 0), leading to a conservative system. Or in case of η = 1 the load follows the direction of the beam θ2 (t), which makes the system nonconservative. This is a fundamental case which serves as a benchmark to present time-periodic analysis techniques, notably in the frequency domain.
For the N degrees of freedom (DoF) beam the finite element formalism is used. The finite element method (FEM) is a well-established numerical technique used in structural engineering. Its equations of motion are derived from virtual work principles and solved mechanics as it exhibits most of the classic bifurcations of dynamical systems, although the perturbed stationary state is spatially trivial.
Balancing the quantity of acceleration of each bar of the bi-articulated elastic system with the applied external moments (the expression of those quantities are given in Ap-pendix A), the nonlinear equation of motion of the Ziegler column, reads, in the physical space (θ1 (t), θ2 (t))
Table of contents :
1 Introduction
1.1 Research Field
1.2 Research goals
1.3 Reading guide
2 State of the art
2.1 Time-periodic systems
2.1.1 Relation between nonlinear and time periodic systems
2.1.2 Time-periodicity in Ordinary Differential Equations
2.1.3 Time-periodic vibrations in engineering
2.2 Examples: vibrations of beams in periodic elastic states
2.2.1 Ziegler column
2.2.2 Finite element discretization of the cantilever beam with periodic prestress
2.3 Stability of time-periodic systems
2.3.1 Floquet theory
2.3.2 Time domain
2.3.3 Frequency domain
2.3.4 Stability types
2.3.5 Stability analysis of the 2-DoF Ziegler column
2.4 Classic Modal Analysis
2.4.1 Modal projection of the cantilever beam with periodic prestress
2.5 Conclusions
3 Frequency domain analysis of Floquet forms
3.1 Introduction
3.2 Time-domain method
3.2.1 STM eigenvectors
3.3 Hill Matrix
3.3.1 Complex Hill Matrix Derivation
3.3.2 Real Hill Matrix Derivation
3.4 Periodically conservative case (η = 0)
3.4.1 Constant elastic state (β = 0)
3.4.2 Periodic elastic state (β 6= 0)
3.4.3 Asymptotic cases (β → +∞) and (β → 0)
3.5 Non-Conservative case (η = 1)
3.5.1 Constant elastic state (β = 0)
3.5.2 Periodic elastic state (β 6= 0)
3.6 Spectral convergence of the stability analysis
3.7 Discrete dynamical stabilization above buckling load
3.7.1 Conservative case
3.7.2 Finding stability zones
3.7.3 Stability Zone Characteristics
3.7.4 Nonconservative case
3.8 Conclusions
4 Time-periodic modal analysis
4.1 Modal-Floquet Transformation
4.2 Free vibration
4.2.1 Floquet form computation and visualisation
4.2.2 Projection on Floquet Forms
4.2.3 Numerical application
4.3 Forced vibrations of time-periodic systems
4.3.1 Projecting the force vector
4.3.2 Example of a harmonic external force
4.3.3 Frequency Response Spectrum
4.4 Conclusions
5 Conclusions
Appendices
A Equation of motion of the Ziegler column
B Hill matrix implementation
B.1 Floquet transform
B.2 Fourier transform
B.3 Harmonic Balance
B.4 Hill matrices
B.5 Numerical implementation
C High Frequency Averaging of the statically diverging Ziegler Column in periodic elastic state
C.1 Averaged equations of motion
