Chapter 2 Simple models for structures

Sometimes, a structure may be approximated simply as a rigid point mass $m$ attached to a spring with stiffness $k$ and a damper with coefficient $b$, as shown in fig. 2.1.

A generic fluid dynamic force on the mass is denoted $f$. The displacement $x(t)$ is the sole state variable describing this system. It obeys

Note that this equation is already linear, so a linearization of this equation around any steady state will not alter it. Also, note that an energy equation can be constructed by multiplying this equation by $\text{d}x/\text{d}t$ as

The rate of energy removed by the damper is $b{(\text{d}x/\text{d}t)}^2$ and the work done by the fluid is $f\text{d}x/\text{d}t$.

The spring in the damped mass-spring system can be nonlinear, such as in question 2.2, and so can be the damper.

Determining the scales for $x$ and $t$, as well as the dimensionless parameters from eq. 2.6 is a worthy exercise in dimensional analysis. Performing this analysis and following it with the linear stability analysis in dimensionless terms is left for the astute reader.

The next simplest structure has two degrees of freedom. An example is depicted schematically in fig. 2.3, where they are the rotational degree called pitch, $\theta$, and the translational degree of freedom called heave, $h$. While fig. 2.3 shows a symmetric airfoil, the object could be of arbitrary shape. It has mass $m$ and moment of inertia about the centre of mass $I=m{R}_g^2$, where $R_g$ is the radius of gyration. The object is supported at the centre of support, which is a distance $x_s$ away from the centre of mass, as shown in the figure. A translational spring with linear stiffness $k$, and a torsional spring with rotational stiffness ${\kappa }^{\prime }$ support the object at the centre of support. And the object also experiences an aerodynamic force of lift, $L$, in the direction of heave at the centre of pressure, which is a distance $x_a$ from the centre of mass. We ignore any damping that may be present in this structure for this analysis.

The state variables $h$ and $\theta$ describe the dynamical state of the object. The evolution of the state is governed by force and torque balance around the centre of mass and satisfies the following equations

where the double-dot decoration above $h$ and $\theta$ means the second derivative with time $t$. For what follows, we will ignore the external force of lift, i.e. take $L=0$, and examine the structural dynamics. For this purpose, we determine the steady state to be $h=0$ and $\theta =0$. Perturbation about the steady state are made as $h=0+h^{\prime }$ and $\theta =0+{\theta }^{\prime }$. Assuming the perturbations to be small (i.e. infinitesimal), they satisfy

where $\kappa ={\kappa }^{\prime }+k{x}_s^2$.

These equations may also be expressed in terms of the state vector $𝐱={[h,\theta ]}^T$ as

where $𝑴$ is the mass matrix and $𝑲$ is the stiffness matrix. Note that both are symmetric and positive definite.

An equation for energy can be constructed by multiplying eq. 2.17a with $\dot{h^{\prime }}=\text{d}{h}^{\prime }/\text{d}t$ and eq. 2.17b with $\dot{{\theta }^{\prime }}=\text{d}{\theta }^{\prime }/\text{d}t$ and adding them to yield

or in matrix notation, where $\dot{𝐱}=\text{d}𝐱/\text{d}t$,

The energy equation guides the rescaling towards a dimensionless formulation with the rescaling

The dimensionless equations governing the perturbations are

The parameter $\delta$ serves as the coupling between the two degrees of freedom. If $\delta =0$, pitch and heave are uncoupled and each have a natural frequency of ${\omega }_a$ and 1, respectively. Because energy neither grows nor decays in this system, we expect pure oscillations for the two degrees of freedom, even when they are coupled. Since these equations are linear, we may seek exponentially growing solutions $e^{i\omega \tilde{t}}$, so that $s=i\omega$ satisfies

Solving the eigenvalue equation yields for the two values of ${\omega }^2$

The first property we note is that ${\omega }^2$ is real, so the growth rate is purely imaginary and thus there is neither growth nor decay of the perturbations. When $\delta =0$, the frequencies agree with our expectation for the uncoupled degrees of freedom. A real physical system will show decay of these perturbations owing to some damping.

The corresponding eigenvectors are

where $c$ is a constant, which ensures that the lengths of the two forms of the eigenvector are equal to each other. It can be readily seen from eq. 2.26 that when $\delta =0$, the eigenvectors have either the $h$ component or $\theta$, but not both, thus verifying that in this case the two modes are indeed decoupled. In this case, naturally the dot product between the two independent eigenvectors is zero, so they are orthogonal. The orthogonality between the two eigenvectors for the case $\delta \ne 0$ can be easily verified using the forms in eq. 2.26 and the property that ${\omega }_1^2+{\omega }_2^2=1+{\omega }_a^2$. This nifty property is left for the reader to discover.

In general, an abstract structure with a finite degrees of freedom may be represented using the state variable vector $𝐱$, where the vector has as many components as the degrees of freedom. The state vector can obey an equations for its evolution written as eq. 1.3 from section 1.4. Let us consider the case where the undeformed state is given by $𝐱=0$ is the steady state, where there are no external forces, and all displacements and internal forces vanish. However, in the absence of damping, the dynamics must conserve energy, and therefore, the following form for the evolution emerges for the perturbation ${𝐱}^{\prime }$ when linearized about the undeformed steady state.

where $𝑴$ is the mass matrix and $𝑲$ the stiffness matrix. Both $𝑲$ and $𝑴$ are symmetric and positive-definite matrices. These matrices in general may not be diagonal or sparse. The system conserves the energy $E$ defined as

For this reason, any perturbation made around the steady state must oscillate, say with frequency $\omega$. The frequency satisfies the eigenvalue equation

where the state is assumed to evolve as ${𝐱}^{\prime }=\widehat{𝐱}{e}^{i\omega t}$. Solution of this equation yields both the many natural frequencies ${\omega }_i$ and the corresponcing mode shapes ${\widehat{𝐱}}_i$, for $i=1,2,\mathrm{\dots },n$, where $n$ is the number of degrees of freedom. Because of the symmetry and positive-definiteness of $𝑲$ and $𝑴$, we are guaranteed that the frequencies are real and that the various modes are mutually orthogonal, i.e.

Using the properties of ${\widehat{𝐱}}_i$ and ${\omega }_i$, we may construct the general evolution of the perturbation in the presence of an external forcing, say from the surrounding fluid. If the infinitesimal forcing on the structure is written as $𝐟$, it responds according to

It is possible to look for a solution to eq. 2.31 of the form

where $a_1(t)$, $a_2(t)$, …, $a_n(t)$ are amplitudes of the various modes, which are to be determined. (Here we assume that the mode shapes ${\widehat{𝐱}}_1$, ${\widehat{𝐱}}_2$, …, ${\widehat{𝐱}}_n$ have been precomputed.) Substituting eq. 2.32 into eq. 2.31, then yields

where we have used eq. 2.29 and the double-dot decoration denoted second derivative in $t$. Now, exploit orthogonality of the eigenvectors from eq. 2.30, and take a dot product with one of the eigenvectors ${\widehat{𝐱}}_j$. Noting that all the terms in the sum vanish, except for the $i=j$, then yields the equation for the amplitude $a_i(t)$ as

In this manner, the response of a structure to an external forcing can be determined by projection on the natural modes of oscillation of the structure.

A string or a wire that is stretched along it length (say, in the direction of the $x$-axis) presents a simple model for a continuum elastic structure. The wire length is $L$, tension in the wire is $T$ and it has a mass $\mu$ per unit length, as shown in fig. 2.4. The point on this string at the coordinate $x$ displaces in a direction perpendicular to the length (the $Y$-direction) by a distance $y(x,t)$. If an external force $f$ per unit length is applied to the string along the $y$-direction, the displatement obeys

The displacement of the string is zero at the two ends $x=0$ and $x=L$.

Figure 2.4 also shows a membrane stretched along $x$ with tension $T$ per unit length. The tension perpendicular to plane of the page (i.e. along the $z$ direction) does not enter the formulation because ${\displaystyle \frac{\partial }{\partial z}}=0$. In this case, $\mu$ is the mass of the membrane per unit area, and $f$ is the external force per unit area perpendicular to the membrane. The membrane displacement $y(x,t)$ is also governed by eq. 2.35. Because of the analogy between the one-dimensional membrane and a stretched string, we will only consider the string from now on.

The stretched string satisfies the energy equations

Here the integral proportional to $\mu$ is the kinetic energy, the one proportional to $T$ is the elastic potential energy, and the right-hand side is the work done by the external force. For the stretched membrane a trivial integral along $z$ may also be included.

Let us first determine the natural modes of vibrations of the string (or the membrane, as the case may be). For the natural modes, we set $f=0$. The equilibrium shape of the string in this case is $y=0$. Perturbing the displacement from the steady state as $y=0+y^{\prime }$ yields the equation for the perturbation

This equation is already linear so no further approximation for an infinitesimal perturbation needs to be made.

To make the formulation dimensionless, we define $\tilde{t}={\displaystyle \frac{t}{L}}\sqrt{{\displaystyle \frac{T}{\mu }}}$ and $\tilde{x}=x/L$, so eq. 2.37 becomes

and $y^{\prime }(\tilde{x}=0,\tilde{t})=y^{\prime }(\tilde{x}=1,\tilde{t})=0$.

When $f=0$, energy is conserved, so we expect perturbations to neither grow nor decay. Hence we seek a solution to eq. 2.38 of the form $y^{\prime }=\widehat{y}(\tilde{x})e^{i\omega \tilde{t}}$. The mode shape $\widehat{y}(\tilde{x})$ satisfies

with $\widehat{y}=0$ at $\tilde{x}=0$ and 1. A non-trivial solution only exists if $\omega ={\omega }_n=n\pi$ for $n=1,2,\mathrm{\dots }$, in which case the mode shape is

The mode shapes satisfy the orthogonality condition

which can be easily verified. (Note that the notion of dot products is replaced by integrals in this case.)

If an external force acts on the string, we can determine the response of the string in the following manner. First, we transform eq. 2.35 to a dimensionless form by using $\tilde{f}=f{L}^2/T$ to yield

Let us write the perturbation as a linear combination of the modes as

Substituting eq. 2.43 in eq. 2.42, with the intention of determining how the amplitudes $a_n(t)$ respond to the forcing $\tilde{f}$, yields

where, as before, the double-dot decoration denotes second derivative with $\tilde{t}$. We use orthogonality to decouple the evolution of distinct modes. For doing so, we multiply with ${\widehat{y}}_m(\tilde{x})$ and integrate along $\tilde{x}$ as

The orthogonality condition from eq. 2.41 then facilitates the simplification

The stretched string serves as an example that applies the general formalism of normal modes to determine the response of a continuum structure to external forcing. To test their understanding, the reader is invited to apply the formalism to the example of an Euler beam. The state of the beam is again described by the variable $y(x,t)$, which satisfies the governing equation

Here $\mu$, $T$, and $f$ are the same as before, and $B$ is the bending stiffness or bending rigidity of the beam cross section. In what follows, we will take the tension $T$ to be zero. This equations must be supplemented with two boundary conditions at each end, which are of the form

1. 1.

   A given linear combination of displacement and shear force is specified,

   $$ay-Bb\frac{\partial {}_{}{}^{3}y}{\partial x^3}=r_1(t),$$

   (2.48)

   and

2. 2.

   A given linear combination of the slope and the bending moment is specified,

   $$c\frac{\partial y}{\partial x}+Bd\frac{{\partial }^{2}y}{\partial x^2}=r_2(t),$$

   (2.49)

where $a$, $b$, $c$, $d$, $r_1$ and $r_2$ are specified constants. Without loss of generality, $b$ and $d$ may be taken to be unity so long as they are non-zero. These boundary conditions may be interpreted as the ends of the beam being supported by linear and torsional springs of stiffness $a$ and $c$, respectively, and an external force ($r_1$) or torque ($r_2$) being applied there.

In this chapter, we considered simple models for some structures and analyzed their behaviors in terms of perturbations about the steady state. In cases where we expected harmonic oscillations, we were able to apply the formalism of linear theory to determine the mode shape and natural frequency by assuming a time dependence of $e^{i\omega t}$, for some real $\omega$. In cases where we expected an instability, we were also able to apply linear theory to determine the exponential growth of the unstable perturbations.

In this process, we found that while the deformation of the structure itself may be most complicated, it may be decomposed into mode shapes, which exhibit a simple harmonic behaviour for their amplitude. We also saw the method of projection, which exploits orthogonality of modes, to decouple the influence of a general external forcing on individual modes.

We also constructed the energy conservation principle for the structures we considered, including the influence of the external forcing on the rate of change of energy. The energy of a perturbation is a good indicator of its growth or decay, and therefore the derivation of an energy equation is instrumental in stability analyses.

The matter in this chapter is only but an introduction to the otherwise rich subject of structures and elasticity, and especially to the topic of simplified descriptions of them. Prime topic in this subject, which we omitted, is the theory of rods, shells and plates, which presents a one- and two-dimensional approximation to three-dimensional objects which are thin along the remaining dimension(s). Furthermore, structural instabilities are a formidable topic in their own right, without the need for the complicated coupling with the fluid flow.

We conclude with the following question on the buckling of elastic beams.