Abstract
We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring.
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Notes
For an isotropic naturally straight rod, any static solution has a constant twist density as shown by Eq. (11).
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This research is supported by USPHS Research Grant GM 34809.
Appendices
Appendix A: Elastic Ring Material Frame Directors
We derive in this appendix explicit expressions for the material frame directors \(\hat{\boldsymbol {d}}_{1}(\hat{s},\hat{t})\) and \(\hat{\boldsymbol {d}}_{2}(\hat{s},\hat{t})\) of the elastic ring.
We choose, without any loss of generality, the initial condition \(\hat{\boldsymbol {d}}_{1}^{0}(-\pi,\hat{t})=\boldsymbol {Z}\), and with the use of Eq. (4a) and the condition \(\hat{\boldsymbol {d}}_{1}\cdot\hat{\boldsymbol {d}}_{3}=0\), we obtain
where \(g_{1}(\hat{s},\hat{t})\) and \(g_{2}(\hat{s},\hat{t})\) are two unknown functions.
We now introduce the vector \(\hat{\boldsymbol {q}}(\hat{s},\hat{t})\) such that \(\hat{\boldsymbol {d}}_{i}(\hat{s},\hat{t}) - \hat{\boldsymbol {d}}_{i}^{0}(\hat{s}) = \hat{\boldsymbol {q}}(\hat{s},\hat {t})\times\hat{\boldsymbol {d}}_{i}^{0}(\hat{s})\) (the existence of such a vector is guaranteed since we assume that the material frame of the rod remains orthonormal). The identification with the tangent expansion (Eq. (23)) leads to
where \(\xi(\hat{s},\hat{t})\) is an unknown function. We now compare the expansions for \(\hat{\boldsymbol {d}}_{1}(\hat{s},\hat{t})\) and \(\hat{\boldsymbol {q}}(\hat {s},\hat{t})\times\hat{\boldsymbol {d}}_{1}^{0}(\hat{s})\) and find
In addition, we also have from the definition of the curvature vector (Eq. (4a)) the relation \(\hat{\varOmega}_{3}(\hat{s},\hat{t}) = \hat{\boldsymbol {d}}_{1}'(\hat{s},\hat {t}) \cdot ( \hat{\boldsymbol {d}}_{3}(\hat{s},\hat{t})\times\hat{\boldsymbol {d}}_{1}(\hat {s},\hat{t}) )\), which at the first order is
In the static circular configuration we do not require \(\hat{\boldsymbol {d}}_{1}^{0}(-\pi,t) = \hat{\boldsymbol {d}}_{1}^{0}(\pi,t)\), that is, \(\hat{\varOmega}_{3}^{0}\) is not necessarily an integer. In other words, when the static elastic ring is formed by bringing the two ends of the rod together, the material frame directors do not need to coincide. The difference between those vectors is imposed and constant such that \(\hat{\boldsymbol {d}}_{1}(-\pi,t)-\hat{\boldsymbol {d}}_{1}(\pi,t) = \hat{\boldsymbol {d}}_{1}^{0}(-\pi,t)-\hat{\boldsymbol {d}}_{1}^{0}(\pi,t)\). This implies that \(\xi(-\pi,\hat{t})=\xi(\pi,\hat{t})\) for both periodic and clamped boundary conditions; in the case of clamped boundary conditions this condition becomes \(\xi(-\pi,\hat{t})=\xi(\pi,\hat{t})=0\). In particular, it implies that
for periodic and clamped boundary conditions (see Eqs. (32a), (32b), (32c) and (33a)–(33f)).
Appendix B: Second-Order Perturbation
We give in this appendix the details of a second-order perturbation, that is, the expansions of the rod kinematical quantities up to O(ϵ 3). These expansions are needed to assess the topology of the normal modes of vibration as well as to calculate the energies of the modes.
We first introduce the second-order perturbation of the rod centerline as
We obtain from the definition of the tangent (Eq. (1)) and the condition of inextensibility (Eq. (2)) the following expansion
The square of the curvature vector is given by \(\hat{\boldsymbol {K}}(\hat{s},\hat{t})^{2}=\hat{\boldsymbol {d}}_{3}'(\hat{s},\hat {t})\cdot\hat{\boldsymbol {d}}_{3}'(\hat{s},\hat{t})\), that is,
The twist density in the rod up to second order is defined as
and the axial spin velocity as
As for the first-order perturbation, we can use the compatibility condition (Eq. (8)) to relate the axial spin velocity and the twist density and we find
Finally, the periodic boundary conditions at the second order are similar to those at the first order (Eqs. (32a), (32b) and (32c)) for the functions \(\varLambda(\hat{s},\hat{t})\), \(\varUpsilon(\hat{s},\hat{t})\) and \(\varDelta (\hat{s},\hat{t})\). The clamped conditions can easily be extended to second order. We have, in this case,
Appendix C: Elastic Ring Energy
We derive in this appendix the expression of the energy of the elastic ring. The energy can be calculated up to second order of the perturbation, that is, it can be obtained as an O(ϵ 3) expansion.
We recall that within the Kirchhoff elastic rod theory, the energy of the rod is defined as
where \(\mathcal{E}_{\mathrm{el}}\) is the potential energy of elastic deformation and \(\mathcal{E}_{\mathrm{kin}}\) is the kinetic energy.
The elastic energy is defined as
where \(\hat{\boldsymbol {K}}(\hat{s},\hat{t})\) is the binormal curvature vector (see Eq. (6)) and \(\hat{\varOmega}_{3}(\hat{s},\hat{t})\) is the twist density in the rod (we recall that the undeformed state corresponds to a straight and untwisted rod, that is, a rod configuration with zero curvature and zero twist density).
The kinetic energy of the rod is given by
We express the energy of elastic deformation and the kinetic energy as
It follows from the expansion of the squared curvature (Eq. (B.3)), that the zero-order and first-order terms of the elastic energy for periodic and clamped boundary conditions are, respectively,
and
The second-order term of the energy of elastic deformation is given by
For the kinetic energy we directly have \(\hat{\mathcal{E}}_{\mathrm{kin}}^{[0]}=0\) and \(\hat{\mathcal{E}}_{\mathrm{kin}}^{[1]}=0\), since the equilibrium solution around which we linearize the Kirchhoff equations is static. For the second-order term we find
The periodic and clamped boundary conditions imply that the integrated terms (i.e., boundary terms) in the previous expansions are zero. It follows that the total energy of the elastic ring is
where \(\mathcal{E}^{0}\) is the energy of the static elastic ring.
3.1 C.1 Normal Modes Energy Time Average
We now calculate the time average (see Eq. (46)) of the elastic ring energy for a given normal mode. We omit the dependence of the modes on the mode index n for the sake of clarity. We start with the potential energy of elastic deformation
where we use Eq. (43) to express the second order of the twist density. For the kinetic energy we obtain
With the use of the conservation law for the twist density (given by Eq. (15)) the last term in the previous expression can be written as
We can substitute in the last integral the third linearized Kirchhoff equation (see Eqs. (30a), (30b) and (30c)) to obtain
It follows for the time average of the kinetic energy that
The time averages of the potential and kinetic energies can be rewritten with the help of integrations by parts in such a way that the equality \(\langle\hat{\mathcal{E}}_{\mathrm{kin}}^{[2]}(\hat{t})\rangle= \langle\hat{\mathcal{E}}_{\mathrm{el}}^{[2]}(\hat{t})\rangle\) directly follows from the linearized Kirchhoff equations (Eqs. (30a), (30b) and (30c)). The details of such calculations are left to the reader.
Appendix D: Normal Mode Analysis for Periodic Boundary Conditions
This appendix presents detailed results for the normal mode analysis of an elastic ring subjected to periodic boundary conditions.
4.1 D.1 Normal Mode Frequencies
The normal-mode frequencies for the periodic case are the roots of the equation \(\det\underline{\boldsymbol{M}}(\hat{\sigma}_{n},n)=0\), where \(\underline{\boldsymbol{M}}\) is obtained by substituting the expressions given in Eqs. (48a), (48b) and (48c) into the linearized Kirchhoff equations (Eqs. (30a), (30b) and (30c)), yielding
The determinant of \(\underline{\boldsymbol{M}}\) is given by a polynomial of degree six containing only even powers of \(\hat{\sigma}_{n}\),
where the coefficients C i are
Since the determinant is a cubic equation in \(\hat{\sigma}_{n}^{2}\) it can be solved with the help of Cardano’s formulae. We introduce the following auxiliary quantities
where Θ is the discriminant, which depending on its sign, determines the quality (real or complex) and the degeneracy of the roots. The three roots are given by
The normal-mode frequencies are then obtained by taking the positive square root of each expression. If a frequency happens to have a non-zero imaginary part, the associated normal mode will grow exponentially, that is, the normal mode is unstable.
Frequencies for the Zero Rotational Inertia Approximation
For the zero rotational inertia approximation, the same method can be used to obtain the normal-mode frequencies. In this case, the linearized equations (Eqs. (31a) and (31b)) are simpler and the linear system is reduced to a two-dimensional problem. Moreover, in the limit \(\hat{U}^{0}\rightarrow0\) the two equations are uncoupled. One equation describes the in-plane motion of the ring and the other equation the out-of-plane motion. It follows that in this limit, the identification of the modes, in-plane or out-of-plane, is direct (see the discussion in Sect. 5.1). The frequencies for the in-plane and out-of-plane modes are given by the following expressions
where the superscript “ip” and “op” designate the in-plane and out-of-plane modes, respectively. Note that analytical expressions for the case \(\hat{U}^{0}=0\) and η≠0 are given in [14, 15].
Classification of the Modes
We take the limit η→0 in Eqs. (D.5a), (D.5b) and (D.5c) to identify the modes. We first find that \(\hat{\sigma}_{n}^{1} \rightarrow+\infty\), which shows that this frequency is related to the twisting mode. Then, we identify the limits of \(\hat{\sigma}_{n}^{2}\) and \(\hat{\sigma}_{n}^{3}\) with the expressions given in Eqs. (D.6a) and (D.6b). Finally, we obtain the following classifications
where the superscript “tw” designates the twisting mode.
4.2 D.2 Normal Mode Amplitudes
Once the normal-mode frequencies are known we obtain the amplitudes by solving the linear system given by Eq. (49), together with the normalization condition for the kinetic energy as written in Eq. (47). The outcome of these calculations is the set of amplitudes \((A^{\alpha}_{n}, A^{\gamma}_{n}, A^{\delta}_{n})\) as functions of the imposed twist density, i.e., the imposed linking number, \(\hat{U}^{0}\). The amplitude of the radial component \(\beta(\hat {s},\hat{t})\) (Eq. (22)), denoted \(A^{\beta}_{n}\), is obtained with the use of the relation \(\beta(\hat{s},\hat{t})+\gamma'(\hat{s},\hat{t})=0\).
Appendix E: Normal Mode Analysis for Clamped Boundary Conditions
Here we present the details of our numerical method for the clamped boundary conditions.
5.1 E.1 Differential System
We start by rewriting the linearized Kirchhoff equations (Eqs. (30a), (30b) and (30c)) as a first-order differential system. It is convenient for the sake of clarity, to write this system in terms of the cylindrical components of the kinematical and mechanical variables. For example, we will use \(\hat{r}_{z}\), \(\hat{r}_{r}\) and \(\hat{r}_{\theta}\) to denote the first-order components of the centerline path, that is, \(\hat{r}_{z}=\tilde{\alpha}\), \(\hat{r}_{r}=-\tilde {\gamma}'\) and \(\hat{r}_{\theta}=\tilde{\gamma}\). It follows that
where we have again omitted labeling these functions and the frequency in terms of the wavenumber n. Note that this differential system is of dimension 12, which is also the dimension of the system given by Eqs. (30a), (30b) and (30c). The twelve boundary conditions simply read as
5.2 E.2 Numerical Strategy
The first step is to identify the different solutions for \(\hat{U}^{0}=0\) and then each of these solutions can be continued for different values of \(\hat{U}^{0}\) (see [41] for an introduction to numerical continuation of ODE solutions). Each solution corresponds to a normal mode of vibration of the clamped ring. The integration and the continuation of the solutions of this differential system have been carried out using the AUTO-07p software (see [42–45]). The main difficulty of our problem is that the solutions are subjected to an integral constraint given by the normalization condition (see Eq. (47)). The AUTO-07p software ensures that this constraint is satisfied along the numerical path obtained for each solution (we found that the relative error along the different paths was of the order of 10−8–10−9).
The successive continuation steps are as follows:
-
we start with the trivial solution (every function is zero for every value of \(\hat{s}\)),
-
we perform a first continuation along the parameters \(\hat{\sigma}_{n}\) and \(\mathcal{E}_{\mathrm{kin}}\): the bifurcations occurring along this path correspond to the normal-mode solutions,
-
for each bifurcation we then perform a new continuation for which the frequency is fixed and the kinetic energy parameter is increased until the normalization condition is satisfied,
-
the previous paths are continued with the frequency and η as free parameters until a fixed value of η is reached,
-
finally, the last solution of the previous continuation is used to start a new path on which the energy is held constant and only the frequency and the imposed torsional moment \(\hat{U}^{0}\) can change; this last path corresponds to the set of solutions for distinct values of \(\hat{U}^{0}\).
The input files used to generate the solutions are available upon request to the authors.
Appendix F: Additional Results
6.1 F.1 Twisting Amplitudes for Flexural Modes
We give in Fig. 9 the twisting amplitudes \(\tilde{\delta}_{n}^{\mathrm {RMS}}\) for the flexural modes obtained with periodic and clamped boundary conditions. These amplitudes can be compared to those given in Fig. 5. It is clear that the twisting amplitudes are much smaller than the flexural amplitudes. In addition, the twisting amplitudes scale as η 2, which means that for smaller values of η these amplitudes will also become much smaller. We note that in the clamped case, the twisting amplitudes always reach zero when the modes become unstable. Also of interest is the fact that the twisting amplitudes increase with the mode index, unlike the flexural amplitudes.
6.2 F.2 Twisting Modes for Periodic Boundary Conditions
The frequencies and flexural RMS amplitudes for the twisting modes of the periodic ring are given in Fig. 10. We see that even for large values of η, the twisting mode frequencies are significantly higher than those of the flexural modes (compare with the frequencies in Fig. 3). In addition, the in-plane and out-of-plane RMS amplitudes are always much smaller than the twisting RMS amplitude, which for periodic boundary conditions is close to 0.65147.
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Clauvelin, N., Olson, W.K. & Tobias, I. Effect of the Boundary Conditions and Influence of the Rotational Inertia on the Vibrational Modes of an Elastic Ring. J Elast 115, 193–224 (2014). https://doi.org/10.1007/s10659-013-9453-2
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DOI: https://doi.org/10.1007/s10659-013-9453-2