Stability analysis of entrained solutions of the non-autonomous modified hybrid Van der Pol/Rayleigh oscillator: theory and application to pedestrian modelling

Andrea Trovato; Anil Kumar; Silvano Erlicher

doi:10.1007/s12356-014-0034-2

Annals of Solid and Structural Mechanics

December 2014, Volume 6, Issue 1–2, pp 1–16 | Cite as

Stability analysis of entrained solutions of the non-autonomous modified hybrid Van der Pol/Rayleigh oscillator: theory and application to pedestrian modelling

Authors
Authors and affiliations

Andrea Trovato
Anil Kumar
Silvano ErlicherEmail author

Original Article

First Online: 08 July 2014

Received: 30 July 2013

Accepted: 19 June 2014

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1 Citations

Abstract

In this article, the entrained response of the modified hybrid Van der Pol/Rayleigh (MHVR) oscillator undergoing a periodic excitation is analyzed. Based on a large experimental database, this self-sustained oscillator was originally proposed by the authors to model the lateral ground force of a pedestrian walking on a rigid floor. In this situation, there is no external excitation on the oscillator (autonomous regime). In a successive development, the authors used the MHVR oscillator in the non-autonomous regime to model the lateral oscillations of a pedestrian walking on a periodically moving floor. In the same work, the MHVR oscillator was analyzed in terms of amplitude of the entrained response, i.e. a solution having constant amplitude and the same frequency as the one of the given periodic excitation. The main goal of the present paper is the stability analysis of entrained responses. Some theoretical results are first discussed. Then, these theoretical notions are applied to the pedestrian modelling problem: the conditions allowing stability of the solution are used to compute the percentage of pedestrians of a given population that can synchronize their walk with a given periodic floor motion. Finally, these model predictions are compared with experimental results concerning pedestrians walking on a periodically moving floor.

Keywords

Pedestrian lateral force Self-sustained non-autonomous oscillator Stability analysis Synchronization Frequency entrainment Modified hybrid Van der Pol/Rayleigh oscillator

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Notes

Acknowledgments

The Present work has been partially supported by the funds of European Project HITUBES: “ Design and Integrity Assessment of High Strength Tubular Structures for Extreme Loading Conditions”, Grant No. RFSR-CT-2008-00035.

Appendix 1: Perturbation analysis of the steady entrained solution

It has been shown by [4] that by replacing the periodic solution (Eq. 7) into Eq. (5), one obtains the following equations:

$$\begin{aligned} \left\{ \begin{array}{l} \epsilon \tilde{\omega }\left( {-}\frac{\tilde{\omega }^{2}-1}{\epsilon \tilde{ \omega }}R{+}\frac{\gamma }{4}\tilde{\omega }R^{3}\right) =A_{d}\tilde{\omega } ^{2}\cos \left( \theta \right) =\frac{A_{acc}}{\omega _{0}^{2}}\cos \left( \theta \right) \\ \epsilon \tilde{\omega }\left( R{-}R^{3}\frac{1}{4}\left( \beta +3\delta \tilde{\omega }^{2}\right) \right) =A_{d}\tilde{\omega }^{2}\sin \left( \theta \right) =\frac{A_{acc}}{\omega _{0}^{2}}\sin \left( \theta \right) \end{array} \right. \end{aligned}$$

(27)

Moreover, by inserting the expression of the perturbed solution (Eq. (12)) into Eq. (5), one obtains

$$\begin{aligned}&\ddot{{\mathrm{v}}}\left( \tau \right) +{\mathrm{v}}\left( \tau \right) \nonumber \\&-\epsilon \dot{\mathrm{v}}\left( \tau \right) \left( \begin{array}{l} 1{-}\beta \left( R^{2}\left( \frac{1}{2}+\frac{1}{2}\cos {\left( 2\tilde{ \omega }\tau +2{\theta }\right) }\right) {+}{\mathrm{v}}^{2}\left( \tau \right) +2{\mathrm{v}}\left( \tau \right) R{\cos \left( \tilde{\omega }\tau +{\theta } \right) }\right) \\ {-}\gamma \left( -R^{2}\tilde{\omega }\frac{1}{2}{\sin \left( 2\tilde{\omega } \tau +2{\theta }\right) -}R\tilde{\omega }{\sin \left( \tilde{\omega }\tau +{ \theta }\right) }{\mathrm{v}}\left( \tau \right) \right) \\ {-}\gamma \left( R{\cos \left( \tilde{\omega }\tau +{\theta }\right) }\dot{\mathrm{v}}\left( \tau \right) +{\mathrm{v}}\left( \tau \right) \dot{\mathrm{v}} \left( \tau \right) \right) \\ {-}\delta \left( R^{2}\tilde{\omega }^{2}\left( \frac{1}{2}-\frac{1}{2}\cos { \left( 2\tilde{\omega }\tau +2{\theta }\right) }\right) {+}\dot{\mathrm{v}} ^{2}\left( \tau \right) -2\dot{\mathrm{v}}\left( \tau \right) R\tilde{\omega } {\sin \left( \tilde{\omega }\tau +{\theta }\right) }\right) \end{array}\right) \nonumber \\&+\epsilon R\tilde{\omega }\left( \begin{array}{l} -\beta \left( 2R\frac{1}{2}{\sin \left( 2\tilde{\omega }\tau +2{\theta } \right) }{\mathrm{v}}\left( \tau \right) +{\sin \left( \tilde{\omega }\tau +{ \theta }\right) }{\mathrm{v}}\left( \tau \right) ^{2}\right) \\ -\gamma \left( -R\tilde{\omega }\left( \frac{1}{2}-\frac{1}{2}\cos {\left( 2 \tilde{\omega }\tau +2{\theta }\right) }\right) {\mathrm{v}}\left( \tau \right) +R\frac{1}{2}{\sin \left( 2\tilde{\omega }\tau +2{\theta }\right) }\dot{\mathrm{v}}\left( \tau \right) \right) \\ -\gamma \left( {\sin \left( \tilde{\omega }\tau +{\theta }\right) } \dot{\mathrm{v}}\left( \tau \right) {\mathrm{v}}\left( \tau \right) \right) \\ -\delta \left( -2R\tilde{\omega }\left( \frac{1}{2}-\frac{1}{2}\cos {\left( 2 \tilde{\omega }\tau +2{\theta }\right) }\right) \dot{\mathrm{v}}\left( \tau \right) +{\sin \left( \tilde{\omega }\tau +{\theta }\right) } \dot{\mathrm{v}} \left( \tau \right) ^{2}\right) \end{array} \right) \nonumber \\&=0 \end{aligned}$$

(28)

The external excitation term related to $A_{acc}$ does not appear in Eq. ( 28), since it cancels out together with the terms fulfilling the equilibrium equations (27). The third order harmonics have also been neglected, according to the harmonic balance method and to the assumption of first type stability entailing the particular form of ${\mathrm{v}}\left( \tau \right)$ given in Eq. (13). The linearization of Eq. (28) leads to

$$\begin{aligned} \begin{array}{l} \ddot{{\mathrm{v}}}\left( \tau \right) +{\mathrm{v}}\left( \tau \right) \left( 1-\epsilon \beta R^{2}\tilde{\omega }{\sin \left( 2\tilde{\omega }\tau +2{ \theta }\right) +}\gamma \epsilon R^{2}\tilde{\omega }^{2}\left( \frac{1}{2}- \frac{1}{2}\cos {\left( 2\tilde{\omega }\tau +2{\theta }\right) }\right) \right) \\ -\epsilon \dot{\mathrm{v}}\left( \tau \right) \left( 1{-}\beta R^{2}\left( \frac{1}{2}+\frac{1}{2}\cos {\left( 2\tilde{\omega }\tau +2{\theta }\right) } \right) {+}\gamma R^{2}\tilde{\omega }{\sin \left( 2\tilde{\omega }\tau +2{ \theta }\right) } \right) \\ -\epsilon \dot{\mathrm{v}}\left( \tau \right) \left( -3\delta R^{2}\tilde{ \omega }^{2}\left( \frac{1}{2}-\frac{1}{2}\cos {\left( 2\tilde{\omega }\tau +2{ \theta }\right) }\right) \right) \\ =0 \end{array} \end{aligned}$$

(29)

We rewrite here Eq. (13) and make a Van der Pol transformation between the variables $\left( {\mathrm{v}},\dot{\mathrm{v}} \right)$ and $\left( {B}_{1c},{B}_{1s}\right)$:

$$\begin{aligned} \left\{ \begin{array}{l} {\mathrm{v}}\left( \tau \right) ={B}_{1c}{\cos \left( \tilde{\omega }\tau +{ \theta }\right) +}B_{1s}{\sin \left( \tilde{\omega }\tau +{\theta }\right) }\\ \dot{\mathrm{v}}\left( \tau \right) ={-B}_{1c}\tilde{\omega }{\sin \left( \tilde{\omega }\tau +{\theta }\right) +}\tilde{\omega }B_{1s}{\cos \left( \tilde{\omega }\tau +{\theta }\right) } \end{array} \right. \end{aligned}$$

(30)

with the compatibility condition

$$\begin{aligned} \dot{B}_{1s}{\sin \left( \tilde{\omega }\tau +{\theta }\right) =-}\dot{B}_{1c} {\cos \left( \tilde{\omega }\tau +{\theta }\right) } \end{aligned}$$

(31)

This identity is obtained imposing that the velocity defined by Eq. (30)$_{2}$ is equal to the velocity computed by differentiating Eq. (30)$_{1}$. The acceleration, obtained by differentiating (30)$_{2}$, reads:

$$\begin{aligned} \ddot{{\mathrm{v}}}\left( \tau \right) =-B_{1s}\tilde{\omega }^{2}{\sin \left( \tilde{\omega }\tau +{\theta }\right) -}B_{1c}\tilde{\omega }^{2}{\cos \left( \tilde{\omega }\tau +{\theta }\right) +}\tilde{\omega }\dot{B}_{1s}{\cos \left( \tilde{\omega }\tau +{\theta }\right) -}\dot{B}_{1c}\tilde{\omega }{ \sin \left( \tilde{\omega }\tau +{\theta }\right) } \end{aligned}$$

(32)

By replacing (30) and (32) into Eq. (29) and after some trigonometric developments, one has

$$\begin{aligned} \begin{array}{l} -{B}_{1s}\tilde{\omega }^{2}{\sin \left( \tilde{\omega }\tau +{\theta }\right) -B}_{1c}\tilde{\omega }^{2}{\cos \left( \tilde{\omega }\tau +{\theta }\right) + }\tilde{\omega }{\dot{B}}_{1s}{\cos \left( \tilde{\omega }\tau +{\theta } \right) -\dot{B}}_{1c}\tilde{\omega }{\sin \left( \tilde{\omega }\tau +{\theta }\right) } \\ +{B}_{1s}\left( {\sin \left( \tilde{\omega }\tau +{\theta }\right) }-\epsilon \beta R^{2}\tilde{\omega }\frac{1}{2}\cos \left( \tilde{\omega }\tau +{\theta } \right) {+}\gamma \epsilon R^{2}\tilde{\omega }^{2}\left( \frac{1}{2}{\sin \left( \tilde{\omega }\tau +{\theta }\right) }+\frac{1}{4}\sin \left( \tilde{ \omega }\tau +{\theta }\right) \right) \right) \\ +{B}_{1c}\left( {\cos \left( \tilde{\omega }\tau +{\theta }\right) }-\epsilon \beta R^{2}\tilde{\omega }\frac{1}{2}\sin \left( \tilde{\omega }\tau +{\theta } \right) {+}\gamma \epsilon R^{2}\tilde{\omega }^{2}\frac{1}{4}{\cos \left( \tilde{\omega }\tau +{\theta }\right) }\right) \\ -\epsilon \tilde{\omega }{B}_{1s}\left( {\cos \left( \tilde{\omega }\tau +{ \theta }\right) -}\beta R^{2}\frac{3}{4}{\cos \left( \tilde{\omega }\tau +{ \theta }\right) +}\gamma R^{2}\tilde{\omega }\frac{1}{2}\sin \left( \tilde{ \omega }\tau +{\theta }\right) {-}\frac{3}{4}\delta R^{2}\tilde{\omega }^{2}{ \cos \left( \tilde{\omega }\tau +{\theta }\right) }\right) \\ +\epsilon {B}_{1c}\tilde{\omega }\left( {\sin \left( \tilde{\omega }\tau +{ \theta }\right) -}\beta R^{2}\frac{1}{4}{\sin \left( \tilde{\omega }\tau +{ \theta }\right) +}\gamma R^{2}\tilde{\omega }\frac{1}{2}\cos \left( \tilde{ \omega }\tau +{\theta }\right) {-}\frac{9}{4}\delta R^{2}\tilde{\omega }^{2}{ \sin \left( \tilde{\omega }\tau +{\theta }\right) }\right) \\ =0 \end{array} \end{aligned}$$

where the third order harmonic components have been neglected like in the previous expressions. This equation and the compatibility condition (Eq. 31) constitute a system of two first order equations, which can be rewritten, after some simplifications, as follows:

$$\begin{aligned} \left\{ \begin{array}{l} -2\tilde{\omega }{\dot{B}}_{1s}={B}_{1s}\left( \epsilon \beta R^{2}\tilde{ \omega }\frac{1}{4}-\epsilon \tilde{\omega }+\epsilon \tilde{\omega }\delta R^{2}\tilde{\omega }^{2}\frac{3}{4}\right) +{B}_{1c}\left( -\left( \tilde{ \omega }^{2}-1\right) +\gamma \epsilon R^{2}\tilde{\omega }^{2}\frac{3}{4} \right) \\ 2\tilde{\omega }{\dot{B}}_{1c}={B}_{1s}\left( -\left( \tilde{\omega } ^{2}-1\right) +\gamma \epsilon R^{2}\tilde{\omega }^{2}\frac{1}{4}\right) +{B} _{1c}\left( \epsilon \tilde{\omega }-\epsilon \tilde{\omega }\beta R^{2}\frac{3 }{4}-\frac{9}{4}\epsilon \tilde{\omega }R^{2}\tilde{\omega }^{2}\delta \right) \end{array} \right. \end{aligned}$$

The even superharmonics have been neglected. By using the normalized parameters and variables defined by Eqs. (6), (8) and (9), one obtains the equivalent system:

$$\begin{aligned} \left\{ \begin{array}{l} 2{\dot{B}}_{1s}=\epsilon \left( 1-r^{2}\right) \quad {B}_{1s}+\epsilon \left( \nu -3\alpha r^{2}\right) {B}_{1c} \\ 2{\dot{B}}_{1c}=\epsilon \left( -\nu +\alpha r^{2}\right) {B}_{1s}+\epsilon \left( 1-3r^{2}\right) {B}_{1c} \end{array} \right. \end{aligned}$$

(33)

The corresponding matrix form is given in Eq. (14).

Appendix 2: The condition $\nu =\alpha$

According to the conditions given in Table 5, only the boundary $B_{S}$ ($\lambda =\lambda _{Q}\left( \nu ,\alpha \right)$) of the stability domain can fulfil the condition $\lambda =0$. Hence, using Eq. (24) and setting $\lambda _{Q}\left( \nu ,\alpha \right) =0$, one obtains $\nu =\alpha$. The physical meaning of the condition $\nu =\alpha$ can be explained by rewriting it in terms of non-normalized variables. For the sake of simplicity, this is done in the case where the parameter $\mu$ associated with the nonlinear terms of the MHVR oscillator representing a pedestrian (Eq. 2) is very small and assuming that $\omega$ is very close to the frequency $\omega _{0}$ of the underlying linear system:

$$\begin{aligned} \tilde{\omega }=\frac{\omega }{\omega _{0}}=1+s\hbox { with }s=O\left( \mu \right) \ll 1 \end{aligned}$$

(34)

In this case, the definition of $\nu$ given in Eq. (9) leads to:

$$\begin{aligned} s=\nu \mu \end{aligned}$$

(35)

Under the same assumptions, the parameter $\alpha$ defined in Eq. (8) becomes $\alpha =\frac{\gamma }{\beta +3\delta }.$ By using this expression of $\alpha$ and the identity $\nu =\alpha$, Eq. (35) entails

$$\begin{aligned} s=\alpha \mu =\frac{\mu \gamma }{\beta +3\delta } \end{aligned}$$

(36)

It follows from Eq. (34)

$$\begin{aligned} \omega =\omega _{0}\left( 1+\frac{\gamma \mu }{\beta +3\delta }\right) {:=}\omega _{1} \end{aligned}$$

(37)

Eq. (37) shows that the non-normalized counterpart of the condition $\nu =\alpha$ is $\omega =\omega _{1}.\,\omega _{1}$ is the natural frequency of the autonomous MHVR oscillator [3], i.e. the pedestrian walking frequency on a rigid floor. The approximated expression (37) of the natural frequency $\omega _{1}$ can be found by means of a suitable perturbation technique applied on the MHVR in the autonomous case [3]. Due to the assumptions $\omega _{0}>0,\,\mu >0$ and using Eqs. (34–37), it can be also proven that $\nu -\alpha >0$ implies $\omega >\omega _{1}$ and vice versa.

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Andrea Trovato
- 1
Anil Kumar
- 2
Silvano Erlicher
- 3
Email author

1.Department of StructuresUniversity of CalabriaRendeItaly
2.Department of Mechanical and Industrial EngineeringIndian Institute of Technology RoorkeeRoorkeeIndia
3.Research and Development DepartmentEGIS IndustriesMontreuilFrance

Stability analysis of entrained solutions of the non-autonomous modified hybrid Van der Pol/Rayleigh oscillator: theory and application to pedestrian modelling

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