1 Switching Dynamics of Shallow Arches Priyabrata Maharana

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Switching Dynamics of Shallow Arches

Priyabrata Maharana

Department of Mechanical Engineering, Indian Institute of Science, CV Raman Road,

Bengaluru, 560012, Karnataka, India.

priyabratam@iisc.ac.in

Prof. G. K. Ananthasuresh

Department of Mechanical Engineering, Indian Institute of Science, CV Raman Road,

Bengaluru, 560012, Karnataka, India.

suresh@iisc.ac.in

ABSTRACT

This paper presents an analytical method to predict the delayed switching dynamics of nonlinear

shallow arches while switching from one state to other state for different loading cases. We study

an elastic arch subject to static loading and time-dependent loading separately. In particular, we

consider a time-dependent loading that evolves linearly with time at a constant rate. In both the

cases, we observed that the switching does not occur abruptly when the load exceeds the static

switching load, rather the time scale of the dynamics drastically slows down, hence there is a delay

in switching. For time-independent loading this delay increases as the applied load approaches the

static switching load. Whereas, for a time-dependent loading the delay is proportional to the rate of

the applied load. Other than the loading parameters, the delay switching time also depends on the

local curvature of the force-displacement function at the static switching point and the damping co-

efficient of the arch material. The delay switching occurs due to the flatness of the energy curve at

static switching load. Therefore, we linearize the arch near to the static switching point and get a

reduced nonlinear ordinary differential equation to study the switching dynamics of the arch. This

reduced equation allows us to derive analytical expressions for the delay switching time of the. We

further compare the derived analytical results with the numerical solutions and observed a good

agreement between them. Finally, the derived analytical formulae can be used to design arches for

a self-offloading dynamic footwear for the diabetics.

Keywords: Critical slowingdown, linearly varying load, tipping point, pullback attractor.

1. Introduction

Arches are the mechanical structures that shows nonlinear force-displacement characteristics

when a transverse load is applie. This nonlinear behavior is due to the mutual effect of

bending and axial thrust due to the arch deformation. Based on the force-displacement

characteristics, arches are of two kinds: bistable and snap-through. For an bistable arch, the

force-displacement curve has three zero-force configurations, out of which two are stable

states and one is unstable state. On the other hand, the arch is called snap-through when it

has only one zero-force state corresponds to the as-fabricated configuration. In both the

cases, the arch shows spring softening effect initially, becomes unstable by showing negative

stiffness region, and switches to its inverted configuration when the force exceeds a certain

threshold value. In past few decades, the use of arches has been increased drastically in fields

like micro-relays, electromagnetic actuators, micro-valves, mechanical memory components,

retractable devices, consumer products, circuit breakers, and easy-chairs [1-7] due to its

unique force-displacement characteristic. Another application that we are interested in is

designing a self-offloading footwear for the diabetics using snap-through and bistable arches

[8]. The most important factor in designing this kind of footwear is to customize them

according to the person’s weight and the gait cycle. This customization involves the dynamic

analysis of the arches for a certain switching force, and switching and switchback time.

Designing of arches taking a pulse load and the subsequent switchback time from the

deformed shape to the original undeformed state has already been mentioned in our previous

article [wearable]. In this paper, we have covered the switching time of the arch when the

load exceeds the threshold value and the behavior of the arch when the load is time varying.

The force-time relation of human walking is characterized by a double-hump curve [9], in

which the maximum force occurs during flat foot and heel off phase (see Fig. 1) and the lowest

during the mid-stance. Though this a nonlinear curve for a complete gait cycle (black-solid

curve), every individual gate phase can be approximated as a linear curve of slope that

depends on the person’s walking speed (magneta dot curve). So, doing these dynamic

analyses are important in deciding the arch dimensions more precisely to customize the

footwear according to person’s weight and gait cycle speed. We show that for a time-

dependent load, the critical switching load changes and the switching of the arch delay

significantly compared to the static load case.

Fig.1. The vertical force component of the ground reaction forces during gait.

In past few decades several researchers had found out the switching load by

considering it to be quasi-static [10-12], a constant function in time [13] or as an impulse

function of time. In their analysis, they assume the arch switches abruptly to its inverted shape

when the load slightly higher than the threshold value. Despite of applications of bistable

arches for different precission engineering applications, the dynamic behavior is not well

studied, specially the dynamics at the time of switching. Therefore, in this paper we analize

the local behavior of the arch at the static switching point for different loads, and show that

switching of the arch from one stable path to other slows down near the static switching point.

Through analytical modeling we show this slowness of switching of arches and verify them

using numerical methods. When the applied load is time dependent and vary linearly with

time, the switching point of the arch delays by taking more load than the static switching load.

This delaying nature of the arch depends on the rate of change of the applied load and the

arch geometry. This means, the arch does not switch at the static switching load no matter

how slowly we vary the external applied load.

This delay switching occurs at the saddle-node (switching point) due to the nonlinear

character of the arch and can also be observed in fields such as, biological science [14-16],

statistical mechanics [17], chemical science [18], atmospheric science [19], economics [20,

21], optics [22, 23], ecology [24], in mechanics by end shortening oftlat beam [25], and so on.

This phenomena is called the critical slowing down (CSD) [26], and first coined by Racz [17]

where he considered the difference between the critical slowing down of linear and nonlinear

kinetic Ising model. Bonifacio and Meystre [23], taking the advantage of CSD, thought of novel

applications to delay compact optical lines using optical bistability. By taking the controlling

parameter independent of time, they showed that the transition of stability slows down

drastically when the controlling parameter just above the threshold value and this slowness

depends on the closeness of the control parameter to the critical value. Also, when the

controlling parameter is a function of time, particularly a ramp function, the switching point

of the system gets delayed before it finally switches to another stable point. The point to

which the system jumps is called the “tipping point” [25]. This is the point after which the

system change its stability by causing a catastrophic failure. Many researchers use this critical

transition of the system as an early signal of system failure [25, 27] and can be avoided by the

reversal of the control parameter before it reaches tipping point. The time a system takes to

reach the tipping point is called the point of no return [28].

In this paper, we focus on the delaying nature of nonlinear shallow arches in two

different cases, when force is quasistatic; and in another it is ramp function of time. Through

analytical framework, we show that the timescale of the arch decreases compared to the

elastic time scale by obeying a power law relation, and the switching delay as the arch

approaches the static switching point. We discuss the CSD in detail in Section 2, followed by

the theoretical background for the dynamic analysis of the arch in Section 3. In Section 4 we

develop a mathematical framework for the switching dynamics, get the analytical formulation

for the switching time of the arch for time-independent and time-dependent loading. We

extend the method for higher mode approximation in Section 5. The results from both the

analytical and numerical methods are presented and discussed in Section 6 followed by a final

summary in Section 7.

2. Understanding critical slowing down

Due to the nonlinear behavior of arch, there exist multiple displacement values for a paricular

force value. When the applied force is less than a threshold force value,

, there coexist

three equilibrium configurations for a force value out of which one is unstable and other two

are stable configurations. This leads to one unstable branch and two stable branches on the

force-displacement equilibrium plane. The stable equilibrium points corresponds to the

minima on the energy-displacement landscape, whereas the maixima corresponds to a

unstable equilibrium configurations As the load increases, the stable and unstable point

approach each other, coincide, and exchangeing stability at the switching load load

due to

the saddle node bifurcation. As the force approaches

, the curvature of the energy-

displacement landscape becomes flatter gradually, and finally gets zero at the switching point.

When the arch moves through this platue region, its time scale slows down drastically. So,

when

FF

the switching of the arch delay and the arch takes longer time to switch from

one stable point to the other. This phenomena is called the critical slowing down of the arch.

This occurs due to the zero-stiffness value of the arch at the switching point. It results a higher

relaxation time that slows down or delay the dynamics compared to the elastic time scale of

the arch. This slowness increases as the arch approaches the static switching point. When the

force is time-dependent, this delay in the switching of the arch depends on the rate at which

the applied force changes with time. No matter how slowly we vary the load this delay

phenomena occurs at the static switching point of the arch.

Fig. 2. (a) Potential energy-displacement plot for an bistable arch for different loading at the mid-span length of the arch,

FF

(orange-dash curve),

FF=

(black-solid curve), and

FF

(blue-dash curve). The black ball indicates the state of

the arch, (b) static force-displacement behavior of the arch when a time-independent load is applied with red arrow indicating

the switching of the arch after the static switching point, (c) slowdown of arch while switching for different load value when

time-independent load is applied. Here



indicates the closeness of the external applied load to the static switching force,

i.e.,



−=

, (d) behavor of the arch when a time-dependent load is applied. The red curve indicates the path of the

dynamic switching curve w.r.t. the static force-displacement curve (black curve), and the blue colour arrow indicates the

delay in the switching for particular load rate. The yellow patches indicates the state of two stable states (black dot) and one

unstable state (white dot). Exchange of stability occurs at theasddel node bifurcation point (half black and half white dot)

and is called the static switching point.

3. Theoretical formulation

Though our analysis is valid for any kind of arches with any boundary conditions, we

only consider the bistable arches with fixed-fixed boundary conditions for the analysis (see

Fig. xx).We consider an arch of the span

, the in-plane thickness

, out-of-plane width

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1SwitchingDynamicsofShallowArchesPriyabrataMaharanaDepartmentofMechanicalEngineering,IndianInstituteofScience,CVRamanRoad,Bengaluru,560012,Karnataka,India.priyabratam@iisc.ac.inProf.G.K.AnanthasureshDepartmentofMechanicalEngineering,IndianInstituteofScience,CVRamanRoad,Bengaluru,560012,Karnataka,Ind...

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