# Modeling Nonlinear Hydroelastic Response for the Endwall of the Plane Channel Due to Its Upper-Wall Vibrations

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

_{0}, and, since the channel is narrow, 2ℓ >> δ

_{0}. The channel is completely filled with a viscous fluid, the compressibility of which is neglected. The amplitude of the oscillations of the upper wall is z

_{m}<< δ

_{0}. At the left end of the channel, it is assumed that the pressure p

_{0}is constant, which is equivalent to the free outflow of fluid into a large cavity. Starting at this point in time, the value of this pressure is taken as the reference value and set equal to zero. The right channel-end is a mechanical seal in the form of a rigid wall supported by a spring with rigid cubic nonlinearity. That means the spring has the nonlinear characteristic of the restoring force, which changes symmetrically depending on its tension–compression, and the spring stiffness increases during compression. The rigid wall at the end of the right channel can move in the direction of the axis with the amplitude x

_{m}<< ℓ. At the right end, we assume that the fluid flow rate coincides with the flow rate due to its displacement by the endwall, so no leaks are present. Moreover, we take into account that the transient processes decay due to the viscosity of the liquid. So, we study the steady, nonlinear, forced vibrations of the endwall of the channel, i.e., the anharmonic vibrations [25,26].

_{x}, u

_{z}are the projections of the fluid velocity vector on the coordinate axes, p is the pressure, ρ is the density, and ν is the fluid kinematic viscosity.

_{0}is the reference pressure value.

_{1}is the coefficient of rigidity of the support spring attached to the linear term, and n

_{3}> 0 is the stiffness coefficient of the support spring attached to the nonlinear cubic term.

## 3. Determination of Endwall Response

_{m}cos(ωt − φ). When we perform the linearization procedure for Equation (15) by the harmonic balance method [25,26], we obtain the following algebraic system:

_{∗}in Equation (19) defines the so-called skeletal curve, which corresponds to the Duffing equation, from which the damping term and driving force are excluded, i.e., for the problem under consideration, in Equation (15) (or in Equation (19)) ${K}_{z}={K}_{x}=0$ is set. In other words, the skeletal curve is the curve of the natural undamped oscillations of a nonlinear conservative system with cubic nonlinearity.

_{3}= 0. In this case, the endwall will undergo harmonic oscillations excited by the vibration of the upper wall of the channel, and expression (18) will be a linear amplitude characteristic that unambiguously connects the amplitude and frequency of the oscillations of the endwall.

_{m}ω = 1 m/s, k = 1.5 with the following geometric and physical–mechanical parameters: ℓ = 0.1 m, δ

_{0}= 0.05 m, b = 0.5 m, m = 0.5 kg, n

_{1}= 10

^{7}kg/s

^{2}, n

_{3}= 9 × 10

^{12}kg/(m

^{2}s

^{2}), ρ = 1.84 × 10

^{3}kg/m

^{3}, and ν = 2.53 × 10

^{−4}m

^{2}/s. The calculated curves of the hydroelastic response of the channel wall are shown in Figure 2.

## 4. Discussion

_{0}= 0.0065 m, b = 0.038π m, n

_{1}= 10.13 × 10

^{3}kg/s

^{2}, ρ = 9.03 × 10

^{2}kg/m

^{3}, ν = 10

^{−4}m

^{2}/s.

_{x}= 2 Ns/m (see Formula (15)) for a system with the parameters presented above. The experimentally determined damping coefficient in the longitudinal direction was 3.547 Ns/m.

_{3}= 0, to reduce the proposed model to a linear one. The calculation result is shown in Figure 3.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Hydroelastic response of the channel wall: 1: linear support spring: black line (n

_{3}= 0), 2: spring with strong cubic nonlinearity: red line (n

_{3}> 0); the dotted line shows the skeletal curve ω

_{;}A and B—frequencies at which the amplitudes of nonlinear oscillations can change abruptly; green lines with arrows—directions of amplitude change.

**Figure 3.**Hydroelastic response of the channel wall. The theoretical curve is based on experimental data [23].

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**MDPI and ACS Style**

Barulina, M.; Santo, L.; Popov, V.; Popova, A.; Kondratov, D.
Modeling Nonlinear Hydroelastic Response for the Endwall of the Plane Channel Due to Its Upper-Wall Vibrations. *Mathematics* **2022**, *10*, 3844.
https://doi.org/10.3390/math10203844

**AMA Style**

Barulina M, Santo L, Popov V, Popova A, Kondratov D.
Modeling Nonlinear Hydroelastic Response for the Endwall of the Plane Channel Due to Its Upper-Wall Vibrations. *Mathematics*. 2022; 10(20):3844.
https://doi.org/10.3390/math10203844

**Chicago/Turabian Style**

Barulina, Marina, Loredana Santo, Victor Popov, Anna Popova, and Dmitry Kondratov.
2022. "Modeling Nonlinear Hydroelastic Response for the Endwall of the Plane Channel Due to Its Upper-Wall Vibrations" *Mathematics* 10, no. 20: 3844.
https://doi.org/10.3390/math10203844