Mathematical Problems in Engineering

Volume 2016, Article ID 7074206, 13 pages

http://dx.doi.org/10.1155/2016/7074206

## Pneumatic Adaptive Absorber: Mathematical Modelling with Experimental Verification

Institute of Fundamental Technological Research, Ulica Pawinskiego 5B, 02-106 Warszawa, Poland

Received 15 April 2015; Accepted 30 November 2015

Academic Editor: Zhongdong Duan

Copyright © 2016 Grzegorz Mikułowski and Rafał Wiszowaty. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Many of mechanical energy absorbers utilized in engineering structures are hydraulic dampers, since they are simple and highly efficient and have favourable volume to load capacity ratio. However, there exist fields of applications where a threat of toxic contamination with the hydraulic fluid contents must be avoided, for example, food or pharmacy industries. A solution here can be a Pneumatic Adaptive Absorber (PAA), which is characterized by a high dissipation efficiency and an inactive medium. In order to properly analyse the characteristics of a PAA, an adequate mathematical model is required. This paper proposes a concept for mathematical modelling of a PAA with experimental verification. The PAA is considered as a piston-cylinder device with a controllable valve incorporated inside the piston. The objective of this paper is to describe a thermodynamic model of a double chamber cylinder with gas migration between the inner volumes of the device. The specific situation considered here is that the process cannot be defined as polytropic, characterized by constant in time thermodynamic coefficients. Instead, the coefficients of the proposed model are updated during the analysis. The results of the experimental research reveal that the proposed mathematical model is able to accurately reflect the physical behaviour of the fabricated demonstrator of the shock absorber.

#### 1. Introduction

Mechanical energy dissipation is an important task desired in many industry applications [1, 2]. Currently, efforts are given to increase productivity of automated plants and the speed of transportation on production lines. In parallel to increasing the transportation speed, effective means of stopping the objects on the lines are required, which is especially evident in the production processes, where the braking distance is limited due to packaging reasons [3]. The most popular technique is based on hydraulic dampers due to their effectiveness, durability, and favourable volume to force ratio [4–9]. However, in some applications the utilization of fluid-based devices is undesirable due to the possibility of toxic contamination of the goods being produced, for example, in the pharmaceutical or food industry [10, 11]. In such cases damping solutions based on pneumatics can be applied with chemically inactive gases [10, 12–14].

There are known techniques for pneumatic cylindrical shock absorbers used in aeronautical or food industries [10, 15]. However, due to the low viscosity of gases and their compressibility, the energy dissipation efficiency of these devices does not exceed 40%, while the hydraulic dampers are characterized by the efficiency of 80% [5, 15]. There exist a number of patents that propose ways to increase the effectiveness of the cylindrical pneumatic shock absorbers [16–18]. Most of the solutions are based on a double-stage algorithm of operation. After the initial compression of the gas in the cylinder, a mechanically operated valve releases the medium out of the cylinder to the surroundings. By this way the energy accumulated in the compressed gas is dissipated and the spring-back effect is diminished. These solutions increase the effectiveness of the pneumatic absorbers, but they are limited to a single, strictly defined impact energy. When the impact energy is too low, the absorber does not release the gas at the proper moment and the absorption does not take place. Another disadvantage is related to the fact that the compressed gas is released to the surroundings, which introduces the necessity of refilling the device after each working cycle.

An improved solution considered here is based on introduction of a controlled flow between the chambers in the cylinder via the piston. In this way it is possible to dissipate energy of various magnitudes with the efficiency comparable to hydraulic devices (80%). Moreover, the gas is not released to the surroundings, which allows the device to be operated in a repeatable way. Recent developments in functional materials technology allow us to consider a novel approach to adaptive pneumatic shock absorber with utilization of a piezoelectric material for actuation of the device. A piezoelectric multilayered actuator is applied in a miniature valve positioned in the pneumatic cylinder piston [19]. In this paper we focus on mathematical modelling of the cylindrical pneumatic shock absorber with a controlled flow between the internal volumes.

Mathematical modelling of pneumatic actuators is a demanding task due to the necessity of taking into account the thermodynamic properties of the gas and the nonlinearities present in this kind of mechanical system. The nonlinearities exhibit themselves mostly due to the compressibility of the gas, internal friction, and energy transfer by heat.

Pneumatic systems are typically utilized in three domains of applications: suspensions for vibration isolation, actuation in automatics, and mechanical absorbers. Methods of the modelling are strictly related to the field of application.

Many pneumatic systems for isolation and vibration mitigation are developed for suspension of precise measuring instrumentation [20], as well as for large structures: seismic protection of buildings or large installations [21, 22]. The principle for these systems is to suspend the protected object on double chamber interconnected pneumatic springs. In these cases the devices are capable of eliminating or limiting vibrations of small amplitudes in comparison to the scale of the entire structure [23]. Since the devices can be assumed to operate in vibration of small amplitudes, several simplifications to the modelling approach can be assumed; for example, many authors investigate pneumatic systems oscillating with small amplitudes around the equilibrium position, which allows them to assume linearity of the mechanical response [24]. The second important physical phenomenon modelled in these pneumatic structures is the gas flow between the internal volumes of the structures, which has a direct influence on the dissipation properties. In many cases it is acceptable to assume a simplified model of the capillary flow based on the Poiseuille model, which is derived for viscous fluid [24]. This model assumes very low mass flow rates of the fluid, laminar flow, and low average velocity of the fluid.

In contrast to the mentioned analyses, the mathematical model of the PAA investigated in this paper must consider the state of the gas and the internal flow between the volumes enforced in the conditions of PAA under large displacements and high velocities. Such a process is nonstationary and includes large deflections of the piston and time-variant subsonic flow through the valve.

When the pneumatic actuators are to be utilized as actuators in control of applications, the mathematical models tend to be simplified in order to find a linearised version of the plant representation, which allows the further analysis and development of a controller to be based on the classical control theory that operates most efficiently with linear, time invariant plants [25–28]. In these cases the simplification of the models is an advantage. In contrast to utilization of the pneumatic devices as actuators in automation systems, here we consider them as dampers of energy, and we need to precisely analyse the dissipation process from the point of view of its effectiveness. Therefore, a precise thermodynamic model of the structure is developed.

The thermodynamic systems are commonly described with polytropic relation and an assumption of a constant value of the polytropic coefficient. This coefficient is strongly related to the heat exchange in the system. Therefore, a constant value of the polytropic coefficient can be assumed only if the temperature of the object is stabilized, which was not the case for the considered pneumatic shock absorber.

For these reasons, in this paper we propose a numerical method for mathematical modelling of a cylindrical pneumatic dissipater with a controlled flow between internal chambers where the heat transfer, energy balance, and orifice flow are taken into account and thermodynamic state of gas is updated every calculation step.

The paper is divided into six sections, which are organised as follows. Section 2 introduces the structure of the absorber and the principle of its operation. Then in Section 3 analysis of the system is presented and a mathematical model based on thermodynamic analysis is proposed. Experimental methods and hardware are introduced in Section 4, and in Section 5 the results of an experimental verification are given before the conclusions are stated in Section 6.

#### 2. Structure and Principle of Operation of the Adaptive Pneumatic Absorber

The conceptual pneumatic adaptive shock absorber is considered as a piston-cylinder device equipped with a fast operated valve positioned in the piston. A schematic structure of the considered device is presented in Figure 1.