Advances in Acoustics and Vibration

Volume 2017, Article ID 7091425, 8 pages

https://doi.org/10.1155/2017/7091425

## Experimental Investigation on Flutter Similitude of Thin-Flat Plates

^{1}Department of Mechanical Engineering, Bali State Polytechnics, Badung, Bali 80361, Indonesia^{2}Department of Mechanical Engineering, Brawijaya University, Malang, East Java 65144, Indonesia

Correspondence should be addressed to I. P. G. Sopan Rahtika; moc.oohay@akithar_napos

Received 28 December 2016; Revised 20 February 2017; Accepted 27 February 2017; Published 12 March 2017

Academic Editor: Marc Thomas

Copyright © 2017 I. P. G. Sopan Rahtika et al. 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

This paper shows the experimental results of the flutter speed of thin-flat plates with free leading edge in axial flow as a function of plates’ geometry, fluid densities, and viscosities, as well as natural frequencies of the plates. The experiment was developed based on similitude theory using dimensional analysis and Buckingham Pi Theorem. Dimensional analysis generates four dimensionless numbers. Experiment was conducted by placing the thin-flat plates in a laminar flow wind tunnel in order to obtain the relationship among those dimensionless numbers. The flutter speed was measured by varying the flow velocity until the instability occurred. The dimensional analysis gives a map of the flutter Reynolds number as a function of a new type of dimensionless number that is hereby called flutter fluid structure interaction number, thickness-to-length, and aspect ratios as the correcting factors. This map is a very useful tool for predicting the flutter speed of thin-flat plates in general. This investigation found that the flutter Reynolds number is very high at the region of high flutter fluid structure and thickness-to-length ratios numbers; however, it is very sensitive to the change of those two dimensionless numbers. The sensitivity is higher at lower aspect ratio.

#### 1. Introduction

Flutter is a potentially damaging dynamic aeroelastic phenomenon where aerodynamic forces with the natural modes of vibration cause a periodic motion of a structure going unstable. In a certain fluid structure interaction, the aerodynamic forces serve as input energy to the structural vibration. When the system is not damped by the aerodynamic damping, the vibration amplitude will increase and eventually will lead to structural failure. Flutter can occur in a variety of structures, such as in aircraft wings and turbine blades or even on the bridge. Most previous studies on flutter were focused on the prediction of the flutter speed using numerical methods. Very few literatures are available that discuss flutter using experimental similitude method as a tool to predict flutter phenomenon, despite the fact that similitudes offer cost savings in the investigation of the fluid flow phenomena. This scarcity was one that motivated this study.

The flutter phenomenon observed in this research focused on the flutter of thin-flat plates with free leading edge in axial flow. There were numerous studies which have been done on the flutter of the thin-flat plates due to the canonical characteristics of the problem, such as researches by Chad Gibbs et al. [1], Tang et al. [2], Tang and Païdoussis [3], Tang and Dowell [4], Howell et al. [5], and Zhao et al. [6].

Chad Gibbs et al. [1] performed experimental and theoretical work on the flutter of a flat plate with a fixed leading edge using the three-dimensional vortex-lattice method. Gibbs’ report comprehensively described the characteristics of plate flutter as a function of the mass ratio and the aspect ratio. Tang and Païdoussis [3] discussed the flutter of two flat plates which were positioned in parallel with the axial direction of the fluid flow.

The work of Tang and Dowell [4] was about nonlinear flutter and Limit Cycle Oscillation (LCO) of two-dimensional panels in low subsonic flow and later was extended to three-dimensional panels by Tang et al. [2]. The dynamics of the system was built to produce nonlinear models. Zhao et al. [6] discussed both theory and experiment of the flutter of a flat plate. The discussion emphasizes nonlinear analysis to produce Poincaré maps. The nonlinear analysis results were compared to experimental results. Howell et al. [5] used fluid flow interactions to discuss the nature of a cantilevered plate with ideal flow.

The aeroelastic instability of a flexible plate has been investigated using weakly nonlinear analyses by Eloy et al. [7]. Later, a deeper investigation was focused on the origin of the instability hysteresis [8].

Despite the detrimental effect of flutter, there are new research trends to utilize flutter for wind harvesting. The utilization of flutter phenomena for energy harvesting has been explored by Doaré and Michelin [9], Makihara and Shimose [10], and Dunmon et al. [11]. The other works on energy harvesting using a slender structure in the wake of a bluff body were also conducted by Allen and Smits [12]. As a matter of fact, this research is oriented as a theoretical base to explore this application further.

This research chose to explore the free leading edge instead of a fixed leading edge because the chosen configuration experiences flutter at a lower wind speed. A free leading edge plate will experience flutter in its first mode shape as shown later on the experimental results, while the fixed leading edge will experience flutter at the second mode [1, 5, 7]. This advantage of free leading edge is the reason of the selection of this configuration in this research.

Furthermore, the utilization of thin-flat plates for wind harvester requires a map of flutter speed as a function of plates’ geometry for design optimization. One significant part of this research is to generate a flutter speed map for wind harvester design optimization. The authors currently have an on-going development of a micropower generator utilizing flutter of a free leading edge configuration. This is new expansion of flutter energy conversion utilization into the field of microelectromechanical systems. This paper puts a benchmark for this research.

Flutter phenomena of plates continuously have been observed in the last several years. More recent studies on plate or panel flutter were also done by Fernandes and Mirzaeisefat [13], Cunha-Filho et al. [14], Peng and DeSmidt [15], and Yaman [16].

Similitude theory has been well developed and widely used in the field of fluid dynamic. The use of similitude theory continues to grow in the field of structural vibration, for example, Torkamani et al. [17], and acoustic, for example, De Rosa et al. [18]. A work has also been done on structural similitude for flutter of composite plates by Yazdi and Rezaeepazhand [19].

In this study, a testing method is developed using dimensional similitude analysis based on the Buckingham Pi Theorem. Dimensional analysis generated four dimensionless numbers. Experiment was conducted on thin-flat plate which is placed in a wind tunnel. The problem in this research was to formulate relationships between the flutter speed and the affecting parameters. Flutter speed is defined as the velocity of fluid flow at which flutter started happening. The plates’ parameters that affect the flutter speed that were taken into account in this experiment were their length, width, thickness, and natural frequencies, while the fluid’s parameters (in this case, fluid is air) that were taken into account were its density and viscosity. By using the similitude principle, the experimental results were used to obtain the relationship between the dimensionless numbers , , , and .

Later on, it has been observed during the analysis that the first dimensionless number is the Reynolds number measured during flutter and then called flutter Reynolds number. The second dimensionless number is called fluid structure interaction number since it contains the interacting forces. The third and fourth numbers are the thickness-to-length and aspect ratios, consecutively. This study finally discovered the relationship of the flutter Reynolds number as a function of the fluid-structure interaction number and the geometric ratios.

#### 2. Theory of Flutter and Similitude

Theoretically, this research was about implementing the similitude theory to observe the flutter phenomenon of thin-flat plates which then was used to generate a map for predicting the flutter speed. The conceptual theory of this research was developed from the theory of flutter and the similitude theory.

##### 2.1. Flutter

Theoretical work on flutter has been recognized as early as 1878 by Rayleigh [20]. However, the practical work on flutter was later on reported by Theodorsen in 1934. Theodorsen explained the flutter mechanism theoretically and experimentally of the aircraft wing and also the combination of wing-aileron-tab [21–23].

Flutter is an unstable fluid and structure interaction. The dynamics of a thin-flat plate’s structure is stable by itself. However, when it is placed in a moving air, the aerodynamic forces shift the stability of the system. The system will become unstable when the air speed reaches a certain speed. This speed is called the flutter speed.

The dynamics of a thin-flat plate can be modeled as a matrix equation of motion where the plate’s structural dynamics is subjected to aerodynamic force . In this case, are the element nodal displacement, velocity, and acceleration vectors, consecutively. , , and are the mass, damping, and stiffness matrices.

The aerodynamic force is nonlinear in nature. For the purpose of predicting the flutter speed, is often linearized to beThe denotation “” on and represents the aerodynamic contribution to the damping and stiffness matrices.

Substituting the linearized into the full plate’s aeroelastic equation of motion yields the plate’s linearized equation of motion:Equation (3) shows how the flutter can occur. The flutter will occur if (3) is unstable. The* eigenvalue* analysis of this linearized equation can give the value of the flutter speed of the plate.

##### 2.2. Similitude Requirements for Modeling in Fluid Mechanics

The similitude requirements for the fluid dynamic problems are already well developed. Similitude is usually used in analyzing fluid dynamic problems such as lift and drag forces. In this study, the similitude method is used for analyzing the fluid structure interaction.

Similitude deals with the similarity of the actual system with its lab-scaled model or prototype. The similitude requirements for fluid dynamic problems have been well defined by Wolowicz et al. [24]. Similarity in geometric configuration is a fundamental requirement. Prototype and actual objects have to be geometrically congruent.

Another requirement is kinematic similarity. Two flows are kinematically similar if the associated velocities at the same point are related to a constant factor in the direction and magnitude [24]. This means that two streams are equal in their kinematics when streamline pattern associated with a constant factor.

Further requirements that must be met are the dynamics similarity. Two flows are dynamically similar when the associated forces at the same point are related to a constant factor in the direction and magnitude.

For the purpose of this research, an extra similitude requirement should be considered. Flutter problem is a moving boundary problem. The shape of the boundary varies with time. Ideally, in a steady case, the change is periodic or constant in frequency spectrum. Flutter will occur in a certain shape that is associated with the structural natural mode. Due to this reason, it is required to add a fourth similarity requirement when similitude theory is implemented to flutter problem. This fourth requirement is that the mode shape of the actual and the prototype must be congruent.

##### 2.3. Buckingham Pi Theorem

The development of a similitude method relays very much on the Buckingham Pi Theorem. In this section, the Beckingham Pi Theorem is recalled as the base for the dimensional analysis.

For every physical phenomenon dependent parameter which can be expressed by a function of independent parameters, we can express the relationship between the parameters to formwhich is dependent parameters and is the independent parameters. In mathematics the above functional relationship can be expressed by an equivalent functionwhere is an unspecified function, different from .

Buckingham Pi Theorem states that [25] if there are parameters in the functionthen parameters can be grouped into different dimensionless ratio, or parameters, which can be expressed in the form of the functionor

Theorem does not predict the form of functional s or . The form of the functional relationship between dimensionless independent parameters must be determined experimentally.

##### 2.4. Group for Flutter Similitude Thin-Flat Plate

In this experimental study, the dependent variable is the flutter speed . Measurements were performed in SI units; thus, unit is m/s. Variable definitions for the plate’s dimensions can be referred to Figure 1. Independent parameters that are expected to affect the value of are = plate length (m), = plate thickness, = plate width, *μ* = fluid viscosity (N·s/m^{2} = kg/(m·s)), = fluid mass density (kg/m^{3}), = natural frequency of the* thin*-*flat plate* (rad/s).