# Flutter of a Plate at High Supersonic Speeds
## 1 Introduction  
Fluid-elastic structure interactions are ubiquitous in many engineering disciplines. There are historically famous examples that demonstrate that the interaction between fluid flow and the vibrations of the structure can cause catastrophic failure. It is observed that the character of vibrations typically depend on a characteristic flow velocity that, if it exceeds a certain value (the flutter velocity), will cause instability. The main thrust of fluid-elasticstructure interaction research is the determination of the flutter velocity. This study's innovation lies in the correlation between the flutter velocity and other model properties. The literature defines supersonic speeds as M > 1.2, and thus, the speed range pertains to Mach numbers greater than 1.2. The type of problem considered in this study is especially important in aerospace structures and is usually given the name panel flutter. Panel flutter has been an active area of research, starting with the development of high-speed projectiles and rockets around the 1950s. A well-known publication that summarizes the theory and the general methods of solution is [1]. A more recent text is [2]. The AGARD (Advisory Group for Aerospace Research and Development--NATO) publication is an engineering manual dealing with the detection and alleviation of aeroelasticity problems, including experimental methods and various collected tables and graphs [3]. The literature on panel and plate flutter is immense; some of the recent work relevant to the problems considered in this study is summarized here: [PERSON] et al. dealt with incompressible and irrotational flow in a cylindrical channel to analytically find the flutter velocity [4]. [PERSON] et al. investigated vibration-based damage with their chances in material and/or stiffness properties of structures. [PERSON] et al. investigated the membrane vibrations located at the stagnation point of the flow and they give the flutter velocity for the system [5]. [PERSON] considered panel flutter at low supersonic speeds by using piston theory [6]. [PERSON] gave flutter velocity graphs for an infinite plate lying in the same direction as the fluid motion by using the potential theory for different Mach numbers [7]. [PERSON] investigated plate vibrations by using potential flow theory in his PhD thesis [8]. [PERSON] et al. tried to find a similar equation to predict the flutter velocity for a membrane with a stagnation flow [9]. [PERSON] et al. give an analytical solution for a plate placed in a rigid channel that fluid flows in [10]. Also, some of the researchers are also trying to avoid this flutter point by applying a force or moving one boundary to control the motion of the plate. [PERSON] and [PERSON] controlled the resonance frequencies of a rectangular plate's vibrations by applying a discrete force by measuring the displacement of the plate at a point [11]. [PERSON] et al. give a boundary backstepping control method to stabilize the flow-induced vibrations for a membrane [12]. [PERSON] et al. investigated a periodically supported flexible plate under flowing fluid axial flow that is in a channel bounded by a rigid wall [13]. [PERSON] et al. investigated how to control the oscillations of a two-dimensional ribbed plate with free boundary conditions by using a point-wise control method [14]. [PERSON] used the Rayleigh-Ritz method to determine the modal characteristics of a rectangular plate with elastic boundary supports [15]. The effects of the system parameters on the stability of the plate are discussed. In this study, we considered the vibrations of infinite strip plates and rectangular finite plates under the effect of a high-speed flow parallel to the plate. The plate vibrations were modeled by the well-known linear (infinite strip) and two-dimensional (finite rectangle) plate vibration equations. The infinite strip was infinite in the direction perpendicular to the flow and the infinite sides were assumed to be simply supported. The finite rectangular plate was simply supported at all its sides.