# Flutter of a Plate at High Supersonic Speeds
## 4 Infinite Strip Plate  
Figure 2 shows a schematic of the first problem. The flow occupies the region \(z>0\) and is in the \(x\)-direction with uniform velocity \(U\). The region in the \(xy\)-plane with \(-b/2<x<b/2\) and \(-\infty<y<\infty\) is an elastic plate (infinite strip) of width \(b\); the rest of the \(xy\)-plane is rigid. The vibrations of the plate cause small perturbations in the flow velocity. Some different equations can be found in the literature to define the vibrations of the plates [18; 19; 20; 21; 22; 23; 24]. The vibrations of the plate are governed by [18; 19; 20]  
\[D\frac{\partial^{4}w}{\partial x^{4}}+\rho_{p}h_{p}\frac{\partial^{2}w}{ \partial t^{2}}+p|_{z=0}=0 \tag{11}\]  
where \(w=w(x,t)\) is the displacement, \(h_{p}\) is the thickness, \(\rho_{p}\) is the density of the plate, and  
\[D=\frac{Elh_{p}^{3}}{12(1-v^{2})} \tag{12}\]  
is the flexural rigidity, where E is [PERSON]'s modulus and \(v\) is [PERSON]'s ratio. The last term in (11) denotes the fluid pressure on the plate. Although the plate is moving, the fluid pressure can be assumed to have its value at \(z=0\) within the linear theory. The fluid flow is assumed to be inviscid since viscous effects are negligible due to the lack of flow separation. In general, the linearized form of the compressible potential equation can be used, but here a simpler approximation called piston theory is adopted. Piston theory is widely used and basically states that the local pressure is proportional to the local slope of the plate [16; 25; 26]. \[p|_{z=0}=\frac{\rho_{f}U}{M}\bigg{(}\frac{\partial w}{\partial t}+U\frac{ \partial w}{\partial x}\bigg{)} \tag{13}\]  
Thus, the coupled fluid-plate vibrations obey  
\[D\frac{\partial^{4}w}{\partial x^{4}}+\rho_{p}h_{p}\frac{\partial^{2}w}{ \partial t^{2}}+\frac{\rho_{f}U}{M}\bigg{(}\frac{\partial w}{\partial t}+U \frac{\partial w}{\partial x}\bigg{)}=0 \tag{14}\]  
where \(\rho_{f}\) is the density of the fluid, \(U\) is the velocity of the fluid, and \(M\) is the Mach number \(M=\frac{U}{c}\). The boundary conditions on the plate were taken as  
\[w(-b/2)=w(b/2)=0\] \[\frac{\partial^{2}w}{\partial x^{2}}(-b/2)=\frac{\partial^{2}w}{ \partial x^{2}}(b/2)=0 \tag{15}\]which state that the ends of the plate are simply supported. The problem was non-dimensionalized as follows:  
\[x^{*}=\frac{x}{b},w^{*}=\frac{w}{b},z^{*}=\frac{z}{b},t^{*}=\frac{t}{b^{2}\sqrt{ \frac{\rho_{p}h_{p}}{D}}},p^{*}=\frac{p}{b^{3}},u=\frac{U}{\frac{1}{b}\sqrt{ \frac{D}{\rho_{p}hp}}},c_{s}^{*}=\frac{u}{M} \tag{16}\]  
Starred quantities are non-dimensional. Substituting in (14) and (15) and simplifying, the result is, after getting rid of the stars since dimensional quantities will not be needed,  
\[\frac{\partial^{4}w}{\partial x^{4}}+\frac{\partial^{2}w}{\partial t^{2}}+\mu \bigg{(}\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}\bigg{)}=0 \tag{17}\]  
and the boundary conditions are  
\[w(-1/2)=w(1/2)=\frac{\partial^{2}w}{\partial x^{2}}(-1/2)=\frac{\partial^{2}w }{\partial x^{2}}(1/2)=0 \tag{18}\]  
where  
\[\mu=\frac{\rho_{f}b}{\rho_{p}h_{p}}c_{s} \tag{19}\]  
is a non-dimensional number, usually called the mass ratio, and  
\[D=\frac{Elh_{p}^{3}}{12(1-v^{2})} \tag{12}\]0  
is the dimensionless sound velocity. To investigate the stability, the plate displacement was assumed to be  
\[D=\frac{Elh_{p}^{3}}{12(1-v^{2})} \tag{12}\]1  
Then, (17) and (18) become  
\[D=\frac{Elh_{p}^{3}}{12(1-v^{2})} \tag{12}\]2  
\[D=\frac{Elh_{p}^{3}}{12(1-v^{2})} \tag{12}\]3  
where  
\[D=\frac{Elh_{p}^{3}}{12(1-v^{2})} \tag{12}\]4  
\[D=\frac{Elh_{p}^{3}}{12(1-v^{2})} \tag{12}\]5  
Equations (22) and (23) give an eigenvalue problem for \(\omega\). The vibrations of the plate do not grow as long as the imaginary part of \(\omega\) is positive; the stability boundary is \(\mathrm{Im}(\omega)=0\). Since \(\omega\) depends on \(u\) (as well as \(\mu\)), the condition \(\mathrm{Im}(\omega)=0\) gives the non-dimensional flutter velocity. The problem defined by (22) and (23) were solved using the collocation method.