# What is the physical and mathematical meaning of nonsymmetric mass, damping and stiffness matrix of a linear fluid-solid interaction modeling problem?

Suppose the problem of the subsonic flow over a simply-supported rectangular plate as bellow. According to the assumptions of incompressible, inviscid and irrotational flow, and using perturbation potential function, the applied fluid pressure on the plate is as below:

$$P=\sum_{m=1}^{M}\sum_{n=1}^{N}\rho\frac{ab}{\sqrt{{n^2a^2+m^2b^2}}}[W_{mn}(x,y)\frac{d^2q_{mn}(t)}{dt^2}+2U_{\infty}\frac{\partial W_{mn}(x,y)}{\partial x}\frac{dq_{mn}(t)}{dt}+U_{\infty}^2\frac{\partial W_{mn}^2(x,y)}{\partial x^2}q_{mn}(t)]$$

Where $$a$$ and $$b$$ are the length and width of the plate, respectively, $$\rho$$ is the fluid density, $$U_{\infty}$$ is the fluid speed in x direction and $$W_{mn}(x,y)$$ are the free vibration mode shapes of the plate defined as:

$$W_{mn}(x,y)=sin(\frac{m\pi x}{a})sin(\frac{n\pi y}{b})$$

which m and n are natural numbers. The total generalized fluid force can be calculated as:

$$F_{ij}=\int_{0}^a\int_{0}^bW_{ij}(x,y)Pdxdy$$

which i and j are natural numbers similar to m and n. Substituting P from the first equation into the above equation one can get the following equation: $$F=M\frac{d^2q(t)}{dt^2}+C\frac{dq(t)}{dt}+Kq(t)$$ where M, C and K are mass, damping and stiffness matrices, respectively, expressed as follows: $$M_{ij}= \frac{\rho ab}{\sqrt{{n^2a^2+m^2b^2}}}\int_{0}^a\int_{0}^b W_{mn}(x,y)W_{ij}(x,y)dxdy$$ $$C_{ij}=\frac{2\rho ab U_{\infty}}{\sqrt{{n^2a^2+m^2b^2}}}\int_{0}^a\int_{0}^b\frac{\partial W_{mn}(x,y)}{\partial x}W_{ij}(x,y)dxdy$$ $$K_{ij}=\frac{\rho ab U_{\infty}^2}{\sqrt{{n^2a^2+m^2b^2}}}\int_{0}^a\int_{0}^b\frac{\partial W_{mn}(x,y)^2}{\partial x^2}W_{ij}(x,y)dxdy$$ As you can see because of the term $$\sqrt{{n^2a^2+m^2b^2}}$$ none of the above matrices are symmetric. What is the physical and mathematical meaning of non-symmetric mass, damping and stiffness matrix?