<abstract>
<p>The authors introduce a finite variable quadratic functional equation and derive its solution. Also, authors examine its Hyers-Ulam stability in Random Normed space(RN-space) by means of two different methods.</p>
</abstract>
In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation:
\phi (st-uv)+\phi (sv+tu)={[}\phi (s)+\phi (u)]{[}\phi (t)+\phi (v)]
by using the direct method and the fixed point method.
Wind noise is a significant source of interior noise in automobiles at cruising conditions, potentially creating dissatisfaction with vehicle quality. While wind noise contributions at higher frequencies usually originate with transmission through greenhouse panels and sealing, the contribution coming from the underbody area often dominates the interior noise spectrum at lower frequencies. Continued pressure to reduce fuel consumption in new designs is causing more emphasis on aerodynamic performance, to reduce drag by careful management of underbody airflow at cruise. Simulation of this airflow by Computational Fluid Dynamics (CFD) tools allows early optimization of underbody shapes before expensive hardware prototypes are feasible. By combining unsteady CFD-predicted loads on the underbody panels with a structural acoustic model of the vehicle, underbody wind noise transmission could be considered in the early design phases. This paper describes a numerical process capable of simulating underbody contributions to interior wind noise. A CFD model based on the Lattice Boltzmann Method is applied to generate fluctuating pressure loads on the underbody. Structural and acoustic loads are then calculated and applied to a vehicle noise model based on Statistical Energy Analysis, with certain parameter updates from a finite element model to predict accurately the noise transmission through the underbody panels to the driver's ear location. Interior noise validation has been achieved on a series production sedan in an aeroacoustic wind tunnel, making use of acoustic insulation materials to block greenhouse transmission. Variation in flow speed and microphone location is included to verify prediction capability of this completely numerical process.
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