Plate, Plate-type Sound Source Radiation Impedance

In the field of high power ultrasonic,circular and rectangular radiator are widely used.There are two common ways,piston vibration and bending vibration,and the radiation impedance is different at different vibration modes.The radiation efficiency is up to radiation resistance,so it is very important to study the radiation impedance.Circular and rectangular radiator are the planar sound source.What are the similarities and differences of their radiation impedance,so it is obtained whether they are equivalent in certain conditions.Eespecially the calculation of the radiation impedance of rectangular plate is more complex so that the study is less. Based on these problems,the radiation impedance are studied deeply,for example,the radiation impedance of rectangular plate in simply supported boundary conditions.The results of rectangular plate are compared with the circular sound source to get the final guiding significative results.In this paper, specific work is as follows:(1) The radiation impedance expressions of rectangular and circular piston are derived with four integrals.Numerical results of the radiation impedance of rectangular and circular piston are obtained by Gauss numerical integration method.The radiation impedance of different ratio rectangular piston are compared.The curve of radiation impedance of circular piston versus ka is calculated.The radiation impedance of circular, square,rectangular piston are compared. Some conclusions can be obtained.The radiation impedance of rectangular piston is about k(?) function,and the radiation impedance in middle or high frequency is higher than in the low frequency.When the radiation impedance of rectangular plates with different length-width ratio are compared, the closer the length is to the width, the more the value of r (r=a/b,aspect ratio) approximates to1,the greater the radiation impedance is,the greater the radiation sound power is,the higher radiation efficiency is.For circular piston,the radiation impedance is a function about ka, and at high frequency the radiation resistance is large,the radiation resistance is low,in other words radiation efficiency is high;In the comparison of radiation impedance of the circular,square, rectangular piston vibration,the radiation impedance of circular piston is large,and the radiation impedance of square piston is close to circular piston,so the square piston can replace circular piston in the same area.(2) The four integral expressions of the radiation impedance of rectangular plate with flexural vibrations in simply supported boundary conditions are derived.The radiation impedance versus k(?) of rectangular plate are obtained by Gauss numerical integration method at m=n=1mode in two references of surface average velocity and the center of vibration speed.From the angle of radiation sound power,the two definitions of the radiation impedance are united.The radiation impedance of rectangular piston are calculated by the definition of the radiation impedance about the velocity inhomogeneous distribution.The radiation impedance are calculated at m=n=0mode.The radiation impedance of different vibration modal are compared.The radiation impedance of rectangular plates with different ratios in bending vibration are compared.For the resonance condition it is discussed how the length,width and thickness of rectangular plate influence radiation impedance. Two special vibration modes are compared,which are (1,n) mode and (n,l) mode.We can get some results.The radiation impedance of rectangular plate with bending vibration is a function on k(?).At the different reference velocity,radiation impedance numerical results is also different,but its change rule is same.The definition of the radiation impedance at non-uniform vibration speed apply to the calculation of the radiation impedance of rectangular piston,which has general applicability.The radiation impedance of m=n=0just is that of rectangular piston.For the different vibration modes of rectangular plate in the same area,at lower mode radiation impedance is higher than high mode. For different length-width ratio rectangular plates in the same area, the more length-width ratio is close to l,the greater the radiation impedance is.For resonant state,at the same mode, the radiation impedance increase with the increase of the rectangular plate size.At the same size,at the different modal radiation impedance is not determined, and the radiation impedance is more complex at high mode,so we need decide on each specific matter.In the comparison of horizontal and vertical mode,the radiation impedance of (1, n) is higher than (n,1),in other words rectangular plate in horizontal stripe vibration modal has high radiation efficiency.(3) The expression of the radiation impedance of circular plate with simply supported boundary conditions is deduced.The curves versus ka of the radiation impedance of the circular plate are calculated at three modes.The radiation impedance of the circular plate, square plate, and rectangle plate (r=2or0.5) are compared. We compare the radiation impedance of the rectangular plate with r=2(or1/2), r=3(or1/3),r=4(or1/4) with the circular plate.Some conclusions are obtained.For the circular plate,at the high frequency radiation impedance is larger. Low modal radiation impedance is higher than the high mode.For circular plate, square plate, rectangle plate (r=2or0.5),the radiation impedance of square plate is largest, circular plate is smallest.The radiation impedance of rectangular plate with r=3(or1/3) approximates to circular plate.