Vibrations Of Revolution Shells In Turning-Point Range And Applications In Loudspeakers

An overview of the research conducted in the area of linear and nonlinear vibrations in the turning-point frequency range of loudspeakers and revolution shells is given in chapter 1. It shows that some problems concerning vibrations of shells in the turning-point range have to further investigate. The linear vibrations of truncated revolutions shells with the first-order turning point are investigated systematically in the turning-point range from chapter 2 to chapter 6, including the general solutions for the free vibration, the eigenvalues under various boundary conditions, the forced vibrations driven by an edge force or an edge displacement and some special effects, and the applications in loudspeaker vibrations. The nonlinear autoparametric vibration is studied in chapter 7. Main studies are listed as follows:1. We re-define the first and the second category of generalized related functions, present the power series representations and the asymptotic representations for these special functions, and obtain the uniformly valid solutions for free vibrations of revolution shells. These solutions are valid not only in the turning-point range but also in both the low and high frequency intervals. This not only makes the solutions in the three frequency intervals posses a uniform expression but also eliminates the solution gaps in the two boundary regions between the three frequency intervals. The connection formula of the solutions exhibits a coupling symmetric structure between the membrane and bending solutions. The obtained solutions are in good agreement with FEM results.2. The nature frequencies and modes in the turning-point range are studied for revolution shells under 64 cases of boundary conditions by applying these general solutions, showing the coupling between the membrane and bending solutions for any boundary condition. The expressions for the nature frequencies and modes under various boundary conditions can come down to those under four cases of boundary conditions. Simple expressions for the bending-edge-condition effect and nature frequency spacing are presented.3. The forced vibration in the turning-point range is studied for the truncated revolution shell subjected to an edge drive. Three interesting effects emerge from the analysis: the inner-quiescent effect, the inner-membrane-motion-and-outer-bending-motion effect and the non-bending effect. The first two effects found in the present paper are new characteristics of the forced vibration in the turning-point range, and the mechanism of the non-bending effect in the turning-point range is analytically revealed, which is different from that occurring in other frequency intervals4. The vibrations for loudspeaker diaphragm in the frequency interval of loudspeaker are described analytically and numerically along with characteristic frequency equations and the axial admittance. For practical applications, the resonance spacing formula is derived as well as the expression for the first non-bending frequency. The dependence of the geometric parameters of loudspeakers on the frequency characteristics is also discussed. Three conclusions are obtained: first, the first non-bending frequency may be considered the theoretical upper limit of the loudspeaker frequency response; second, there is no means to eliminate the bending waves in general due to the coupling between the membrane and bending solutions; third, the loudspeaker diaphragm of the outside of the turning point radiates much sound. The measured sound pressure level agrees well the calculated one.5. Using the perturbation method and FEM, derived and determined from a loudspeaker shell is the nonlinear evolution equation of two modes with 1:2 subharmonic internal resonance and harmonic external excitation, describing an aotoparametric system. The axisymmetric mode of the loudspeaker shell is driven directly but the non-symmetric mode is excited through the nonlinear coupling between the two modes. The steady-state solutions and their local stabilities are analyzed analytically. The Hopf bifurcation sets and the Pitchfork sets, on the driving frequency and driving force plane, are determined theoretically and experimentally, showing they agree with each other qualitatively. The complex nonlinear phenomena are discovered in the dynamic system, such as, 1/2 subharmonics, Hopf bifurcation, limit circle and chaos. Energy sharing between the two modes leads to an amplitude-modulated response that may become chaotic. The period-doubling route to the chaos is observed in both the theoretic analyses and the experimental measurements.