Analysis of the ocsillating system with combined Coulomb and viscous friction shown in Fig. 1 has already been studied by Dr. Den Hartog, but this paper introduces some new results of calculaiton on the responses from transient to steady state by the method of connecting two linear equations alternately, with a computer. Eq. (1) is the equation of motion, and Eq. (2), Eq. (3) and Eq. (4) are the general solutions of displacement. Eq. (6), the equation of velocity, is obtained by differentiating Eq. (2), whereas Eq. (5) is derived from the condition for balance of forces, and it is necessary for knowing the time at which a mass sets up motion. Integral constants in Eq. (2) and Eq. (6) can be decided by giving initial conditions, and a certain part of the motion as shown in Fig. 2 or Fig. 3 can be culculated precisely. But, whenever the sign of velocity is changed, it is necessary that the sign of Coulomb friction is inverted, and the decision of the integral constants must make fresh start, then various motions can be computed by considering Eq. (5). Consequently, the transinent responses of some sorts of motions are shown in Fig. 7, Fig. 8, Fig. 9 and Fig. 10, and the new type motions, which were able to be obtained from this numerical analysis for the first time, are shown in Fig. 11 and Fig. 12. The motions in Fig. 11 and Fig. 12 are irregular stop motion and unsymmetrical motion, respectively, and differ from already well known motions essentially, but are not existent in case of large viscous damping factor. Fig. 6 is a distribution chart showing the existence region of all kinds of motions including the new type motions, and Fig. 14 is an amplitude diagram of displacement without viscous damping, in which insufficient region is complemented by the numerical computations. Finally the new type motions were examined with an analog computer, and the results are shown in Fig. 16 and Fig. 17.
So called "Kirchhoff-Huygens Fomula" Φ≡φ/[(2<Q>^^^. exp[-irk])/r]=1+Φ_a (a) Φ=1-1/(2π)lmoust^^<2π>__0exp[ikr(φ)]dφ (b) (where φ: velocity potential, k: wave number, and other notations are shown in Fig. 1) is useful for the calculation of the sound field produced by a baffle having a point source, because it is applicable to the baffles of various shapes, moreover, numerical calculation is simple. However, this Fomula is derived under some assumptions which are satisfied only at high signal frequency. It is known that the error due to these assumptions cannot be ignored in practical use. In this paper the correction of the error is discussed. The assumptions concern the sound pressure on the plane containing the front of a baffle. The normalized sound pressure distribution on a circular plate baffle is shown in Fig. 4. As suggested in this example, improved approximation is given by using the mean value of the normalized pressure instead of the conventional assumption "6 dB". As the result of examination on the correction of error, following conclusions were obtained. 1) For the circular plate baffle (radius a) Φ=1+f_aΦ_a (c) is useful, where f_a is shown in Fig. 6. The error of this corrected formula is less than ±1 dB. 2) For a plate baffle of arbitrary shape, the mean radius a^^^-=1/(2π)⎰^^<2π>__0a(φ)dφ (d) should be used for the abscissa of Fig. 6 to obtain f_a. 3) For cylindrical and box baffles, it seems that uncorrected Kirchhoff-Huygens Formula is practically useful in present state.
The random signals (e. g. , street noise, machine or structure vibration) often appearing in the actual engineering fields exhibit various kinds of prbability distributons apart from the usual Gaussian distribution due to the diversified causes of the fluctuations. When a general expression of the cumulative distribution function (agr, c. d. f. ) of such a random fluctuation is sought, we must give attention, particularly from fundamental and practical viewpoints, to the following considerations: i) A unified expression of the c. d. f. is required, which is not influenced too much on the whole but is concretely reflected in its internal parameters by the fluctuation mode with time of the individual random phenomenon under consideration. ii) In order to bring the above generality to the expression, it is better to choose a statistical expansion series expression whose expansion coefficients reflect the first and higher order statistical consepts which are necessary to explain the phenomenon. iii) From the standpoint of the convergence property of expansion expression to be used, the kind of c. d. f. which is chosen for the first term of the expansion expression is of vital importance, since this term describes the principal part of the random fluctuation. iv) In practical applications, since a statistical expansion expression will inevitably be employed in the form of a finite number of expansion terms, the exact correction to the truncation error is always important. From the above essential considerations, when a random noise or vibration Z(t) of arbitary distribution can be considered to be the sum of two different random processes X(t) and U(t) as a result of the natural internal structure of the fluctuation or the analytically artificial classification of the fluctuation, a unified statistical treatment for the c. d. f. of the resultant random fluctuation Z(t)(≜X(t)+U(t)) is introduced exactly in the form of finite expansion terms (here, X(t) and U(t) may be mutually correlated and need not always have a Gaussian type distribution). First, let us introduce an arbitary function φ(Z) with the property of Eq. (2) and consider its expectation value Eq. (3). Carrying out the Taylor series expansion with a remainder term for Eq. (4), Eqs. (5) and (6) can be obtained. Our main problem is how to derive the probability density function P(Z) in the form of finite expansion terms based on statistical information of X(t) and U(t). After a somewhat complicated derivation, we obtain the two expansion expressions, Eq. (16) when X(t) is statistically correlated with U(t) and Eq. (17) when X(t) is statistically independent of U(t). Needless to say, we can easily show that these results satisfy the above-mentioned properties (i)-(iv). Introducing the dimensionless variable Y≜(Z-μ)/σ, Eq. (16) can be rewritten in the universal form of Eq. (18). In the practical study of level fluctuations of random noise and vibration, the c. d. f. expressions (19), (20) and (21) are more important than the probability density expression (18). Also, our results contain the well-known expression of the c. d. f. for the non-stationary random process with a mean value fluctuation as a special case. Furthemore, we have experimentally confirmed the validity of our theory not only by means of digital simulation but also by experimentally obtained road traffic noise data in Hiroshima City. The experimental results clearly show the usefulness of the theory, especially the importance of the exact correction to the truncation error.
Piezoelectric plates are widely used as electromechanical resonators and filters and their multi-mode utilization is now in fashion. Lloyd and Redwood have applied the well-known finite difference method to the analysis of square and rectangular plates. Their treatment, however, becomes troublesome for plates of irregular shape or having complicated boundary conditions. The technique of applying the finite elememt method to coupled electromechanical systems has been developed, and has been used to successfully analyze electromechanical resonators and filters of lexure type. The present paper describes a simular technique which can be applied to the viblation simulation of two-dimensioal electromechanical thin plates which vibrate in plane(Fig. 1). Finite element formulation is made with the inclusion of electromechanical coupling. The division of a plate is approximated by the assemblage of traingular elements, so that the formulation is applicable to piezoelectric or electrostrictive thin plate resonators and filters of arbitrary shape. A linear approximation is employed to describe the displacements of a triangular element of the plate, as illustrated in Fig. 2. To demonstrate the validity and the availability of the procedure, the natural frequencies of planarly isotropic rectangular plates are calculated for the symmetrical modes (Fig. 5). The results are discussed in conjunction with those of other investigators [(a) in Fig. 6: the finite element method using a 2nd order polynomial to approximate the displacement functions, (b): the finite element method using a linear displacement approximation, (c): the finite difference method, (d): the series expansion method]. Good agreement is obtained for the L1 (longitudinal dominant) mode. For the other modes, discrepancies are found within the range of a few percent. Effect on the natural frequencies due to electrical termination is less than 1% for ordinary electrostrictive materials such as barium titanite. The vibrational modes including the asymmetrical ones, are then calculated for the plates having length-to-width ratios (r_<ab>)of 1 : 2 and 1 : 1. The model shapes are illustrated in Figs. 7 and 8. For plates of arbitrary shape, a calculation is also made for the plates with one of the corners rounded off. The modal shapes and the corresponding natural frequencies are illustrated in Figs. 9 and 10. The effect of rounding off is clearly seen which, as shown in Fig. 10, resolves the degenerate modes (F1/L2, F3/L2') as expected. In the present analysis, the input admittance at the electrical terminals can be directly obtained, because electromechanical coupling is included. In Fig. 11, the frequency characterristics of motional admittance are shown for a rectangular plate fully electroded on both surfaces (r_<ab>=2). For the plates with partial electrodes and those with one rounded corner, they are shown in Figs. 12 and 13, respectively. The motional admittance for the square plates are shown in Figs. 15 to 18. Full electrode arrangements are likely to excite only the symmetrical modes while partial arrangements excite all associated modes. From the curve, the excitation strength of each mode is predicted, and the equivalent stiffness and mass in its vicinity can also be evaluated from the slope. In the numerical demonstration, the convergence does not appear to be complete. This is due to the limited capacity of the available computer, and not an essential defect of the procedure. This can be overcome by making the divisions smaller or employing a higher order polynomial to approximate the displacements. The technique introduced proves to be a powerful means for simulating and analysing two-dimensional electromechanical resonater problems. Once the computer program is developed, wide applications are possible with little modification.
A steady background noise is necessary to mask interference which originates from adjacent working areas. With the help of loundspeakers a sound field is produced which is, as far as possible, diffuse and steady but with adjustable loudness and frequency range. The intention is to obtain the greatest possible attenuation of sound with increase of distance from the source, so that the masking is more effective. The acoustic design and equipment installation in office blocks with open plan areas will be described.