English translation
Analyse von natürlichem Modellseegang
Complete English translation of the original German-language document (9 pages).
Analysis of Natural Model Sea States
Dipl.-Ing. S. Kastner Schriftenreihe Schiffbau Nr. 223, 1968 (Schiffstechnik, Vol. 15, 1968, Issue 75, pp. 15–22)
[Preceding context — continuation from prior article]
From the four curves, one reads for “PT 50” (displacement D = 65 t) a wave height of H = 1.3 m for following seas, which emerges from the author’s calculations as the upper limit. Since this diagram by Lacey takes no account of the Froude number or submergence depth, one reads for “H. S. Denison” at 95 t only H = 1.5 m as the permissible wave height. This value lies close to the lower limit of the calculated wave height of 1.55 m.
In following seas the permissible wave height for “H. S. Denison”, owing to the different Froude numbers, is improved relative to “PT 50” by a greater margin than the ratio of the submergence depths alone would suggest. Compared with the submergence depth of the “Denison” craft of a = 1.3 m, the pivot point of the bow foil lies relatively close to the waterline (1.2 m), and it has therefore occurred that the entire bow foil leaps clear of the waves.
4.2.1. Accelerations
The incident just mentioned is naturally associated with large vertical accelerations, particularly in the forward part of the craft. Case 4 of Table 2 shows that medium-length waves of only H = 0.55 m height already impose centre-of-gravity accelerations of p_Gz = ±0.15 g on “H. S. Denison”, for which H = 0.3 m suffices for “PT 50”. Significantly larger waves can be handled when running against waves of 120/125 m length, with wave heights of H = 3.0/3.2 m being nearly equal for both craft. With respect to accelerations, the Froude numbers 1.1/2.2 act in the opposite sense to their effect on the heaving oscillations.
If one were to fully exploit the submergence depths of the craft in medium-length steep waves, p_Gz would double for “PT 50” (±0.30 g) and triple for “H. S. Denison” (±0.45 g, Case 5). The wave heights given by R. Lacey for heading into the sea lie between the calculated values for Cases 1 through 3.
Acceleration measurements above the foil of “PT 50” yielded, in waves of 30–40 m length and 2 m height amidships, a maximum of ±0.8 g from ahead, whereby brief impacts caused by light emergence of the hull were not included. This figure of ±0.8 g appears high, but is exceeded up to tenfold by planing craft of similar Froude number. The measurements on “PT 50” confirm that accelerations when heading into the sea are generally greater than when running with the sea.
When one takes into account the uncertainty in sea-state estimation, the simplifications of the theory, and the deviations of the hull and foil parameters between theory and full-scale execution, the agreement between calculated and measured values of seakeeping for hydrofoil craft can be described as satisfactory.
Further improvements in the full-scale vessel can be achieved through stabilisation aids. According to Lacey, last row of Table 2, this raises the permissible wave height by 1/3 to 2/3 depending on wave direction. In the present case of symmetric oscillations, this applies above all to the pitch stabilisation acting at the stern foil, whose superior effectiveness the author was able to observe in February 1964 during a demonstration run with “H. S. Denison” in heavy seas off Miami.
(Received 14 November 1966)
Analysis of Natural Model Sea States
Dipl.-Ing. S. Kastner
For stability investigations of ships in sea states, model tests in natural sea states have been carried out at the suggestion of Roden [3]–[8]. For this purpose it is necessary to measure the sea state as an exciting quantity and to present it in a suitable form for further processing. This paper sets out the considerations that led to the construction of a special analyser. Furthermore, the developed instrument is described and an example of a result is shown.
I. The Task
Natural sea states arise at the water surface through the action of wind, depending essentially on wind speed w, the duration of wind action d, and the fetch length F over water. They can be described by a function ζ = ζ(x, y) for a specific point in time.
This representation results from a spatial fixing of the sea-state picture at a specific fixed instant t₀, for example by means of a stereophotogrammetric exposure. Such sea-state recordings at sea by means of series of instantaneous exposures in temporal sequence have already been carried out for oceanographic and naval-architectural research [9, 10]. The evaluation is laborious and little suited to the present task.
The representation ζ(x, y) thus corresponds to an instantaneous exposure of the wave picture. In reality the picture changes continuously, so that time t must be considered as an additional parameter and the general form of the sea-state function becomes ζ(x, y, t). The simplest measurement of the so-called sea-state function ζ over time at a fixed point (x₀, y₀) is in the form ζ(x₀, y₀, t) = ζ(t).
For the model sea states under consideration, because of the correspondingly shorter wavelengths and shallower water depths compared with maritime sea states, a wave probe on a fixed mast can be used to good effect. This measurement method at a wave staff is particularly well suited for shallow depths up to about 25 m. In previous tests on the Großer Plöner See, measurements were made at a point with approximately 10 m water depth [3, 11]. A staff probe can also be used in shallow coastal waters. For sea-state measurements in the open ocean this is no longer possible owing to the greater water depth. Pressure and acceleration measurements on ships [12] or buoys [13, 14] are therefore generally employed for capturing the sea state. However, direct measurements of the sea-state function ζ(t) by means of the probe method are also conceivable even in the open sea, provided the probe can be given appropriate fixation relative to the horizontal reference frame without touching the bottom. This will not, however, be pursued further in the present context.
By means of the electrical probe measurement, the fluctuations of the wave ordinate at a fixed point in the sea area are obtained as a voltage fluctuation u_w(t) ≅ ζ(t), which can be carried by cable to shore and recorded there. At sea a wireless transmission via radio would be required, increasing the complexity.
Such a recording of ζ at a fixed point in the sea area over time t, for example with a loop oscillograph, delivers the image of an irregular sea state from which, in this form, only with difficulty can general statements be made about the constituent frequencies, amplitude distributions, or the energy content of the waves. A statistical evaluation of the recordings can, however, yield characteristic values. In this way a sea state can be represented by transformation of the sea-state function ζ(t) in the time domain as a spectral function S_ζζ(f) in the frequency domain. The so-called sea-state spectrum S_ζζ(f) provides far-reaching statistical information on:
a) the frequencies present in the sea state b) the distribution of power over the individual frequencies c) the frequency of maximum power density d) the total power content of the sea state
An installation has now been developed which processes the sea-state function ζ(t), available as an electrical voltage, electrically on-site immediately and delivers the spectrum S_ζ(f), so that the recording of ζ(t) as well as any subsequent laborious analysis can be dispensed with. This simplifies the evaluation on the one hand, and above all increases the informational value of the measured data, since a judgement on the existing spectrum can be formed immediately.
II. Mathematical Representation of the Problem
The mathematical relationships are given here only briefly; the rigorous derivation and detailed explanation can be found in adequate scope in the literature [15, 16, 17].
The sea-state function ζ(x, y, t) in its most general form described above represents a random process. Such a process is called a stochastic process. The individual recording of all values, for example ζ(x, y, t₀) or ζ(x₀, y₀, t), reveals no unambiguous regularity. If, however, the values are considered over a larger region (X_m, Y_m) or time interval T_m and statistical mean values of some kind are formed, these hold—under the same external conditions—for another region (X_n, Y_n) or another time interval T_n as well.
From the apparently random process in a small domain there has thus emerged a process with fixed characteristic values over a large domain. Such a process is then called stationary, and it is assumed here, in agreement with customary sea-state investigations, that the sea state in the measurement area and during the measurement period is such a stationary stochastic process. This means for the time function ζ(t) that any finite time shift τ leaves the statistical properties of the function unchanged. For example, the expected values of different function segments are equal:
$$E[\zeta(t_n)] = E[\zeta(t + \tau_n)]$$ (1)
The measurement of the sea state at a single point therefore presupposes ergodicity. This is assumed here in agreement with customary sea-state investigations. In practice one must be content with finite measurement durations, and even these are very limited in magnitude, since otherwise, owing to changes in wind speed etc., the condition of a stationary sea state would no longer be satisfied.
Under the assumptions made for the sea state—namely the existence of a stationary stochastic process in the form of the sea-state function ζ(t), for which the ergodic theorem is to hold—the so-called autocorrelation function¹ suffices for a statistical description from the standpoint of mathematical statistics:
$$\Phi_{\zeta\zeta}(\tau) = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{+T/2} \zeta(t) \cdot \zeta(t + \tau), dt$$ (5)
The autocorrelation function possesses two important limiting cases for τ = 0 and τ → ∞:
$$\Phi_{\zeta\zeta}(0) = \overline{\zeta^2}$$ (6)
and
$$\lim_{\tau \to \infty} \Phi_{\zeta\zeta}(\tau) = \bar{\zeta}^2 = [E[\zeta(t)]]^2$$ (7)
i.e., for τ = 0 the mean-square value is obtained; for τ → ∞ the square of the linear mean value is obtained.
The sea state consists in the region XY of ordinate fluctuations ζ that yield a different picture at every instant of time. In turn, all points in the x-y plane yield different functions ζ_i over time t. One now selects from the totality of functions ζ_i(t) one function measured at a specific point (x₀, y₀) of the sea area. Applying the ergodic theorem, one can state that a single function ζ(t) can yield the same statistical information about the sea state as the function ζ(x, y), provided the statistical characteristic values are formed over a sufficiently long, theoretically infinite, time interval or a sufficiently large region. The corresponding expected values of both functions are then equal:
$$E[\zeta(x, y)] = E[\zeta(t)]$$ (2)
where
$$E[\zeta(x, y)] = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \zeta(x, y) \cdot w_2(x, y), dx, dy$$ (3)
and
$$E[\zeta(t)] = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{+T/2} \zeta(t), dt$$ (4)
where E denotes the expected value, p is an arbitrary absolutely integrable function of ζ, w₂ is the second-order probability density function, and W is the distribution function of the wave field ζ(x, y).
From the autocorrelation function the sought power spectrum follows according to the theorem of Wiener and Khinchin as the Fourier transform, the so-called power spectral density function:
$$S_{\zeta\zeta}(f) = \int_{-\infty}^{+\infty} \Phi_{\zeta\zeta}(\tau) \cdot e^{-i\omega\tau}, d\tau = \mathcal{F}{\Phi_{\zeta\zeta}(\tau)}$$ (8)
Conversely:
$$\Phi_{\zeta\zeta}(\tau) = \mathcal{F}^{-1}{S_{\zeta\zeta}(f)} = \int_{-\infty}^{+\infty} S_{\zeta\zeta}(f) \cdot e^{i\omega\tau}, df$$ (8a)
where F⁻¹ denotes the inverse Fourier transform.
The power spectrum is informationally completely equivalent to the autocorrelation function. For the engineer it is often more intuitive, and for the present sea-state analysis this representation is customary. The spectrum can be determined not only via the route of autocorrelation but directly from the definition of the power spectrum:
$$S_{\zeta\zeta}(f) = \lim_{T \to \infty} \frac{1}{T} |C(f)|^2$$ (9)
where
$$C(f) = \int_{-T/2}^{+T/2} \zeta(t) \cdot e^{-i\omega t}, dt = \mathcal{F}{\zeta(t)}$$ (10)
represents the Fourier transform of the process ζ(t).
Thus the time function ζ(t) bears the same relation to the spectral function C(f) as the autocorrelation function Φ(τ) does to the power spectrum S(f), with the relationship established by the Fourier transformation. In both cases the same power spectrum can be calculated as the spectral power density of the process, i.e. the distribution density of power over frequency f. This provides a frequency-domain representation derived from the process ζ(t) in the time domain with loss of phase information. The process is therefore irreversible: from the spectrum S_ζζ(f) the original function ζ(t) cannot be recovered.
The spectrum does, however, provide comprehensive and clearly arranged information on statistical characteristic values of the random function ζ(t). If the Fourier coefficients C(f) can be determined, the power spectrum can readily be computed from them. From equation (10) for the complex Fourier coefficient C(f), the determining equations for the corresponding Fourier coefficients a(f) and b(f) can be developed:
$$C(f) = \int_{-T}^{T} \zeta(t) \cdot e^{-i\omega t}, dt = \int_{-T}^{T} \zeta(t) \cos\omega t, dt - i \int_{-T}^{T} \zeta(t) \sin\omega t, dt = a(f) + ib(f)$$ (11)
Hence:
$$a(f) = \Re{C(f)} = \int_0^T \zeta(t) \cdot \cos 2\pi f t, dt$$ (12)
$$b(f) = \Im{C(f)} = \int_0^T \zeta(t) \cdot \sin 2\pi f t, dt$$ (13)
The magnitude of the spectral value S_ζζ(f) then becomes:
$$S_{\zeta\zeta}(f) = \frac{1}{M}|C(f)|^2 = \frac{1}{M}\bigl(a^2(f) + b^2(f)\bigr)$$ (14)
In the following, the practical methods used to determine the power spectrum are briefly summarised. In principle, the use of any of the above relations is permitted, but in practice the choice depends on the specific conditions prevailing, such as frequency range, the form in which the process is available, accuracy requirements, and possible expenditure, etc. It will be seen that for the present case relations (12), (13), and (14) offer the best possibility of realisation.
According to Parseval’s equation:
$$\lim_{T \to \infty} \frac{1}{T} \int_{-\infty}^{+\infty} |C(f)|^2, df = \lim_{T \to \infty} \frac{1}{T} \int_{-T/2}^{+T/2} \zeta^2(t), dt$$ (15)
a simple relationship between the area under the spectrum and the r.m.s. value of the process is given. Equation 15 states that the zeroth moment of the spectrum equals the second central moment of the sea-state function:
$$m_0{S_{\zeta\zeta}(f)} = m_2{\zeta(t)}$$ (15a)
Since for real time functions ζ(t) the spectrum S(f) is an even function of frequency, integration is carried out, as is customary in the engineering domain, over positive frequencies only. One can then write:
$$\int_{-\infty}^{+\infty} S(\omega), df = \int_0^{\infty} 2 S_{\zeta\zeta}(f), df = 2\pi \cdot \int_0^{\infty} S_{\zeta\zeta}(f), df$$
With this:
$$\int_0^{\infty} S_{\zeta\zeta}(f), df = R = \overline{\zeta_{eff}^2}$$ (16)
Equation (16) allows calibration of the analyser by measurement of the mean-square value of the time function.
¹ If ζ(t) is understood as the fluctuation about the linear mean value, this particular autocorrelation function Φ_ζζ(τ) is called the autocovariance function.
² The spectrum S(f) of frequency f is used here for reasons of clarity rather than the spectrum S(ω) of angular frequency ω, where f is given in the generally customary unit of Hertz and ω = 2πf gives angular displacement per second. This notation is also useful for logical connection to the Laplace transform.
III. Practical Analysis of Natural Sea States
The waves to be investigated are so-called gravity waves—waves at a water surface whose angular frequency ω is related to their wavelength L_w by the relation:
$$\omega = 2\pi f = \sqrt{\frac{2\pi g}{L_w}}$$ (17)
The angular frequency referred to 2π is shown in Fig. 1 as a function of wavelength L_w.
For stability investigations on ship models in sea states, characteristic wavelengths approximately equal to the model length are preferred:
$$L_w \approx L_{model}$$ (18)
The models investigated hitherto are between 2 and 3 m long. Larger model lengths are also conceivable for this purpose, in order to reduce the scale factor and to accommodate more units in terms of weight. It is assumed here that the characteristic wavelengths in the range 1 m < L_w < 10 m should be covered by the analysis installation, since tests with ratios L_ship/L_w ≠ 1 are also of interest. According to Fig. 1, this would correspond to a frequency range of approximately 1.25 Hz > f > 0.1 Hz. Since adjacent components are also present in the sea state, it is advisable to extend the range somewhat to capture them. The desired frequency range of the analyser is therefore set at:
$$1.5,\text{Hz} > f \geq 0.25,\text{Hz}$$ (19)
corresponding to a wavelength range of approximately:
$$0.7,\text{m} < L_w < 25.0,\text{m}$$ (19a)
Fig. 1: Wave frequency as a function of wavelength for gravity waves (double-logarithmic scale)
1. Known Analysis Methods
1.1 Mechanical Methods
The customary drum analysers for determining the Fourier coefficients are ruled out, since the capturable time range is very limited and does not suffice for the present purpose. The admittedly possible joining of partial segments is very time-consuming. The sea-state function ζ(t) must be available as a paper recording (trace).
1.2 Electronic Methods
a) Electronic analysis by variable narrow-band filter
The process, available as an electrical quantity, is fed through a variable narrow-band filter, which delivers narrow-band noise of corresponding intensity. An electronic multiplication stage forms the mean-square value thereof, and one thus obtains a quantity proportional to the respective spectral density (Fig. 2). Filters from passive elements can, however, be realised only for relatively high resonance frequencies. Instruments of this type available on the market are developed for the purposes of high-frequency technology and do not permit the analysis of the low frequencies present here. Processes in the low-frequency range would have to undergo a corresponding frequency transformation to higher frequencies. This is frequently achieved by recording on magnetic tape at lower tape speed and playback at high tape speed. This means, however, high measurement-technical and financial outlay.
b) Analysis with the analogue computer
Direct power-density measurement for low frequencies is, however, also possible on the analogue computer [20]. The construction of a filter with very low resonance frequencies from analogue-computer elements is shown in Fig. 3. The filter possesses approximately the desired ideal rectangular form for its transfer function. Damping and various resonance frequencies can be set easily at three potentiometers. Only one summing element, two integrators, and three potentiometers are required. However, to be able to scan through various frequencies, the process must be stored.
Furthermore, on the analogue computer the direct simulation of equations (12) and (13) for the Fourier coefficients is also possible, as illustrated in Fig. 4. From an oscillatory circuit according to the differential equation ÿ + αy = 0, the sine and/or cosine function is tapped and multiplied by the function to be analysed in a multiplier. The integrand is thus formed, whose integration over time delivers the sought Fourier coefficients.
Fig. 2: Circuit diagram for variable narrow-band filter (after [15])
Fig. 3: Computing circuit on the analogue computer for measurement of power density (after [20])
Fig. 4: Computing circuit for direct formation of Fourier coefficients on the analogue computer (after [20])
If one wishes to manage with the units sketched, ζ(t) must again be available in stored form in order to repeat the computation for various frequencies. In principle it is conceivable to cover the expected frequency range with such computing circuits having various fixed preset frequencies. In that case the storage of the input function could be dispensed with. On customary analogue computers, the simultaneous appearance of the various Fourier coefficients for the chosen frequencies f_n would make display difficult, thus requiring additional auxiliary equipment. If approximately 20 required frequencies are assumed, the purely numerical outlay in multipliers and amplifiers would be too large and the method thus too expensive. In principle, however, this approach underlies the developed installation; only the execution is based on different components.
c) Light scanning on a drum
The recorded sea-state function ζ(t) is mounted as a black-and-white profile on the circumference of a drum [21]. A photocell slit perpendicular to the time axis scans the drum, which has been brought to maximum rotational speed and then, owing to its large moment of inertia and low bearing friction, runs down only slowly. The photocell delivers an alternating electrical voltage, which is amplified and fed to a fixed filter. As the frequency transformation changes during the drum’s spin-down, all oscillations contained in the trace successively come into resonance with the filter frequency. The alternating voltage at the frequency passed by the filter is rectified and recorded. One thus obtains in continuous succession the Fourier coefficients. By the device of the drum, only a single fixed filter is needed. The first analyses of model sea states [?] were carried out with an instrument of this type developed by Christoph [22] for other purposes. The preparation of the black-and-white profile proved, however, to be too laborious.
1.3 Digital Methods
If the values of the sea-state function ζ(t), appearing as an analogue electrical voltage quantity, are converted to digital values at equidistant time intervals and these values are stored, the various statistical characteristic values can be computed from these value sequences on customary digital computers with appropriate programmes, including the spectrum, for example via the autocorrelation function. This approach appears to be the most precise method, but it is also associated with a certain outlay. In apparatus terms, an analogue-to-digital converter with card or punched-tape reader or a magnetic tape for storage would be required. The values would then have to be supplied to a large-scale computer. In no case is a picture of the sea state available in a short time even while the test is being run. Even digital small computers for such specialised computing tasks are available on the market, but they are ruled out on cost grounds.
All the methods described so far have in common the fact that they cannot provide an immediate picture of the spectrum of the sea state prevailing on the test water. This is, however, necessary for the conduct of the model tests. Only the interesting statistical characteristic values of the sea state, such as the spectrum and various mean values, are retained.
2. Electro-Mechanical Method
The Fourier coefficients of the sea-state function ζ(t) are formed according to equations (12) and (13) in an electro-mechanical system. The block diagram in Fig. 5 shows the fundamental construction of the analyser. The sea-state function, available as a modulated direct voltage, is processed electrically immediately. The product formation for the integrand is accomplished via sine-cosine potentiometers rotating at the respective filter frequency; the integration is performed via suitable RC circuits.
By arranging all filters of the various frequencies required for adequate spectral resolution in parallel, the storage of ζ(t) can be dispensed with. The Fourier coefficients are sequentially read from storage locations via a rotating switch and recorded with a compensation writer. Using relatively simple electrical components, the outlay can be kept within limits.
An electro-mechanical analyser is therefore a special analogue computer. For many tasks the accuracy of analogue technology is sufficient. If one sensibly weighs the accuracy requirements of the task against the cost-effectiveness of an installation, a wide field remains for analogue application even today. However, the number of transmission stages must not exceed an acceptable measure, and particular attention must be paid to limiting transmission errors. Such errors have been reduced here to an acceptable level by selection of components and circuit measures. This is discussed in more detail below. Statistical errors arising theoretically from the finite practical integration times and the practically possible spectral resolution are not treated in general here; the relevant literature should be consulted for this [23–26].
IV. Description of the Electro-Mechanical Analyser
1. Electrical Construction
At the input lies the measurement voltage to be analysed, in this case the sea-state function ζ(t). The input voltage u is composed of a DC component U_G and an oscillating voltage U_w. To apply only the fluctuations of the sea-state ordinate ζ(t) = U_w to the potentiometers, the DC component U_G is compensated by means of a transistor circuit. This compensation is important because of the necessary manufacturing tolerances of the potentiometer winding. The deviation caused by tolerances produces asymmetries in the two parallel potentiometer branches across which the measurement voltage drops. If a large DC voltage component is present at the input, this means that, owing to the asymmetry, the proportion of charging voltage added to or drawn from one branch is not again compensated when the wiper traverses the other branch. Because of the integration over relatively large time intervals, the summation of the DC component would render the measured value unusable. In addition, the asymmetries are largely compensated by series-connected trimming resistors that can shift the centre tap to the mid-potential, in order to reduce the errors.
The potentiometers have a wire winding whose resistance variation over the angle of rotation corresponds to a sine function (see Fig. 6).
Two wipers are arranged at an angle of 90° relative to each other. The voltage tap via the first wiper now delivers, at a given position φ, the value of the input voltage u_w corresponding to the sine winding, i.e. the product u_w(t) sin φ. Since cosine can be represented as a phase-shifted sine,
$$\cos\varphi = \sin(\varphi + \pi/2)$$ (20)
the second wiper delivers simultaneously the product u_w cos φ. If the potentiometer is now rotated with angular velocity ω, the tap occurs at each instant t at the angle φ = ε + ωt, where ε is the initial phase of the potentiometer. The two taps therefore deliver the respective instantaneous values of the products:
$$\zeta(t) \cdot \sin(\omega t + \varepsilon) \quad \text{and} \quad \zeta(t) \cdot \cos(\omega t + \varepsilon)$$
The initial phase ε at the beginning of the integration, i.e. the position of the wiper on the potentiometer circumference, is purely arbitrary. The relative frequency of the phases is constant from 0 to 2π, i.e. all phases are equally probable. This applies then also to the phase remainder at limited integration time:
$$0 < (\omega T + \varepsilon - n \cdot 2\pi) < 2\pi$$
With increasing integration time the percentage proportion of the phase remainder in the total angle becomes ever smaller (max. ca. 2%). In practice, however, all other phase remainders with smaller percentage proportions occur equally frequently, so that the influence of the phase is small.
Fig. 5: Block diagram of the electro-mechanical analyser
Fig. 6: Function of the potentiometers for sine and cosine tapping
Important on the other hand is an accurate and as lossless as possible storage and readout of the integration values. Integration and storage are accomplished by charging capacitors through series resistors.
The requirement of obtaining at readout a value directly proportional to the integrated value demands a linear characteristic of the storage element. For the capacitor used, this is satisfied only in the lower portion of its characteristic curve (Fig. 7). The deviation of the practical storage element from the ideal characteristic is:
$$\frac{u_C}{U_0} = 1 - \exp\left(-\frac{t}{\tau}\right)$$ (21)
Fig. 7: Charging characteristic of the integration storage elements
The quantity t denotes an arbitrary integration time, which has been chosen as the fixed value T for the instrument, defined by the duration of one readout cycle. T represents here the so-called time constant, which results from R and C as τ = RC [sec]. In order to remain in the linear portion of the storage characteristic with sufficient potential for the measured values, a high time constant must be aimed for. The instrument now contains 40 storage elements with 1200 sec each. Capacitors of 24 µF and resistors of 50 MΩ were used. The resistors can be trimmed via series-connected potentiometers by approximately ±5% in order to achieve a uniform time constant for all storage locations and to avoid distortion of the measured values. The individual capacitor storage elements are read out in sequence, so that only one recording trace is needed, whereas simultaneous readout would require approximately 40 traces, which would represent an unreasonably large outlay. In principle it would now be necessary, while the data of all storage elements are being recorded with a single recording trace—since the analysis must not be interrupted—for the storage to take place in a second set of capacitors. Both sets of storage elements would then be read out alternately. Besides the outlay for a second storage set, this would mean that the capacitors must hold their charge for different lengths of time before they can be read out. For the last capacitors, impermissibly large leakage losses could not be avoided. A continuous operation with only one storage set was therefore provided. Once a storage element has been read out and discharged, it is immediately available again for charging. This does, however, entail a time offset of the individual components relative to one another during a readout cycle, shown in schematic form in Fig. 8. For each filter with frequency f_n, the integration time T is constant. If the analysis is continued continuously, the spectra for longer time intervals of approximately 10 to 20 minutes can be easily determined by simple averaging—thus for times that are practically sufficient for the formation of statistical mean values. If m is the number of analysis cycles running without interruption and T is the time for one cycle, the fraction of the time offset between the first and last readout will be:
$$\frac{T}{m \cdot T} = \frac{1}{m}$$ (22)
amounting to approximately 10% at 10 passes, i.e. the missing time at the beginning is correspondingly made up at the end of the overall analysis, so that on the whole the same integration time results for all filters, albeit with the described phase shift. For the statistical determination of the characteristic data, this procedure appears permissible.
One integration cycle of the instrument is now 4 minutes, switchable to 2 minutes. Shorter times are unfavourable because of the minimum required readout time; longer times would reduce the accuracy of the storage elements. With these values one is practically in a favourable range, particularly when one additionally considers that in the model domain a time compression occurs for equal amounts of information.
Fig. 8: Schematic representation of the readout cycle
A programme-switching mechanism permits, via push-button operation before and after the measurement operation, the automatic execution of a calibration and functional check of the installation (see the view of the instrument with the compensation writer in Fig. 9).
2. Mechanical Construction
The potentiometers are mechanically connected in series. The drive is applied at the potentiometer with the highest frequency by a synchronous motor. From potentiometer to potentiometer, a reduction of i = 0.9 takes place. If the frequency of the fastest potentiometer driven by the motor is designated as the upper frequency f₁, the frequency for the following potentiometers is obtained from the relation:
$$f_n = f_1 \cdot i^{n-1}$$ (23)
where n is the ordinal number of the potentiometer, since each gear reduction reduces the rotational speed by a factor of 0.9.
In the built version with 18 potentiometers, this means an 17-fold reduction of the initial rotational speed of the first potentiometer. The covered frequency range is:
$$\frac{f_1}{f_{18}} = i^{-17} \approx 6.006$$ (24)
If the filter frequencies resulting from the chosen gear ratio are plotted against the ordinal number of the potentiometers, a straight line results in a logarithmic representation (see Fig. 10).
Fig. 10: Filter frequency as a function of the ordinal number of the filters
With this gear arrangement a progressively finer filter spacing towards lower frequencies is achieved, namely logarithmically graded. In a representation of the spectrum over the logarithm of frequency, an equidistant graduation is thus present. In this way a common drive for all potentiometers is found, which moreover provides a favourable coverage (increasingly more measuring points towards lower frequencies) of the frequency range to be analysed.
The drive speed of the potentiometer set is designed for the range given in equation (19). An additional change-speed gear allows the frequency range to be changed very conveniently. Towards higher frequencies, the frequency is limited by the service life and noise errors of the potentiometers. The highest partial component possible is 2 Hz. The analyser is designed for a maximum frequency of 1.5 Hz. Towards lower frequencies (larger wavelengths), the application of the analyser is predestined.
Fig. 11: Filter function of the potentiometers (theoretical)
For practical purposes, however, calibration is most quickly achieved via the Parseval relation given in equation (16), on the basis of the measured r.m.s. value of the sea-state function. This is possible because all filters have the same integration duration.
V. Analyses Carried Out
In recent years, wave measurements and analyses have been carried out on the Großer Plöner See in Schleswig-Holstein. The lake extends up to 7 km in the longitudinal direction and approximately 2 km in the transverse direction. Inland waters of this size have proved suitable for measurements with ship models in natural sea states. After appropriate adaptation and testing of the new analyser, it fully met the expectations set for it. During the tests on the lake, the existing sea state could be assessed immediately on-site from the recording of the Fourier coefficients. Fig. 12 shows an example of an analysis recording for one integration cycle. An example of a spectrum derived from such recordings is shown in Fig. 13.
Fig. 12: Example of an analysis recording (excerpt from a recording, measured on the Großer Plöner See 1963)
Caption to Fig. 13: Example of a determined spectrum. Model scale 1:12. Date: 9.8.1963. Time: 11– . Duration of analysis: 7 min. Wind speed: 4–5 m/s. Fetch: 1.5 km. Wave height: 2.5–3 m [model scale]. Water depth: 1.5 m [model scale]. Estimated values per protocol.
Various tests were carried out successfully. Fig. 14 shows the result of an approximately half-hour analysis of the values of a random-voltage generator [28], compared with the theoretical spectrum delivered by the generator at this setting. The agreement can be considered good.
An extension of the instrument is planned, which shall be briefly outlined here. It is possible to send the read-out values a_n, b_n through a computing circuit that immediately delivers the sought C²_n values according to equation (14). The recording instrument then displays directly the sought partial spectrum at 18 nodes. Suitable analogue elements with adequate accuracy and smallest construction size are offered by industry.
Fig. 14: Analysis of a theoretical noise voltage
An analyser that has been tested in principle is now available for further model investigations in natural sea states; it captures the sea state statistically as an exciting quantity and permits its assessment. The task has thus been solved of providing a suitable evaluation instrument to facilitate the statistical evaluation of exciting quantities in sea-state tests with ship models.
Acknowledgements
The original concept of this analyser is due to Dipl.-Ing. S. Roden; Elektro-Ing. R. Grundmann carried out the electrical construction of the installation; and H. Pöhlsen developed the instrument into the present form. Thanks are due to the Deutsche Forschungsgemeinschaft, which made this work possible.
(Received January 1967)
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