DE2439236C3 - Arrangement for determining the natural frequencies of a test object - Google Patents

Description

zen of a test object; rather, this type of determination of the natural frequencies is assumed to be known

The invention is based on the problem of the accuracy in determining the natural frequencies in particular through sharper separation of the poles in the complex frequency range display and suppression to increase unwanted signal components.

Based on an arrangement of the type mentioned at the beginning, the invention proposes to solve this The task is that a differentiator for differentiating the frequency domain representation of the response signal of the test object is connected to the output of the converter and that a peaks (extreme points) of a differentiated signal in a complex plane determining detector with an output of the Differentiator is coupled. By differentiating the response signal of the test object in a complex frequency domain representation the signal peaks used to determine the natural frequencies are resolved and sharpened to a greater extent and those between the Signal peaks located signal tail, which the position of the signal peaks and thus the measurement result affect, more suppressed.

According to the invention, an alternative solution is that a multiplier circuit with one input to the device sampling the response signal h (t) and another input to a function generator generating a predetermined weighting function x (t) for forming a product signal from the response signal h (t) and the weighting function x (t) is switched on, so that the multiplier circuit is followed by the converter which receives the output from the multipli- F (j

The product signal supplied by the decorative circuit is displayed in a frequency range converts, and that the peaks (extreme points) of the output signal of the converter in a complex plane determining detector is connected to the converter. The multiplication in the The time domain corresponds to a differentiation or folding in the frequency domain, so that this is also an alternative Arrangement for the same advantageous increase in the accuracy in determining the natural frequencies leads.

In the following, the invention will be described with reference to the exemplary embodiments shown schematically in the drawing explained in more detail. It shows

F i g. 1 is a graphic representation to illustrate the interaction of the poles in the s-plane, where the difficulties in localizing the resonance frequencies from a frequency domain representation the response signal of the device under test should be made clear along the j (ü)) axis;

F i g. 2 shows a general block diagram of a first embodiment of the arrangement for determining the Natural frequencies of a test object;

3 shows a block diagram of a second embodiment the arrangement for determining the natural frequencies of a test object;

F i g. 4a is a graph of a function which represents the Fourier transform "w:. ·· Response signal of a device under test to an excitation; FIG.

Fig.4b shows the function of Fig.4a after Tighten or differentiate and

Fig.4c shows the function of Fig.4a after the second Differentiation.

In the following, reference is made to the determination of the poles in the s-plane. The »s« denotes the symbol commonly used for the exponent in the Laplace transformation:

CO

F iß) = J / We "" dt

where s is σ + ja The position of a pole relates to the special resonance frequency (f) or the angular frequency ω = 2; r / and σ, the damping rate at which the tip or pole occurs, and the residual relates to the size \ K \ of the pole, which denotes the initial size or gain of the resonance, and the phase angle < K, which denotes the initial phase of the impulse response associated with this pole. Although the following explanation relates specifically to the determination of the poles in the s-plane It is readily apparent to the person skilled in the art that the measures taken in this connection can also be used in the same way to determine the zero positions. The following description is primarily aimed at determining the positive frequencies or positive ω'ε instead of dL negative frequencies, since the poles associated with negative frequencies are the complex-conjugate values of the poles in the upper half (positive frequency) of the s-plane. It is clear that σ can be positive as in the case of an unstable state or negative as in the case of a damped state. In the following part of the description, reference is also made to the Fourier transform defined below:

OD
⁼ J

f (t) e ~ "dt

In order to make the problem easier to understand, the following first refers to FIG. 4a is referred to. This represents the Fourier transformation of the response signal of a test object to the excitation. The abscissa 13 represents the ju> axis and the ordinate 14 the amplitude of the Fourier coefficients. As will be explained in more detail below, the amplitudes are due to the limited dynamic range of the display media changed to scale before display (by a logarithmic function). At first glance, the frequency at which the peaks, e.g. B. the tip 15, can be read directly from the abscissa 13. However, this is generally not the case, since the interaction of the peaks shown 4a via the offshoots leads in Fig. To a displacement of the tip 15, whereby the actual yaj-position of the tip 15 of the associated pole from the value ju> on the abscissa 13 deviates below the tip (extreme point).

In Fig. 1 the shift of the tip points (extreme points) along the ju> axis 10 as a result of the interaction of the poles is shown graphically. The poles 16 and 17 are shown within the coordinate system which includes the jiu axis 10, the σ axis 11 and the K axis 12. For explanatory purposes it is assumed that both poles 16 and 17 have a position which corresponds to the negative values of σ. The pole 16 seems to be on the ycu axis at ωο '. The influence of the extension of the pole 17, shown as a dashed area 18, causes a change in the position of the pole 16 along the ./ω axis in the direction shown by the arrow 19. Similarly, the tail of the pole calls

16 shows a shift in the position of the pole 17 along the ./ω axis in the direction of the arrow 20. Because of This interaction via the foothills (18) of neighboring extreme points leads to the investigation of the Fourier transformation of a time domain function only for an imprecise determination of the position of the poles lengthways the yco axis. With the arrangement, the Fourier transform "exacerbated" by differentiation, with the foothills of the extreme points largely suppressed, the interaction between the poles minimized and a precise determination of the resonance frequencies is made possible. In Fig. 4b is the first derivative of the signal curve shown in FIG. 4a shown, and it can be seen that the peaks or extreme points have much steeper tails, thereby eliminating some of the interaction between the poles. In Fig. 4c is the second derivative of the signal curve shown in Figure 4a. In of this figure, the flanks and poles are considerably steeper than those in figure 4b, which makes the interaction between the poles over their foothills largely eliminated and a more precise determination the frequency of the poles is made possible.

To facilitate understanding of the described arrangement, those systematic principles on which the invention is based are first given below. As is known, the magnitude of a pole at any point in the s-plane is equal to the maximum magnitude of the residual | K | of a pole divided by the distance from the pole. Therefore, the amount of the Fourier transform of a pole that occurs at σο + jcüo results from the following system of equations:

Formulation, the / -th derivative can be written as follows:

/ 'J

where μ \ again represents a correction factor.

As previously mentioned and shown in FIGS. 4b and 4c, the sharpening process decreases at the differentiation of the frequency domain representation, the interaction between the poles. For the first Derivative, the apparent position of the pole along the yiu axis essentially coincides with the actual pole position together along this axis. Hence, the expression (ω - ωο) approaches zero because the frequency at which the Expression is calculated, the ωο corresponding frequency is. Neglecting the expression (ω —ωο) (with ω = ωο) in equations (2) and (3) the following two equations result:

I / r ₍ , " ₀ ) | =

similarly, expressions can be written for all poles.

If the value of the first derivative Η '(ω _α ) is divided by Η "(ωα), the second derivative is divided as follows:

IH

IK _n

I / o · ² + {o, - r ~ " ₀ -) ²

^a ) 35 so the σο position of the pole can be determined.

The ratio of the cube of the first

Derivative and the second power of the second

j,) derivative leads to the following equation:

where Oo equals the pole position along the σ-axis and ωο the Is the angular frequency at which the expression is determined. A similar expression can be given for each pole, which occurs along the jcü axis The first derivative the equation (1 a) can be written as

₄₀

-JK ₀

la + j (f -.- r.-o)] ²

then:

ri ι - -υ t

(J ₀ + (f. -

(2 a)

(2 B)

(, "-," o) ² ] ³ ' ²

P) -

45 so

where μ.ι represents a correction factor that may be necessary if Η {ω) is not a continuous function, but represents a finite number of samples. For the theoretical case in which Η {ω) is a continuous function, μι would be equal to 1 ropes. The second derivative of equation (la) can be used in a similar way

60

can be obtained, where μι in turn represents a correction factor that may be necessary if Η (ω) is not a continuous function In general IH "(

/ 4

From this equation, the size or the value of the pole and for this reason the size or the Value of all poles along the yia axis can be determined.

In equations (7) and (8), the ratios of the first and second derivatives were used, although any other two derivatives were used to calculate | Ko | and σο, so z. B. the first and third derivatives could be used. In the exemplary embodiment, two successive derivatives, e.g. The first and second derivatives are used as the use of consecutive derivatives makes the calculation easier. Other consecutive derivatives, e.g. B. the third and fourth derivatives or the fourth and fifth derivatives could be used in the same way.

The remainder of the residual remains for the calculation, this is the one assigned to the amount Phase angle.

Similar to the analysis previously discussed in connection with equations (la) through (2b), the following Equation can be written:

65 where Äo is equal to that of the pole at ω ₀ and σ ₀

associated phase angle and where / Η '(ω) is the phase angle of the differentiated Fourier transform at the observed ω. The phase of the second derivative can be written in a similar way:

\ h "(ω) = JK ₀ -π + 2arctane (
\ ° o

(10)

and the / -th derivative can be written as follows:

'("Ή = j K ₀ - I ~ + / arctan Λ-— ° Λ (11) 1 2 \ σ ₀ j

Equations (9) and (10} result in:

K ₀ = 2

(12)

The arctan expressions in equations (9) and (10) are omitted, so that these expressions do not need to be calculated to determine the phase angle. All phase angles can be calculated in a similar way. In the calculation described above, the first and second derivatives of the frequency domain representation were again used, and as in the case of the calculation of K and σ, any other two consecutive derivatives can be used, e.g. B. the fourth and fifth derivatives can be used.
• The previous theoretical analysis shows that the position of the pole in the direction of the yiu and σ axes can be determined in addition to the residual (\ K \ and / K) of the pole.

In Fig. 2 is a generalized block diagram of FIG Arrangement shown. The impulse or transition response of the test object is measured by an impulse sensor 22 scanned and fed to a Zek-Z frequency range converter 23. The pulse counter 22 can be known in a Be designed such. B. as an electromechanical or optical device. The time-to-frequency converter can be any known means for converting a time domain representation to a frequency domain representation exhibit. A computing device to perform this function and to implement the Time domain information in Fourier coefficients is described in US Pat. No. 3,638,004. The frequency domain representations at the output of the converter 23 are fed to a differentiator 24, where the input information is in the form shown in FIGS. 4b and 4c is "tightened" in a recognizable manner. Numerous known circuits are available for performing the Differentiation function available in both analog and digital form. The results everyone Differentiations are transmitted to a storage device 25 via a line 26. The storage device 25 may be a random access memory, e.g. B. a magnetic core memory or a semiconductor memory be. To transfer the data stored in the memory 25 to the differentiator 24 is a Line 27 is provided.

The memory device 25 is also connected to an arithmetic device 28 such that Information from the memory device via a line 29 to the arithmetic device 28 and from the latter can flow via a line 30 to the storage device 25. The arithmetic device 28 can be designed as a multipurpose digital computer that is programmed to the required calculations, or they can be formed by a suitable hardwired computer.

A peak detector 33 is connected to the memory device 25 so that an information flow from the memory device via line 35 to the peak detector and similarly from the peak detector 33 can be done via a line 34 back to the memory. The peak detector 33 is used for Determination of the deryiu position of the poles, for example in Fig. 4c are shown. Hence, a derivative, e.g. B. the second, third or fourth derivative of the Frequency domain representation of the signal sampled by the sensor 22 from the storage device 25 to the Peak detector 33 transmitted. The peak detector determines the /.o-I.age of each of the peaks and is1 on the output side connected to the memory 25 via the line 34. Various circuits are known which determine the peak amplitudes according to the function of the peak detector and the required information procure. Because of the tightening resulting from the differentiation, it is not difficult to find the individual peaks or maxima'stellen from each other distinguish and precisely localize their frequency.

The functioning of the in F i g. 2 is best understood in conjunction with the aforementioned equations. The impulse sensor 22 first scans the impulse response of the system and transmits this response to the converter 23. The converter 23 carries out a Fourier transformation or another transformation on the data and forms! the quantity Η (ω). This information is transmitted to the differentiator 24, in which it is differentiated, and the results of the differentiation are then stored in the memory 25. The results of the first differentiation are fed back to the differentiator 24 via the line 27, and the differentiator 24 carries out a second differentiation, the results of which are sent to the memory 25 via the line 26 and are then stored. As previously explained, this differentiation process can be continued until any two successive differentiations e.g. B. the fourth or fifth are executed. As soon as the results of the differentiations, ζ. Β. Η '(ω) and Η "(ω, are stored in the memory 25, the calculation of the I K | values, σ values and the phase angle can begin.

With the results of the differentiations stored in the memory 25, the σ values and [ K

then suitably determine values according to equation (7) or (8) with the aid of the arithmetic device 28! will. These calculations only require relatively simple mathematical operations. As in detail in connection with FIG. 3 it will be explained how the σ-values and # -values are calculated and stored in the memory device 25. The correction factors {μ. \ And μ.2) can optionally be stored in the same way in the memory device 25 and used to correct the 0 and Ti values. Thereafter, the phase angles assigned to each of the K values can be solved in the arithmetic device 28 the equation (12) are calculated again this is a relatively simple calculation process, and the phase angle for each value of / Cbestimmt After the calculation of all values of a, K and the phase angle is a derivative of Η {ω) on the line 35 to the Peak detector 33 supplied The highest order derivative, e.g. that in Fig. 4 (c)

10

= F {t ²

The second derivative shown is transmitted to the peak detector 33. In the peak detector 33, the frequencies at which the peaks or extreme points occur can be determined in the manner explained above. The Χω), which corresponds to these frequencies, is fed back to the storage device 25 via the line 34. The σ, [ K | and the phase angle corresponding to the values determined by the detector 33 represent the position of the poles and the residuals for the signal sensed by the sensor 22 and are transmitted to the output line 37.

As mentioned above, the values for | K |, σ and the phase angle are determined for each ω, and then the values for σ, ω and the phase angle for each pole are retrieved from the storage device 25. The arithmetic operations can be shortened in that, if necessary, only those values of | K |, σ and the phase angle are calculated for the values of yco at which peaks occur. This would make it necessary to determine the peaks with the aid of the peak detector 33 before calculating σ, \ K \ and the phase angle in the arithmetic device 28. In those cases in which all values of K, 0 and the phase angle are calculated, this calculation process in the arithmetic device 28 can take place simultaneously with the determination of the peaks in the peak detector 33.

In order to explain the mode of operation of the exemplary embodiment of the arrangement even better, a supplementary mathematical explanation should first be sent in advance. In the arrangement described, the input signal is sampled, e.g. B. the impulse response, so that a continuous function is not used. For purposes of explanation it is assumed that the input signal is sampled / V times and the sampled signal is converted into digital form, the arithmetic operations in the arrangement being carried out in digital form. ψ (1) = sin

If a finite number of samples of the sampled signal is used, the described differentiation in the frequency domain is actually a convolution in the frequency domain or a multiplication in the time domain. For example, assume that the time domain input function, e.g. B. the impulse response or the transient response a finite number of samples N and by h (t) _{n =} \ _n

' and that the Fourier transform of this signal is represented by Η {ω) . It can be seen that the first derivative of Η (ω) can be written as follows:

Therefore the Fourier transform of th (t) is equal to the first derivative in the frequency domain of Η (ω). Similarly, the second derivative is to be written as follows:

(14)

or the / -t derivative can be written in the following form:

H ' (ω) = F {-j't'-h (c) \

(15)

The above equations show that the differentiation operation becomes superfluous by multiplying h (t) by t or a ramp function and that the poles can also be sharpened as described above by multiplication in the time domain. Many different known functions can be used as the weight function in place of t to achieve the convolution in the frequency domain; In the exemplary embodiment, a trigonometric function (sine function) is used for this purpose. It has been found that when a sine function is used, the correction factor is close to 1 and, if necessary, can be suitably calculated. This is because a finite subtraction is used in calculating the values of σ, K and the phase angle.

When the trigonometric or weight function is represented by x (t) , the following equation can be written:

where Δ "{... ) represents the / th" differentiation "or convolution and Δ the known operator. r (t) is represented by

π η τ

(Π)

where η is 0, 1, 2, 3 ... / VI, Γ is the period of the input signal and τ is dps sampling interval. From the above equations it can be shown that using the first and second convolutions, the following equation holds

I ² iff Μι + Γ Iff Mi sin π / 2 I ² {ff Ml - I 'iff M) sin π / 2

(18)

TT 'ι \ Cf: *. / " Ι * \\

11 \ ·· ΐ) - ι , - j ι it \ r; i

where F is the Fourier transform where Z is a correction factor and In is the natural logarithm. Similarly, an equation for K can be written in complex form, ie the magnitude and phase angle associated with the poles.

κ = \ k \ Ik_ =

+ 2e "cos j- + e

^Sin i

Γ \ H (<")} (19)

In Fig. 3 shows an exemplary embodiment of the arrangement described, in which a pulse sensor 40 similar to the pulse sensor 22 of FIG. 2 is used to sample the impulse response of the sample to transmit the signal response to the sample and hold device 41. In some cases the impulse or transition response is pre-recorded and applied directly to the sample and hold device when in analog form. In the case of a response signal prerecorded in digital form, this can be applied to line 46. The sample and hold device 41 samples the impulse response with a finite number (N) of samples and transmits each of the samples to an analog / digital converter 42. Im

Converter 42 converts each of the samples into digital form and puts them on line 46. The output of converter 42 corresponds to h (t) in the above equations.

An additional input line 54 is shown in FIG. 3 is connected to a data conditioner 53. The data conditioner 53 is used in cases where the input signal of the arrangement is in a other form than an impulse response or another transition response or transient response exists, and is used to condition the data and convert it into a time domain function. That The output signal of the conditioner 53 is instead of the data from the converter 42 via the line 54 a Multiplier circuit 45 supplied. Examples of the type of data supplied to the conditioner 54 are those for the autocorrelation function, the autospectral function, the cross-power spectrum or a transfer function are representative. In situations where the transfer function or If the cross spectrum is used, this data can be converted into an impulse response. In the These functions can be used in cases where the auto spectrum or the auto correlation function are supplied can be implemented in a similar manner in a suitable time domain function by a physically appropriate phase function of the square root of the auto spectrum or the square root of the Fourier transform is assigned to the autocorrelation function. A criterion for assigning such a Phase function is by H. Bode in Network Analysis and Feedback Amplifier Design, Van Nostrand Co .. Inc., New York, 1945 stated.

The multiplier 45, which is called ordinary digital multiplier can be formed, multiplies two digital signals and developed on the Line 49 is a digital signal representing the product. The multiplier 45 is used for multiplication of two digital signals which are supplied via lines 46 and 47 or 47 and 48. A storage device 52, which is connected to the output of the multiplier circuit 45, supplies stored signals via line 48 back to the multiplier circuit 45 and via line 55 to a Fourier transform converter 58. The memory device is, for example, a semiconductor or magnetic core memory designed that digital information can be stored in it.

A function generator 43 is used to generate the weight function, which, as explained above, in described embodiment is a sine function. The function generator 43 generates this information in digital form, and its output is connected to the multiplier 45 via line 47.

The Fourier transform converter 58 is a digital circuit for converting a time domain signal in a frequency domain signal and, in the exemplary embodiment, has a hardwired computer according to US-PS 36 38 004. In Fig.6 of this patent and in the associated description are a sample and hold device and an analog / digital converter indicated which for blocks 41 and 42 according to FIG. 3 can be used.

However, various other known circuits as a sample and hold device 41 and ^ s converters are used 42nd In the aforementioned US-PS 36 38 004, a trigonometric function generator is also indicated in FIG. 7, which can form the function generator 43 in this embodiment. However, other known circuits are also suitable for generating the sine functions in digital form for the purposes of the present invention.

The storage device 60 may be any of various commercially available storage devices, e.g. B. a Semiconductor or magnetic core memory can be used. Although in the block diagram according to FIG Storage device 60 is shown as a separate memory, it can be combined with storage device 52 be summarized. The storage device 60 receives the results of the converter 58 performed Fourier transformation on line 59 and stores the complex Fourier coefficients. The memory device 60 is also available via lines 68 and 69 with an arithmetic Device 71 in connection and is designed so that they stored signals via a line 6fi to a Digital / log-to-analog converter 62 can transmit. Information that includes the resonance frequencies in the described embodiment can be via a Line 67 are read into the memory. The line 67 is connected to a display 63. the Output information is output from memory 60 via an output line 72.

In the embodiment described, the position of the poles along the yiw axis is obtained from a display device selected by the "derivative" or highest order convolution of the input signal Usually the digital data used for the calculations have a significantly greater dynamic Area as a display device, e.g. B. a cathode ray tube. The digital / log-to-analog converter 62 inserts the digital information to be displayed on the display device 63 in Analog signal around. whose value is a logarithmic function of the digital signal A circuit for Implementation of the digital / log-to-analog conversion, especially for use in connection with a Display device is suitable is given in US Pat. No. 3,670,326. The practice * · the line 66 with the Display device 63 connected to converter 62 may be an ordinary cathode ray display device or be designed in some other way. As soon as the ad, e.g. B. the diagram according to FIG. 4 (c) is present, The frequencies at which the poles occur can be transmitted via line 67 to storage device 60 be transmitted. To determine the frequencies of the poles, usual Scale devices may be provided.

The arithmetic device 71 can be either a hard-wired digital computer or a general-purpose digital computer with one to solve the equations (18) and (19) appropriate program. These equations are relatively easy to use for the digital computer solve so that the arithmetic means 71 easily using known technologies can be produced

The mode of operation of the arrangement shown schematically in FIG. 3 is as follows. After determining h (t) , the variable h (t) is multiplied by the sine function generated by generator 73 in multiplier 45 saved. The results of this first multiplication are transmitted to the Fourier transform converter 58, where the Fourier transform is carried out.

The results of the first multiplication x (t) · h (t) are then fed back via the line 48 to the multiplier circuit 45 and multiplied again by the trigonometric function generated by the generator 43, the quantity x ² (t) h (t) is formed. The results of this multiplication are transferred to the memory device 52 and stored there. The results of this second multiplication are transferred, similar to the first, to the Fourier transform converter 58, the Fourier coefficients are recalculated and stored in the memory device 60. This process is repeated until the in defined by the above equations / equal to the desired number. For the exemplary embodiment, only two such multiplications are carried out, and for purposes of explanation it is assumed that the frequency domain representation of the first two multiplications is stored in the memory device 60.

As soon as the complex Fourier coefficients resulting from the two multiplication processes are stored in the memory device 60, the σ values are determined by solving equation (18) in the arithmetic device 71 for each discrete value of ω. the magnitude and the phase angle for each discrete value of ω are determined by solving equation (19) in the arithmetic device 7t. All values of σ and K, including the phase angle information, are then stored in memory device 60. As mentioned before, all possible values of these quantities are calculated when calculating the values of σ and / (T) because of the low computational effort.

Next, the Fourier coefficients resulting from the second multiplication are transferred to the digital / log-analog converter 62 and, after a suitable scale conversion and conversion into analog form, they are displayed on the display device 63. The display can show the resonance frequencies or the values for π or ω at which the poles appear can be determined appropriately. These values are then transmitted to the memory 60 and used to select the σ values and the complex K values which correspond to the resonance frequencies. This information can then be obtained on the output line 72.

In equation (18), the correction factor Z was assumed to be one for the purposes of the above calculations. It has been shown that when a trigonometric function is used as a weight function and with a relatively large number N of samples, the correction factor is in the immediate vicinity of one.The correction factor Z can be expressed as follows:

Z =

1 - e-itf-i

(20)

If the correction factor Z is important, it can be determined by first assuming a correction factor of 1 and calculating σ using equation (18). Then, the value of the correction factor Z can be calculated using equation (20). Using this calculated value, the values for σ are re-determined, again using equation (18). This process can be repeated according to the accuracy required for the correction factor. It should be noted that correction factors other than that according to equation (20) can be used.

For this purpose 3 sheets of drawings

Claims (7)

Patent claims: 1. Arrangement for determining the natural frequencies of a test object from a response signal of the Test specimen to an excitation, with a device that scans the response signal and one of these Device downstream converter for obtaining a frequency range representation, thereby characterized in that a differentiator (24) for differentiating the frequency domain representation the response signal of the device under test is connected to the output of the converter (23) and that a peaks (extreme points 15) of a differentiated signal in a complex plane is more decisive Detector (33) is coupled to an output of the differentiator (24) 2. Arrangement according to claim 1, characterized in that the differentiator (24) with a Memory (25) connected and a return (27) from the memory to the differentiator (24) for implementation of at least two successive differentiations is provided and that one is the Position of the poles and their residuals in the complex plane from the two consecutive derivatives determining arithmetic means (28) is coupled to the differentiator (24). 3. Arrangement for determining the natural frequencies of a test object from a response signal of the test object to an excitation, with a device that scans the response signal and a converter downstream of this device for obtaining a frequency range representation, characterized in that a multiplier circuit (45) with an input (46) ün to the device (40, 41, 42, 53 ) scanning the response signal h (t) and with another input (47) to a function generator (43) generating a predetermined weight function x (t ) for forming a product signal from the response signal h ( t) and the weight function x (t) is switched on, that the multiplier circuit (45) is followed by the converter (58) which converts the product signal supplied by the multiplier circuit into a frequency domain representation, and that the peaks (extreme points 15) of the output signal of the Converter in a complex plane determining detector is connected to the converter (58). 4. Arrangement according to claim 3, characterized in that the converter as a Fourier transform converter (58) associated with this, the position of the poles (16, 17) of the response signal h (t) of the. DUT and their residues in the complex plane determining arithmetic device (71) is formed. 5. Arrangement according to claim 4, characterized in that a storage device (60) between the arithmetic device (71), the converter (58) and the peak detector (62,63) is arranged. 6. Arrangement according to one of claims 3 to 5, characterized in that the multiplier circuit (45) a memory unit (52) is connected downstream from which a connection (48) to the multiplier circuit is returned. 7. Arrangement according to one of claims 3 to 6, characterized in that the function generator (43) is designed as a sine generator that generates a sine function as a weight function xfζ). The invention relates to an arrangement for determining the natural frequencies of a test object a response signal of the device under test to an excitation, with a device that scans the response signal and a converter downstream of this device for obtaining a frequency range display. In many areas of technology it is desirable to carry out a model analysis of a test object, where the natural frequencies of the test object and their attenuation, which are determined by the position of the poles in the complex s-plane is given, and the state of the system determined by the frequencies assigned to these complex residuals are of particular importance. Obtaining a theoretical transfer function and in particular the theoretical position of the Poles and their residuals is generally only practicable for the simplest structures and systems. in the Ideally, the poles and their residuals should be determined from empirically derived functions or quantities which can be measured in a suitable form or derived from the test object. So should, for example, willow the position of the poles in the s-plane and their residuals ideally from some time domain function, e.g. B. the impulse or transition function of the test object can be determined. Some methods are known for obtaining the position of the poles in the s-plane and their residuals empirical information, e.g. B. the impulse response or from a frequency domain representation of the impulse response, z. B. the Fourier transform of the answer. Most of these methods or systems - albeit not all - require a special assessment or evaluation and do not provide an overall arrangement Determination of the required s-level information is available. The following are known to be relevant Methods and Techniques: (1) Kennedy-Pancu method, using curve approximation techniques Journal of Aeronautical Science, Vol. 14, 603-625 (1947). (2) James C. Davis method, published in an article entitled Modal Modeling Techniques for Vehicle Shake Analysis, Automotive Engineering Congress, Detroit, Michigan, January 10-14 1972, No. 720 045. (3) Method for the experimental determination of the poles of a structure by multiple excitation of a shaker system by R. Lewis and D. Wrisley in an article entitled A System For The Excitation of Pure Natural Modes of Complex Structure, Journal of Aeronautical Sciences, Volume 17, No. 11, November 1950, pages 705-723. (4) A general contribution to the discussion is in an article by Bishop and Gladwell entitled An Investigation into the Theory of Resonance Testing, published in Philosophical Transactions of the Royal Society of London, Volume 255, January 1963, pages 241-280. From "Freiberg Research Books", C 130, Akademie-Verlag-Berlin, 1962. Pages 14-19 and 91-93 it is known the motion characteristics of a seismometer and one with this coupled galvanometer with the help of differential equation systems determine. The natural frequencies of the two coupled vibration systems are also important. In the present invention, however, it is not about obtaining the position of the poles in the complex s-plane to determine the natural frequency