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General
A complete vibration testing system, whether electro-dynamic or servo-hydraulic, consists of essential components such as the power amplifier, vibrator, loaded test fixture, and control system.
Before initiating the test or during the testing phase, it is essential to verify the basic and cross axis motions using an additional input monitor channel of the controller The relevant specifications must outline the test levels and procedures to be followed for the investigation.
The standardized test method involves a sequence of tests applied along each mutually perpendicular axis of the test specimen It begins with an initial vibration response investigation using low-level sinusoidal or random excitation This is followed by a mixed mode excitation, serving as the load or stress test Finally, a concluding vibration response investigation is conducted to compare results with the initial findings and identify any mechanical failures resulting from changes in dynamic behavior.
However, the relevant specification may renounce the requirement for a response investigation, or part thereof, if the dynamic behaviour of the test specimen is known or not of interest.
Control systems
Specialized software control packages are essential for control systems that need to analyze and manage tests involving a combination of random on random or sine on random excitations and specifications.
Basic motion
The fixing points of the specimen must follow a rectilinear motion as specified by relevant guidelines, ensuring that their movements are substantially identical In cases where achieving identical motions proves challenging, the implementation of multipoint control is recommended.
The characteristics of the basic motion shall be nominally a Gaussian distribution for the random waveform and sinusoidal for the periodic components.
Cross axis motion
Before applying the test, it is essential to check cross axis motion through a sine or random investigation as specified in the relevant guidelines, or alternatively, during testing by using an additional monitoring channel.
The signal values at check points in any axis perpendicular to the specified axis must not exceed the defined limits, with frequencies above 500 Hz and below 500 Hz restricted to –3 dB of the specified values Additionally, the total root mean square (r.m.s.) acceleration in any perpendicular axis should not surpass 50% of the r.m.s value for the specified axis For instance, in the case of a small specimen, the allowable cross motion signal value may be capped at –3 dB of the basic motion, as outlined in the relevant specifications.
Achieving the required specifications can be challenging at certain frequencies or with large or high-mass specimens, particularly when a wide dynamic range is mandated In these instances, the relevant specification must clarify whether: a) cross-axis motion exceeding the specified limits should be monitored and reported, or b) monitoring of cross-axis motion is not necessary.
Mounting
The specimen must be mounted following the guidelines of IEC 60068-2-47 It is essential to square the transmissibility curve selected from IEC 60068-2-47 before multiplying it with the ASD spectrum, or alternatively, to multiply it directly with the sine amplitudes.
Measuring systems
The measuring system must be designed to ensure that the true value of vibration, measured along the specified axis at the reference point, falls within the required tolerance for accurate testing.
The accuracy of measurements is significantly influenced by the frequency response of the entire measuring system, which comprises the transducer, signal conditioner, and data acquisition and processing device It is essential for the frequency range of the measuring system to span from at least 0.5 times the lowest frequency (\$f_1\$) to 2.0 times the highest frequency (\$f_2\$) of the test frequency range Additionally, the frequency response must remain flat within ±5% throughout this specified frequency range.
A cc el erat ion s pe ct ral dens ity d B
Figure 1 – Boundaries for acceleration spectral density
5 Requirements for testing mixed mode
This standard outlines test methods for implementing random vibration alongside narrow band random or sinusoidal vibration, or a combination of both The narrow band random and sinusoidal elements can be adjusted across a specified frequency range as detailed in the relevant specifications It is essential to consider specific factors when conducting mixed mode testing.
The relevant specification shall state whether the narrow band random profiles are the maximum (MAX) spectral levels or shall be added to the background spectral profile (SUM)
The acceleration spectrum can be classified into two types: a) a superpositional acceleration spectrum that combines broadband random, narrowband random, and sine tones for control systems where the sine wave is generated at Fourier spectral lines, or b) a superpositional acceleration spectrum consisting of broadband random and narrowband random with independent sine tones, applicable to control systems where the sine wave is continuously generated in the frequency domain.
Vibration tolerances – Random
The acceleration spectral density at the reference point and check points between frequencies f1 and f2, as shown in Figure 1, must remain within ±3 dB of the specified acceleration spectral density, accounting for instrument error It is important to note that random and bias errors are not included in these tolerances, although the random error can be calculated.
The root mean square (r.m.s.) value of acceleration, whether calculated or measured, must fall within ±10% of the r.m.s value linked to the specified acceleration spectral density This requirement applies to both the reference point and the fictitious reference point.
Achieving specified values can be challenging at certain frequencies or with large or heavy specimens, leading to the expectation that the relevant specifications will allow for a broader tolerance.
The initial slope shall be not less than +6 dB/octave and the final slope shall be –24 dB/octave or steeper (see also B.2.3)
In swept narrow band random tests, the tolerances for the swept components must align with those of the wide band component However, certain sweep rates may render these tolerances unattainable Consequently, the specific tolerance requirements for these components should be clearly outlined in the relevant specification.
The instantaneous acceleration values at the reference point exhibit an approximately normal (Gaussian) distribution, as illustrated in Figure 2 Validation will be conducted during the normal system calibration process For mixed mode signals that include sine waves, refer to Figure 4.
Figure 2 – Stochastical excitation, representation of signal clipping and Gaussian (normal) probability
The drive signal clipping must be a minimum of 2.5, as referenced in section 3.18 Additionally, the crest factor of the acceleration waveform at the reference point should be analyzed to confirm that the signal includes peaks that are at least three times the specified root mean square (r.m.s) value, unless stated otherwise in the applicable specification.
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If a fictitious reference point is used for control, the requirement for the crest factor applies to all the check points used to form the control acceleration spectral density
The probability density function shall be computed for the reference point for a duration of
2 min at the beginning, middle and end of the test duration
The statistical accuracy is determined from the statistical degrees of freedom N d and the confidence level (see also Figure 3) The statistical degrees of freedom are given by:
T a is the effective averaging time;
N d shall not be less than 120, unless otherwise specified by the relevant specification
If the relevant specification states confidence levels to be met during the test, Figure 3 should be used to calculate statistical accuracy
Sta tis tical a ccu ra cy
Figure 3 – Statistical accuracy of acceleration spectral density versus degrees of freedom for different confidence levels
Sine 120 Hz, 50 m/s 2 + Random 20 200 Hz , different ASD values (overall rms in m/s 2 )
Figure 4 – Distribution (probability density) of sine, sine-on-random and random signals
To minimize the discrepancy between the actual and indicated acceleration spectral density, the required frequency resolution \( B \) in Hz should be determined by dividing the digital controller's frequency range by the number of spectral lines \( n \).
The equation \$B_e = \frac{f_{high}}{n}\$ defines the relationship between the frequency range of a digital vibration controller, \$f_{high}\$, measured in hertz, and the number of spectral lines, \$n\$, which are evenly distributed across the frequency bandwidth up to \$f_{high}\$ It is essential that \$f_{high}\$ is greater than or equal to twice the value of \$f_2\$, expressed as \$f_{high} \geq 2f_2\$, as illustrated in Figure 1.
The frequency resolution shall be given in the relevant specification, (see also Clause 13, item h)
B e shall be chosen such that:
The minimum frequency line aligns with frequency \$f_1\$ as shown in Figure 1, while the first frequency line is positioned at 0.5 of \$f_1\$ Additionally, these two frequency lines establish the initial slope of the first sweeping narrowband.
If this gives two different values then the smallest B e shall be chosen
There is a trade-off between achieving a finer bandwidth resolution, which leads to slower loop control times, and obtaining a clearer representation of the Rate of Return (RoR) spectrum Additionally, faster sweep rates for the sweeping bands may necessitate higher frequency resolution to effectively manage the sweeping bandwidths.
B e shall be chosen such that: as a minimum, a frequency line coincides with the frequency f 1 in Figure 1 and the first frequency line is at 0,5 of f 1
The sine sweep should where possible be continuous For control systems where the sine wave generation jumps from one frequency line to the next, B e should be less than 0,1 % f high
Vibration tolerances – Sine
For swept sine on random testing, a digital tracking filter is normally employed to estimate the sinusoidal amplitude A tracking filter will also reduce the random portion of the signal
The estimated value of the sinusoidal amplitude incorporates contributions from the random components of the signal at the sinusoidal frequency Additionally, an increase in the ratio of the random signal's amplitude spectral density (ASD) to the square of the sinusoidal root mean square (r.m.s.) value is observed.
A higher "power ratio" leads to increased random error, while reducing the bandwidth of the tracking filter decreases this error However, a narrower bandwidth necessitates a greater number of averages to maintain accuracy.
High-quality resonances in a specimen can lead to increased bias error when a larger number of averages is used This bias error represents the discrepancy between the averaged sine amplitude and the actual response.
Vibration tolerances for sine tones in swept sine random testing must exceed the total of random, bias, control, and instrument errors.
Figure 5 shows the recommended sweep rate as a function of power ratio with the following assumptions:
– using digital tracking filter by Fourier integration;
– using exponential averaging to estimate the sinusoidal amplitude;
– damping ratio of a specimen is 0,01;
– is a combined error of random error and bias error, other errors such as control error and instrument error are not included;
– indicated value of combined error is assumed as standard deviation
The total error E t is expressed in equation (3):
K is 2 when using 2 sigma deviation;
E i is an instrument error as standard deviation;
E c is a control error as standard deviation
The following frequency tolerances apply:
– for swept frequencies ±1 Hz from 5 Hz to 50 Hz, ±2 % above 50 Hz;
Control strategy
When multipoint control is specified or necessary, the control strategy shall be specified
The relevant specification shall state whether single point or multipoint control shall be used
When multipoint control is implemented, the specification must clarify whether the average signal values at the checkpoints or the signal value at a specific point, such as the one with the highest amplitude, will be regulated to meet the designated level.
If single point control is unattainable, multipoint control should be implemented by managing the average or extreme values of signals at designated checkpoints In such cases, the reference point is considered fictitious, and the method employed must be clearly outlined in the test report.
The following strategies are available
In this method, the control value is computed from the signal from each check point
A composite control value is created by averaging the signal values at various frequencies from designated check points This averaged control value is subsequently compared to the specified signal values for each frequency.
The control value of each frequency a c is formed by averaging the signal value from the check points a 1 to a n according to their weighting w 1 to w n :
This control strategy offers the possibility that different check point signals contribute a different portion to the control value of each frequency
This method calculates a composite control value using the maximum (MAX) or minimum (MIN) extreme values of the signal at each frequency line measured at designated checkpoints By employing this strategy, a control value is generated for each frequency, reflecting either the envelope of the signal values (MAX) or the lower limit of the signal values (MIN) as a function of frequency from each checkpoint.
Multiple reference spectra can be defined for various checkpoints, measuring points, or types of controlled variables, as specified by the relevant standards, such as in force-limited vibration testing.
When multi-reference control is specified, the control strategy shall be prescribed by one of the following:
– limiting: all control signals have to be beneath their appropriate reference spectrum;
– superseding: all control signals have to be above their appropriate reference spectrum.
Vibration response investigation
The requirements for sinusoidal excitation are given in Test Fc (IEC 60068-2-6) and those for random excitation are given in Test Fh (IEC 60068-2-64) See also IEC 60068-3-8
The test severity is determined by a combination of the following parameters:
– shape of acceleration spectral density curve;
Each parameter must be defined according to the applicable specification, either by selecting values from sections 6.1 to 6.3 or by deriving them from the known environment when this results in significantly different values.
When analyzing measured data, it is crucial to exercise caution in determining the levels of random and sinusoidal signals, as the data reduction techniques used can significantly impact the amplitudes of these signals For further guidance, refer to IEC 60068-3-8.
Broadband random vibration
The test frequency range must be specified in the relevant documentation, adhering to frequency limits that closely align with the series: 1, 2, 5, 10, 20, 50, etc The lower frequency limit, denoted as \$f_1\$, should begin at a minimum of 1 Hz, while the upper frequency limit, \$f_2\$, should not exceed 5,000 Hz.
The acceleration spectral density, as illustrated in Figure 1, should be specified within the range of frequencies f 1 and f 2, using values in either (m/s²)²/Hz or m²/s³ It is important to select values that closely align with the series: 1, 2, 5, 10, etc The specified range must have a minimum value of 0.01 and a maximum value of 100.
NOTE For those wishing to continue giving values in g n , the value of 10 m/s 2 is ascribed to g n for the purposes of this standard
6.1.3 Shape of acceleration spectral density curve
This test defines an acceleration spectral density spectrum featuring a flat horizontal section In certain instances, it may be necessary to outline a shaped acceleration spectral density curve, with the specification detailing the shape as a function of frequency The break points, which correspond to different levels and their frequency ranges, should ideally be chosen from the values provided in sections 6.1.1 and 6.1.2 Furthermore, the specification must also define the slopes between these levels.
The testing duration should be specified in the relevant documentation, using values that closely align with the series: 1, 2, 5, 10, etc., measured in minutes, hours, or days, allowing for a tolerance of +5%.
Random narrowbands
The relevant specification shall specify the number of random narrowbands to be added to the background acceleration spectral density
Each narrowband must adhere to specific criteria: a) the bandwidth should range from a minimum of 0.5% to a maximum of 10% of the background random bandwidth, with the lower limit being no less than two frequency resolutions; b) the start and end frequencies of the sweep must be clearly defined; c) the sweep rate should be specified in octaves per minute or Hz per second, or the total sweep time for one cycle should be indicated; d) the number of sweep cycles or the duration of the narrowbands must be stated; e) it should be specified whether the sweep is logarithmic or linear; f) the initial direction of the sweep, either up or down, should be noted for each band; g) the specified spectrum for each narrowband must fall within the defined bandwidth limits, from frequency f1 to f2; h) finally, the chosen strategy for combining all narrowbands with the background acceleration spectral density value should be either SUM or MAX.
Sine tones
The relevant specification shall specify the number of sine tones to be added to the broadband acceleration spectral density
It shall state: a) whether they are harmonically related to each other or not and their phase relationship;
The phase relationship pertains to the output from the controller, while the phase of acceleration signals may be altered by the transfer function of the vibration generator, fixture, and specimen Additionally, it is important to consider the start and end frequency of the sweep, the sweep rate measured in octaves per minute or Hz per second, and the duration of one complete sweep cycle.
NOTE It is recommended to use a sweep rate that is as slow as possible according to 5.2.1 and Figure 5
High sweep rates can lead to inaccurate control of sine tones Key factors include the initial direction (up/down) and the on/off timing for each tone, the amplitude of each tone relative to frequency, the number of sweep cycles or tone duration, the type of sweep (logarithmic or linear), and the frequencies and amplitudes of fixed sinusoids.
In some instances, sinusoids may not cover a frequency range, meaning that the parameters outlined in items b), c), d), f), and g) of section 6.3 do not need to be defined The specification should clearly indicate the appropriate method to be utilized.
Figure 5 – Recommended sinusoidal sweep rate as a function of power ratio for sine on random depending on E sor
The relevant specification shall call for preconditioning and shall then prescribe the conditions
The specimen shall be submitted to visual, dimensional and functional checks and any others as prescribed by the relevant specification
General
Testing follows the sequence prescribed by the relevant specification The different steps are as follows:
– initial vibration response investigation, if prescribed;
– low-level excitation for equalization prior to testing;
– final vibration response investigation, if prescribed
The specimen must be tested in each preferred axis sequentially, unless specified otherwise by the relevant guidelines The sequence of testing along these axes is flexible unless dictated by the applicable specification If testing is to occur solely in the specimen's normal service position, this must be outlined in the relevant procedure.
The control value for each frequency at the reference point is determined from a single check point when single-point control is applied, or from multiple check points in the case of multipoint control.
In the latter case, the relevant specification shall state whether
– the averaged value of the signal of each check point (average control),
– the weighted average value of the signals at the check points (weighted average control),
– or the maximum or minimum extreme values of each frequency of all check points (extremal control) shall be controlled to the specified level
In either of these cases of multipoint control, the control spectrum becomes a fictitious one without a reference to an existing check point
Special action is necessary when a specimen normally intended for use with vibration isolators needs to be tested without them, see also IEC 60068-2-47.
Initial vibration response investigation
The dynamic response of at least one point on the specimen must be examined within the specified frequency range, as outlined in the relevant specification It is essential to clearly define both the number and the location of the response points in the specification.
Vibration response testing can be conducted using either sinusoidal or random vibration within a specified frequency range and test level, as outlined in relevant specifications For sinusoidal vibration, refer to IEC 60068-2-6, while IEC 60068-2-64 provides guidance for random vibration excitation Additional insights, including the advantages and disadvantages of each method, can be found in IEC 60068-3-8.
The response investigation will be conducted at a test level that ensures the specimen's response is lower than that observed during mixed mode testing, while still being high enough to identify critical frequencies.
The investigation of response with sinusoidal excitation should utilize a logarithmic sweep rate of no more than one octave per minute, and it may be necessary to reduce this rate for more accurate determination of response characteristics Additionally, excessive dwell time should be avoided.
The random vibration response investigation must ensure that the test duration is sufficiently long to reduce stochastic variations in the results.
To accurately identify response peaks, it is essential to achieve a high frequency resolution, ideally including at least five spectral lines within the narrowest –3 dB bandwidth.
The specimen must operate during the investigation as specified If mechanical vibration characteristics cannot be evaluated while functioning, a separate investigation with the specimen non-operational will be conducted This phase will focus on identifying critical frequencies, which will be documented in the test report.
Low-level excitation for equalization prior to testing
Before conducting mixed mode vibration testing at the designated level, it may be essential to perform a preliminary random excitation at lower levels using the actual specimen This step helps to equalize the signal and facilitates preliminary analysis It is crucial to maintain a minimal level of acceleration spectral density during this phase.
The permitted durations for preliminary random excitation are the following:
Below –12 dB of the specified acceleration r.m.s level: no time limit
From –12 dB to –6 dB of the specified acceleration r.m.s level: not more than 1,5 times the specified exposure
Between –6 dB and 0 dB of the specified acceleration r.m.s level: not more than 10 % of the specified exposure
The duration of the preliminary random excitation shall not be subtracted from the specified duration of exposure for mixed mode vibration testing.
Mixed mode testing
In specific scenarios, the vibration environment is influenced by quasi-periodic excitations from reciprocating or rotating components, including rotor blades, gears, propellers, and pistons When this type of excitation is dominant, source dwell vibration becomes relevant Source dwell is defined by broadband vibration, which features a higher level of narrowband random or sinusoidal vibration superimposed on it.
9.4.2 Testing with random on random
Swept frequency narrowband random vibration refers to one or more narrowbands of random signals that are varied over a specific frequency range, layered on top of a background of wideband random vibration.
A composite vibration severity of swept narrowband vibration superimposed on a wideband random background is defined by the parameters in 6.1 and 6.2
In some instances, narrowband random vibration may not cover a specified bandwidth, making it comparable to wideband random vibration as outlined in IEC 60068-2-64 The applicable specification will indicate the appropriate method to employ.
9.4.3 Testing with sine on random
Swept frequency sinusoidal vibration on wideband random vibration is defined as one or more sinusoids swept over a frequency range and superimposed on random vibration
A composite vibration severity, consisting of swept frequency sinusoidal components on a random background, is defined by the parameters of 6.1 and 6.3
In some instances, sinusoids may not cover a frequency range, meaning that the parameters outlined in items b), c), d), f), and g) of section 6.3 do not need to be defined The specification should clearly indicate the appropriate method to be utilized.
9.4.4 Testing with sine on random on random
Effectively this is some combination of 9.4.2 and 9.4.3 The relevant specification shall state which combinations are required.
Final vibration response investigation
If the specification mandates an initial response investigation, it may also necessitate a subsequent vibration response investigation after mixed mode testing to assess any changes or failures since the initial assessment The final response investigation must be conducted using the same response points and parameters as the initial investigation The specification will outline the actions to be taken if discrepancies arise between the two investigations.
When prescribed by the relevant specification, the specimen shall function during the prescribed mixed mode tests and its performance shall be checked
When specified, it is essential to allow a period after testing and before final measurements for the specimen to stabilize, ensuring it reaches the same conditions, such as temperature, as during the initial measurements.
The specimen shall be submitted to visual, dimensional and functional checks and any others as prescribed by the relevant specification
The relevant specification shall provide the criteria upon which the acceptance or rejection of the specimen shall be based
13 Information to be given in the relevant specification
When incorporating this test into a relevant specification, it is essential to provide specific details, especially for the items marked with an asterisk (*), as this information is always mandatory.
Clause or subclause a) Basic motion* 4.3 b) Fixing points* 4.3 c) Cross axis motion 4.4 d) Mounting of the specimen* 4.5 e) Tolerances 5.1 et 5.2
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The EN 60068-2-80:2005 standard outlines several critical parameters for testing, including the crest factor and drive signal clipping, statistical accuracy, and frequency resolution It specifies the test frequency range and broadband acceleration spectral densities, as well as the shape of the acceleration spectral density curve and the duration of exposure The standard also addresses random narrowbands, sine tones, and sweep rates, along with the importance of preconditioning and initial measurements Multipoint control and preferred testing axes are highlighted, alongside the investigation of initial and final vibration responses Performance and functional checks, intermediate measurements, recovery processes, and final measurements are also essential components of the testing protocol.
14 Information to be given in the test report
As a minimum the test report shall show the following information:
2) Test laboratory (name and address)
3) Test report identification (date of issue, unique number)
5) Type of test (SoR, RoR, SoRoR)
6) Purpose of the test (development test, qualification, etc)
7) Test standard, edition (relevant test procedure)
8) Test specimen description (unique identity, drawing, photo, quantity, etc)
9) Mounting of test specimen (fixture identity, drawing, photo, etc)
10) Performance of test apparatus (cross motion, etc)
11) Measuring system, sensor location (description, drawing, photo, etc)
12) Uncertainties of measuring system (calibration data, last and next date)
13) Control strategy (multipoint control, multireference control,
14) Initial, intermediate or final measurements
16) Test severities with documentation (measuring points, degrees of freedom, test spectra)
17) Test results (comment on status of test specimen)
18) Observations during testing and actions taken
20) Test manager (name and signature)
21) Distribution (list of those receiving report)
A test log is essential for documenting testing processes, including a chronological list of test runs, parameters, observations, actions taken, and measurement data This log can be attached to the test report for comprehensive record-keeping.
A.1 Random-on-random and sine-on-random
When working with signals that combine random and sinusoidal waveforms, it is essential to consider specific signal-processing challenges outlined in IEC 60068-2-6, IEC 60068-2-64, and IEC 60068-3-8 Modern digital control systems can execute intricate strategies, allowing for the integration of both random and sine signals Additionally, sweeping sine tones can intersect with sweeping random bands, adding to the complexity Despite the advanced mathematics involved, there are fundamental limitations in the overall process that necessitate finding a compromise solution.
A.1.2 Fixed frequency narrowband random on wideband random
This type of vibration is essentially the same as the wideband random vibration discussed in IEC 60068-2-64 and no extra special techniques are required
Tolerances remain consistent across the narrowband spectrum; however, adjustments may be necessary at the transition points from broadband to narrowband If the transition occurs over just one or two spectral lines and the difference between the two ASD values is significant, it may be essential to modify the tolerances to facilitate testing, with these adjustments documented in the test report.
A.1.3 Swept frequency narrowband random on wideband random
If the sweep rate for narrowbands is excessively fast while the control loop times are prolonged, spectral smearing can occur This phenomenon causes energy from one spectral line to spread into adjacent lines, resulting in the loss of the rectangular shape of the sweeping band Consequently, the control system may struggle to maintain accuracy and could abort the test if it identifies an excessive number of spectral lines that fall out of tolerance.
The control system enhances stability by averaging multiple time frames, often using an exponential method, with previous values as it updates the control ASD The degrees of freedom (DOF) considered are influenced by an averaging factor; a larger DOF results in a slower response time to changes, although it contributes to greater stability in the controlled acceleration spectral density.
When narrowband sweeps occur, the processing algorithm may incorporate high previous values, leading to potential control loss as these values trigger abort conditions To mitigate this issue, it is essential to reduce the averaging factor, which decreases the number of averages considered and shortens the control loop response time However, excessive reduction of the averaging factor can destabilize the control loop, resulting in a loss of control once more.
Therefore, a suitable compromise has to be found for these parameters for each individual situation
If the test laboratory is equipped with the necessary tools, recording the time signal from the control point can facilitate off-line analysis using advanced spectral analysis techniques, such as overlap processing While this will not alter the test levels achieved, it will provide a more detailed demonstration of the results, which can be included in the test report.
A.1.4 Fixed frequency sinusoidal vibration on wideband random
Extracting a sinusoidal waveform from a complex mixture of sine and random noise poses a significant challenge When the amplitude of the sine wave is much larger than the random root mean square (r.m.s.) value, this task becomes easier However, as the ratio of sine amplitude to random r.m.s decreases, the accuracy of the sine extraction diminishes, as demonstrated by the results presented below.
Three types of modern digital control systems were used for the investigation The test parameters for each control system were as follows:
Frequency range: 10 – 2000 Hz ASD level: 0,005 / 0,01/0,05 g n 2 /Hz (flat) Frequency resolution: 1 Hz (or maximum available) Degree of freedom: 120 (or maximum available) Sine:
Each combination of ASD level and sine frequency was recorded for a minimum of 60 s at constant sine frequency
The control system's output, operating in a closed loop, was recorded using a digital tape recorder at a sampling frequency of 12.5 kHz The recorded data was then transferred to a computer for analysis, where ASD spectra were calculated based on specified parameters.
Example plots of the resulting ASD spectra for one control system at different sine frequencies are shown in Figures A.1 and A.2
Table A.1 presents three ASD values near the center frequency for each measurement An r.m.s magnitude was calculated from these values, with the last column indicating the percentage deviation from the theoretical value This deviation can be used as a parameter to assess the quality of the sine level in the signal However, it is important to note that no conclusions can be drawn about the 'shape' of the sine wave, as only r.m.s values are being compared.
To analyze the periodicity of the sine wave within the signal, an autocorrelation function was utilized on a 5-second segment of each signal Example plots illustrating this analysis for two distinct random background levels are presented in Figure A.3.