Measuring rotational degrees of freedom using laser Doppler vibrometer

Transcription

1 Measuring rotational degrees of freedom using laser Doppler vibrometer 1. Abstract: Rotational degrees of freedom are essential part of dynamic modeling of the structures in modal analysis. Due to some limitations to the measurement of rotational degrees of freedom, they are usually neglected which causes some serious errors in the results obtained from dynamic modeling. To overcome this difficulty, various methods and approximations have been introduced. In many of these methods the pure moment is necessary to excite the structure but creating this pure moment is difficult. In this study, a new method of measuring rotational degrees of freedom of the structure is presented. In this method, by using continuous laser scanning on the structure surface the rotational degrees of freedom is obtained without having to measure the applied forces and FRF. In order to evaluate the simulations are done and the results are compared with the FRF method. Also the sensitivity of the presented method to the presence of noise in the response signal is investigated. The obtained results demonstrate the applicability (efficiency) of the presented method. Then, an experimental study is performed to analyze the presented method. To do this, a system is set up to form (do) a linear scanning on the surface of the structure. Using the measured scanning signal of the structure surface, rotational degrees of freedom are

2 calculated with both FRF and the presented methods. The results show that the presented method has a high degree of precision and applicability (efficiency). Key Words: rotational degrees of freedom, Doppler vibrometer, linear scanning 2. Introduction Understanding the vibrational behavior of industrial products is of significant importance. Generally, prior to the production of a specific product, a prototype is produced and all the required tests are performed on this prototype. One of the biggest limitations of analyzing this model is the lack of accurate (valid) information of rotational vibrations. Every point of the structure has six degrees of freedom. Three components are related to translational movement in X, Y and Z direction and the other three components are the rotational movement around the above-named axes. Rotational displacements are seen in many elements of coefficient matrix of the frequency response functions. Therefore, measuring the rotational degrees of freedom (RDOFs) are important for validity of the frequency response functions (FRFs) and also plays an important role in the accuracy of the analysis results. However, due to the limitations in measuring rotational degrees of freedom, it is generally neglected in the analysis which causes error in the results. To overcome this difficulty, many methods and approximated are introduced [1]. Smith et al. [2] used special combination of two electromagnetic shakers to create two equal but oppositely directed forces. In his studies, excitation of the pure moment is used to measure the moment motion, but this method could only

3 be applied for limited range of frequencies. Sanderson et al. [3] introduced excitation of the moment using two shakers to measure the moment motion of the beam. They studied the details of different inherent errors of the beam and purposed a method to correct the experimental results. However to do this correction measurement of other quantities were required. Sanderson and Fredo [4] introduced the direct method in which two excitation forces are simultaneously applied to the rigid body attached to the structure. The rigid body is considered to be in both I and T shaped blocks. Petersson [2] introduced the method of magnetically varied rods to excite moment. However in some cases of high motion difference, the excitation was not suitable. Su and Gibs [5] purposed a method which improved the problem of motion difference. Su et al. [3] theoretically explained the effect of external forces impedance in the measurement of the moment motion. Stanbridge and Ewins [6] studied the application and implementation of LASER to measure the rotation response. In this case, velocity of the measured point is calculated by comparing Doppler frequencies between radiation and reflection rays (beams) coming from the vibrated body (object). Single point (discrete) measurement is a method for measuring rotational degrees of freedom in which laser Doppler accelerometer and vibrometer (LDV) is used for a single point [7]. LDV method is capable of measuring point velocity of the structure surface by focusing laser light in a point using Doppler phenomena (varying radiation and reflection frequencies). One of the biggest

4 advantages of LDV method in comparison to accelerometer method is that it has non-contact measurement capability which has great application in warm and highly illuminated environments and also in rotating machines. Moreover, simplicity of changing the point of measurement, measurement of light bodies and wide frequency range are among other benefits of LDV method. However there are some disadvantages regarding LDV measurement method. Expensiveness and producing noise in low frequencies are among its disadvantages. The first application of LDV method was reported on measuring vibration of turbine s blade [8]. Analysis of magnetic disks and modal analysis of rotating disks are among the recent applications of laser. However using LDV method instead of accelerometer increases the accuracy of the measurement, but due to its pointwise measurement and in cases that there are plenty of points to be measured, it requires long time for the testing to complete. Therefore, a more improved method of LDV method, called Continuous Scanning Laser Doppler Vibrometer )CSLDV) is introduced [9]. In this method, laser beam moves over the structure in a continuous and harmonic manner and can do the measurement of many points in a very short period of time. To create the laser beam movement over the surface structure, many methods are introduced [10,11]. One of the best methods of measuring the rotational degrees of freedom is through using output frequency response functions of Doppler laser vibrometer. In this method, by dividing the response (results) obtained from the laser by the

5 forces acting on the system, the frequency response function for the structure is calculated. Using this FRF, angular rotation is obtained. Since in many structures, there exists no means of measuring the applied forces, in this paper a new method of measuring rotational degrees of freedom for the structure is introduced which does not require the information about applied forces and can measure the system rotational degrees of freedom directly from some mathematical calculations. Therefore, there is no need to have the information about the applied force. Moreover, using this method, laser signal can be completely retrieved in a complete parametric from which could not be done in FRF method. To evaluate the presented method, simulations are done the results are compared to the FRF method. To measure the sensitivity of the presented method to the noise, some input noises are randomly entered to the system and the results are compared to that of FRF method. Then, an experimental study is done to analyze the presented method. To do that, a system that creates a linear scanning on the structure surface is used. Using the scanning signal of the surface structure, rotational degrees of freedom are calculated with both FRF and the presented method. The obtained results show that the presented method has a high degree of precision.

6 3. Linear scanning theory with sinusoidal excitation of the structure In this section of the paper, assuming that the structure is rigid, equations of motion for the structure which are measured with Linear Scanning are derived. And with the derived equations, relations to determine the rotational degrees of freedom for the structure are presented [12]. System (structure) is assumed to be suspended in the air with a flexible strip and then is excited with a shaker, which is eccentrically attached to the cube, under the frequency of. According to Figure 1, laser beam starts off from point A and repeatedly scans the distance between points B and C with the frequency of [13]. Assuming that the structure surface moves linearly after the excitation, line BAC still remains straight after the excitation. Figure 1 shows the structure surface after being excited with the shaker. Structure surface displacements for points A, B and C in Z direction are denoted as, and, respectively. Figure 1. Surface angular and translational vibration

7 Therefore the angular deviation of point A,, is approximated by the following equation[12]: (1) Z amplitude of a point located x cm (mm) away from point A is equal to. Now, if LDV starts a sinusoidal scanning in the direction of line BAC with the frequency of, location of the laser beam at any moment on line BAC can be calculated according to the calculations done in reference [14]: ( ) ( ) (2) where δ is the angular phase equal to phase change between location of LDV beam and LDV mirror signal and can be neglected by matching the mirror excitation signal. Considering the excitation force to be in the following form: ( ) ( ) (3) Angular and translational deviations at point A can be expressed as: ( ) ( ) (4) (5)

8 where α and β are angular phase with respect to the input force. Hence, displacement for every point on line BAC using LDV scanning signal is equal to: ( ) ( ) [ ( )] ( ) (6) Using some mathematical calculations, the above-mentioned equation can be rewritten as: ( ) ( ) (( ) ) (( ) ) (7) According to equation (7), LDV response consists of three components. In another word, there is one response component at frequency which describes the translational response and the two remaining responses which are at frequency ± describes the rotational responses [14]. By dividing the response value by the input force, FRF is obtained. However this can be easily done for measurement of the translational displacement, but the angular vibration depends on. In this case, a reference input signal is introduced according to the following equation [12]: ( ) ( ) ( ) (8) where ( ) is the input signal for movement (motion) of the mirror. Equation 8 can be rewritten as:

9 ( ) ( ) ( ) (9) Therefore, the final form of the equation describing the amplitude and LDV response phase are presented in Table 1 [12,14]. Table 1. Linear Scan Reference Frequency(HZ) magnitude phase f α fv δ fv + δ 4. New method of measuring angle of rotation of the structure Since in many structures, there exists no means of measuring the applied forces to the system, in this paper a new method of measuring rotation of the structure using motion equations of the structure under sinusoidal excitation and linear scanning without measuring the applied forces is introduced. According to equation (7), LDV response consists of three components. In another word, there is one response component at frequency which describes the translational response and the two remaining responses are at frequency ± which describes the rotational responses. Considering the experiment done in this study which would follow in section 5, due to the sinusoidal movement of the beam (bar) which is caused by the shaker to move the mirror,

10 the extra term ( ) ( ) is added to equation (7). Therefore the following equation (10) is obtained for the LDV response. ( ) ( ) (( ) ) (( ) ) ( ) ( ) (10) Taking derivative of the equation (10) with respect to time, velocity equation is obtained according to the following equation: ( ) ( ) ( ) (( ) ) ( ) (( ) ) (11) ( ) ( ) By expanding sinus and cosine functions in equation (11), the following equation for v(t) is obtained: ( ) [ ( ) ( ) ( ) ( )] ( )[ (( ) ) ( ) ( ) (( ) )] (12) ( )[ (( ) ) ( ) ( ) (( ) )] ( ) ( )

11 Constants values in equation (12) which are not time dependant are simplified by a change of variable and are considered according to the following relations: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (13) (14) (15) (16) (17) (18) [ ( ) ( ) ( ) ] [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] [ ] (19) Laser output, which is in the form of time dependant velocity, is obtained by doing an experimental study. According to the obtained velocity in the experiment and equation (19), values for a, b, c, d, e,f,a and B are obtained using equation (20).

12 [ ] [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] [ ( ) ( ) ( ) ] where the sign "+" demonstrates the quasi-inverse of the coefficients matrix. Also the unknown values in equation (20) are in the left hand side of the equation and are obtained by solving equation (20). (20) 5. Simulation 5.1. Simulation of the presented method In this section of the paper, the new method to measure the rotational degrees of freedom that was introduced in section 3 is simulated. The required parameters for the simulation are arbitrary and presented in Table 2. (m) 5e-5 Table 2. The arbitrary parameters for the simulation (r A ad/s) (rad) (rad) (rad) (m) e e-5 4 B (m) -4e- 4 Scanning frequency, excitation frequency and the length of the scanning line are known and presented in Table 3. Table 3. Values for frequencies and length of the scanning line (HZ) (HZ) l (mm)

13 Laser response with respect to time is calculated by solving equation 11 and using parameters in Table 2 and 3. Figure 2 demonstrates the result for both the entire time interval and the time between 2.5 s and 2.7 s. (a) (b) Figure 2. a) simulated signal of the laser response b) simulated signal of the laser response for interval s. Now, assuming that the scanning frequency, excitation frequency and the length of the scanning line are known (Table 3) and the laser signal is also measured, the unknown parameters are calculated using equation 20 and presented in Table 4. (m) 5e-5 Table 4. Obtained parameters using new method (rad/ (rad) (rad) (rad) A (m) s) 1.58e e-4 B (m) -4e-4

14 As can be seen in Table 4, the obtained parameters are the same as those of Table 2. This shows that the new method is valid and has no error. 5.2 Simulating the FRF method In this section of the paper, the conventional method of measuring rotational degrees of freedom using FRF function is simulated. In order to simulate FRF method, the required parameters are considered the same as those in section 4.1 which are presented in Table 2 and 3 and the velocity is calculated using equation (11). LDV output spectrum obtained from this simulation is shown in Figure 3. According to Figure 3, two side bands are formed at distance equal to the scanning frequency on both sides of the excitation frequency ( - =129HZ =150HZ and + =171HZ). As can be seen in Figure 3, besides the excitation frequency and the side bands, there is another frequency at 21 Hz due to excitation frequency of the mirror that has no effect on side bands and excitation frequency of the structure at 150 Hz. In order to obtain FRF, values of the applied forces to the structure should be defined. These values are defined arbitrarily and are set as 56 N and 4 N for amplitude of the excitation force for the structure and amplitude of the excitation force for the mirror, respectively. Using these values, the reference force applied to the system is calculated by solving equation 9. Now, dividing diagram of the laser response by reference force, FRF of the system is obtained (Figure 4).

15 Figure 3. LVD output spectrum obtained from the simulation Figure 4. FRF diagram

16 According to figure 4, values for the amplitude and phase of the cubic structure motion at frequencies 172 Hz and 129 Hz are obtained and presented in Table 5. Table 5. Values obtained by experiment Magnitude Frequency (HZ) Phase (deg) (mm/s ) - = e-7 β δ= = e-7 β + δ=70.3 By calculating amplitude of excitation force for the structure which is F=56 N, and amplitude of excitation force for the mirror which is V=4 N, and considering the presented equations in Table 1, amplitude of the angular motion is obtained: =(5.659e-7) =1.58e-5 (rad/s) (21) Phase angle ( β ) at frequency of 150 Hz is the phase average of the two frequencies 171 Hz and 129 Hz and is equal to 55.5 degrees. 6. Investigation of effect of the noise In experimental studies, usually there are some noises that affect the laser output. Therefore, in order to evaluate applicability (efficiency) of the methods presented in sections 4.1 and 4.2 in practical conditions, effect of noise on these methods should be investigated. So the following (stochastic) term for the noise signal is added to the laser output:

17 V(t)=v(t)(k(2rand-1)+1) (22) where v(t) and k are the velocity without noise and the input noise percentage, respectively. Now, assuming that the scanning frequency, excitation frequency and the length of the scanning line are defined according to Table 3 and the laser signal is also measured according to equations 11 and 22, the unknown parameters are calculated by solving equation 20. Substituting parameters d and c in equations 15 and 16, amplitude of the angular motion is obtained by the following equation: ( ) (23) Also by substituting parameters f and e in equations (17) and (18), amplitude of angular motion is calculated by the following equation: ( ) (24) The values for obtained with equations (23) and (24) are different with each other. This difference is due to the presence of noise in laser output. When there is no noise in the laser output, the values of obtained with the two equations are the same as each other. Table 6. Input noise percentage and variations of Percentage of system input noise The percentage difference between for different input noise

18 Considering equations 23 and 24, due to presence of the noise in laser output, two different amplitudes are calculated for. Therefore amplitude of the angular motion is obtained by averaging equations 23 and 24. (25) Table 7 shows the variation of obtained from equation (25) and also variation of calculated from FRF method, which is presented in section 5.2. in comparison to the assumed which is tabulated in Table 2 with respect to input noise percentage. Table 7. Error with respect to the input noise percentage for the rotation measurement Percent noise Percent error method FRF Percent error method new Tables 7 and 8 demonstrate that increasing the noise in the laser output, error percentage also increases for both methods. Although new method gives more error in comparison with FRF method when increasing noise, there is no need to measure the applied forces to the structure in this new method and rotational degrees of freedom are simply calculated by measuring the response. Moreover, the laser signal in the presented method can be retrieved in a complete parametric form.

19 7. Experimental setup for the scanning system Setup for CSLDV system using single point (point-wise?) LDV is shown in Figure 5. This setup is suitable for those LDVs not having scanning system so that a linear scanning system for measuring rotational degrees of freedom can be formed (set up) for the structure. In order to form (set up) a scanning system, a mirror is placed on the beam (bar) as is shown in Figure 6. This beam (bar) is harmonically excited with the shaker of the model B&K 4808 (connected to a signal generator of the model Hameg and series HM8130 and an amplifier of the model B&K 2719). Beam reflection of the Ometron VH 300 LDV from the mirror on the structure (structure is cubic in this study) which is formed by shaker of the model B&K4808 (connected to amplifier of the model B&K 2719 and a computer with software Pulse v.8 installed in it) is sinusoidally excited in order to measure the rotational degrees of freedom. Therefore, a linear scanning is formed (set up) on the cubic structure. In this case, length of the scanning line can vary with variations of the shaker amplitude.

20 Figure 5. Experimental setup for CSLDV system mirror Figure 6. Cantilever beam (bar) and its mirror to set up the scanning system Figure 7 demonstrates the arrangement of the experimental equipments to set up the linear scanning system. The purpose of this arrangement is to measure rotational degrees of freedom of the system (cube).

21 Figure 7. Arrangement of the experiment LDV output signal is measured with analyzer of the model B&K 3560D. Figure 8 shows the LDV output signal made due to motion of the LDV beam over the cubic structure, which is caused by simultaneous vibration of the cube and the beam (bar) which is attached to the mirror. According to figure 8, in this case LDV output signal consists of two sinusoidal waves which are superimposed. One of the two waves is related to the scanning system with the angular frequency of and the other one is related to the excitation system with the angular frequency of.

22 (a) (b) Figure 8. a) LDV laser output resulting from scanning the cubic structure b) LDV laser output resulting from scanning cubic structure in the interval s. 8. Experimental measurement of the angle of rotation 8.1 Measurement of the rotational angle using FRF method Figure 9 shows a 64*154*132 mm aluminum cube which is suspended with a flexible strip. To excite this cubic structure, the shaker is eccentrically attached to the structure. This shaker starts to vibrate at frequency of 158 Hz and consequently makes the structure move, thus creating a diagram like the one shown in Figure 8. Also, to set up a linear scanning system, the shaker B which is attached to the beam (bar) from behind the mirror, is excited sinusoidally at the frequency of 20 Hz. And with increasing the amplitude of the shaker through the amplifier, the length of the scanning line reaches 16 mm and point A is its center. In another word, radius of the scanning line is 8 mm.

23 Figure 9. Schematic diagram of the linear scanning set-up LDV output spectrum resulting from the experiment on the cubic structure is shown in Figure 10. According to Figure 10, two side bands are formed at distance equal to the scanning frequency on both sides of the excitation frequency ( - =138HZ =158HZ and + =178HZ). As can be seen in Figure 10, besides the excitation frequency and the side bands, there is another frequency at 20 Hz due to excitation frequency of the shaker B which moves the mirror and has no effect on side bands and excitation frequency of the structure at 158 Hz.

24 Figure 10. LDV output spectrum using linear scanning The mirror excitation signal and structure excitation signal are measured with ergometer. Using equation (9), reference input signal (fv) is calculated from signals of the measured forces. In order to calculate FRF, the signal obtained from LDV output is divided by the reference input signal (fv). FRF amplitude and phase for the motion of the cubic structure are shown in figure 11.

25 Figure 11. Frequency response of the cubic structure According to diagrams of figure 11, the value for structure amplitude and phase are presented in Table 8. Table 8. Values obtained from experiment Magnitude (mm/s ) Frequency (HZ) - =138 + = e e-5 Phase (deg) β δ=51.94 β + δ=47.48 Considering equations presented in Table 1, amplitude of the angular motion ( ) is obtained from measuring the amplitude of excitation force for the structure which is F = N and the amplitude of excitation force for the mirror which is V = N:

26 =(1.066e-5) =7.022e-5 (rad/s) (26) Phase angle (β) at frequency of 158 Hz is the phase average of two phase frequencies 138 Hz and 178 Hz and is equal to 55.5 degrees. To validate the values obtained for the angles, measurements are repeated at points B and C using point-wise scanning system as shown in Figure 1. The results obtained for these two points, amplitude of the angular motion for point A is calculated as 6.795e-5 rad/s. The error in the measurement of the angular motion using the linear scanning system is 3.34 percent with respect to the point-wise scanning system Measuring rotational angle using the presented method In this section, by using the presented method, angle of rotation is calculated from analysis of the laser output obtained from the experiment. By doing experiment according to the arrangement described in section 6, the laser output was obtained. Using equation (20), scanning frequency, excitation frequency, length of the scanning line and also laser output signal, the unknown parameters are calculated. By substituting parameters d and c in equations (15) and (16), amplitude of the angular motion is obtained by the following equation: 6.169e-5 (rad/s) (27) ( )

27 And also by substituting parameters f and e in equations (17) and (18), amplitude of the angular motion is calculated by the following equation: ( ) 6.514e-5 (rad/s) (28) Considering equations (27) and (28), two different amplitudes are obtained for which is caused due to the presence of noise in the laser output, as described in section 6. Therefore, amplitude of the angular motion ( ) is obtained by averaging equations (27) and (28) and is equal to rad/s. The angle of rotation calculated from the new method has 7.1 percent deviation in comparison to the point-wise scanning. 9. Discussion As mentioned in section 2.7, two values are obtained for the angle of rotation using the presented method. This is due to the presence of noise in the laser output. Noise in the laser output can be reduced by increasing the precision of the measurement equipment and also reducing environmental effect thus reaching a higher precision in the results. Therefore, in the experiment that the noise value (ampunt) is less, the difference between these two values is also less meaning that the precision in the obtained angle of rotation is higher. Amplitude of the angular motion calculated in section 2.7 shows a 7.1 percent variation between the obtained value and the value obtained from point-wise scanning system. The presented method does not require the amplitudes of the applied

28 forces and is able to measure the degree of rotation without measuring the amplitude of excitation force for the structure and the amplitude of excitation force for the mirror. Moreover in this study, magnitude of the angle of rotation is also calculated with both linear scanning system and FRF function. The results with the latter solution is 3.34 percent different from the linear scanning method which is due to error in measuring the length of the scanning line and also assuming the linear motion of the system. Results obtained from these three methods are tabulated in Table 9. Errors presented in this Table are calculated with respect to the point-wise laser method. Table 9. Comparison of amplitude of the angular motion calculated with three methods Error Method (rad/s) (percent) Method Discrete Point 6.795e Method FRF 7.022e Method new 6.341e Besides presence of noise, the reflected beams from a coarse surface also cause error and noise and therefore scatter the output result. If laser beam reflects from a surface, some spots (dots) (?) are formed in front of the vibrating surface due to the interference of the surface laser radiation and reflection waves. This interference is caused due to the coarseness of the surface. If degree

29 of microscopic coarseness of the surface is equal to or more than the beam wave length, the surface is considered coarse. According to Rayleigh Criterion, a surface is called smooth when ratio of the maximum height of the microscopic coarseness to the length of the radiated beam wave tends (goes) to zero. Therefore, all the surfaces except for those being finely machined (?) can cause spots (dots) to appear in the results. In fact, waves interference, which enables the measurement of Doppler frequency, causes this destructive phenomena. Reflection of the laser beam from coarse surfaces causes the laser beams reflect in different directions. These reflected beams are combined (superimposed) based on the interference phenomena. Depending on the phase and the amplitude of the waves, they support (?) or attenuate each other. When the waves support each other, the spots (dots) are lighter and when they attenuate each other the spots (dots) get darker. The formation of these light and dark spots (dots) decreases precision of the measurement. These spots (dots) also appear in the continuous LDV scanning system. When the vibrating surface being radiated with the laser beam moves, the spots (dots) move too. These movements cause a phase shift (change) in every point of the laser surface making the dots look like they are time-dependant. Spot (dot) formation decreases the precision of the measurement thus causing some limitations for the LDV system. Adjusting the laser setting can decrease the laser Speckle effect to a great extent [13,14]. Also to decrease the negative effect of noise spot (dot), surface of

30 the cube is covered with Retro Reflective in the scan location to gain a better reflection. In the continuous LDV scanning system (CSLDV) scanning speed (rate) can also cause noise and therefore decrease the precision of the measurement. In order to have better results, the frequency of the scanning procedure should be carefully defined [15]. 10. Conclusion Determining the rotational degrees of freedom is an essential part of dynamic modeling of structures. So far various methods have beeb introduced to measure the rotational degrees of freedom. One of the best methods of measuring the rotational degrees of freedom is FRF Doppler laser vibrometer. In this method, dividing the response (results) obtained from the laser by the forces acting on the system, frequency response function for the structure is calculated. Using this FRF, angular rotation is obtained. Since in many structures, there exists no means of measuring the applied forces, in this paper a new method of measuring rotational degrees of freedom for the structure is introduced which does not require the information about applied forces and can measure the system rotational degrees of freedom directly from the laser output. Moreover, using this method, laser signal can be completely retrieved in a parametric from. To evaluate the presented method, simulations are done and sensitivity of the method to the noise is investigated and the results are compared to the FRF method. The results show the applicability (efficiency) of the presented method. Then, an experimental study is done to measure the laser output from the

31 scanning surface of the structure. Using the experimental data, rotational degrees of freedom are calculated with both FRF and the presented method. The obtained results show that the presented method has a high degree of precision.

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