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Optical method for strength test for general industrial products
ARTICLE IN PRESS
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 735–744 www.elsevier.com/locate/jnlabr/ymssp
Optical method for strength test for general industrial products Tomokazu Suzuki, Yusaku Fujii, J.D.R. Valera Department of Electronic Engineering, Faculty of Engineering, Gunma University, 1-5-1 Tenjincho, Kiryu 376-8515, Japan Received 5 April 2004; received in revised form 3 June 2004; accepted 22 June 2004 Available online 11 September 2004
Abstract A method for evaluating the strength of general industrial products is proposed. In the method, a levitated mass encountering negligible friction is made to collide with a general industrial product under test. During the collision, the frequency Doppler shift of a laser beam reflecting from the mass is accurately measured using an optical interferometer. The velocity, position, acceleration and inertial force of the mass are calculated from the measured time-varying Doppler shift. The mechanical response of general industrial products is accurately determined with this method. r 2004 Elsevier Ltd. All rights reserved. Keywords: Strength test; Dynamic force; Optical interferometer pneumatic linear bearing
1. Introduction Recently, the need for strength testing of general industrial products has arisen in various industrial and research applications such as material and crash testing. In such tests, the force acting on the product under test is measured using a force transducer and the position of the applied force is obtained with a position transducer. However, force transducers are typically calibrated by using static weights and under static conditions. At present there are no accepted standard methods available for evaluating the dynamic characteristics of force transducers. This results in two major problems concerning the strength testing of materials. One is the difficulty in Corresponding author. Tel.: +81-277-30-1757; fax: +81-277-30-1707.
E-mail address: [email protected] (T. Suzuki). 0022-460X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2004.06.005
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evaluating the uncertainty in the measured varying force. The other is the difficulty in evaluating the uncertainty in the time at which the varying force is measured. Force, which is a basic mechanical quantity, is defined as the product of mass and acceleration, F ¼ Ma; where F is the force acting on an object, M is the mass of the object, and a is the acceleration of the centre of gravity of the object. This means that an accurately known acceleration is necessary to determine the force accurately and to calibrate force transducers accurately. The acceleration due to gravity, g, is conveniently used for generating and/or measuring constant force. Constant force can be accurately compared using a conventional balance with a knife-edge or a hinge. However, there are no accepted methods for calibrating force transducers under dynamic conditions. The development of methods for the dynamic calibration of force transducers is therefore important. Although the methods for the dynamic calibration of force transducers are not yet well established, there have been several attempts to develop dynamic calibration methods [1–7]. These attempts can be divided into three main categories: methods for calibrating transducers by using an impact force [1,2], methods for calibrating transducers by using a step force [3], and methods that use an oscillation force for calibration [4–7]. In these methods, a mass is levitated using a pneumatic linear bearing [8] in order to obtain a moving mass with negligible friction. This moving mass is used to generate the reference force that is applied to the force transducers. The inertial force of the mass is measured using an optical interferometer. Methods for material testing, such as a method for dynamic three-point bending test [9] and a method for evaluating material viscoelasticity under an oscillating load [10] have also been proposed. In this paper, a method for testing the strength of general industrial products is reported. In the method, a levitated mass encountering negligible friction is made to collide with a general industrial product under test. During the collision, the applied force is accurately determined by using an optical interferometer to measure the Doppler frequency shift of a laser beam that is reflected from the levitated mass. The velocity, position, acceleration and inertial force of the mass are calculated from the measured time-varying Doppler shift. The mechanical response of general industrial products is accurately determined by using this method.
2. Experimental setup Fig. 1 shows the experimental setup used for testing the strength of general industrial products. The test material is attached to the base. An impact force is applied to the test material by colliding the moving part of the pneumatic linear bearing. A pneumatic linear bearing is used to obtain linear motion with negligible friction acting on the mass (i.e., the moving part of the bearing). An initial velocity is given to the moving part manually. The moving part of the linear bearing is made to collide, with the product under test, with a velocity v1 (m/s). It deaccelerates due to the reaction force of the product, and then finally separates from the product with velocity v2 (m/s). A cube-corner prism (CC) (for the interferometer) and an additional mass (for adjusting the collision position) are attached to the moving part; the total mass, M, is approximately 4.4990 kg. The extension block is made of
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Fig. 1. Experimental setup.
stainless-steel, with a tip rounded. The inertial force of the mass is determined very accurately by measuring the velocity of the mass using an optical interferometer. Since, the friction force on the levitated mass is negligible, then the impact force acting on the test specimen from the mass and the reaction force acting on the mass from the test specimen have equal magnitudes but different signs, in accordance to the principle of action and reaction. The inertial force is given by the product of mass and acceleration, and the acceleration is obtained by differentiating the velocity of the moving part. An optical interferometer was used to accurately measure the velocity of the moving part. It consisted of a Michelson interferometer in which the mirrors were replaced with corner cube prisms. One corner cube prism was firmly attached to the moving mass and defined the signal arm of the Michelson interferometer. The other corner cube prism was at rest and defined the reference arm. The light source used was a Zeeman-type He–Ne laser that emits light at two wavelengths in which the two wavelengths had orthogonal polarisation. The light from the He–Ne laser was incident on a polarisation beam splitter. One wavelength was transmitted to the signal arm and then reflected from the corner cube prism that is attached to the moving mass. The other wavelength was reflected from the beam splitter, since it has orthogonal polarisation, and propagates in the reference arm. After propagation in the Michelson interferometer arms, the two beams were made to interfere by transmitting them through a polariser with its transmission axis oriented at 451 to the polarisation of the beams. The interfering beams were then incident on a detector and resulted in a beat signal, since the beams had slightly different wavelengths. The frequency of the measured beat signal, fbeat, which corresponds to the frequency difference between the signal and reference beams, was measured with a frequency counter. When the object was at rest, then f beat ¼ f rest was approximately 2.8 MHz. However, object motion resulted in a Doppler frequency shift in the signal beam which in turn resulted in a variation in fbeat. Using a
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two wavelength source allowed us to distinguish if the velocity of the moving part was in the positive or negative direction. The change in momentum of the moving part is the product of the mass and the change in velocity. The mass velocity was obtained by measuring the induced Doppler frequency shift in the signal beam of the laser interferometer and by using Eqs. (1) and (2), v ¼ lair ðf Doppler Þ=2;
ð1Þ
f Doppler ¼ ðf beat f rest Þ;
ð2Þ
where fDoppler is the Doppler shift, lair is the wavelength of the signal beam, fbeat is the beat frequency, that is the frequency difference between the signal and reference beams, and frest is the rest frequency defined above. The positive direction for the position, the velocity, the acceleration and the force acting on the moving part is towards the right in Fig. 1. An electronic frequency counter (model: R5363; manufactured by Advantest Corp., Japan) continuously measures and records the beat frequency, fbeat, until 14 000 samples are stored with a sampling interval of T=400/ fbeat. This counter continuously measures the time interval taken by 400 periods without dead time. The sampling period of the counter is approximately 0.15 ms at a frequency of 2.8 MHz. Another frequency counter (same model: R5363; manufactured by Advantest Corp., Japan) measures the rest frequency, frest, using an electric signal supplied by a photodiode embedded inside the He–Ne laser. Fig. 2 shows a photograph of the test section of the experimental setup. The height of the collision point and of the centre of gravity of the moving part are set to be equal. This is done in order to avoid a collision induced change in the shape of the air film that exists between the moving part and the guide way of the pneumatic bearing. To set the height of the centre of gravity of the moving part at a convenient position, an additional mass is attached to the moving part. The total mass of the moving part, including the additional mass and the corner cube prism, is approximately 4.4990 kg.
Fig. 2. Photograph of the mechanical part.
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The pneumatic linear bearing, ‘‘Air-Slide TAAG10A-02’’ (NTN Co., Ltd., Japan), is attached to an adjustable tilting stage. The maximum weight that can be supported by the moving part is approximately 30 kg, the thickness of the air film is approximately 8 mm, the stiffness of the air film is more than 70 N/mm, and the straightness of the guideway is better than 0.3 mm/100 mm. The friction characteristics are accurately determined by using the method described in Ref. [8]. The two electric counters (R5363 and TA1100) are triggered by a sharp trigger signal generated by a digital-to-analog converter. This signal was generated by using a light switch, that is a combination of a laser-diode and a photodiode. In this experiment, three sets of collision measurements were obtained by manually changing the initial velocity of the moving mass.
3. Results Fig. 3 shows the procedure for obtaining the force from the measured time-varying beat frequency. During the collision experiment, only the beat frequency is measured. Using the measured value of the time-varying beat frequency, the velocity, position, acceleration and inertial force are numerically calculated with a computer. It was found that the noise in the time-varying beat frequency measured was too big to accurately determine the position of the peak. The measured data was therefore low-pass filtered with a filter that assigned, to every point, the average of the 11 neighbouring points.
Fig. 3. Data processing procedure.
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The inertial force is given by the product of mass and acceleration, and the acceleration is calculated by differentiating the velocity of the moving part. The position of the mass is calculated by integrating the velocity. The velocity is obtained from the Doppler shift frequency of the signal beam of a laser interferometer, fDoppler. The origins of the time and position axes are set to be the time and position where the reaction force from the product under test is detected, respectively. The velocity before and after the collision are approximately, v1 ¼ 0:154226 m=s and v2 ¼ 0:112196 m=s; respectively. The reduction of the kinetic energy is approximately 0.0251895 J. This loss of the kinetic energy is believed to disperse as heat inside the test specimen. In this experiments, a plastic case was used as an example of the general industrial product tested. Three sets of collision measurements were conducted. The collision measurement was done by colliding the moving part to the centre of the lid of the plastic case. The size of plastic case under test was 38.7 mm in width, 67.7 mm in length and 15.2 mm in depth. Its thickness was 1.4 mm. Fig. 4 shows the results of the first measurement. In this first measurement, the impulse was not enough to destroy the tested material. The upper graph shows the change in reaction force against time, and the lower graph shows the change in reaction force against position. From the upper graph in Fig. 4, it can be seen that the applied force is gentle and the maximum force was approximately 30 N with only 35 ms of collision time. From the lower part in Fig. 4, it can be seen that the magnitude of the displacement was approximately 0.8 mm when the force was approximately 30 N. The spring constant, given by the slope of the graph, increases as the displacement is increased. There is a slight area bounded by the curve shown in Fig. 4, representative of hysteresis. This is caused by the slight viscosity of the test specimen. This area corresponds to the loss of the kinetic energy and is believed to disperse as heat inside the test specimen under test.
Fig. 4. Results of 1st collision measurement.
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Fig. 5 shows the results of the second measurement (superimposed on the results from the first experiment, for comparison), in which the test specimen was damaged by the impact applied. The test specimen cracked. Fig. 6 shows a photograph of the crack in the test specimen after the second collision measurement. The length of the crack is approximately 37 mm. The upper graph in Fig. 5 shows a sharp decrease in the force from the peak value of approximately 55 N. At this point, the test specimen cracked. From the lower graph in Fig. 5 it can be seen that the magnitude of the displacement at the peak was approximately
Fig. 5. Results of 2nd collision measurement.
Fig. 6. Photograph of the case after the 2nd collision measurement.
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1.2 mm (when the sample cracked) and then, it increased to approximately 2.0 mm, after the sample cracked. The slope of the graph, after damage, was lower than that before damage. This is because the spring constant of the test specimen has changed after the damage to the sample. The work done by the moving part is given by the integral, Z W¼
ðF Þ dx
ð3Þ
and is approximately 0.025195 J. The absolute value of this work is equal to the area bounded by the curve shown in Fig. 4. This value agrees well with the reduction of the kinetic energy of approximately 0.025190 J calculated by using the velocity before and after the collision, v1 (m/s) and v2 (m/s). The energy dissipation ratio, W/E1, was approximately 47%, where E1 is the initial kinetic energy of the moving mass. The area bounded by the graph shown in Fig. 5 is equal to the energy loss that originated from the damage. The spring constant, or slope of the graph, changes before and after the destruction. Both slopes up to the point where the sample cracked coincide. This indicates high reproducibility. Fig. 7 shows the results of the third measurement, superimposed with the results of the first and second measurements, for comparison. In this last measurement, a small impact that does not cause further damage is applied. From the upper graph in Fig. 7 it can be seen that the applied force was gentle and the impact was characterised by a maximum force is 16 N and 50 ms collision time. From the lower graph in Fig. 7, it can be seen that the spring constants (slope of the curves) before and after sample damage were distinct, and that the slope after damage was smaller than before damage occurred.
Fig. 7. Results of 3rd collision measurement.
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4. Evaluation of uncertainty The force is measured as the inertial force of the moving mass. The inertial force is calculated as the product of acceleration and mass. If the external force acting on the moving mass is negligible, then the inertial force and the force acting on the material under test are the same. The uncertainty sources in the measured instantaneous values of the varying force are as follows. (1) Acceleration: For the acceleration measurement, the vibration of the optical parts of the optical interferometer is dominant. The standard deviations of the measured force just before and after the collision in the second experiment are approximately 0.07 and 0.08 N, respectively. (2) Mass: The uncertainty in measuring the moving mass is approximately 0.1 g, which corresponds to 0.002% of the total mass of approximately 4.4990 kg. (3) External force: For the external force acting on the moving mass, the frictional force acting inside the pneumatic bearing is dominant under the assumption that the air film of approximately 8 mm thickness inside the bearing is not broken. The frictional characteristics of the air bearing are determined using the technique given in Reference [8]. The dynamic frictional force acting on the moving part, Fdf, is estimated by (4)
F df ¼ Av;
ð4Þ 2
1
where A ¼ 8 10 /kg s . This is calculated to be approximately 0.02 N at a velocity of approximately 0.2 ms1. Therefore, the standard uncertainty in measuring the instantaneous value of the varying force in the experiments is estimated to be approximately 0.1 N, which corresponds to 0.2% of the maximum applied force in the experiments of approximately 55 N.
5. Discussion In the proposed method the object velocity is obtained by measuring the time-varying Doppler frequency shift (of a laser beam reflected from the moving object) during the collision. All the other quantities, such as the position, acceleration and force, are numerically calculated from this. In addition, force is directly calculated according to its definition, that is, the product of mass and acceleration. The authors consider that this simplicity is the most significant advantage of the proposed method compared with other conventional methods using a force transducer and a position sensor. The dynamic response of the tested object against an impact force is evaluated highly accurately by measuring the time-varying Doppler frequency shift of a reflected laser beam. If the interferometer is miniaturised by the use of a stabilised laser diode, and the pneumatic linear bearing is replaced by a simple pendulum mechanism, then the whole system could be made portable.
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6. Conclusions A method for testing the strength of general industrial products was proposed. In the method, a mass using a pneumatic linear bearing with negligible friction is made to collide with a general industrial product under test. During the collision, only the time-varying Doppler frequency shift of a laser beam reflecting from the mass is accurately measured using an optical interferometer. The velocity, position, acceleration and inertial force of the mass are calculated from the measured time-varying beat frequency. The experimental setup used in the proposed method is relatively simple. The strength of a plastic case was highly accurately determined by using the proposed method. Any general industrial product, can be tested by attaching it to the base using an appropriate adhesive material or mechanical holder. The advantages and future prospects of the proposed method were discussed.
Acknowledgment This work was supported by a research-aid fund of the Mitsutoyo Association for Science and Technology.
References [1] Y. Fujii, H. Fujimoto, Proposal for an impulse response evaluation method for force transducers, Measurement Science and Technology 10 (4) (1999) N31–N33. [2] Y. Fujii, Measurement of steep impulse response of a force transducer, Measurement Science and Technology 14 (1) (2003) 65–69. [3] Y. Fujii, Proposal for a step response evaluation method for force transducers, Measurement Science and Technology 14 (10) (2003) 1741–1746. [4] R. Kumme, Investigation of the comparison method for the dynamic calibration of force transducers, Measurement 23 (1998) 239–245. [5] Y.-K. Park, R. Kumme, D.-I. Kang, Dynamic investigation of a three-component force-moment sensor, Measurement Science and Technology 13 (2002) 654–659. [6] Y.-K. Park, R. Kumme, D.-I. Kang, Dynamic investigation of a binocular six-component force-moment sensor, Measurement Science and Technology 13 (2002) 1311–1318. [7] Y. Fujii, A method for calibrating force transducers against oscillation force, Measurement Science and Technology 14 (8) (2003) 1259–1264. [8] Y. Fujii, Measurement of force acting on a moving part of a pneumatic linear bearing, Review of Science Instruments 74 (6) (2003) 3137–3141. [9] Y. Fujii, D. Isobe, S. Saito, H. Fujimoto, Y. Miki, A method for determining the impact force in crash testing, Mechanical Systems and Signal Processing 14 (6) (2000) 959–965. [10] Y. Fujii, T. Yamaguchi, A method for evaluating material viscoelasticity, Review of Science Instruments 75 (1) (2004) 119–123.
Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 20 (2006) 735–744 www.elsevier.com/locate/jnlabr/ymssp
Optical method for strength test for general industrial products Tomokazu Suzuki, Yusaku Fujii, J.D.R. Valera Department of Electronic Engineering, Faculty of Engineering, Gunma University, 1-5-1 Tenjincho, Kiryu 376-8515, Japan Received 5 April 2004; received in revised form 3 June 2004; accepted 22 June 2004 Available online 11 September 2004
Abstract A method for evaluating the strength of general industrial products is proposed. In the method, a levitated mass encountering negligible friction is made to collide with a general industrial product under test. During the collision, the frequency Doppler shift of a laser beam reflecting from the mass is accurately measured using an optical interferometer. The velocity, position, acceleration and inertial force of the mass are calculated from the measured time-varying Doppler shift. The mechanical response of general industrial products is accurately determined with this method. r 2004 Elsevier Ltd. All rights reserved. Keywords: Strength test; Dynamic force; Optical interferometer pneumatic linear bearing
1. Introduction Recently, the need for strength testing of general industrial products has arisen in various industrial and research applications such as material and crash testing. In such tests, the force acting on the product under test is measured using a force transducer and the position of the applied force is obtained with a position transducer. However, force transducers are typically calibrated by using static weights and under static conditions. At present there are no accepted standard methods available for evaluating the dynamic characteristics of force transducers. This results in two major problems concerning the strength testing of materials. One is the difficulty in Corresponding author. Tel.: +81-277-30-1757; fax: +81-277-30-1707.
E-mail address: [email protected] (T. Suzuki). 0022-460X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2004.06.005
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evaluating the uncertainty in the measured varying force. The other is the difficulty in evaluating the uncertainty in the time at which the varying force is measured. Force, which is a basic mechanical quantity, is defined as the product of mass and acceleration, F ¼ Ma; where F is the force acting on an object, M is the mass of the object, and a is the acceleration of the centre of gravity of the object. This means that an accurately known acceleration is necessary to determine the force accurately and to calibrate force transducers accurately. The acceleration due to gravity, g, is conveniently used for generating and/or measuring constant force. Constant force can be accurately compared using a conventional balance with a knife-edge or a hinge. However, there are no accepted methods for calibrating force transducers under dynamic conditions. The development of methods for the dynamic calibration of force transducers is therefore important. Although the methods for the dynamic calibration of force transducers are not yet well established, there have been several attempts to develop dynamic calibration methods [1–7]. These attempts can be divided into three main categories: methods for calibrating transducers by using an impact force [1,2], methods for calibrating transducers by using a step force [3], and methods that use an oscillation force for calibration [4–7]. In these methods, a mass is levitated using a pneumatic linear bearing [8] in order to obtain a moving mass with negligible friction. This moving mass is used to generate the reference force that is applied to the force transducers. The inertial force of the mass is measured using an optical interferometer. Methods for material testing, such as a method for dynamic three-point bending test [9] and a method for evaluating material viscoelasticity under an oscillating load [10] have also been proposed. In this paper, a method for testing the strength of general industrial products is reported. In the method, a levitated mass encountering negligible friction is made to collide with a general industrial product under test. During the collision, the applied force is accurately determined by using an optical interferometer to measure the Doppler frequency shift of a laser beam that is reflected from the levitated mass. The velocity, position, acceleration and inertial force of the mass are calculated from the measured time-varying Doppler shift. The mechanical response of general industrial products is accurately determined by using this method.
2. Experimental setup Fig. 1 shows the experimental setup used for testing the strength of general industrial products. The test material is attached to the base. An impact force is applied to the test material by colliding the moving part of the pneumatic linear bearing. A pneumatic linear bearing is used to obtain linear motion with negligible friction acting on the mass (i.e., the moving part of the bearing). An initial velocity is given to the moving part manually. The moving part of the linear bearing is made to collide, with the product under test, with a velocity v1 (m/s). It deaccelerates due to the reaction force of the product, and then finally separates from the product with velocity v2 (m/s). A cube-corner prism (CC) (for the interferometer) and an additional mass (for adjusting the collision position) are attached to the moving part; the total mass, M, is approximately 4.4990 kg. The extension block is made of
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Fig. 1. Experimental setup.
stainless-steel, with a tip rounded. The inertial force of the mass is determined very accurately by measuring the velocity of the mass using an optical interferometer. Since, the friction force on the levitated mass is negligible, then the impact force acting on the test specimen from the mass and the reaction force acting on the mass from the test specimen have equal magnitudes but different signs, in accordance to the principle of action and reaction. The inertial force is given by the product of mass and acceleration, and the acceleration is obtained by differentiating the velocity of the moving part. An optical interferometer was used to accurately measure the velocity of the moving part. It consisted of a Michelson interferometer in which the mirrors were replaced with corner cube prisms. One corner cube prism was firmly attached to the moving mass and defined the signal arm of the Michelson interferometer. The other corner cube prism was at rest and defined the reference arm. The light source used was a Zeeman-type He–Ne laser that emits light at two wavelengths in which the two wavelengths had orthogonal polarisation. The light from the He–Ne laser was incident on a polarisation beam splitter. One wavelength was transmitted to the signal arm and then reflected from the corner cube prism that is attached to the moving mass. The other wavelength was reflected from the beam splitter, since it has orthogonal polarisation, and propagates in the reference arm. After propagation in the Michelson interferometer arms, the two beams were made to interfere by transmitting them through a polariser with its transmission axis oriented at 451 to the polarisation of the beams. The interfering beams were then incident on a detector and resulted in a beat signal, since the beams had slightly different wavelengths. The frequency of the measured beat signal, fbeat, which corresponds to the frequency difference between the signal and reference beams, was measured with a frequency counter. When the object was at rest, then f beat ¼ f rest was approximately 2.8 MHz. However, object motion resulted in a Doppler frequency shift in the signal beam which in turn resulted in a variation in fbeat. Using a
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two wavelength source allowed us to distinguish if the velocity of the moving part was in the positive or negative direction. The change in momentum of the moving part is the product of the mass and the change in velocity. The mass velocity was obtained by measuring the induced Doppler frequency shift in the signal beam of the laser interferometer and by using Eqs. (1) and (2), v ¼ lair ðf Doppler Þ=2;
ð1Þ
f Doppler ¼ ðf beat f rest Þ;
ð2Þ
where fDoppler is the Doppler shift, lair is the wavelength of the signal beam, fbeat is the beat frequency, that is the frequency difference between the signal and reference beams, and frest is the rest frequency defined above. The positive direction for the position, the velocity, the acceleration and the force acting on the moving part is towards the right in Fig. 1. An electronic frequency counter (model: R5363; manufactured by Advantest Corp., Japan) continuously measures and records the beat frequency, fbeat, until 14 000 samples are stored with a sampling interval of T=400/ fbeat. This counter continuously measures the time interval taken by 400 periods without dead time. The sampling period of the counter is approximately 0.15 ms at a frequency of 2.8 MHz. Another frequency counter (same model: R5363; manufactured by Advantest Corp., Japan) measures the rest frequency, frest, using an electric signal supplied by a photodiode embedded inside the He–Ne laser. Fig. 2 shows a photograph of the test section of the experimental setup. The height of the collision point and of the centre of gravity of the moving part are set to be equal. This is done in order to avoid a collision induced change in the shape of the air film that exists between the moving part and the guide way of the pneumatic bearing. To set the height of the centre of gravity of the moving part at a convenient position, an additional mass is attached to the moving part. The total mass of the moving part, including the additional mass and the corner cube prism, is approximately 4.4990 kg.
Fig. 2. Photograph of the mechanical part.
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The pneumatic linear bearing, ‘‘Air-Slide TAAG10A-02’’ (NTN Co., Ltd., Japan), is attached to an adjustable tilting stage. The maximum weight that can be supported by the moving part is approximately 30 kg, the thickness of the air film is approximately 8 mm, the stiffness of the air film is more than 70 N/mm, and the straightness of the guideway is better than 0.3 mm/100 mm. The friction characteristics are accurately determined by using the method described in Ref. [8]. The two electric counters (R5363 and TA1100) are triggered by a sharp trigger signal generated by a digital-to-analog converter. This signal was generated by using a light switch, that is a combination of a laser-diode and a photodiode. In this experiment, three sets of collision measurements were obtained by manually changing the initial velocity of the moving mass.
3. Results Fig. 3 shows the procedure for obtaining the force from the measured time-varying beat frequency. During the collision experiment, only the beat frequency is measured. Using the measured value of the time-varying beat frequency, the velocity, position, acceleration and inertial force are numerically calculated with a computer. It was found that the noise in the time-varying beat frequency measured was too big to accurately determine the position of the peak. The measured data was therefore low-pass filtered with a filter that assigned, to every point, the average of the 11 neighbouring points.
Fig. 3. Data processing procedure.
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The inertial force is given by the product of mass and acceleration, and the acceleration is calculated by differentiating the velocity of the moving part. The position of the mass is calculated by integrating the velocity. The velocity is obtained from the Doppler shift frequency of the signal beam of a laser interferometer, fDoppler. The origins of the time and position axes are set to be the time and position where the reaction force from the product under test is detected, respectively. The velocity before and after the collision are approximately, v1 ¼ 0:154226 m=s and v2 ¼ 0:112196 m=s; respectively. The reduction of the kinetic energy is approximately 0.0251895 J. This loss of the kinetic energy is believed to disperse as heat inside the test specimen. In this experiments, a plastic case was used as an example of the general industrial product tested. Three sets of collision measurements were conducted. The collision measurement was done by colliding the moving part to the centre of the lid of the plastic case. The size of plastic case under test was 38.7 mm in width, 67.7 mm in length and 15.2 mm in depth. Its thickness was 1.4 mm. Fig. 4 shows the results of the first measurement. In this first measurement, the impulse was not enough to destroy the tested material. The upper graph shows the change in reaction force against time, and the lower graph shows the change in reaction force against position. From the upper graph in Fig. 4, it can be seen that the applied force is gentle and the maximum force was approximately 30 N with only 35 ms of collision time. From the lower part in Fig. 4, it can be seen that the magnitude of the displacement was approximately 0.8 mm when the force was approximately 30 N. The spring constant, given by the slope of the graph, increases as the displacement is increased. There is a slight area bounded by the curve shown in Fig. 4, representative of hysteresis. This is caused by the slight viscosity of the test specimen. This area corresponds to the loss of the kinetic energy and is believed to disperse as heat inside the test specimen under test.
Fig. 4. Results of 1st collision measurement.
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Fig. 5 shows the results of the second measurement (superimposed on the results from the first experiment, for comparison), in which the test specimen was damaged by the impact applied. The test specimen cracked. Fig. 6 shows a photograph of the crack in the test specimen after the second collision measurement. The length of the crack is approximately 37 mm. The upper graph in Fig. 5 shows a sharp decrease in the force from the peak value of approximately 55 N. At this point, the test specimen cracked. From the lower graph in Fig. 5 it can be seen that the magnitude of the displacement at the peak was approximately
Fig. 5. Results of 2nd collision measurement.
Fig. 6. Photograph of the case after the 2nd collision measurement.
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1.2 mm (when the sample cracked) and then, it increased to approximately 2.0 mm, after the sample cracked. The slope of the graph, after damage, was lower than that before damage. This is because the spring constant of the test specimen has changed after the damage to the sample. The work done by the moving part is given by the integral, Z W¼
ðF Þ dx
ð3Þ
and is approximately 0.025195 J. The absolute value of this work is equal to the area bounded by the curve shown in Fig. 4. This value agrees well with the reduction of the kinetic energy of approximately 0.025190 J calculated by using the velocity before and after the collision, v1 (m/s) and v2 (m/s). The energy dissipation ratio, W/E1, was approximately 47%, where E1 is the initial kinetic energy of the moving mass. The area bounded by the graph shown in Fig. 5 is equal to the energy loss that originated from the damage. The spring constant, or slope of the graph, changes before and after the destruction. Both slopes up to the point where the sample cracked coincide. This indicates high reproducibility. Fig. 7 shows the results of the third measurement, superimposed with the results of the first and second measurements, for comparison. In this last measurement, a small impact that does not cause further damage is applied. From the upper graph in Fig. 7 it can be seen that the applied force was gentle and the impact was characterised by a maximum force is 16 N and 50 ms collision time. From the lower graph in Fig. 7, it can be seen that the spring constants (slope of the curves) before and after sample damage were distinct, and that the slope after damage was smaller than before damage occurred.
Fig. 7. Results of 3rd collision measurement.
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4. Evaluation of uncertainty The force is measured as the inertial force of the moving mass. The inertial force is calculated as the product of acceleration and mass. If the external force acting on the moving mass is negligible, then the inertial force and the force acting on the material under test are the same. The uncertainty sources in the measured instantaneous values of the varying force are as follows. (1) Acceleration: For the acceleration measurement, the vibration of the optical parts of the optical interferometer is dominant. The standard deviations of the measured force just before and after the collision in the second experiment are approximately 0.07 and 0.08 N, respectively. (2) Mass: The uncertainty in measuring the moving mass is approximately 0.1 g, which corresponds to 0.002% of the total mass of approximately 4.4990 kg. (3) External force: For the external force acting on the moving mass, the frictional force acting inside the pneumatic bearing is dominant under the assumption that the air film of approximately 8 mm thickness inside the bearing is not broken. The frictional characteristics of the air bearing are determined using the technique given in Reference [8]. The dynamic frictional force acting on the moving part, Fdf, is estimated by (4)
F df ¼ Av;
ð4Þ 2
1
where A ¼ 8 10 /kg s . This is calculated to be approximately 0.02 N at a velocity of approximately 0.2 ms1. Therefore, the standard uncertainty in measuring the instantaneous value of the varying force in the experiments is estimated to be approximately 0.1 N, which corresponds to 0.2% of the maximum applied force in the experiments of approximately 55 N.
5. Discussion In the proposed method the object velocity is obtained by measuring the time-varying Doppler frequency shift (of a laser beam reflected from the moving object) during the collision. All the other quantities, such as the position, acceleration and force, are numerically calculated from this. In addition, force is directly calculated according to its definition, that is, the product of mass and acceleration. The authors consider that this simplicity is the most significant advantage of the proposed method compared with other conventional methods using a force transducer and a position sensor. The dynamic response of the tested object against an impact force is evaluated highly accurately by measuring the time-varying Doppler frequency shift of a reflected laser beam. If the interferometer is miniaturised by the use of a stabilised laser diode, and the pneumatic linear bearing is replaced by a simple pendulum mechanism, then the whole system could be made portable.
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6. Conclusions A method for testing the strength of general industrial products was proposed. In the method, a mass using a pneumatic linear bearing with negligible friction is made to collide with a general industrial product under test. During the collision, only the time-varying Doppler frequency shift of a laser beam reflecting from the mass is accurately measured using an optical interferometer. The velocity, position, acceleration and inertial force of the mass are calculated from the measured time-varying beat frequency. The experimental setup used in the proposed method is relatively simple. The strength of a plastic case was highly accurately determined by using the proposed method. Any general industrial product, can be tested by attaching it to the base using an appropriate adhesive material or mechanical holder. The advantages and future prospects of the proposed method were discussed.
Acknowledgment This work was supported by a research-aid fund of the Mitsutoyo Association for Science and Technology.
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