Abstract

The first fiber Bragg grating (FBG) accelerometer using direct transverse forces is demonstrated by fixing the FBG by its two ends and placing a transversely moving inertial object at its middle. It is very sensitive because a lightly stretched FBG is more sensitive to transverse forces than axial forces. Its resonant frequency and static sensitivity are analyzed by the classic spring-mass theory, assuming the axial force changes little. The experiments show that the theory can be modified for cases where the assumption does not hold. The resonant frequency can be modified by a linear relationship experimentally achieved, and the static sensitivity by an alternative method proposed. The principles of the over-range protection and low cross axial sensitivity are achieved by limiting the movement of the FBG and were validated experimentally. The sensitivities 1.333 and 0.634nm/g were experimentally achieved by 5.29 and 2.83 gram inertial objects at 10 Hz from 0.1 to 0.4 g (g=9.8m/s2), respectively, and their resonant frequencies were around 25 Hz. Their theoretical static sensitivities and resonant frequencies found by the modifications are 1.188nm/g and 26.81 Hz for the 5.29 gram one and 0.784nm/g and 29.04 Hz for the 2.83 gram one, respectively.

© 2013 Optical Society of America

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2013

2012

A. Stefani, S. Andresen, W. Yuan, N. Herholdt-Rasmussen, and O. Bang, “High sensitivity polymer optical fiber-Bragg-grating-based accelerometer,” IEEE Photon. Technol. Lett. 24, 763–765 (2012).
[CrossRef]

Q. P. Liu, X. G. Qiao, J. L. Zhao, Z. A. Jia, H. Gao, and M. Shao, “Novel fiber Bragg grating accelerometer based on diaphragm,” IEEE Sens. J. 12, 3000–3004 (2012).
[CrossRef]

P. F. C. Antunes, C. A. Marques, H. Varum, and P. S. Andre, “Biaxial optical accelerometer and high-angle inclinometer with temperature and cross-axis insensitivity,” IEEE Sens. J. 12, 2399–2406 (2012).
[CrossRef]

Y. X. Guo, D. S. Zhang, H. Meng, X. Y. Wen, and Z. D. Zhou, “Metal packaged fiber Bragg grating accelerometer,” Proc. SPIE 8421, 84213V (2012).

N. Basumallick, I. Chatterjee, P. Biswas, K. Dasgupta, and S. Bandyopadhyay, “Fiber Bragg grating accelerometer with enhanced sensitivity,” Sens. Actuators A 173, 108–115 (2012).
[CrossRef]

2011

2009

2008

M. Jones, “Structural-health monitoring: a sensitive issue,” Nat. Photonics 2, 153–154 (2008).
[CrossRef]

2007

A. Laudati, F. Mennella, M. Giordano, G. D’Altrui, C. C. Tassini, and A. Cusano, “A fiber-optic Bragg grating seismic sensor,” IEEE Photon. Technol. Lett. 19, 1991–1993 (2007).
[CrossRef]

D. Graham-Rowe, “Sensors take the strain,” Nat. Photonics 1, 307–309 (2007).
[CrossRef]

2006

Y. Zhang, S. Li, Z. Yin, B. Chen, H.-L. Cui, and J. Ning, “Fiber-Bragg-grating-based seismic geophone for oil/gas prospecting,” Opt. Eng. 45, 084404 (2006).

2003

B. Lee, “Review of the present status of optical fiber sensors,” Opt. Fiber Technol. 9, 57–79 (2003).
[CrossRef]

2000

1999

J. Jung, H. Nam, B. Lee, J. O. Byun, and N. S. Kim, “Fiber Bragg grating temperature sensor with controllable sensitivity,” Appl. Opt. 38, 2752–2754 (1999).
[CrossRef]

P. Touboul, B. Foulon, and E. Willemenot, “Electrostatic space accelerometers for present and future missions,” Acta Astronaut. 45, 605–617 (1999).
[CrossRef]

1998

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett. 10, 1605–1607 (1998).
[CrossRef]

1997

P. L. Walter, “The history of the accelerometer,” J. Sound Vib. 31, 16–22 (1997).

1996

T. A. Berkoff and A. D. Kersey, “Experimental demonstration of a fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett. 8, 1677–1679 (1996).
[CrossRef]

Althouse, B. A.

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett. 10, 1605–1607 (1998).
[CrossRef]

Andre, P. S.

P. F. C. Antunes, C. A. Marques, H. Varum, and P. S. Andre, “Biaxial optical accelerometer and high-angle inclinometer with temperature and cross-axis insensitivity,” IEEE Sens. J. 12, 2399–2406 (2012).
[CrossRef]

P. Antunes, H. Varum, and P. S. Andre, “Uniaxial fiber Bragg grating accelerometer system with temperature and cross axis insensitivity,” Measurement 44, 55–59 (2011).
[CrossRef]

Andresen, S.

A. Stefani, S. Andresen, W. Yuan, N. Herholdt-Rasmussen, and O. Bang, “High sensitivity polymer optical fiber-Bragg-grating-based accelerometer,” IEEE Photon. Technol. Lett. 24, 763–765 (2012).
[CrossRef]

Aneesh, R.

Antunes, P.

P. Antunes, H. Varum, and P. S. Andre, “Uniaxial fiber Bragg grating accelerometer system with temperature and cross axis insensitivity,” Measurement 44, 55–59 (2011).
[CrossRef]

Antunes, P. F. C.

P. F. C. Antunes, C. A. Marques, H. Varum, and P. S. Andre, “Biaxial optical accelerometer and high-angle inclinometer with temperature and cross-axis insensitivity,” IEEE Sens. J. 12, 2399–2406 (2012).
[CrossRef]

Au, H. Y.

Baldwin, J. N. C.

J. N. C. Baldwin, J. Kiddy, and T. Salter, “Review of fiber optic accelerometers,” in 23rd Conference and Exposition on Structural Dynamics 2005 (IMAC XXIII), Orlando, Florida, January 31–February 3, 2005.

Bandyopadhyay, S.

N. Basumallick, I. Chatterjee, P. Biswas, K. Dasgupta, and S. Bandyopadhyay, “Fiber Bragg grating accelerometer with enhanced sensitivity,” Sens. Actuators A 173, 108–115 (2012).
[CrossRef]

Bang, O.

A. Stefani, S. Andresen, W. Yuan, N. Herholdt-Rasmussen, and O. Bang, “High sensitivity polymer optical fiber-Bragg-grating-based accelerometer,” IEEE Photon. Technol. Lett. 24, 763–765 (2012).
[CrossRef]

Basumallick, N.

N. Basumallick, I. Chatterjee, P. Biswas, K. Dasgupta, and S. Bandyopadhyay, “Fiber Bragg grating accelerometer with enhanced sensitivity,” Sens. Actuators A 173, 108–115 (2012).
[CrossRef]

Berkoff, T. A.

T. A. Berkoff and A. D. Kersey, “Experimental demonstration of a fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett. 8, 1677–1679 (1996).
[CrossRef]

Biswas, P.

N. Basumallick, I. Chatterjee, P. Biswas, K. Dasgupta, and S. Bandyopadhyay, “Fiber Bragg grating accelerometer with enhanced sensitivity,” Sens. Actuators A 173, 108–115 (2012).
[CrossRef]

Braga, A. M. B.

S. R. K. Morikawa, A. S. Ribeiro, R. D. Regazzi, L. C. G. Valente, and A. M. B. Braga, “Triaxial Bragg grating accelerometer,” in OFS 2002: 15th Optical Fiber Sensors Conference, Technical Digest (2002), pp. 95–98.

Byun, J. O.

Chan, T. H. T.

Chatterjee, I.

N. Basumallick, I. Chatterjee, P. Biswas, K. Dasgupta, and S. Bandyopadhyay, “Fiber Bragg grating accelerometer with enhanced sensitivity,” Sens. Actuators A 173, 108–115 (2012).
[CrossRef]

Chen, B.

Y. Zhang, S. Li, Z. Yin, B. Chen, H.-L. Cui, and J. Ning, “Fiber-Bragg-grating-based seismic geophone for oil/gas prospecting,” Opt. Eng. 45, 084404 (2006).

Chung, W. H.

Cui, H.-L.

Y. Zhang, S. Li, Z. Yin, B. Chen, H.-L. Cui, and J. Ning, “Fiber-Bragg-grating-based seismic geophone for oil/gas prospecting,” Opt. Eng. 45, 084404 (2006).

Cusano, A.

A. Laudati, F. Mennella, M. Giordano, G. D’Altrui, C. C. Tassini, and A. Cusano, “A fiber-optic Bragg grating seismic sensor,” IEEE Photon. Technol. Lett. 19, 1991–1993 (2007).
[CrossRef]

D’Altrui, G.

A. Laudati, F. Mennella, M. Giordano, G. D’Altrui, C. C. Tassini, and A. Cusano, “A fiber-optic Bragg grating seismic sensor,” IEEE Photon. Technol. Lett. 19, 1991–1993 (2007).
[CrossRef]

Dasgupta, K.

N. Basumallick, I. Chatterjee, P. Biswas, K. Dasgupta, and S. Bandyopadhyay, “Fiber Bragg grating accelerometer with enhanced sensitivity,” Sens. Actuators A 173, 108–115 (2012).
[CrossRef]

Demokan, M. S.

Feng, Z.

Feng, Z. Y.

Foulon, B.

P. Touboul, B. Foulon, and E. Willemenot, “Electrostatic space accelerometers for present and future missions,” Acta Astronaut. 45, 605–617 (1999).
[CrossRef]

Fu, H. Y.

Gao, H.

Giordano, M.

A. Laudati, F. Mennella, M. Giordano, G. D’Altrui, C. C. Tassini, and A. Cusano, “A fiber-optic Bragg grating seismic sensor,” IEEE Photon. Technol. Lett. 19, 1991–1993 (2007).
[CrossRef]

Graham-Rowe, D.

D. Graham-Rowe, “Sensors take the strain,” Nat. Photonics 1, 307–309 (2007).
[CrossRef]

Guo, Y. X.

Y. X. Guo, D. S. Zhang, H. Meng, X. Y. Wen, and Z. D. Zhou, “Metal packaged fiber Bragg grating accelerometer,” Proc. SPIE 8421, 84213V (2012).

Herholdt-Rasmussen, N.

A. Stefani, S. Andresen, W. Yuan, N. Herholdt-Rasmussen, and O. Bang, “High sensitivity polymer optical fiber-Bragg-grating-based accelerometer,” IEEE Photon. Technol. Lett. 24, 763–765 (2012).
[CrossRef]

Hu, M.

Hu, M. L.

Jia, Z. A.

Q. P. Liu, X. G. Qiao, J. L. Zhao, Z. A. Jia, H. Gao, and M. Shao, “Novel fiber Bragg grating accelerometer based on diaphragm,” IEEE Sens. J. 12, 3000–3004 (2012).
[CrossRef]

Johnson, G. A.

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett. 10, 1605–1607 (1998).
[CrossRef]

Jones, M.

M. Jones, “Structural-health monitoring: a sensitive issue,” Nat. Photonics 2, 153–154 (2008).
[CrossRef]

Jung, J.

Kersey, A. D.

T. A. Berkoff and A. D. Kersey, “Experimental demonstration of a fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett. 8, 1677–1679 (1996).
[CrossRef]

Khijwania, S. K.

Kiddy, J.

J. N. C. Baldwin, J. Kiddy, and T. Salter, “Review of fiber optic accelerometers,” in 23rd Conference and Exposition on Structural Dynamics 2005 (IMAC XXIII), Orlando, Florida, January 31–February 3, 2005.

Kim, N. S.

Laudati, A.

A. Laudati, F. Mennella, M. Giordano, G. D’Altrui, C. C. Tassini, and A. Cusano, “A fiber-optic Bragg grating seismic sensor,” IEEE Photon. Technol. Lett. 19, 1991–1993 (2007).
[CrossRef]

Lee, B.

Li, K.

Li, S.

Y. Zhang, S. Li, Z. Yin, B. Chen, H.-L. Cui, and J. Ning, “Fiber-Bragg-grating-based seismic geophone for oil/gas prospecting,” Opt. Eng. 45, 084404 (2006).

Liu, A.

Liu, Q. P.

Q. P. Liu, X. G. Qiao, J. L. Zhao, Z. A. Jia, H. Gao, and M. Shao, “Novel fiber Bragg grating accelerometer based on diaphragm,” IEEE Sens. J. 12, 3000–3004 (2012).
[CrossRef]

Maharana, M.

Marques, C. A.

P. F. C. Antunes, C. A. Marques, H. Varum, and P. S. Andre, “Biaxial optical accelerometer and high-angle inclinometer with temperature and cross-axis insensitivity,” IEEE Sens. J. 12, 2399–2406 (2012).
[CrossRef]

Meng, H.

Y. X. Guo, D. S. Zhang, H. Meng, X. Y. Wen, and Z. D. Zhou, “Metal packaged fiber Bragg grating accelerometer,” Proc. SPIE 8421, 84213V (2012).

Mennella, F.

A. Laudati, F. Mennella, M. Giordano, G. D’Altrui, C. C. Tassini, and A. Cusano, “A fiber-optic Bragg grating seismic sensor,” IEEE Photon. Technol. Lett. 19, 1991–1993 (2007).
[CrossRef]

Morikawa, S. R. K.

S. R. K. Morikawa, A. S. Ribeiro, R. D. Regazzi, L. C. G. Valente, and A. M. B. Braga, “Triaxial Bragg grating accelerometer,” in OFS 2002: 15th Optical Fiber Sensors Conference, Technical Digest (2002), pp. 95–98.

Munendhar, P.

Nam, H.

Ning, J.

Y. Zhang, S. Li, Z. Yin, B. Chen, H.-L. Cui, and J. Ning, “Fiber-Bragg-grating-based seismic geophone for oil/gas prospecting,” Opt. Eng. 45, 084404 (2006).

Qiao, X.

Qiao, X. G.

Q. P. Liu, X. G. Qiao, J. L. Zhao, Z. A. Jia, H. Gao, and M. Shao, “Novel fiber Bragg grating accelerometer based on diaphragm,” IEEE Sens. J. 12, 3000–3004 (2012).
[CrossRef]

J. H. Zhang, X. G. Qiao, M. L. Hu, Z. Y. Feng, H. Gao, Y. Yang, and R. Zhou, “Flextensional fiber Bragg grating-based accelerometer for low frequency vibration measurement,” Chin. Opt. Lett. 9, 090607 (2011).
[CrossRef]

Regazzi, R. D.

S. R. K. Morikawa, A. S. Ribeiro, R. D. Regazzi, L. C. G. Valente, and A. M. B. Braga, “Triaxial Bragg grating accelerometer,” in OFS 2002: 15th Optical Fiber Sensors Conference, Technical Digest (2002), pp. 95–98.

Ribeiro, A. S.

S. R. K. Morikawa, A. S. Ribeiro, R. D. Regazzi, L. C. G. Valente, and A. M. B. Braga, “Triaxial Bragg grating accelerometer,” in OFS 2002: 15th Optical Fiber Sensors Conference, Technical Digest (2002), pp. 95–98.

Salter, T.

J. N. C. Baldwin, J. Kiddy, and T. Salter, “Review of fiber optic accelerometers,” in 23rd Conference and Exposition on Structural Dynamics 2005 (IMAC XXIII), Orlando, Florida, January 31–February 3, 2005.

Shao, M.

Q. P. Liu, X. G. Qiao, J. L. Zhao, Z. A. Jia, H. Gao, and M. Shao, “Novel fiber Bragg grating accelerometer based on diaphragm,” IEEE Sens. J. 12, 3000–3004 (2012).
[CrossRef]

Stefani, A.

A. Stefani, S. Andresen, W. Yuan, N. Herholdt-Rasmussen, and O. Bang, “High sensitivity polymer optical fiber-Bragg-grating-based accelerometer,” IEEE Photon. Technol. Lett. 24, 763–765 (2012).
[CrossRef]

Tam, H. Y.

Tassini, C. C.

A. Laudati, F. Mennella, M. Giordano, G. D’Altrui, C. C. Tassini, and A. Cusano, “A fiber-optic Bragg grating seismic sensor,” IEEE Photon. Technol. Lett. 19, 1991–1993 (2007).
[CrossRef]

Thambiratnam, D.

Todd, M. D.

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett. 10, 1605–1607 (1998).
[CrossRef]

Touboul, P.

P. Touboul, B. Foulon, and E. Willemenot, “Electrostatic space accelerometers for present and future missions,” Acta Astronaut. 45, 605–617 (1999).
[CrossRef]

Valente, L. C. G.

S. R. K. Morikawa, A. S. Ribeiro, R. D. Regazzi, L. C. G. Valente, and A. M. B. Braga, “Triaxial Bragg grating accelerometer,” in OFS 2002: 15th Optical Fiber Sensors Conference, Technical Digest (2002), pp. 95–98.

Varum, H.

P. F. C. Antunes, C. A. Marques, H. Varum, and P. S. Andre, “Biaxial optical accelerometer and high-angle inclinometer with temperature and cross-axis insensitivity,” IEEE Sens. J. 12, 2399–2406 (2012).
[CrossRef]

P. Antunes, H. Varum, and P. S. Andre, “Uniaxial fiber Bragg grating accelerometer system with temperature and cross axis insensitivity,” Measurement 44, 55–59 (2011).
[CrossRef]

Vohra, S. T.

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett. 10, 1605–1607 (1998).
[CrossRef]

Walter, P. L.

P. L. Walter, “The history of the accelerometer,” J. Sound Vib. 31, 16–22 (1997).

Weiss, D. E.

D. E. Weiss, “Design and application of accelerometers,” Proc. SESA (now SEM) (Addison-Wesley, 1947), Vol. IV, pp. 89–99.

Wen, X. Y.

Y. X. Guo, D. S. Zhang, H. Meng, X. Y. Wen, and Z. D. Zhou, “Metal packaged fiber Bragg grating accelerometer,” Proc. SPIE 8421, 84213V (2012).

Willemenot, E.

P. Touboul, B. Foulon, and E. Willemenot, “Electrostatic space accelerometers for present and future missions,” Acta Astronaut. 45, 605–617 (1999).
[CrossRef]

Yang, Y.

Yau, M. H.

Yin, Z.

Y. Zhang, S. Li, Z. Yin, B. Chen, H.-L. Cui, and J. Ning, “Fiber-Bragg-grating-based seismic geophone for oil/gas prospecting,” Opt. Eng. 45, 084404 (2006).

Yu, Y. L.

Yuan, W.

A. Stefani, S. Andresen, W. Yuan, N. Herholdt-Rasmussen, and O. Bang, “High sensitivity polymer optical fiber-Bragg-grating-based accelerometer,” IEEE Photon. Technol. Lett. 24, 763–765 (2012).
[CrossRef]

Zhang, D. S.

Y. X. Guo, D. S. Zhang, H. Meng, X. Y. Wen, and Z. D. Zhou, “Metal packaged fiber Bragg grating accelerometer,” Proc. SPIE 8421, 84213V (2012).

Zhang, J.

Zhang, J. H.

Zhang, Y.

Y. Zhang, S. Li, Z. Yin, B. Chen, H.-L. Cui, and J. Ning, “Fiber-Bragg-grating-based seismic geophone for oil/gas prospecting,” Opt. Eng. 45, 084404 (2006).

Zhao, J. L.

Q. P. Liu, X. G. Qiao, J. L. Zhao, Z. A. Jia, H. Gao, and M. Shao, “Novel fiber Bragg grating accelerometer based on diaphragm,” IEEE Sens. J. 12, 3000–3004 (2012).
[CrossRef]

Zhou, R.

Zhou, Z. A.

Zhou, Z. D.

Y. X. Guo, D. S. Zhang, H. Meng, X. Y. Wen, and Z. D. Zhou, “Metal packaged fiber Bragg grating accelerometer,” Proc. SPIE 8421, 84213V (2012).

Acta Astronaut.

P. Touboul, B. Foulon, and E. Willemenot, “Electrostatic space accelerometers for present and future missions,” Acta Astronaut. 45, 605–617 (1999).
[CrossRef]

Appl. Opt.

Chin. Opt. Lett.

IEEE Photon. Technol. Lett.

A. Stefani, S. Andresen, W. Yuan, N. Herholdt-Rasmussen, and O. Bang, “High sensitivity polymer optical fiber-Bragg-grating-based accelerometer,” IEEE Photon. Technol. Lett. 24, 763–765 (2012).
[CrossRef]

A. Laudati, F. Mennella, M. Giordano, G. D’Altrui, C. C. Tassini, and A. Cusano, “A fiber-optic Bragg grating seismic sensor,” IEEE Photon. Technol. Lett. 19, 1991–1993 (2007).
[CrossRef]

T. A. Berkoff and A. D. Kersey, “Experimental demonstration of a fiber Bragg grating accelerometer,” IEEE Photon. Technol. Lett. 8, 1677–1679 (1996).
[CrossRef]

M. D. Todd, G. A. Johnson, B. A. Althouse, and S. T. Vohra, “Flexural beam-based fiber Bragg grating accelerometers,” IEEE Photon. Technol. Lett. 10, 1605–1607 (1998).
[CrossRef]

IEEE Sens. J.

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Figures (19)

Fig. 1.
Fig. 1.

Diagrams of an undamped (a) free-vibrated and (b) forced spring-mass system.

Fig. 2.
Fig. 2.

Systematic errors of the measured acceleration at different frequencies.

Fig. 3.
Fig. 3.

Diagrams of the proposed FBG accelerometer in undamped (a) free and (b) forced vibrations.

Fig. 4.
Fig. 4.

Relationship between an FBG’s resonant wavelength shift/strain change and the transverse deflection at its middle.

Fig. 5.
Fig. 5.

Design and experimental setup of the proposed FBG accelerometer.

Fig. 6.
Fig. 6.

Frequency responses of the (a) 5.29 gram and (b) 2.83 gram accelerometers, and (c) some of their original records.

Fig. 7.
Fig. 7.

Experimental sensitivities of the 5.29 and 2.83 gram accelerometers.

Fig. 8.
Fig. 8.

Frequency responses of the 2.83 gram accelerometer after being lubricated.

Fig. 9.
Fig. 9.

Comparisons between the accelerations observed by the piezo accelerometer and the lubricated 2.83 gram one.

Fig. 10.
Fig. 10.

Setup of the resonant frequency investigation experiment.

Fig. 11.
Fig. 11.

Time domain records and their FFT at 0.1 nm prestretch, 0.18 gram weight, and two knocks.

Fig. 12.
Fig. 12.

Resonant frequency at the different prestretches of the FBG and weights of the inertial object.

Fig. 13.
Fig. 13.

Linear relationship found for modifying the resonant frequency.

Fig. 14.
Fig. 14.

Inconsistent resonant frequency records of the 5.29 gram accelerometer by the knock incitation method.

Fig. 15.
Fig. 15.

Theoretical static sensitivities of the 5.29 and 2.83 gram accelerometers found by the classic method.

Fig. 16.
Fig. 16.

Relations between an FBG’s resonant wavelength shift and its subjected transverse force at its different prestretches.

Fig. 17.
Fig. 17.

Theoretical static sensitivity of the 5.29 gram accelerometer.

Fig. 18.
Fig. 18.

Over-range protection experiment records of the 5.29 gram accelerometer when its central cylinder was pulled by hand to each side of its shell three times.

Fig. 19.
Fig. 19.

Over-range protection experiment records of the 5.29 gram accelerometer at 25 Hz, 5 g.

Tables (1)

Tables Icon

Table 1. Resonant Frequency at the Different Prestretches and Weights of the Inertial Objects

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

F=ma=md2xdt2,
F=kx,
d2xdt2+kmx=0,
x=cosk/mt=cosω0t=cos2πf0t,
F=ma=md2xdt2=k(xz)=k(xz0cosωt),d2xdt2+kmx=kmz0cosωtd2xdt2+ω02x=ω02z0cosωt.
x0=ω02ω02ω2z0.
a=d2xdt2=x0ω2cosωt.
a=d2zdt2=z0ω2cosωt.
aa=ω02ω02ω2.
|xza|=mK=1ω02.
ma=2Fex+yL/2+mg=4FexL.
f0=Fe/(mL)/π.
ma=2Fexz+yL/2+mg=4FexzL.

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