Resonant column method - Revision history

Designing Buildings at 16:03, 3 October 2022

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	In theory, to model the perfectly clamped support condition of the resonant column, the spring constants theta and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to theta and K to give significant values of the root rigidity parameters theta and Z. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.40) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.		In theory, to model the perfectly clamped support condition of the resonant column, the spring constants theta and K should be made to approach infinity. However, in Matlab, for simplicity, extreme values have been assigned to theta and K to give significant values of the root rigidity parameters theta and Z. As mentioned in Farghaly (1992), when using the matrix determinant equation (3.40) to compute the resonant frequencies, inaccurate results might be obtained for large values of the slenderness ratio. According to Liu in Author’s Reply (1992), from his own practical point of view, if one can accept the idea of treating a complicated cantilever structure as a Timoshenko beam, then the discrepancies caused by non-exact boundary conditions might be considered as tolerable.

−	~~= Find out more =~~	+	= External references =
−		+
−	=== External references ===	+

	* Alan, K. (2011). Stiffness and damping of sand at small strain using a resonant column. Civil Engineering 3rd year Individual Project, University of Southampton.		* Alan, K. (2011). Stiffness and damping of sand at small strain using a resonant column. Civil Engineering 3rd year Individual Project, University of Southampton.

Designing Buildings at 12:44, 20 October 2020

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	* White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. Journal of Applied Mechanics Vol. 62		* White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. Journal of Applied Mechanics Vol. 62

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Editor at 16:05, 9 August 2018

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	== Young modulus (E) ==		== Young modulus (E) ==

−	The resonant column can also be used in flexural excitation to determine the material’s Young modulus (E). The conventional method with long samples allowing the application of Rayleighod’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration. To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century and takes into account shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (E).	+	The resonant column can also be used in flexural excitation to determine the material’s Young modulus (E). The conventional method with long samples allowing the application of Rayleighod’s energy method and Euler-Bernoulli beam theory, disregarded the shear strain energy and rotary inertia effect. When the tested specimen is short in length compared to its diameter, the effects of rotation and shear deformation of the samples during flexure can be substantial. These effects can be significant in interpreting data from flexural test, especially at high frequencies. Therefore, the Euler-Bernoulli theory of flexural vibration of elastic beam is found to be inadequate for short specimens and also for the prediction of higher modes of vibration.
		+
		+	To be more accurate, Timoshenko beam theory is used as a model for this interpretation. The theory was developed by Ukrainian scientist Stephen Timoshenko in the 20th century and takes into account shear deformation and rotary inertia. Different frequency equations for the clamped-free Timoshenko beam with an end mass in flexural vibration are solved to compute the value of elastic stiffness (E).

	= The resonant column method =		= The resonant column method =
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	[[File:Eq3.JPG\|link=File:Eq3.JPG]]		[[File:Eq3.JPG\|link=File:Eq3.JPG]]

−	~~<br />~~	+	Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass. By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.
−	Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass. By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.	+

	Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended:		Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended:
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	The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of K was given in terms of the Poisson ratio as:		The shear coefficient in Timoshenko’s beam theory is a dimensionless quantity, dependent on the shape of the cross section, which accounts for the fact that the shear stress and shear strain are not uniformly distributed over the cross section of the specimen. Cowper (1966) developed a new formula for the shear coefficient from the derivation of the equations of Timoshenko beam theory. For a circular cross-section, the value of K was given in terms of the Poisson ratio as:

−	[[File:K.JPG\|link=File:K.JPG]]~~<br />~~	+	[[File:K.JPG\|link=File:K.JPG]]
−	Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity.	+
		+	Farghaly (1993) offered suggestion to extend Liu’s work by applying Timoshenko beam theory in treating the boundary conditions. He realised that the use of Euler-Bernoulli theory in the boundary conditions could result in inaccurate natural frequencies calculated, particularly for high slenderness ratios and higher modes of vibration. Farghaly’s model also includes the root flexibilities and the tip mass’s eccentricity.

	The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.		The work of Bruch and Mitchell (1987), Liu (1989) and Farghaly (1993) were in an attempt to simulate the motion of a flexible robot arm modeled as a cantilevered Timoshenko beam with a lumped mass and lumped moment of inertia at the free end. However, for the purpose of this essay, their resonant frequency equations were considered to be adequate for use in computing the material’s Young modulus from the flexural resonant column test, if the angular natural frequency is known.
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	* Timoshenko, S. P. (1955). Vibration Problems in Engineering. In S. P. Timoshenko. New York: D. Van Nostrand Company, third edition.		* Timoshenko, S. P. (1955). Vibration Problems in Engineering. In S. P. Timoshenko. New York: D. Van Nostrand Company, third edition.
	* White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. Journal of Applied Mechanics Vol. 62		* White, M. W. D., and Heppler, G. R. (1995). Vibration Modes and Frequencies of Timoshenko Beams with Attached rigid bodies. Journal of Applied Mechanics Vol. 62
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	=== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ===		=== Finding the Young modulus by Euler-Bernoulli beam theory (short samples) ===

−	The RCA can also be used to measure the Young modulus (E) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode.	+	The RCA can also be used to measure the Young modulus (E) of the material. Cascante et al. (1998) modified the standard Stokoe torsional resonant column (Stokoe cell SBEL D1128) to include flexural vibration mode.

	In the original configuration, four pairs of excitation coils were connected in series to produce a net torque at the top of the sample. In Cascante’s modified version, the coils are reconnected so that only two magnets are used to produce a net horizontal force on top of the specimen.		In the original configuration, four pairs of excitation coils were connected in series to produce a net torque at the top of the sample. In Cascante’s modified version, the coils are reconnected so that only two magnets are used to produce a net horizontal force on top of the specimen.
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	<br/>Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass. By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.		<br/>Bruch and Mitchell (1987) investigated a particular case of a cantilevered Timoshenko beam with a tip mass. By applying the boundary conditions and using Huang’s non-dimensional variables, the solutions to the coupled equations are determined as functions of the Young modulus (E), the Shear modulus (G), material’s density (rho), the angular natural frequency (omega n) and the geometry of the specimen. Bruch and Mitchell derived the frequency equation of the beam in flexural excitation by inserting the solutions to the coupled equations (2.6) and (2.7) into the boundary conditions, from which the matrix equation can be determined. By taking the determinant of the coefficient matrix equation, the resonant frequency equation was found from which the Young modulus (E) can be calculated.

−	Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended:	+	Liu (1989) suggested three ways in which the work of Bruch and Mitchell could be further extended:

−	(i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support),	+	(i) The base condition for the beam-mass system considered in [3] should be modeled as an imperfect clamped support (or elastic support),

−	(ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip,	+	(ii) The tip mass’s centre of gravity is not practically right at the top of the beam but usually at a distance from the beam tip,

	(iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.		(iii) the shear coefficient depends on both the shape of the cross-section and the Poisson ratio. Liu added springs at the hub to simulate the imperfect clamped support therefore the boundary condition also includes the spring’s properties which are the rotational spring constant and translational spring constant. By substituting the general solution into the new boundary conditions, Liu gave the improvement of Bruch and Mitchell’s frequency equation for the mass-loaded clamped-free Timoshenko beam.
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