Single degree of freedom damped free vibrations - Revision history

Ok: Text replacement - "" to ""

2024-10-22T00:34:30Z

Text replacement - "</tex>" to "</math>"

← Older revision		Revision as of 02:34, 22 October 2024
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>m\ddot{u}(t) + c\dot{u}(t) + ku(t) = 0</~~tex~~>		\| <math>m\ddot{u}(t) + c\dot{u}(t) + ku(t) = 0</math>
	\| align = "right" \| (1)		\| align = "right" \| (1)
	\|}		\|}
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>\ddot{u}(t) + \frac{c}{m}\dot{u}(t) + \frac{k}{m}u(t) = 0</~~tex~~>		\| <math>\ddot{u}(t) + \frac{c}{m}\dot{u}(t) + \frac{k}{m}u(t) = 0</math>
	\| align = "right" \| (2)		\| align = "right" \| (2)
	\|}		\|}


	After substitution of <math>\omega = \sqrt{\frac{k}{m}}</~~tex~~> Eqn. (2) becomes		After substitution of <math>\omega = \sqrt{\frac{k}{m}}</math> Eqn. (2) becomes


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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>\ddot{u}(t) + \frac{c}{m} \dot{u}(t) + \omega^2 u(t) = 0</~~tex~~>		\| <math>\ddot{u}(t) + \frac{c}{m} \dot{u}(t) + \omega^2 u(t) = 0</math>
	\| align = "right" \| (3)		\| align = "right" \| (3)
	\|}		\|}
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</~~tex~~>		\| <math>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</math>
	\| align = "right" \| (4)		\| align = "right" \| (4)
	\|}		\|}
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>\lambda_{1,2} = \frac{-\frac{c}{m} \pm \sqrt{(\frac{c}{m})^2 - 4 \omega^2}}{2} = -\frac{c}{2m} \pm \sqrt{(\frac{c}{2m})^2 - \omega^2}</~~tex~~>		\| <math>\lambda_{1,2} = \frac{-\frac{c}{m} \pm \sqrt{(\frac{c}{m})^2 - 4 \omega^2}}{2} = -\frac{c}{2m} \pm \sqrt{(\frac{c}{2m})^2 - \omega^2}</math>
	\| align = "right" \| (5)		\| align = "right" \| (5)
	\|}		\|}
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	=== Critically-Damped Systems ===		=== Critically-Damped Systems ===

	If the radical term in Eqn. (5)is set equal to zero then <math>c/2m = \omega</~~tex~~> and the critical value of damping coefficient, <math>c_c</~~tex~~> can be expressed as		If the radical term in Eqn. (5)is set equal to zero then <math>c/2m = \omega</math> and the critical value of damping coefficient, <math>c_c</math> can be expressed as


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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>c_c = 2 m \omega</~~tex~~>		\| <math>c_c = 2 m \omega</math>
	\| align = "right" \| (6)		\| align = "right" \| (6)
	\|}		\|}
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	=== Undercritically-Damped Systems ===		=== Undercritically-Damped Systems ===

	If damping is less than critical, i.e. if <math>c < c_c</~~tex~~> (in other words if <math>c < 2 m \omega</~~tex~~>), it is convenient to define the damping in the terms of damping ratio, which is defined as		If damping is less than critical, i.e. if <math>c < c_c</math> (in other words if <math>c < 2 m \omega</math>), it is convenient to define the damping in the terms of damping ratio, which is defined as


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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}</~~tex~~>		\| <math>\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}</math>
	\| align = "right" \| (7)		\| align = "right" \| (7)
	\|}		\|}


	Introducing <math>\xi</~~tex~~> into Eq. (5) yields		Introducing <math>\xi</math> into Eq. (5) yields


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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm i \omega \sqrt{1 - \xi^2} = - \xi \omega \pm i \omega_D</~~tex~~>		\| <math>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm i \omega \sqrt{1 - \xi^2} = - \xi \omega \pm i \omega_D</math>
	\| align = "right" \| (8)		\| align = "right" \| (8)
	\|}		\|}
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <math>\omega_D \equiv \omega \sqrt{1 - \xi^2} </~~tex~~>		\| <math>\omega_D \equiv \omega \sqrt{1 - \xi^2} </math>
	\| align = "right" \| (9)		\| align = "right" \| (9)
	\|}		\|}

Ok: Text replacement - "" to " $"$

2024-10-22T00:13:05Z

Text replacement - "<tex>" to "<math>"

← Older revision		Revision as of 02:13, 22 October 2024
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>m\ddot{u}(t) + c\dot{u}(t) + ku(t) = 0</tex>		\| <math>m\ddot{u}(t) + c\dot{u}(t) + ku(t) = 0</tex>
	\| align = "right" \| (1)		\| align = "right" \| (1)
	\|}		\|}
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>\ddot{u}(t) + \frac{c}{m}\dot{u}(t) + \frac{k}{m}u(t) = 0</tex>		\| <math>\ddot{u}(t) + \frac{c}{m}\dot{u}(t) + \frac{k}{m}u(t) = 0</tex>
	\| align = "right" \| (2)		\| align = "right" \| (2)
	\|}		\|}


	After substitution of <~~tex~~>\omega = \sqrt{\frac{k}{m}}</tex> Eqn. (2) becomes		After substitution of <math>\omega = \sqrt{\frac{k}{m}}</tex> Eqn. (2) becomes


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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>\ddot{u}(t) + \frac{c}{m} \dot{u}(t) + \omega^2 u(t) = 0</tex>		\| <math>\ddot{u}(t) + \frac{c}{m} \dot{u}(t) + \omega^2 u(t) = 0</tex>
	\| align = "right" \| (3)		\| align = "right" \| (3)
	\|}		\|}
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</tex>		\| <math>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</tex>
	\| align = "right" \| (4)		\| align = "right" \| (4)
	\|}		\|}
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>\lambda_{1,2} = \frac{-\frac{c}{m} \pm \sqrt{(\frac{c}{m})^2 - 4 \omega^2}}{2} = -\frac{c}{2m} \pm \sqrt{(\frac{c}{2m})^2 - \omega^2}</tex>		\| <math>\lambda_{1,2} = \frac{-\frac{c}{m} \pm \sqrt{(\frac{c}{m})^2 - 4 \omega^2}}{2} = -\frac{c}{2m} \pm \sqrt{(\frac{c}{2m})^2 - \omega^2}</tex>
	\| align = "right" \| (5)		\| align = "right" \| (5)
	\|}		\|}
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	=== Critically-Damped Systems ===		=== Critically-Damped Systems ===

	If the radical term in Eqn. (5)is set equal to zero then <~~tex~~>c/2m = \omega</tex> and the critical value of damping coefficient, <~~tex~~>c_c</tex> can be expressed as		If the radical term in Eqn. (5)is set equal to zero then <math>c/2m = \omega</tex> and the critical value of damping coefficient, <math>c_c</tex> can be expressed as


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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>c_c = 2 m \omega</tex>		\| <math>c_c = 2 m \omega</tex>
	\| align = "right" \| (6)		\| align = "right" \| (6)
	\|}		\|}
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	=== Undercritically-Damped Systems ===		=== Undercritically-Damped Systems ===

	If damping is less than critical, i.e. if <~~tex~~>c < c_c</tex> (in other words if <~~tex~~>c < 2 m \omega</tex>), it is convenient to define the damping in the terms of damping ratio, which is defined as		If damping is less than critical, i.e. if <math>c < c_c</tex> (in other words if <math>c < 2 m \omega</tex>), it is convenient to define the damping in the terms of damping ratio, which is defined as


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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}</tex>		\| <math>\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}</tex>
	\| align = "right" \| (7)		\| align = "right" \| (7)
	\|}		\|}


	Introducing <~~tex~~>\xi</tex> into Eq. (5) yields		Introducing <math>\xi</tex> into Eq. (5) yields


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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm i \omega \sqrt{1 - \xi^2} = - \xi \omega \pm i \omega_D</tex>		\| <math>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm i \omega \sqrt{1 - \xi^2} = - \xi \omega \pm i \omega_D</tex>
	\| align = "right" \| (8)		\| align = "right" \| (8)
	\|}		\|}
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <~~tex~~>\omega_D \equiv \omega \sqrt{1 - \xi^2} </tex>		\| <math>\omega_D \equiv \omega \sqrt{1 - \xi^2} </tex>
	\| align = "right" \| (9)		\| align = "right" \| (9)
	\|}		\|}

Ok: /* Undercritically-Damped Systems */

2009-10-06T01:40:28Z

Undercritically-Damped Systems

← Older revision		Revision as of 03:40, 6 October 2009
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	is the free vibration frequency of damped system.		is the free-vibration frequency of damped system.

	=== Overcritically-Damped Systems ===		=== Overcritically-Damped Systems ===

Ok: /* Undercritically-Damped Systems */

2009-10-06T01:39:38Z

Undercritically-Damped Systems

← Older revision		Revision as of 03:39, 6 October 2009
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <tex>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm \omega i \sqrt{1 - \xi^2}</tex>		\| <tex>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm i \omega \sqrt{1 - \xi^2} = - \xi \omega \pm i \omega_D</tex>
	\| align = "right" \| (8)		\| align = "right" \| (8)
	\|}		\|}

Ok: /* Governing Equation */

2009-10-06T01:33:42Z

Governing Equation

← Older revision		Revision as of 03:33, 6 October 2009
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	This is a [[second-order linear ordinary differential equation with constant coefficients]] that can be rearranged as follows:		This is a [[homogeneous second-order linear ordinary differential equation with constant coefficients]] that can be rearranged as follows:

Ok: /* Overcritically-Damped Systems */

2009-10-06T01:33:04Z

Overcritically-Damped Systems

← Older revision		Revision as of 03:33, 6 October 2009
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	=== Overcritically-Damped Systems ===		=== Overcritically-Damped Systems ===

	{{To Do \| who = user:ok \| comment = Provide solution for single degree of freedom damped free vibrations, over-critically damped system		{{To Do \| who = user:ok \| comment = Provide solution for single degree of freedom damped free vibrations, over-critically damped system}}


	== References ==		== References ==

Ok at 01:32, 6 October 2009

2009-10-06T01:32:44Z

← Older revision		Revision as of 03:32, 6 October 2009
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	=== Overcritically-Damped Systems ===		=== Overcritically-Damped Systems ===

	== ~~Derivation ==~~		{{To Do \| who = user:ok \| comment = Provide solution for single degree of freedom damped free vibrations, over-critically damped system

	~~{{To Do \| who = user:ok \| comment = Provide solution for single degree of freedom damped free vibrations}}~~

	== References ==		== References ==

Ok: /* Undercritically-Damped Systems */

2009-10-06T01:29:45Z

Undercritically-Damped Systems

← Older revision		Revision as of 03:29, 6 October 2009
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	\|-		\|-
	\| width = "50px" \|		\| width = "50px" \|
	\| <tex>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm \omega i \sqrt{1 - \~~omega~~}</tex>		\| <tex>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm \omega i \sqrt{1 - \xi^2}</tex>
	\| align = "right" \| (8)		\| align = "right" \| (8)
	\|}		\|}


			where


			{\| width="100%"
			\|-
			\| width = "50px" \|
			\| <tex>\omega_D \equiv \omega \sqrt{1 - \xi^2} </tex>
			\| align = "right" \| (9)
			\|}


			is the free vibration frequency of damped system.

	=== Overcritically-Damped Systems ===		=== Overcritically-Damped Systems ===

Ok: /* Undercritically-Damped Systems */

2009-10-06T01:23:31Z

Undercritically-Damped Systems

← Older revision		Revision as of 03:23, 6 October 2009
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	\|}		\|}


			Introducing <tex>\xi</tex> into Eq. (5) yields


			{\| width="100%"
			\|-
			\| width = "50px" \|
			\| <tex>\lambda_{1,2} = - \xi \omega \pm \sqrt{(\xi \omega)^2 - \omega^2} = - \xi \omega \pm \omega i \sqrt{1 - \omega}</tex>
			\| align = "right" \| (8)
			\|}

	=== Overcritically-Damped Systems ===		=== Overcritically-Damped Systems ===

Ok at 01:16, 6 October 2009

2009-10-06T01:16:45Z

← Older revision		Revision as of 03:16, 6 October 2009
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	After substitution of <tex>\omega = \sqrt{\frac{k}{m}}</tex> (Eqn. 2) becomes		After substitution of <tex>\omega = \sqrt{\frac{k}{m}}</tex> Eqn. (2) becomes


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	\| width = "50px" \|		\| width = "50px" \|
	\| <tex>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</tex>		\| <tex>\lambda^2 + \frac{c}{m} \lambda + \omega^2 = 0</tex>
	\| align = "right" \| (3)		\| align = "right" \| (4)
	\|}		\|}

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	\| width = "50px" \|		\| width = "50px" \|
	\| <tex>\lambda_{1,2} = \frac{-\frac{c}{m} \pm \sqrt{(\frac{c}{m})^2 - 4 \omega^2}}{2} = -\frac{c}{2m} \pm \sqrt{(\frac{c}{2m})^2 - \omega^2}</tex>		\| <tex>\lambda_{1,2} = \frac{-\frac{c}{m} \pm \sqrt{(\frac{c}{m})^2 - 4 \omega^2}}{2} = -\frac{c}{2m} \pm \sqrt{(\frac{c}{2m})^2 - \omega^2}</tex>
	\| align = "right" \| (3)		\| align = "right" \| (5)
	\|}		\|}


	=== Critically-Damped Systems ===		=== Critically-Damped Systems ===

			If the radical term in Eqn. (5)is set equal to zero then <tex>c/2m = \omega</tex> and the critical value of damping coefficient, <tex>c_c</tex> can be expressed as


			{\| width="100%"
			\|-
			\| width = "50px" \|
			\| <tex>c_c = 2 m \omega</tex>
			\| align = "right" \| (6)
			\|}


	=== Undercritically-Damped Systems ===		=== Undercritically-Damped Systems ===

			If damping is less than critical, i.e. if <tex>c < c_c</tex> (in other words if <tex>c < 2 m \omega</tex>), it is convenient to define the damping in the terms of damping ratio, which is defined as



			{\| width="100%"
			\|-
			\| width = "50px" \|
			\| <tex>\xi = \frac{c}{c_c} = \frac{c}{2 m \omega}</tex>
			\| align = "right" \| (7)
			\|}


	=== Overcritically-Damped Systems ===		=== Overcritically-Damped Systems ===