User Defined Functions - Revision history

Nairnj: /* Cosine Ramp to Desired Velocity */

2024-11-16T18:31:52Z

Cosine Ramp to Desired Velocity

← Older revision		Revision as of 18:31, 16 November 2024
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	In terms of defined functions, this displacement can be written		In terms of defined functions, this displacement can be written

	~~    <math>d(t) = {V\over2}\left[t + (t-t_r){\rm sign}(t-t_r) - {\rm sinbox}\left({t_r\over\pi},{t\over t_r}\right)\right]</math>~~

	<math>d(t) = {Vt_r\over2}\left[\frac{t}{t_r} + \left(\frac{t}{t_r}-1\right){\rm sign}\left(\frac{t}{t_r}-1\right) - {\rm sinbox}\left({1\over\pi},{t\over t_r}\right)\right]</math>		<math>d(t) = {Vt_r\over2}\left[\frac{t}{t_r} + \left(\frac{t}{t_r}-1\right){\rm sign}\left(\frac{t}{t_r}-1\right) - {\rm sinbox}\left({1\over\pi},{t\over t_r}\right)\right]</math>

Nairnj: /* Cosine Ramp to Desired Velocity */

2024-11-16T18:17:37Z

Cosine Ramp to Desired Velocity

← Older revision		Revision as of 18:17, 16 November 2024
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	<math>d(t) = {V\over2}\left[t + (t-t_r){\rm sign}(t-t_r) - {\rm sinbox}\left({t_r\over\pi},{t\over t_r}\right)\right]</math>		<math>d(t) = {V\over2}\left[t + (t-t_r){\rm sign}(t-t_r) - {\rm sinbox}\left({t_r\over\pi},{t\over t_r}\right)\right]</math>

			<math>d(t) = {Vt_r\over2}\left[\frac{t}{t_r} + \left(\frac{t}{t_r}-1\right){\rm sign}\left(\frac{t}{t_r}-1\right) - {\rm sinbox}\left({1\over\pi},{t\over t_r}\right)\right]</math>

	For example, to ramp to V and then hold constant velocity until reaching desired displacement d<sub>max</sub>, the simulation time should be set to:		For example, to ramp to V and then hold constant velocity until reaching desired displacement d<sub>max</sub>, the simulation time should be set to:

Nairnj: /* Cosine Ramp to Desired Displacement */

2023-11-06T22:56:41Z

Cosine Ramp to Desired Displacement

← Older revision		Revision as of 22:56, 6 November 2023
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	<math>d(t) = {\rm cosramp}\left(d,{t\over t_r}\right)</math>		<math>d(t) = {\rm cosramp}\left(d,{t\over t_r}\right)</math>

	Differentiating this displacement to get velocity that ~~much~~ applied as a boundary condition gives		Differentiating this displacement to get velocity that must be applied as a boundary condition gives

Nairnj: /* Periodic Boundary Condition Function */

2023-03-28T22:43:10Z

Periodic Boundary Condition Function

← Older revision		Revision as of 22:43, 28 March 2023
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	=== Periodic Boundary Condition Function ===		=== Periodic Boundary Condition Function ===

	The <tt>mod(A,x)</tt> function is useful for defining any periodic boundary condition function of time. First create any function of time that is well defined over a given cycle time <tt>t<sub>c</sub></tt>. To define a periodic function replace every "<tt>t</tt>" in that function with "<tt>mod(t,t<sub>c</sub>)</tt>". The result will be a periodic function that will repeat the same defined cycle as long as the simulations keeps running. For example, a sequence of triangular displacements from the previous section's example could be defined with:		The <tt>mod(A,x)</tt> function is useful for defining any periodic boundary condition function of time. First create any function of time that is well defined over a given cycle time <tt>t<sub>c</sub></tt>. To define a periodic function replace every "<tt>t</tt>" in that function with "<tt>mod(t,t<sub>c</sub>)</tt>". The result will be a periodic function that will repeat the same defined cycle as long as the simulations keeps running. For example, a sequence of triangular displacements from the previous section's example staring at the beginning (t<sub>0</sub>=0) could be defined with:


	<math>v(t) = {\rm box}\left(v_r,{mod(t,2t_r)~~-t_0~~\over t_r}\right)-{\rm box}\left(v_r,{mod(t,2t_r)~~-t_0~~-t_r\over t_r}\right)</math>		<math>v(t) = {\rm box}\left(v_r,{{\rm mod}(t,2t_r)\over t_r}\right)-{\rm box}\left(v_r,{{\rm mod}(t,2t_r)-t_r\over t_r}\right)</math>

	=== Rigid Rotation ===		=== Rigid Rotation ===

	See [[Rigid Material#Rotation of Rigid Object\|Rigid Material]] documentation for details on moving a plot of rigid particles in rigid rotation about a point.		See [[Rigid Material#Rotation of Rigid Object\|Rigid Material]] documentation for details on moving a plot of rigid particles in rigid rotation about a point.

Nairnj: /* Notes */

2023-03-28T22:41:52Z

Notes

← Older revision		Revision as of 22:41, 28 March 2023
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	=== Periodic Boundary Condition Function ===		=== Periodic Boundary Condition Function ===

	The <tt>mod(A,x)</tt> function is useful for defining any periodic boundary condition function of time. First create any function of time that is well defined over a given cycle time <tt>t<sub>c</sub></tt>. To define a periodic function replace every "<tt>t</tt>" in that function with "<tt>mod(t,t<sub>c</sub>)</tt>". The result will be a periodic function that will repeat the same defined cycle as long as the simulations keeps running.		The <tt>mod(A,x)</tt> function is useful for defining any periodic boundary condition function of time. First create any function of time that is well defined over a given cycle time <tt>t<sub>c</sub></tt>. To define a periodic function replace every "<tt>t</tt>" in that function with "<tt>mod(t,t<sub>c</sub>)</tt>". The result will be a periodic function that will repeat the same defined cycle as long as the simulations keeps running. For example, a sequence of triangular displacements from the previous section's example could be defined with:


			<math>v(t) = {\rm box}\left(v_r,{mod(t,2t_r)-t_0\over t_r}\right)-{\rm box}\left(v_r,{mod(t,2t_r)-t_0-t_r\over t_r}\right)</math>

	=== Rigid Rotation ===		=== Rigid Rotation ===

	See [[Rigid Material#Rotation of Rigid Object\|Rigid Material]] documentation for details on moving a plot of rigid particles in rigid rotation about a point.		See [[Rigid Material#Rotation of Rigid Object\|Rigid Material]] documentation for details on moving a plot of rigid particles in rigid rotation about a point.

Nairnj: /* Loading and Unloading Cycles */

2023-03-28T22:38:55Z

Loading and Unloading Cycles

← Older revision		Revision as of 22:38, 28 March 2023
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	<math>v(t) = {\rm box}\left(v_r,{t-t_0\over 2t_r}\right){\rm sgn}(t_0+t_r-t)</math>		<math>v(t) = {\rm box}\left(v_r,{t-t_0\over t_r}\right)-{\rm box}\left(v_r,{t-t_0-t_r\over t_r}\right)
			= {\rm box}\left(v_r,{t-t_0\over 2t_r}\right){\rm sgn}(t_0+t_r-t)</math>

	The displacement corresponding to this velocity is		The displacement corresponding to this velocity is

Nairnj: /* Loading and Unloading Cycles */

2023-03-28T22:35:12Z

Loading and Unloading Cycles

← Older revision		Revision as of 22:35, 28 March 2023
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	=== Loading and Unloading Cycles ===		=== Loading and Unloading Cycles ===

	To ramp at velocity v<sub>r</sub> from t<sub>0</sub> and t<sub>0</sub>+t<sub>r</sub> then unload using the reverse velocity, the velocity is:		To ramp at velocity v<sub>r</sub> from t<sub>0</sub> and t<sub>0</sub>+t<sub>r</sub> then unload using the reverse velocity from t<sub>0</sub>+t<sub>r</sub> to t<sub>0</sub>+2t<sub>r</sub>, the velocity is:


	<math>v(t) = {\rm box}\left(v_r,{t-t_0\over ~~t_r~~}\right){\rm sgn}(t_0+t_r-t)</math>		<math>v(t) = {\rm box}\left(v_r,{t-t_0\over 2t_r}\right){\rm sgn}(t_0+t_r-t)</math>

	The displacement corresponding to this velocity is		The displacement corresponding to this velocity is

Nairnj: /* Loading and Unloading Cycles */

2023-03-28T22:30:57Z

Loading and Unloading Cycles

← Older revision		Revision as of 22:30, 28 March 2023
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	<math>d(t) = v_rt_r {\rm tri}\left({t-t_0\over t_r}-1\right)</math>		<math>d(t) = v_rt_r {\rm tri}\left({t-t_0\over t_r}-1\right)</math>

	A sequence of loading and unloading steps can be created by ~~summation of these functions with various values of v<sub>r</sub>, t<sub>0</sub>, and t<sub>r</sub>~~.		A sequence of loading and unloading steps can be created by defining periodic function as explained in the next section.

	=== Periodic Boundary Condition Function ===		=== Periodic Boundary Condition Function ===

Nairnj: /* Integrals and Derivatives */

2023-03-28T22:28:46Z

Integrals and Derivatives

← Older revision		Revision as of 22:28, 28 March 2023
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	<math>{d\bigl({\rm sinbox}(A,x)\bigr)\over dx} = \left\{\begin{array}{ll} 0 & x\le0 \\ {A\pi}\cos(\pi x) & 0<x<1 \\		<math>{d\bigl({\rm sinbox}(A,x)\bigr)\over dx} = \left\{\begin{array}{ll} 0 & x\le0 \\ {A\pi}\cos(\pi x) & 0<x<1 \\
	0 & x \ge 1 \end{array}\right. = \pi A {\rm sign}(x) - {\rm cosramp}(2\pi A,x)</math>		0 & x \ge 1 \end{array}\right. = \pi A\thinspace {\rm sign}(x) - {\rm cosramp}(2\pi A,x)</math>

	=== Cosine Ramp to Desired Velocity ===		=== Cosine Ramp to Desired Velocity ===

Nairnj: /* Notes */

2023-03-28T22:27:39Z

Notes

← Older revision		Revision as of 22:27, 28 March 2023
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	== Notes ==		== Notes ==

			=== Integrals and Derivatives ===

	The cosramp(A,x) and sinbox(A,x) are useful for gradually applying velocity or displacement at the start of the simulation rather than instantaneously applying some velocity or displacement at time zero. To interchange between velocity and displacement, it is useful to evaluate their integrals and derivatives:		The cosramp(A,x) and sinbox(A,x) are useful for gradually applying velocity or displacement at the start of the simulation rather than instantaneously applying some velocity or displacement at time zero. To interchange between velocity and displacement, it is useful to evaluate their integrals and derivatives: