Stiffness focusing - Revision history

Apm: Category:Incremental path

2026-03-29T19:50:14Z

← Older revision		Revision as of 21:50, 29 March 2026
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	* [[Kinematic loop]]		* [[Kinematic loop]]
	* [[Virtual stiffness]]		* [[Virtual stiffness]]

			[[Category:Incremental path]]

Apm: /* Related */ added link to page Stiffness nesting

2025-06-16T08:56:26Z

Related: added link to page Stiffness nesting

← Older revision		Revision as of 10:56, 16 June 2025
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	== Related ==		== Related ==

			* '''[[Stiffness nesting]]'''
	* '''[[Foldamer technology stiffness nesting]]''', [[Stiffness]]		* '''[[Foldamer technology stiffness nesting]]''', [[Stiffness]]
	* [[High pressure]], [[Piezochemical mechanosynthesis]]		* [[High pressure]], [[Piezochemical mechanosynthesis]]
	* [[Kinematic loop]]		* [[Kinematic loop]]
	* [[Virtual stiffness]]		* [[Virtual stiffness]]

Apm: /* Related / added link to yet unwritten page Virtual stiffness

2025-05-14T12:58:47Z

Related: added link to yet unwritten page * Virtual stiffness

← Older revision		Revision as of 14:58, 14 May 2025
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	* [[High pressure]], [[Piezochemical mechanosynthesis]]		* [[High pressure]], [[Piezochemical mechanosynthesis]]
	* [[Kinematic loop]]		* [[Kinematic loop]]
			* [[Virtual stiffness]]

Apm: /* Math of simplified 1D problem */ added notes on limits to the formula

2024-11-19T11:28:44Z

Math of simplified 1D problem: added notes on limits to the formula

← Older revision		Revision as of 13:28, 19 November 2024
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	k_{total} = \left[\sum_{i=1}^{l} \frac{L_i}{E_i A_i} + \sum_{j=1}^{n} \frac{1}{k'_A{_j} A_j} + \sum_{k=1}^{m} \int_{z_{k,1}}^{z_{k,2}} \frac{dz}{E_k A_k(z)}\right]^{-1}		k_{total} = \left[\sum_{i=1}^{l} \frac{L_i}{E_i A_i} + \sum_{j=1}^{n} \frac{1}{k'_A{_j} A_j} + \sum_{k=1}^{m} \int_{z_{k,1}}^{z_{k,2}} \frac{dz}{E_k A_k(z)}\right]^{-1}
	</math>		</math>

			This formula does not take into account some simple things like say 90° turns of a support structure. <br>
			One could extend for that but at some point numerical finite element simulations are a better approach beyond <br>
			some crude initial back on the envelope estimations. <br>

	== Variables and Units ==		== Variables and Units ==

Apm: removed stub marker

2024-11-11T14:38:03Z

removed stub marker

← Older revision		Revision as of 16:38, 11 November 2024
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	{{site specific term}}		{{site specific term}}
	~~{{Stub}}~~
	{{wikitodo\|Add image illustrating the stiffness focusing effect and math method for calculation}}		{{wikitodo\|Add image illustrating the stiffness focusing effect and math method for calculation}}
	[[File:Foldeamer-printer-stiffness-adapters.jpg\|thumb\|right\|350px\|'''Left:''' Adding small inserts that can be made stiffer in a lower stiffness background framework that can be made bigger. '''Right:''' Application of such a combining of self-assembly technologies technology demonstrated in the context of the [[foldamer printer]] concept. – See: [[Foldamer technology stiffness nesting]] and [[Kinematic loop]].]]		[[File:Foldeamer-printer-stiffness-adapters.jpg\|thumb\|right\|350px\|'''Left:''' Adding small inserts that can be made stiffer in a lower stiffness background framework that can be made bigger. '''Right:''' Application of such a combining of self-assembly technologies technology demonstrated in the context of the [[foldamer printer]] concept. – See: [[Foldamer technology stiffness nesting]] and [[Kinematic loop]].]]

Apm: /* Math of simplified 1D problem */

2024-11-11T12:04:45Z

Math of simplified 1D problem

← Older revision		Revision as of 14:04, 11 November 2024
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	== Math of simplified 1D problem ==		== Math of simplified 1D problem ==

	'''Formula for ~~Stiffness~~ at the tip:''' <br>		'''Formula for stiffness at the tip:''' <br>
	The total absolute stiffness <math>k_{total}</math> can be expressed as: <br>		The total absolute stiffness <math>k_{total}</math> can be expressed as: <br>
	<math>		<math>

Apm at 12:04, 11 November 2024

2024-11-11T12:04:36Z

← Older revision		Revision as of 14:04, 11 November 2024
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	== Math of simplified 1D problem ==		== Math of simplified 1D problem ==

	= Formula for Stiffness at the tip =		'''Formula for Stiffness at the tip:''' <br>

	The total absolute stiffness <math>k_{total}</math> can be expressed as: <br>		The total absolute stiffness <math>k_{total}</math> can be expressed as: <br>
	<math>		<math>

Apm: /* Math of simplified 1D problem */

2024-11-11T12:03:52Z

Math of simplified 1D problem

← Older revision		Revision as of 14:03, 11 November 2024
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	== Math of simplified 1D problem ==		== Math of simplified 1D problem ==

	~~{{todo\|Treat the simplified 1D problems math here, for starters.}}~~
	= Formula for Stiffness at the tip =		= Formula for Stiffness at the tip =

Apm: /* Math of simplified 1D problem */ added basic math

2024-11-11T12:03:39Z

Math of simplified 1D problem: added basic math

← Older revision		Revision as of 14:03, 11 November 2024
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	{{todo\|Treat the simplified 1D problems math here, for starters.}}		{{todo\|Treat the simplified 1D problems math here, for starters.}}
			= Formula for Stiffness at the tip =

			The total absolute stiffness <math>k_{total}</math> can be expressed as: <br>
			<math>
			k_{total} = \left[\sum_{i=1}^{l} \frac{L_i}{E_i A_i} + \sum_{j=1}^{n} \frac{1}{k'_A{_j} A_j} + \sum_{k=1}^{m} \int_{z_{k,1}}^{z_{k,2}} \frac{dz}{E_k A_k(z)}\right]^{-1}
			</math>

			== Variables and Units ==

			* Absolute stiffness <math>k</math>: [N/m]
			* Bulk Young's modulus <math>E</math>: [Pa] = [N/m²]
			* Specific interface stiffness <math>k'_A</math>: [(N/m)/m²] = [N/m³]
			This represents "absolute stiffness per area". <br>
			Also known as "interface stiffness parameter". <br>
			Product with area gives the absolute interface stiffness [N/m].

			== Back to tip interface stiffness (sensible?) ==

			<math>
			k'_{tip} = \frac{k_{total}}{A_{tip}}
			</math>
			Where:
			* <math>k'_{tip}</math>: [(N/m)/m²] = [N/m³] specific interface stiffness at tip
			* <math>k_{total}</math>: [N/m] previously calculated absolute stiffness
			* <math>A_{tip}</math>: [m²] tip area

	== Related ==		== Related ==

Apm: basic page

2024-11-11T10:58:00Z

basic page

New page

{{site specific term}}
{{Stub}}
{{wikitodo|Add image illustrating the stiffness focusing effect and math method for calculation}}
[[File:Foldeamer-printer-stiffness-adapters.jpg|thumb|right|350px|'''Left:''' Adding small inserts that can be made stiffer in a lower stiffness background framework that can be made bigger. '''Right:''' Application of such a combining of self-assembly technologies technology demonstrated in the context of the [[foldamer printer]] concept. – See: [[Foldamer technology stiffness nesting]] and [[Kinematic loop]].]]

Going along the [[kinematic loop]] usually with monotonously in area shrinking cross section (finally through conical tip) 
the same force gets focused on smaller and smaller areas thereby increasing the pressure towards really [[High pressure]]. 
There is more to this though. 

An interesting for [[piezochemical mechanosynthesis]] highly relevant question is 
what the the absolute stiffness at the tip in terms of N/m (as SI units) will be. 

With systems built up out of heterogeneous materials there are different specific stiffnesses (Young modulus). 
For every specific stiffness and cross section at some point in the [[kinematic loop]] 
there is an equivalent absolute stiffness at the very tip (with defined areal cross section). 

To get the absolute stiffness at the very tip of the [[SPM probe]] one 
calculates the equivalent stiffnesses of fine layers (or sections with constant cross section) 
an sums up their inverses (summing up compliances). Possible infinetesimally i.e. integration. 
This physical process and modeling for it is what we shall call here in this wiki 
'''"stiffness focussing"''' 

A higher area can make up for a lower stiffness (to a degree) 
So e.g. what higher upstream the [[kinematic loop]] would only suffice for 
[[topological atomic precision]] can further downstream the [[kinematic loop]] (towards tip) 
suffice for [[positional atomic precision]]. 

To a degree means there are obviously limits. 
E.g. protein assemblies with matching up complementary surfaces that make their side chains crystallize 
may be able to focus to enough stiffness for [[positional atomic precision]] with an insets 
[[crystolecule]] core final tip of the [[kinematic loop]]. 
But just linking proteins together more covalently without matching up surfaces will not provide sufficient stiffness 
for [[positional atomic precision]]. It may still provide enough constraints for other tasks like [[tether assisted assembly]]. 

== Math of simplified 1D problem ==

{{todo|Treat the simplified 1D problems math here, for starters.}}

== Related ==

* '''[[Foldamer technology stiffness nesting]]''', [[Stiffness]]
* [[High pressure]], [[Piezochemical mechanosynthesis]]
* [[Kinematic loop]]

Stiffness focusing - Revision history

Apm: Category:Incremental path

Apm: /* Related */ added link to page Stiffness nesting

Apm: /* Related */ added link to yet unwritten page * Virtual stiffness

Apm: /* Math of simplified 1D problem */ added notes on limits to the formula

Apm: removed stub marker

Apm: /* Math of simplified 1D problem */

Apm at 12:04, 11 November 2024

Apm: /* Math of simplified 1D problem */

Apm: /* Math of simplified 1D problem */ added basic math

Apm: basic page

Apm: /* Related / added link to yet unwritten page Virtual stiffness