Conventional Guns - Revision history

Tshhmon at 21:29, 23 April 2024

2024-04-23T21:29:22Z

← Older revision		Revision as of 14:29, 23 April 2024
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	[[Category:Warfare]][[Category:Physics ~~& Math~~ & Engineering‏‎]]		[[Category:Warfare]][[Category:Military Technology]][[Category:Physics & Engineering‏‎]]

Tshhmon at 19:11, 4 April 2024

2024-04-04T19:11:08Z

← Older revision		Revision as of 12:11, 4 April 2024
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	[[Category:Warfare]]		[[Category:Warfare]][[Category:Physics & Math & Engineering‏‎]]

Phoenix: /* Gun with Finite Rate of Burning */

2023-08-11T07:25:57Z

Gun with Finite Rate of Burning

← Older revision		Revision as of 00:25, 11 August 2023
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	When the equation of burning is added to the interior ballistic system of equation, which now consists of the equation of motion of the projectile, and finally the balance of energy equation, the resulting system is not necessarily analytical (i.e. admits a closed form solution), and certainly not simple. Then, the choice of burn rate law and that of form equation must consider both the accuracy of representation, with regard to the practicality of solving these systems.		When the equation of burning is added to the interior ballistic system of equation, which now consists of the equation of motion of the projectile, and finally the balance of energy equation, the resulting system is not necessarily analytical (i.e. admits a closed form solution), and certainly not simple. Then, the choice of burn rate law and that of form equation must consider both the accuracy of representation, with regard to the practicality of solving these systems.

	~~Consequently~~, a great deal of freedom ~~of choice existed~~ in the exact formulation of the equations, and particular solutions are often associated with its progenitor or practitioner by name.		All told, a great deal of freedom exists in the exact formulation of the equations for interior ballistics, with certain line of attack preferred over others depending on the time, location and resource available, and the nature of insight desired. It is then not surprising that particular solutions are often associated with its progenitor or practitioner by name.

	~~(WIP)~~

	====Burning Described by Quadratic Form Function====		====Burning Described by Quadratic Form Function====

Phoenix: /* Burning Described by Quadratic Form Function */

2023-08-11T07:11:45Z

Burning Described by Quadratic Form Function

← Older revision		Revision as of 00:11, 11 August 2023
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	\|}		\|}

	A special case is when <math>\theta = 0 </math>, the most common example being the tubular propellant geometry, in which case, experience has shown that a internal diameter no less than 3/4th of the annulus for even ignition on both sides is required. The mathematical justification for these shall not be detailed too much here, except by noting that these can be derived from expressing <math>\psi </math> in <math>Z </math> and neglecting the higher order terms in the coefficients.		A special case is when <math>\theta = 0 </math>, the most common example being the tubular propellant geometry, in which case, experience has shown that a internal diameter no less than 3/4th of the annulus for even ignition on both sides is required. Then, the web is simply the difference between the annulus and internal diameter. The mathematical justification for these shall not be detailed too much here, except by noting that these can be derived from expressing <math>\psi </math> in <math>Z </math> and neglecting the higher order terms in the coefficients.

	Since by definition, <math>\psi(1) = 0</math> and psi(0) = 1<math> </math>, that is <math>\theta </math> must take between -1 and 1. Geometries where the regressiveness of burning is higher than allowed in this formulation, that is when the volume is burnt as a higher order function of the linear burnt extent, a particular example being spherical grains, which burns according to <math> \psi = 1-Z^3</math>. These simply cannot be represented in the Charbonnier form. This, although on first sight a serious limitation of this approach, is in practice mitigated by the relative ease of conducting experiments in the few cases where more regressive propellant are used e.g. in small arms and infantry mortars, out of practical concerns for manufacture.		Since by definition, <math>\psi(1) = 0</math> and psi(0) = 1<math> </math>, that is <math>\theta </math> must take between -1 and 1. Geometries where the regressiveness of burning is higher than allowed in this formulation, that is when the volume is burnt as a higher order function of the linear burnt extent, a particular example being spherical grains, which burns according to <math> \psi = 1-Z^3</math>. These simply cannot be represented in the Charbonnier form. This, although on first sight a serious limitation of this approach, is in practice mitigated by the relative ease of conducting experiments in the few cases where more regressive propellant are used e.g. in small arms and infantry mortars, out of practical concerns for manufacture.

Phoenix: /* Burning Described by Quadratic Form Function */

2023-08-11T07:10:32Z

Burning Described by Quadratic Form Function

← Older revision		Revision as of 00:10, 11 August 2023
Line 276:		Line 276:
	Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> conveniently takes 0 for a "neutral" geometry, where the area of combustion does not depend on the extent of burning, and takes 1 for a propellant in "degressive" geometry, that is the area exposed to burning is reduced during burning.		Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> conveniently takes 0 for a "neutral" geometry, where the area of combustion does not depend on the extent of burning, and takes 1 for a propellant in "degressive" geometry, that is the area exposed to burning is reduced during burning.

	The definition of <math>\theta </math> is particular suited to describing propellant of the shapes usually denoted "ribbon" and "flake"~~, if few higher order terms can be neglected~~. The ~~below~~ table ~~summarize some of these properties~~.		The definition of <math>\theta </math> is particular suited to describing propellant of the shapes usually denoted "ribbon" and "flake". The following table summarizes the procedure for a few simple shapes.

	{\| class="wikitable", style="text-align: center;"		{\| class="wikitable", style="text-align: center;"
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	\|}		\|}

	~~Since by definition,~~ <math>\~~psi(1)~~ = 0</math> ~~and psi(0) = 1~~<math> </math>, <math>~~\theta~~ </math> ~~must take between -1~~ and ~~1. Geometries where~~ the ~~regressiveness of burning is~~ higher ~~than allowed in this formulation, that is, when burning can proceeds from all 3 dimensions, with particular examples being the spherical and cube shapes. These simply cannot be represented~~ in the ~~Charbonnier form~~.		A special case is when <math>\theta = 0 </math>, the most common example being the tubular propellant geometry, in which case, experience has shown that a internal diameter no less than 3/4th of the annulus for even ignition on both sides is required. The mathematical justification for these shall not be detailed too much here, except by noting that these can be derived from expressing <math>\psi </math> in <math>Z </math> and neglecting the higher order terms in the coefficients.

	This, although on first sight a serious limitation of this approach, is in practice mitigated by the relative ease of conducting experiments in the few cases where more regressive propellant are used e.g. in small arms and infantry mortars, out of practical concerns for manufacture.		Since by definition, <math>\psi(1) = 0</math> and psi(0) = 1<math> </math>, that is <math>\theta </math> must take between -1 and 1. Geometries where the regressiveness of burning is higher than allowed in this formulation, that is when the volume is burnt as a higher order function of the linear burnt extent, a particular example being spherical grains, which burns according to <math> \psi = 1-Z^3</math>. These simply cannot be represented in the Charbonnier form. This, although on first sight a serious limitation of this approach, is in practice mitigated by the relative ease of conducting experiments in the few cases where more regressive propellant are used e.g. in small arms and infantry mortars, out of practical concerns for manufacture.


	[[Category:Warfare]]		[[Category:Warfare]]

Phoenix at 13:09, 10 August 2023

2023-08-10T13:09:51Z

← Older revision		Revision as of 06:09, 10 August 2023
Line 276:		Line 276:
	Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> conveniently takes 0 for a "neutral" geometry, where the area of combustion does not depend on the extent of burning, and takes 1 for a propellant in "degressive" geometry, that is the area exposed to burning is reduced during burning.		Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> conveniently takes 0 for a "neutral" geometry, where the area of combustion does not depend on the extent of burning, and takes 1 for a propellant in "degressive" geometry, that is the area exposed to burning is reduced during burning.

	The definition of <math>\theta </math> is particular suited to describing propellant of the shapes usually denoted "ribbon" and "flake". The below table summarize some of these properties.		The definition of <math>\theta </math> is particular suited to describing propellant of the shapes usually denoted "ribbon" and "flake", if few higher order terms can be neglected. The below table summarize some of these properties.

	~~(WIP_table)~~		{\| class="wikitable", style="text-align: center;"
			\|+ Propellant Geometry Described by the Form Factor
			\|-
			! Shape !! Length !! Width !! Height !! Form Factor
			\|-
			\| Ribbon<br><math>\lambda \gg \mu \gg 1</math> \|\| <math>\lambda e</math> \|\| <math>\mu e</math> \|\| <math>e</math> \|\| <math>\theta \approx \frac{1}{\mu}</math>
			\|-
			\| Flake<br><math>\lambda = \mu \gg 1</math> \|\| <math>\mu e</math> \|\| <math>\mu e</math> \|\| <math>e</math> \|\| <math>\theta \approx \frac{2}{\mu}</math>
			\|}

	Since by definition, <math>\psi(1) = 0</math> and psi(0) = 1<math> </math>, <math>\theta </math> must take between -1 and 1. Geometries where the regressiveness of burning is higher than allowed in this formulation, that is, when burning can proceeds from all 3 dimensions, with particular examples being the spherical and cube shapes. These simply cannot be represented in the Charbonnier form.		Since by definition, <math>\psi(1) = 0</math> and psi(0) = 1<math> </math>, <math>\theta </math> must take between -1 and 1. Geometries where the regressiveness of burning is higher than allowed in this formulation, that is, when burning can proceeds from all 3 dimensions, with particular examples being the spherical and cube shapes. These simply cannot be represented in the Charbonnier form.

Phoenix: /* Gun with Finite Rate of Burning */

2023-08-10T12:28:29Z

Gun with Finite Rate of Burning

← Older revision		Revision as of 05:28, 10 August 2023
Line 276:		Line 276:
	Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> conveniently takes 0 for a "neutral" geometry, where the area of combustion does not depend on the extent of burning, and takes 1 for a propellant in "degressive" geometry, that is the area exposed to burning is reduced during burning.		Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> conveniently takes 0 for a "neutral" geometry, where the area of combustion does not depend on the extent of burning, and takes 1 for a propellant in "degressive" geometry, that is the area exposed to burning is reduced during burning.

	The definition of <math>\theta </math> is particular suited to describing propellant of the shapes usually denoted "ribbon" and "flake". ~~Then, by simple geometry, <math>\theta</math> is the ratio~~ of ~~web thickness to the length of the ribbon. For flake, <math>\theta</math> is then twice of the same ratio~~. ~~A special case is that for the tube, where <math>\theta = 0</math>~~		The definition of <math>\theta </math> is particular suited to describing propellant of the shapes usually denoted "ribbon" and "flake". The below table summarize some of these properties.

			(WIP_table)

	Since by definition, <math>\psi(1) = 0</math> and psi(0) = 1<math> </math>, <math>\theta </math> must take between -1 and 1. Geometries where the regressiveness of burning is higher than allowed in this formulation, that is, when burning can proceeds from all 3 dimensions, with particular examples being the spherical and cube shapes. These simply cannot be represented in the Charbonnier form.		Since by definition, <math>\psi(1) = 0</math> and psi(0) = 1<math> </math>, <math>\theta </math> must take between -1 and 1. Geometries where the regressiveness of burning is higher than allowed in this formulation, that is, when burning can proceeds from all 3 dimensions, with particular examples being the spherical and cube shapes. These simply cannot be represented in the Charbonnier form.

Phoenix: /* Gun with Finite Rate of Burning */

2023-08-10T09:50:00Z

Gun with Finite Rate of Burning

← Older revision		Revision as of 02:50, 10 August 2023
Line 260:		Line 260:
	It bears mentioning that we take the various ballistic approximation as applicable in this section, although notably propellant combustion of a finite rate violates the Pidduck-Kent (and by extension, Mamontov's) argument. A valid argument can be made that in practical guns, the oscillation of the wave system is not damped out in a negligible time frame. In fact, as stated by Hunt (1951), the rarefaction wave predicted by these above works were never observed in practice. Then, the applicability of these assumptions, and consequently the prediction based on these, is always an approximation, and should be taken with an acute awareness of these limitations. In effect, this reduces the faith that can be placed on these sophisticated approximation to no more than what can be given to the simpler Lagrangian approximation, i.e. they are only as valid as they are a close approximation of reality, and this approximation is held to be good for the solution space where <math>\frac{\omega}{\varphi_1 m} < 1 </math>.		It bears mentioning that we take the various ballistic approximation as applicable in this section, although notably propellant combustion of a finite rate violates the Pidduck-Kent (and by extension, Mamontov's) argument. A valid argument can be made that in practical guns, the oscillation of the wave system is not damped out in a negligible time frame. In fact, as stated by Hunt (1951), the rarefaction wave predicted by these above works were never observed in practice. Then, the applicability of these assumptions, and consequently the prediction based on these, is always an approximation, and should be taken with an acute awareness of these limitations. In effect, this reduces the faith that can be placed on these sophisticated approximation to no more than what can be given to the simpler Lagrangian approximation, i.e. they are only as valid as they are a close approximation of reality, and this approximation is held to be good for the solution space where <math>\frac{\omega}{\varphi_1 m} < 1 </math>.

	Theory that models the finite rate burning of propellant requires some way of relating the bulk burn rate (as explained in the propellant section) to the evolution of variables such as free volume (or volume not occupied by the charge), and energy added to the gun system. The former, in the interest of preserving the scalability of theory, is best described by taking the derivative with regard to time, for the product of linear burnt extent <math>Z</math> and a characteristic length for the grain <math>e</math>. For the latter, ~~it is convenient to describe~~ the fraction of propellant ~~burnt~~ using <math>\psi </math>. The two may then be related through what is known as the form function, so called since its purely a function of the grain geometry. This, together with a choice of bulk burn rate law for the propellant, is collectively referred to as a equation of burning.		Theory that models the finite rate burning of propellant requires some way of relating the bulk burn rate (as explained in the propellant section) to the evolution of variables such as free volume (or volume not occupied by the charge), and energy added to the gun system. The former, in the interest of preserving the scalability of theory, is best described by taking the derivative with regard to time, for the product of linear burnt extent <math>Z</math> and a characteristic length for the grain <math>e</math>, also referred to as the propellant web. For the latter, by convention the the fraction of propellant unburnt using <math>\psi</math>. The two may then be related through what is known as the form function, so called since its purely a function of the grain geometry. This, together with a choice of bulk burn rate law for the propellant, is collectively referred to as a equation of burning.

	When the equation of burning is added to the interior ballistic system of equation, which now consists of the equation of motion of the projectile, and finally the balance of energy equation, the resulting system is not necessarily analytical (i.e. admits a closed form solution), and certainly not simple. Then, the choice of burn rate law and that of form equation must consider both the accuracy of representation, with regard to the practicality of solving these systems. ~~Historically~~, a great ~~many~~ freedom of choice existed in ~~this regard~~, and particular ~~choice is~~ often associated with its progenitor or practitioner by name.		When the equation of burning is added to the interior ballistic system of equation, which now consists of the equation of motion of the projectile, and finally the balance of energy equation, the resulting system is not necessarily analytical (i.e. admits a closed form solution), and certainly not simple. Then, the choice of burn rate law and that of form equation must consider both the accuracy of representation, with regard to the practicality of solving these systems.

			Consequently, a great deal of freedom of choice existed in the exact formulation of the equations, and particular solutions are often associated with its progenitor or practitioner by name.

	(WIP)		(WIP)

	====Burning Described by Quadratic Form Function====		====Burning Described by Quadratic Form Function====
	The quadratic, or the <b>Charbonnier form</b> of the equation of burning was the preferred formulation in the mid ~~to latter part of the~~ last century. Its primary strength lies in its success in capturing the theoretical behaviour of ~~many simpler shapes~~ through the tuning of one variable, the form factor <b>\theta</b>. It takes the form:		The quadratic, or the <b>Charbonnier form</b> of the equation of burning was the preferred formulation up to the mid last century, and remains especially popular in the Western world. Its primary strength lies in its success in capturing the theoretical behaviour of relevant geometries to high-powered interior ballistic work, while keeping the complexity of the resulting solution low, through the tuning of one variable, the form factor <b>\theta</b>. It takes the form:

	:::<math> e \frac{dZ}{dt} = B(P)</math>, and		:::<math> e \frac{dZ}{dt} = B(P)</math>, and
	:::<math> \psi = (1-Z)(1+\theta Z)</math>, i.e., <math> \psi = -\theta Z^2 + (\theta-1)Z + 1</math>		:::<math> \psi = (1-Z)(1+\theta Z)</math>, i.e., <math> \psi = -\theta Z^2 + (\theta-1)Z + 1</math>

	Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> is ~~conveniently~~		Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> conveniently takes 0 for a "neutral" geometry, where the area of combustion does not depend on the extent of burning, and takes 1 for a propellant in "degressive" geometry, that is the area exposed to burning is reduced during burning.

			The definition of <math>\theta </math> is particular suited to describing propellant of the shapes usually denoted "ribbon" and "flake". Then, by simple geometry, <math>\theta</math> is the ratio of web thickness to the length of the ribbon. For flake, <math>\theta</math> is then twice of the same ratio. A special case is that for the tube, where <math>\theta = 0</math>

			Since by definition, <math>\psi(1) = 0</math> and psi(0) = 1<math> </math>, <math>\theta </math> must take between -1 and 1. Geometries where the regressiveness of burning is higher than allowed in this formulation, that is, when burning can proceeds from all 3 dimensions, with particular examples being the spherical and cube shapes. These simply cannot be represented in the Charbonnier form.

			This, although on first sight a serious limitation of this approach, is in practice mitigated by the relative ease of conducting experiments in the few cases where more regressive propellant are used e.g. in small arms and infantry mortars, out of practical concerns for manufacture.


	[[Category:Warfare]]		[[Category:Warfare]]

Phoenix: /* Burning Described by Quadratic Form Function */

2023-08-10T08:48:31Z

Burning Described by Quadratic Form Function

← Older revision		Revision as of 01:48, 10 August 2023
Line 270:		Line 270:

	:::<math> e \frac{dZ}{dt} = B(P)</math>, and		:::<math> e \frac{dZ}{dt} = B(P)</math>, and
	:::<math> \psi = (1-f)(1+\theta f)</math>		:::<math> \psi = (1-Z)(1+\theta Z)</math>, i.e., <math> \psi = -\theta Z^2 + (\theta-1)Z + 1</math>

	(~~WIP~~)		Where <math>B(P)</math> is some burn rate function of pressure. The "form factor", <math>\theta</math> is conveniently
	[[Category:Warfare]]		[[Category:Warfare]]

Phoenix: /* Gun with Finite Rate of Burning */

2023-08-10T08:36:08Z

Gun with Finite Rate of Burning

← Older revision		Revision as of 01:36, 10 August 2023
Line 260:		Line 260:
	It bears mentioning that we take the various ballistic approximation as applicable in this section, although notably propellant combustion of a finite rate violates the Pidduck-Kent (and by extension, Mamontov's) argument. A valid argument can be made that in practical guns, the oscillation of the wave system is not damped out in a negligible time frame. In fact, as stated by Hunt (1951), the rarefaction wave predicted by these above works were never observed in practice. Then, the applicability of these assumptions, and consequently the prediction based on these, is always an approximation, and should be taken with an acute awareness of these limitations. In effect, this reduces the faith that can be placed on these sophisticated approximation to no more than what can be given to the simpler Lagrangian approximation, i.e. they are only as valid as they are a close approximation of reality, and this approximation is held to be good for the solution space where <math>\frac{\omega}{\varphi_1 m} < 1 </math>.		It bears mentioning that we take the various ballistic approximation as applicable in this section, although notably propellant combustion of a finite rate violates the Pidduck-Kent (and by extension, Mamontov's) argument. A valid argument can be made that in practical guns, the oscillation of the wave system is not damped out in a negligible time frame. In fact, as stated by Hunt (1951), the rarefaction wave predicted by these above works were never observed in practice. Then, the applicability of these assumptions, and consequently the prediction based on these, is always an approximation, and should be taken with an acute awareness of these limitations. In effect, this reduces the faith that can be placed on these sophisticated approximation to no more than what can be given to the simpler Lagrangian approximation, i.e. they are only as valid as they are a close approximation of reality, and this approximation is held to be good for the solution space where <math>\frac{\omega}{\varphi_1 m} < 1 </math>.

	Theory that models the finite rate burning of propellant requires some way of relating the bulk burn rate (as explained in the propellant section) to the evolution of variables such as free volume (or volume not occupied by the charge), and energy added to the gun system. The former, in the interest of preserving the scalability of theory, is best described by taking the derivative with regard to time, for the product of linear burnt extent <math>Z</math> and a characteristic length for the grain <math>e</math>. For the latter, it is convenient to describe the fraction of propellant burnt using <math>\~~varPsi~~ </math>. The two may then be related through what is known as the form function, so called since its purely a function of the grain geometry. This, together with a choice of bulk burn rate law for the propellant, is collectively referred to as a equation of burning.		Theory that models the finite rate burning of propellant requires some way of relating the bulk burn rate (as explained in the propellant section) to the evolution of variables such as free volume (or volume not occupied by the charge), and energy added to the gun system. The former, in the interest of preserving the scalability of theory, is best described by taking the derivative with regard to time, for the product of linear burnt extent <math>Z</math> and a characteristic length for the grain <math>e</math>. For the latter, it is convenient to describe the fraction of propellant burnt using <math>\psi </math>. The two may then be related through what is known as the form function, so called since its purely a function of the grain geometry. This, together with a choice of bulk burn rate law for the propellant, is collectively referred to as a equation of burning.

	When the equation of burning is added to the interior ballistic system of equation, which now consists of the equation of motion of the projectile, and finally the balance of energy equation, the resulting system is not necessarily analytical (i.e. admits a closed form solution), and certainly not simple. Then, the choice of burn rate law and that of form equation must consider both the accuracy of representation, with regard to the practicality of solving these systems. Historically, a great many freedom of choice existed in this regard, and particular choice is often associated with its progenitor or practitioner by name.		When the equation of burning is added to the interior ballistic system of equation, which now consists of the equation of motion of the projectile, and finally the balance of energy equation, the resulting system is not necessarily analytical (i.e. admits a closed form solution), and certainly not simple. Then, the choice of burn rate law and that of form equation must consider both the accuracy of representation, with regard to the practicality of solving these systems. Historically, a great many freedom of choice existed in this regard, and particular choice is often associated with its progenitor or practitioner by name.
Line 269:		Line 269:
	The quadratic, or the <b>Charbonnier form</b> of the equation of burning was the preferred formulation in the mid to latter part of the last century. Its primary strength lies in its success in capturing the theoretical behaviour of many simpler shapes through the tuning of one variable, the form factor <b>\theta</b>. It takes the form:		The quadratic, or the <b>Charbonnier form</b> of the equation of burning was the preferred formulation in the mid to latter part of the last century. Its primary strength lies in its success in capturing the theoretical behaviour of many simpler shapes through the tuning of one variable, the form factor <b>\theta</b>. It takes the form:

	:::<math> e \frac{dZ}{dt} = B(P)</math>		:::<math> e \frac{dZ}{dt} = B(P)</math>, and
	:::<math> \~~varPsi~~ = (1-f)(1+\theta f)</math>		:::<math> \psi = (1-f)(1+\theta f)</math>

	(WIP)		(WIP)
	[[Category:Warfare]]		[[Category:Warfare]]