text: # Solving complex equations
  
  • Mar 26th 2011, 01:53 PM
  fin
  Solving complex equations
  I am trying to solve two complex equations, but as I've never really dealt with complex numbers and equations before, I've run into trouble.
  
  The equations are as follows:
  
  $T_{L} = i \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \cdot e^{i\phi} \right]$
  
  $J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ iP\xi_{0} - \xi_{0}^{'}\right]$
  
  For the first one:
  
  $e^{i\phi} = - \sin(\phi) + i\cos(\phi)$
  
  ;thus:
  
  $T_{L} = - \frac{2\pi r \cdot \sin(\phi)}{nd} + \frac{2\pi r \cdot i\cos(\phi)}{nd}$
  
  ;which becomes:
  
  $T_{L} = \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \right]^{2}$
  
  ;and through a wild goose chase the second one simply becomes:
  
  $J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ - (P\xi_{0})^{2} - \xi_{0}^{'}\right]$
  
  I honestly do not know what I am doing, a friend of mine gave me some "hints" but he doesn't really know either, and I wouldn't be surprised if it's all wrong. The solution to the first one makes sense (when you know what it's supposed to do) but I still have a feeling that I'm just making humbug here.
  
  Any help would be very much appreciated!
  • Mar 26th 2011, 02:01 PM
  HallsofIvy
  Well, what, exactly, are you trying to do with them? Are $T_L$ and $J_A$ constants and you want to solve for r and $\phi$?
  • Mar 26th 2011, 02:37 PM
  fin
  $T_{L}$ and $J_{A}$ are ballistic constants, lateral throw-off and aerodynamic jump.
  
  ----------
  
  $T_{L} = i \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \cdot e^{i\phi} \right]$
  
  Lateral throw-off is the dispersion (deviation from the mean path; measured in radians) due to static imbalance in the projectile. Lateral throw-off is always at a right angle to the orientation at which the centre of mass emerges from the muzzle, e.g. if the CM emerges at 12 o'clock, the lateral throw-off is directed towards 3 o'clock for a right-hand spin.
  
  As I understand it, the complex part is to properly model the angle of lateral throw-off AND the angle of orientation, e.g. x rad towards -π/2 rad (3 o'clock). I want to find the absolute value, the angle of dispersion without the second component that tells in what direction (I already know it's at right angles).
  
  In the book it also says: "TL = Lateral throw-off (the tangent of the deflection angle)."
  
  n = twist rate, calibres/turn
  r = radius of gyration, m
  d = projectile diameter, m
  φ = roll orientation angle at emergence from the muzzle, probably deg
  
  The actual yaw angle due to static imbalance is:
  
  $\phi = \frac{\omega r}{v}$
  
  ;so I wasn't surprised to see the familiar form in the "solution" for $T_{L}$.
  
  ----------
  
  $J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ iP\xi_{0} - \xi_{0}^{'}\right]$
  
  $k_{y}^{2} = \frac{I_{y}}{md^{2}}$ (although probably not needed for a solution)
  
  $P = \frac{I_{x}}{I_{y}} \cdot \frac{\omega d}{v}$ (although probably not needed for a solution)
  
  Ix = angular mass, kg·m^2
  Iy = angular mass, kg·m^2
  m = projectile mass, kg
  d = projectile diameter, m
  CLα = lift force constant, k/1
  CMα = overturning moment constant, k/1
  ξ0 = yaw angle at the muzzle, rad
  ξ0' = yaw rate at the muzzle, rad/s
  ω = angular velocity (spin) of projectile, rad/s
  v = velocity relative to wind, m/s
  
  Aerodynamic jump is the dispersion due to initial yaw angle and rate (of tip-off rate) as it emerges from the muzzle. This would also be an angle (rad) that describes the deviation from the mean path.
  
  This is all from the book "Modern Exterior Ballistics: The Launch And Flight Of Symmetric Projectiles" by Robert L. McCoy. Because it's a proper thesis on this particular subject, most of the equations are full models that describe the physics in full mathematical form, even though it could be described in simpler forms.
  
  It is those simpler forms that I am looking for, e.g. the drag force is explained in full vector form with a few other things thrown in, but it is equal to F=kpA if you use a scalar form and make a few things implicit, etc.
explanation: The document is a forum post where the user "fin" is seeking help with solving two complex equations related to ballistic constants, specifically lateral throw-off ($T_{L}$) and aerodynamic jump ($J_{A}$). The user provides the equations and some context about what these constants represent, along with some additional information and their attempts at solving the equations. The post includes a question from another user, "HallsofIvy," asking for clarification on the goal of solving these equations.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
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distilabel_metadata.role: system
distilabel_metadata.content: # Solving complex equations
  
  • Mar 26th 2011, 01:53 PM
  fin
  Solving complex equations
  I am trying to solve two complex equations, but as I've never really dealt with complex numbers and equations before, I've run into trouble.
  
  The equations are as follows:
  
  $T_{L} = i \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \cdot e^{i\phi} \right]$
  
  $J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ iP\xi_{0} - \xi_{0}^{'}\right]$
  
  For the first one:
  
  $e^{i\phi} = - \sin(\phi) + i\cos(\phi)$
  
  ;thus:
  
  $T_{L} = - \frac{2\pi r \cdot \sin(\phi)}{nd} + \frac{2\pi r \cdot i\cos(\phi)}{nd}$
  
  ;which becomes:
  
  $T_{L} = \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \right]^{2}$
  
  ;and through a wild goose chase the second one simply becomes:
  
  $J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ - (P\xi_{0})^{2} - \xi_{0}^{'}\right]$
  
  I honestly do not know what I am doing, a friend of mine gave me some "hints" but he doesn't really know either, and I wouldn't be surprised if it's all wrong. The solution to the first one makes sense (when you know what it's supposed to do) but I still have a feeling that I'm just making humbug here.
  
  Any help would be very much appreciated!
  • Mar 26th 2011, 02:01 PM
  HallsofIvy
  Well, what, exactly, are you trying to do with them? Are $T_L$ and $J_A$ constants and you want to solve for r and $\phi$?
  • Mar 26th 2011, 02:37 PM
  fin
  $T_{L}$ and $J_{A}$ are ballistic constants, lateral throw-off and aerodynamic jump.
  
  ----------
  
  $T_{L} = i \left[ \frac{2\pi}{n} \cdot \frac{r}{d} \cdot e^{i\phi} \right]$
  
  Lateral throw-off is the dispersion (deviation from the mean path; measured in radians) due to static imbalance in the projectile. Lateral throw-off is always at a right angle to the orientation at which the centre of mass emerges from the muzzle, e.g. if the CM emerges at 12 o'clock, the lateral throw-off is directed towards 3 o'clock for a right-hand spin.
  
  As I understand it, the complex part is to properly model the angle of lateral throw-off AND the angle of orientation, e.g. x rad towards -π/2 rad (3 o'clock). I want to find the absolute value, the angle of dispersion without the second component that tells in what direction (I already know it's at right angles).
  
  In the book it also says: "TL = Lateral throw-off (the tangent of the deflection angle)."
  
  n = twist rate, calibres/turn
  r = radius of gyration, m
  d = projectile diameter, m
  φ = roll orientation angle at emergence from the muzzle, probably deg
  
  The actual yaw angle due to static imbalance is:
  
  $\phi = \frac{\omega r}{v}$
  
  ;so I wasn't surprised to see the familiar form in the "solution" for $T_{L}$.
  
  ----------
  
  $J_{A} = k_{y}^{2} \cdot \frac{C_{L_{\alpha}}}{C_{M_{\alpha}}} \left[ iP\xi_{0} - \xi_{0}^{'}\right]$
  
  $k_{y}^{2} = \frac{I_{y}}{md^{2}}$ (although probably not needed for a solution)
  
  $P = \frac{I_{x}}{I_{y}} \cdot \frac{\omega d}{v}$ (although probably not needed for a solution)
  
  Ix = angular mass, kg·m^2
  Iy = angular mass, kg·m^2
  m = projectile mass, kg
  d = projectile diameter, m
  CLα = lift force constant, k/1
  CMα = overturning moment constant, k/1
  ξ0 = yaw angle at the muzzle, rad
  ξ0' = yaw rate at the muzzle, rad/s
  ω = angular velocity (spin) of projectile, rad/s
  v = velocity relative to wind, m/s
  
  Aerodynamic jump is the dispersion due to initial yaw angle and rate (of tip-off rate) as it emerges from the muzzle. This would also be an angle (rad) that describes the deviation from the mean path.
  
  This is all from the book "Modern Exterior Ballistics: The Launch And Flight Of Symmetric Projectiles" by Robert L. McCoy. Because it's a proper thesis on this particular subject, most of the equations are full models that describe the physics in full mathematical form, even though it could be described in simpler forms.
  
  It is those simpler forms that I am looking for, e.g. the drag force is explained in full vector form with a few other things thrown in, but it is equal to F=kpA if you use a scalar form and make a few things implicit, etc.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a forum post where the user "fin" is seeking help with solving two complex equations related to ballistic constants, specifically lateral throw-off ($T_{L}$) and aerodynamic jump ($J_{A}$). The user provides the equations and some context about what these constants represent, along with some additional information and their attempts at solving the equations. The post includes a question from another user, "HallsofIvy," asking for clarification on the goal of solving these equations.
  
  Label: question-and-answer