# The Effect of Manufacturing Quality on Rocket Precision

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{L}, inertia tensor

**I**

_{L}and center of gravity (CG) position ${\rho}_{L}={\left[{x}_{L}\text{}{y}_{L}\text{}{z}_{L}\right]}^{T}$ are obtained from a CAD program and depend on the simulated manufacturing errors for the particular rocket.

#### 2.1. Manufacturing Errors Simulation

_{G}axis coincides with the longitudinal axis of symmetry, and transversal axes y

_{G}and z

_{G}are fixed to rocket fins. The origin is placed at the center of gravity. The attitude of the G-CS is determined relative to the vehicle-carried coordinate system (O-CS, i.e., north-east-down c.s.) by Euler angles: ϕ

_{G}, θ

_{G}, and ψ

_{G}[11]. The transformation matrix from O-CS to G-CS is

**L**

_{GO}=

**L**

_{x}(ϕ

_{G})

**L**

_{y}(θ

_{G})

**L**

_{z}(ψ

_{G}). Manufacturing errors are expressed in regard to G-CS.

#### 2.2. Warhead-to-Engine Chamber Misalignment

_{HC}lies in the plane of disturbance x

_{G}− x

_{H}, while the angle φ

_{HC}defines the rotation of that plane relative to the reference plane x

_{G}− y

_{G}. Both angles δ

_{HC}and φ

_{HC}occur as stochastic parameters in the rocket 3D CAD model.

- δ
_{HC}—the angle of misalignment in the plane of disturbance, dispersed according to the normal distribution (parameters of dispersion defined by the manufacturing quality level); - φ
_{HC}—the radial angle, defining the radial position of angle δ_{HC}, ${0}^{\mathrm{o}}\le {\phi}_{HC}\le {360}^{\mathrm{o}}$ dispersed according to the uniform distribution.

#### 2.3. Warhead and Propellant Errors

_{H}and φ

_{H}for the warhead error, and by δ

_{P}and φ

_{P}for the propellant error. Both δ

_{H}and δ

_{P}angles are dispersed according to the normal distribution, but not necessarily with the same standard deviations, as will be explained later in the case study analysis. The radial angles φ

_{H}and φ

_{P}are dispersed according to the uniform distribution, in a full circle 0–360°.

_{HC}angle, both the warhead and the explosive charge rotate around the warhead-chamber joint. Additionally, if there is a δ

_{H}angle, the explosive charge rotates inside the warhead around the same point. As a result, there is a rotation of both parts relative to the rocket CG (CR in Figure 2) and their CGs are no longer on the x

_{G}axis. This was introduced into the 6DOF model via the modified inertia tensor (obtained from the 3D CAD model) and altered moment of aerodynamic force.

#### 2.4. Thrust Force Misalignment

_{G}axis of symmetry. Here the combustion is assumed to be perfect, so the thrust force misalignment occurs only due to the nozzle manufacturing error.

_{N}. The plane x

_{G}− x

_{N}is called the nozzle disturbance plane. The origin of N-CS is located at the nozzle center with a radial position ${\rho}_{N}^{G}={\left[{l}_{N}00\right]}^{T}$. In analogy with the warhead-to-chamber error, the nozzle error is defined over two angles: δ

_{N}and φ

_{N}, as in Figure 4. The vector of the thrust force ${\overrightarrow{F}}_{T}$ coincides with the x

_{N}axis and is deflected from the x

_{G}by the angle δ

_{N}.

_{T}is the thrust force intensity, variable during the propellant combustion.

## 3. Monte-Carlo Simulation

#### 3.1. Statistical Process Control, Quality, and Process Capability Indices

#### 3.2. Simulating Manufacturing Errors

_{HC}dispersion is then: ${\sigma}_{HC}=0.66/6\xb70.8335=0.132$, since Cp = 0.8335 is chosen as the average for the low-quality manufacturing level. Finally, if µ = 0 and σ

_{HC}= 0.132, the PDF for δ

_{HC}becomes:

_{HC}= 0.094 for the standard quality and σ

_{HC}= 0.060 for the high quality level. PDFs for δ

_{HC}are presented in Figure 6.

_{H}, δ

_{P}, and δ

_{N}angles, giving the probability density function for each error and each of three quality level.

## 4. Flight Model Adapted for Imperfect Projectile

#### 4.1. Governing Equations

- Extension of Newton’s laws of motion for a body with variable mass. Includes the aerodynamic force ${F}_{A}^{G}$, thrust force ${F}_{T}^{G}$, gravity $m{g}^{O}$ and Coriolis force $m{a}_{c}^{G}$. ${\dot{V}}_{K}^{G}$ is a derivation of flight velocity, in G-CS:$$m\left({\dot{V}}_{K}^{G}+{\tilde{\mathsf{\Omega}}}_{G}^{G}\xb7{V}_{K}^{G}\right)={F}_{A}^{G}+{F}_{T}^{G}+{L}_{GO}m{g}^{O}-m{a}_{c}^{G}$$
- Derivation of angular momentum ${\dot{H}}^{G}$. Includes moment of aerodynamic forces ${M}_{A}^{G}$ and moment of thrust force ${M}_{T}^{G}$, all written in G-CS:$${\dot{H}}^{G}+{\tilde{\mathsf{\Omega}}}_{G}^{G}\xb7{H}^{G}={M}_{A}^{G}+{M}_{T}^{G}$$
- Matrix form of kinematic equation connecting attitude ${s}^{G}={\left[{\varphi}_{G}{\theta}_{G}{\psi}_{G}\right]}^{T}$ and angular velocity of the rocket ${\mathsf{\Omega}}_{G}^{G}={\left[pqr\right]}^{T}$:$$\dot{s}={R}^{-1}\xb7\left({\mathsf{\Omega}}_{G}^{G}-{\mathsf{\Omega}}_{O}^{G}\right)$$
**R**is given as$$=\left[\begin{array}{ccc}1& 0& -sin{\theta}_{G}\\ 0& cos{\varphi}_{G}& sin{\varphi}_{G}cos{\theta}_{G}\\ 0& -sin{\varphi}_{G}& cos{\varphi}_{G}cos{\theta}_{G}\end{array}\right]$$ - Coordinates in the geocentric coordinate system (E-CS, Earth-fixed spherical coordinate system):$$\begin{array}{c}{\dot{\varphi}}_{E}=\frac{{V}_{Kx}^{O}}{R+h}\\ \dot{\lambda}=\frac{{V}_{Ky}^{O}}{\left(R+h\right)cos{\varphi}_{E}}-{\mathsf{\Omega}}_{E}\\ \dot{h}=-{V}_{Kz}^{O}\end{array}$$

_{G}is $\varphi =\mathrm{arctan}\left(v/w\right)$. Derivations of σ and ϕ are:

_{S}is the attitude of the airflow around the longitudinal axis.

#### 4.2. Inertia Model

_{G}-axis.

_{L}and its C.G. position ${\overrightarrow{\rho}}_{L}$ are known.

**L**

_{GP}must be used:

#### 4.3. Aerodynamics of an Asymmetrical Rocket

_{G}-z

_{G}plane):

_{H}in the plane of disturbance, and the moment M

_{H}, as shown in Figure 7.

_{H}is x

_{Hn}, and thus its moment in the projectile center of gravity is ${M}_{H}={N}_{H}\left({x}_{R}-{x}_{Hn}\right)$ where x

_{R}is the distance from the apex of the rocket to its center of gravity. The aerodynamic gradient ${C}_{N\delta}$ and the position of the force application x

_{Hn}are determined according to [23]. Since both N

_{H}and M

_{H}are tied to the warhead geometry, they should be transformed into the G-CS:

_{H}, the warhead-to-chamber manufacturing error causes the additional induced drag. The aerodynamic derivative is C

_{x}

_{δ}and it is dependent on the total angle of attack σ, as well as on the angle of misalignment δ

_{HC}, as presented in Figure 8.

_{HC}angle: $\mathsf{\Delta}\alpha ={\delta}_{HC}\xb7sin{\phi}_{HC}$, $\mathsf{\Delta}\beta ={\delta}_{HC}\xb7cos{\phi}_{HC}$. The aerodynamic coefficient of the total drag, incorporating the influence of the misalignment angle δ

_{HC}, would be usually defined as in Equation (18) but now with the application of σ

_{Hn}: ${C}_{x}\left(Ma\right)={C}_{x0}\left(Ma\right)+{C}_{x{\sigma}^{2}}\left(Ma\right){\sigma}_{Hn}{}^{2}$.

## 5. Results and Discussion

- length: L
_{M}= 2.87 m; - initial total mass: m
_{M}_{0}= 66.6 kg; - initial mass of propellant: m
_{P}_{0}= 20.5 kg; - mass of explosive and mass of the entire warhead: m
_{E}= 6.4 kg, m_{H}= 18.4 kg; - initial moments of inertia: I
_{x}_{0}= 0.1499 kgm^{2}, I_{y}_{0}= I_{z}_{0}= 41.58 kgm^{2}.

_{y}

_{0}= I

_{z}

_{0}is no longer valid.

_{x}

_{0}(Ma) from Equation (20) is determined according to the etalon C

_{D}

_{58}(Ma) and adjusted so that the ideal rocket has the same maximum range as stated in the Firing Tables (in the NATO standard atmosphere). For the coefficient of induced drag, it is assumed C

_{x}

_{σ}= C

_{z}

_{α}.

- error angles δ
_{H}, δ_{HC}, δ_{P}and δ_{N}dispersed according to the normal distribution, with standard deviations depending on the production quality as given in Table 2; - ${0}^{o}\le {\phi}_{i}\le {360}^{o}$ for all radial angles, dispersed according to the uniform distribution.

_{N}= 0.151°, δ

_{H}= 0.056°, δ

_{HC}= 0.173°, δ

_{P}= 0.091°). As presented in Figure 11, the angle of attack α and the angle of sideslip β for the non-ideal rocket (Figure 11b) are larger in comparison to the results for the ideal rocket (Figure 11a). The stability of the non-ideal rocket is not endangered, not even if produced with considerable production errors as the one analyzed here. In Figure 11c, the angle of attack for both projectiles is directly compared for the first five seconds, since later the angle of attack diminishes, and differences are difficult to be graphically presented.

- Descriptive location parameters (mean, mode and median) coincide.
- The absolute values of skewness and kurtosis for the analyzed variables are in the range of 0–0.1 and 0–0.5, respectively.
- In addition, the assumption of normality was tested using the normal—probability plots and Shapiro Wilk test.

_{N}= 0). An appropriate Monte Carlo simulation is prepared with 1000 iterations for each of three cases. Results show that in such a case, the dispersion area multifold decreases, regardless of whether other rocket parts are manufactured at a standard (Case No. 8) or even low quality (Case No. 7). If all parts are manufactured at high quality and δ

_{N}= 0 (Case No. 9), a dispersion area matches the one of a guided rocket [26].

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Monte-Carlo Simulation scheme [16].

**Figure 11.**Comparison of the pitching and yawing motion for ideally (

**a**), and non-ideally manufactured projectile (

**b**), and the comparison of angle of attack for both projectiles (

**c**).

Cp | Z-Value | DPMO | Quality Level |
---|---|---|---|

0.33–0.67 | 2–3 | 45500–2700 | Low Quality |

1.00–1.33 | 3–4 | 2700–63.4 | Standard Quality |

1.66–2.00 | 5–6 | 0.57–0.002 | High Quality |

$\frac{\mathbf{USL}\left(\mathbf{deg}\right)}{\mathbf{LSL}\left(\mathbf{deg}\right)}$ | Low Quality | Standard Quality | High Quality | |
---|---|---|---|---|

Warhead inner and outer surface misalignment | −0.500 0.500 | _{H} = 0.200° | _{H} = 0.142° | _{H} = 0.090° |

Warhead to engine chamber misalignment | −0.330 0.330 | _{HC} = 0.132° | _{HC} = 0.094° | _{HC} = 0.060° |

Propellant positioned incorrectly | −0.330 0.330 | _{P} = 0.132° | _{P} = 0.094° | _{P} = 0.060° |

Nozzle misalignment | −0.1146 0.1146 | _{N} = 0.046° | _{N} = 0.033° | _{N} = 0.021° |

Direction | 95% PI Low | 95% PI High | |
---|---|---|---|

Low Quality (LQ) (Case No. 1) | Range, m | 13,458.0 | 13,944.7 |

Drift, m | −141.6 | 140.8 | |

Area, m^{2} | 107,948.3 | ||

Standard Quality (SQ) (Case No. 2) | Range, m | 13,521.7 | 13,868.6 |

Drift, m | −96.9 | 98.6 | |

Area, m^{2} | 53,264.9 | ||

High Quality (HQ) (Case No. 3) | Range, m | 13,590.3 | 13,806.8 |

Drift, m | −67.8 | 65.4 | |

Area, m^{2} | 22,649.2 |

Std. Deviation Range (m) | Std. Deviation Drift (m) | 95% PI Low 95% PI High Range (m) | 95% PI Low 95% PI High Drift (m) | Dispersion Area (m^{2}) | |
---|---|---|---|---|---|

Case No. 1: All LQ | 13,458.0 13,944.7 | −141.6 140.8 | 107,948.3 | ||

Case No. 2: All SQ | 13,521.7 13,868.6 | −96.9 98.6 | 53,264.9 | ||

Case No. 3: All HQ | 13,590.3 13,806.8 | −67.8 65.4 | 22,649.2 | ||

Case No. 4: Nozzle LQ, Other Errors SQ | 13,459.9 13,936.3 | −133.2 135.4 | 100,500.4 | ||

Case No. 5: Nozzle LQ, Other Errors HQ | 13,460.9 13,920.8 | −132.9 132.8 | 95,972.1 | ||

Case No. 6: Nozzle LQ, No Other Errors | 13,469.5 13,926.8 | −130.3 129.8 | 93,418.2 | ||

Case No. 7: No Nozzle Error, Other Errors LQ | 13,611.5 13,787.7 | −52.3 53.9 | 14,696.7 | ||

Case No. 8: No Nozzle Error, Other Errors SQ | 13,637.5 13,759.7 | −37.8 38.9 | 7361.4 | ||

Case No. 9: No Nozzle Error, Other Errors HQ | 13,660.4 13,741.9 | −23.9 22.8 | 2989.3 |

Factor | Coefficient Estimate | df | Standard Error | 95% CI Low | 95% CI High | VIF |
---|---|---|---|---|---|---|

Intercept | 14,897.06 | 1 | 3.32 | 14,888.52 | 14,905.61 | |

A-Warhead Error δ_{H} | −1.69 | 1 | 3.32 | −10.23 | 6.86 | 1.00 |

B-Head-to-Chamber Error δ_{HC} | 21.94 | 1 | 3.32 | 13.39 | 30.48 | 1.00 |

C-Propellant Error δ_{P} | 21.31 | 1 | 3.32 | 12.77 | 29.86 | 1.00 |

D-Nozzle Error δ_{N} | −144.94 | 1 | 3.32 | −153.48 | −136.39 | 1.00 |

F-Head-to-Chamber Error, Radial Angle φ_{HC} | −13.81 | 1 | 3.32 | −22.36 | −5.27 | 1.00 |

G-Propellant Error, Radial Angle φ_{HC} | −26.44 | 1 | 3.32 | −34.98 | −17.89 | 1.00 |

H-Nozzle Error, Radial Angle φ_{N} | 170.69 | 1 | 3.32 | 162.14 | 179.23 | 1.00 |

Factor | Coefficient Estimate | df | Standard Error | 95% CI Low | 95% CI High | VIF |
---|---|---|---|---|---|---|

Intercept | 43.09 | 1 | 0.39 | 42.09 | 44.10 | |

A-Warhead Error δ_{H} | −2.92 | 1 | 0.39 | −3.92 | −1.92 | 1.00 |

B-Head-to-Chamber Error δ_{HC} | 13.23 | 1 | 0.39 | 12.23 | 14.23 | 1.00 |

C-Propellant Error δ_{P} | −19.52 | 1 | 0.39 | −20.52 | −18.52 | 1.00 |

D-Nozzle Error δ_{N} | 131.12 | 1 | 0.39 | 130.12 | 132.12 | 1.00 |

F-Head-to-Chamber Error, Radial Angle φ_{HC} | 16.27 | 1 | 0.39 | 15.27 | 17.27 | 1.00 |

G-Propellant Error, Radial Angle φ_{HC} | −15.81 | 1 | 0.39 | −16.81 | −14.80 | 1.00 |

H-Nozzle Error, Radial Angle φ_{N} | 103.42 | 1 | 0.39 | 102.42 | 104.42 | 1.00 |

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**MDPI and ACS Style**

Trzun, Z.; Vrdoljak, M.; Cajner, H.
The Effect of Manufacturing Quality on Rocket Precision. *Aerospace* **2021**, *8*, 160.
https://doi.org/10.3390/aerospace8060160

**AMA Style**

Trzun Z, Vrdoljak M, Cajner H.
The Effect of Manufacturing Quality on Rocket Precision. *Aerospace*. 2021; 8(6):160.
https://doi.org/10.3390/aerospace8060160

**Chicago/Turabian Style**

Trzun, Zvonko, Milan Vrdoljak, and Hrvoje Cajner.
2021. "The Effect of Manufacturing Quality on Rocket Precision" *Aerospace* 8, no. 6: 160.
https://doi.org/10.3390/aerospace8060160