Projectile Motion Research Articles

Analytical Study on the Trajectory of a Projectile Motion with the Quadratic Resistant Law

In this paper, the trajectory of a projectile in a resistant medium with quadratic resistant law is investigated. Based on the elementary analysis method, the equation for the trajectory in Cartesian coordinates system is derived. By retaining the first order derivative of the slope of the trajectory, the trajectory equation is integrated by two different schemes, the Riccati eqation expansion method and the exp-function method. The solutions obtained in this paper present new viewpoint for future investigations on the traditional mechanical problem.

Numerical Simulation on the Muzzle Shock Wave Pressure for Aircraft Gun

The muzzle shock wave has a great impact on the airborne structure, airborne personnel and airborne equipment. Therefore, the muzzle flow field of the aircraft gun is used as the research object, based on the three-dimensional reynolds-averaged Navier-Stokes control equation and the standard k − ε turbulence model, a finite element simulation model of the powder gas motion process in the after-effect period without considering the projectile motion is established, the adaptive grid technique is used to divide the mesh inside the barrel and the muzzle brakes, and the The finite volume method is applied to numerically calculate the flow field control equations at the muzzle to obtain the muzzle shock wave pressure with the muzzle brakes and its development law, and to build an experimental verification platform to verify the accuracy of the simulated shock wave pressure. The research results show that the established finite element simulation model of the muzzle flow field can accurately predict the muzzle shock wave pressure.

Projectile motion with quadratic drag

Two-dimensional coupled nonlinear equations of projectile motion with air resistance in the form of quadratic drag are often treated as inseparable and solvable only numerically. However, when they are recast in terms of the angle between the projectile velocity and the horizontal, they become completely uncoupled and possess analytic solutions for projectile velocities as a function of that angle. The equations relating the time and position coordinates to this angle are not integrable in terms of elementary functions but are easy to integrate numerically. Additionally, energy equations explicitly including dissipation terms can be developed as integrals of the equations of motion. One-dimensional numerical integrations can be treated in a pedagogically straightforward way using numerical analysis software or even within a spreadsheet, making this topic accessible to undergraduates. We present this approach with sample numerical results for velocity components, trajectories, and energy-balance of a baseball-sized projectile.

Research on spin reduction mechanism of the projectile sliding belt in bore

The sliding belt assembly plays a key role in spin reduction to protect the electronic components and smoothly control the projectile posture. The complete guided projectile motion in the bore was reproduced by LS-DYNA software, where the gunpowder pressure was loaded on the projectile bottom. The dynamic parameters of the projectile obtained by simulation agree well with the test, including the axial velocity, acceleration and rotation speed. Great spin reduction occurs in the bore by the sliding belt assembly, which is indicated by the rotation speed and the motion of the sliding belt and projectile, respectively. From the interaction of the belt assembly at typical times, it is the escaping from the bases periodically of the belt ring so that greatly spin reduction of the projectile body is achieved.

Analysis Model for Damage of PBO Composite Laminate Target by High-Velocity Projectile Penetration

Poly-p-phenylene benzobisoxazole (PBO) fiber is a new fiber with high ballistic performance, such as excellent heat and flame resistance and super mechanical properties. Consequently, increasing attention has been paid to its applications in the field of ballistic resistance. To identify the damage mechanism of PBO composite target caused by the high-velocity penetration of cylindrical projectiles and the energy conversion mode involved in the penetration process, this paper constructs a target damage engineering analysis model using the energy method in combination with stress wave propagation and projectile motion analysis. The residual velocity of the steel cylindrical projectile after perforating a PBO composite laminate is measured through ballistic tests, and it is compared with the value predicted by the analysis model. Comparison results show that the predicted value by the model is in good agreement with the measurement, with an error of less than 2.6%, thereby proving the validity of the constructed analysis model.

Guidance and Control design for CubeSat small body landing using discrete firings of sliding mode control

Small body exploration has seen a renewed interest in recent years. Landing a spacecraft on the body’s surface enhances the mission’s scientific return, yet it is a high-risk operation. CubeSats offer a cheaper alternative to bigger spacecraft, and when used in tandem with a mothercraft, they may perform risky operations without exposing the mission to a complete loss. Nonetheless, their small size as well as their limited operational and computational capabilities pose hard limitations to their efficient exploitation. Historically, powered and controlled landing has often been key to soft landing success. In an asteroid complex dynamical environment, manoeuvre execution benefits from a dedicated Guidance and Control (G&C) scheme that is robust to dynamical uncertainty, autonomous and reliable. To robustly and reliably soft-land a CubeSat on a small body, we propose a novel Guidance & Control strategy. It alternates closed-loop control segments using the existing Multiple Surface Sliding Guidance algorithm with intervals of ballistic trajectory. We show that very good landing performance in terms of landing ellipse, collision velocity and fuel consumption can be achieved. The method has been tested for Hera’s Milani CubeSat landing attempt on Dimorphos, the moon of (65803) Didymos binary asteroid, using Monte Carlo analysis. The Guidance & Control strategy is shown to comply with stringent requirements for targeted soft landing.

Effect of Wave Phases and Heights on Supercavitation Flow Field and Dynamic Characteristics of Successively Fired High-Speed Projectiles

The study of the water entry of successively fired projectiles under a wave environment is of great significance for the development and application of supercavitation weapons. In this paper, the supercavitating flow field of two successively fired projectiles entering water under different wave conditions is numerically simulated by the volume of the fraction model considering the cavitation of water. The motion of projectiles is handled by the overlapping grid technology and the simulated projectiles have six degrees of freedom. The effects of different wave phases and wave heights on the supercavitating flow field and the dynamic loads of the projectiles are studied. The research results show that the wave phase has an effect on the evolution and size of the supercavitation and the effect of the wave phase on the water splash above the free surface is more obvious. The peak of the drag force of the first projectile under conditions of different wave phases with 0.12 m wave height can be reduced by about 50% compared with that under the no-wave condition. The wave phases have an effect on the peak of the drag coefficient, and for the first projectile the peak under the condition of the 180° phase is about 40% lower than that of the 0° phase. The peak of the drag coefficient of the first projectile decreases with the increase in wave height. When the wave height increases from 0.0 m to 0.05 m, the peak value decreases by about 45%. For all conditions, regardless of wave phases or wave heights, the peak of the drag coefficient of the second projectile is obviously much lower than that of the first projectile. Accordingly, the decrease in the velocity of the second projectile is far slower than that of the first one. Negative values of the drag coefficient on the second projectile are observed when the second projectile enters the cavity of the first one.

High-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer based on probability density evolution method

This paper proposed an efficient research method for high-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer. Firstly, the dynamic model of projectile motion is established considering the flexible deformation of the barrel and the interaction between the projectile and the barrel. Subsequently, the accuracy of the dynamic model is verified based on the external ballistic projectile attitude test platform. Furthermore, the probability density evolution method (PDEM) is developed to high-dimensional uncertainty quantification of projectile motion. The engineering example highlights the results of the proposed method are consistent with the results obtained by the Monte Carlo Simulation (MCS). Finally, the influence of parameter uncertainty on the projectile disturbance at muzzle under different working conditions is analyzed. The results show that the disturbance of the pitch angular, pitch angular velocity and pitch angular of velocity decreases with the increase of launching angle, and the random parameter ranges of both the projectile and coupling model have similar influence on the disturbance of projectile angular motion at muzzle.

The Influence of Magnus Effect on The Flying Distance of Baseball and Golf

The Magnus effect is also very common in ball sports, so the study of the causes and applications of the Magnus effect in ball sports is of great significance and practical importance to the teaching level, training effect, and competition performance of ball sports. The effect of the Magnus effect on baseball and golf will therefore be discussed. In the case of baseball, we will consider the effects of drag and the Magnus effect on its motion and establish a three-plane system consisting of gravity, drag, and the Magnus force, thus demonstrating the interaction of the three forces in the trajectory of the baseball. The concept of projectile motion is introduced in mathematics, physics, and engineering disciplines in a vacuum environment, and when applied to real-life situations, the theory differs significantly from the ideal state. This is why baseball's trajectory in a non-ideal situation is studied. In the case of golf balls, the results are subject to large errors because only the air resistance is taken into account in the discussion of the golf ball range problem, and the effect of the Magnus force on its range is ignored. According to fluid dynamics, a golf ball moving through the air is subjected to bypassing (air) resistance and Magnus force. The article will therefore also look at the effect of the bending (air) resistance and the Magnus force on its movement.

Interactive STEM educational resource regarding to the projectile motion phenomenon on inclined plane

In this manuscript it is proposed a simple and low-cost computer-based simulation regarding to the concept of projectile motion (including also a game mode) on its more general case, that is, on inclined plane; with the aim of easing the learning-teaching process regarding to this physical phenomenon. The novelty of this proposal relies on the use of an external control that changes the projectile shooting angle, the initial velocity and inclination of the plane. This educational resource is coded in Scratch for Arduino (S4A), while its electronic interface is based on Arduino UNO.

Research on the Roundness Approximation Search Algorithm of Si3N4 Ceramic Balls Based on Least Square and EMD Methods

This paper aims to obtain the best shape accuracy evaluation algorithm for silicon nitride ceramic balls after lapping, and to extract the initial signal of the ball surface to improve the accuracy and reliability of the algorithm. The research methods of this paper are as follows: Firstly, an analysis of the uniform envelope of the lapping trajectory of ceramic balls is carried out to verify whether the lapping trajectory after processing can achieve a consistent envelope on the balls' surface. On this basis, it is found through experiments that the standard deviation SD between the roundness deviations of different contour sections is small. The value is maintained at approximately 0.03 μm, and the roundness deviation can approximately replace the spherical deviation. Then the different contour sections of the sphere are sampled by the Taylor roundness instrument. Considering the uncertainty, the sampling points of different contour sections are averaged and used as the original signal of the sphere surface. Then the EMD method is used to process the signal to be detected on the sphere surface. The initial signal of the sphere surface is extracted by judging whether the number of ripples Kc obtained by decomposition is greater than the critical value. Then the initial signal is used as the input value of the approximation algorithm. Through the roundness deviation approximation algorithm based on the least square method, the given minimum approximation domain range is finely processed. The divided fine points are used as the center of the circle to intersect with the initial signal. The maximum, minimum, and range of each circle are calculated to obtain the roundness error based on the minimum circumscribed circle, the maximum inscribed circle, and the minimum region method. Finally, the calculated values are compared with those obtained by the traditional algorithm. The experimental results of this paper show that the algorithm is consistent with the roundness error measured by the instrument, compared with the mainstream evaluation criteria. In summary, the conclusions can be drawn as follows: Through a large number of experimental cases and comparative experiments, the algorithm has high accuracy and reliability. The research results of this paper have essential reference significance for accurately evaluating the shape accuracy of ceramic balls in actual production.

Object motion influences feedforward motor responses during mechanical stopping of virtual projectiles: a preliminary study

An important window into sensorimotor function is how humans interact and stop moving projectiles, such as stopping a door from closing shut or catching a ball. Previous studies have suggested that humans time the initiation and modulate the amplitude of their muscle activity based on the momentum of the approaching object. However, real-world experiments are constrained by laws of mechanics, which cannot be manipulated experimentally to probe the mechanisms of sensorimotor control and learning. An augmented-reality variant of such tasks allows for experimental manipulation of the relationship between motion and force to obtain novel insights into how the nervous system prepares motor responses to interact with moving stimuli. Existing paradigms for studying interactions with moving projectiles use massless objects and are primarily focused on quantifying gaze and hand kinematics. Here, we developed a novel collision paradigm using a robotic manipulandum where participants mechanically stopped a virtual object moving in the horizontal plane. On each block of trials, we varied the virtual object's momentum by increasing either its velocity or mass. Participants stopped the object by applying a force impulse that matched the object momentum. We observed that hand force increased as a function of object momentum linked to changes in virtual mass or velocity, similar to results from studies involving catching free-falling objects. In addition, increasing object velocity resulted in later onset of hand force relative to the impending time-to-contact. These findings show that the present paradigm can be used to determine how humans process projectile motion for hand motor control.

Ground Crash Area Estimation of Quadrotor Aircraft Under Propulsion Failure

Small unmanned aircraft systems or drones are expected to be used for different applications, such as parcel delivery, inspection, and aerial photography, in urban areas. However, drones usually use an electric system to power up the propulsion, communications, navigation, and flight control system, which means that it is not as reliable as the manned aircraft system and may result in failure during operation and then crash to the ground. At present, there is almost no extensive publication about the high-fidelity modeling used by drones to calculate the crash trajectory and point of crash. The experimental data for modeling and simulation verification of multirotor aircraft are limited. So far, crash trajectory prediction has been limited to point mass or ballistic methods, and these methods are usually only suitable for complete power failure and without any control system. This study intends to investigate the effects of different multirotor drones’ failure modes on its crash trajectory and crash area compared to the ballistic model by using ADAMS and MATLAB cosimulation methods. Conclusions from the study show the crash trajectory, flight distance, and impact speed of the drones under four failure modes, which are quite different from the ballistic trajectory. The findings can potentially contribute to better risk assessment of the multirotor drones for the urban environment operation.

Nonlinear Dynamic Characteristics of Deep Groove Ball Bearings with an Improved Contact Model

In this paper, the nonlinear dynamic response of deep groove ball bearings with clearance was studied numerically. The imperfect connections with the clearance of raceways and rolling balls were established by the contact elements. In order to describe the contact characteristics accurately, a hysteresis damping coefficient was introduced into the Hertz contact model, which represented the dissipative term during the contact–impact process. The tangential force of the contact bodies was obtained based on a modified Coulomb friction model. Then, the dynamic analysis model of the deep groove ball bearings with clearance was built. Meanwhile, the experimental test platform of the deep groove ball bearings with various operation conditions was built and the dynamic simulation was utilized as the demonstrative case to conduct the investigation. The numerical results revealed that the existence of clearance would change the motion trajectory of a rolling ball and the appearance of the different movement states (free, contact, and penetration). In addition, the contact characteristics and sliding features would be changed with the variations in the operation conditions and structural characteristics.

Dynamic modeling and analysis of compound pendulum motion of projectile

The influence of air lift on the stability of inaccurate precision strike ammunition is studied, and the mechanical analysis of projectile and its flight stage is carried out. The dynamic equation of projectile under the action of air lift is established and simulated. The angular velocity and angular time-history curve of the compound pendulum motion of the projectile are obtained, and the correctness of the mechanical analysis of compound pendulum motion during flight is verified.

Limit detection sensitivity analysis and maximum detection height prediction for the photoelectric detection target

Photoelectric detection target (PDT) is a kind of non-contact photoelectric detection systems widely used in high-velocity projectile testing for ballistic trajectory measurements. However, invalid measurement data are often acquired during the experiments due to the flying heights of a measured object exceeding its maximum detectable height which is determined by limit detection sensitivity (LDS). As quantitative LDS is difficult to be analyzed and hence the maximum detectable height is usually estimated empirically. Here, we present a model to evaluate the LDS and the maximum detection height in a PDT to ensure the reliability of measurement results. With the model, the maximum detection height of a PDT can be quantitatively predicted for the first time and the LDS can also be analyzed. Experiments are conducted and the predicted maximum detection heights are in good agreement with the experimental results, which verify the effectiveness of the presented model. The proposed method can be used for calculating the LDS and predicting the maximum detection height exactly that ensuring the validity of measurement results.

A semi-empirical formula based on a modified general penetration resistance for predicting the motion of the rigid projectile

The semi-empirical formula is an effective method for predicting the motion of rigid projectiles in practical applications due to the simplicity of its theory and the convenience of parameter calibration. The commonly used semi-empirical formula is Forrestal’s form, combining several specific experimental cases that have been published, we find it exists deficiencies in predicting deceleration histories and the penetration depths of high velocities. To solve this problem, the general penetration resistance is employed to formulate the semi-empirical formula due to the ‘general’ characteristic of the general penetration resistance, and also make an assessment of this semi-empirical through experimental data. The results show that this semi-empirical method, like Forrestal’s form, is not good at predicting high-velocity penetration depth. Thus, it propels us to develop a new semi-empirical formula. To this end, the general penetration resistance is modified with the assumption that the additional mass should be increased with the penetrating velocity and the projectile mass, based on which a new semi-empirical formula is developed. Then, the proposed semi-empirical formula is employed in individual published experimental data of different projectiles and striking velocities as well as different targets. The predictions of the proposed semi-empirical formula show good agreement with the experimental data both in penetration depths and deceleration histories, which also demonstrate the reasonableness of the assumption that the additional mass of rigid projectile increases with penetrating velocity and the projectile mass.

Motor outcomes congruent with intentions may sharpen metacognitive representations

We can monitor our intentional movements and form explicit representations about our movements, allowing us to describe how we move our bodies. But it is unclear which information this metacognitive monitoring relies on. For example, when throwing a ball to hit a target, we might use the visual information about how the ball flew to metacognitively assess our performance. Alternatively, we might disregard the ball trajectory — given that it is not directly relevant to our goal — and metacognitively assess our performance based solely on whether we reached the goal of hitting the target. In two experiments we aimed to distinguish between these two alternatives and asked whether the distal outcome of a goal-directed action (hitting or missing a target) informs the metacognitive representations of our own movements. Participants performed a semi-virtual task where they moved their arm to throw a virtual ball at a target. After each throw, participants discriminated which of two ball trajectories displayed on the screen corresponded to the flight path of their throw and then rated their confidence in this decision. The task included two conditions that differed on whether the distal outcome of the two trajectories shown matched (congruent) or differed (incongruent). Participants were significantly more accurate in discriminating between the two trajectories, and responded faster in the incongruent condition and, accordingly, were significantly more confident on these trials. Crucially, we found significant differences in metacognitive performance (measured as meta-d’/d’) between the two conditions only on successful trials, where the virtual ball had hit the target. These results indicate that participants successfully incorporated information about the outcome of the movement into both their discrimination and confidence responses. However, information about the outcome selectively sharpened the precision of confidence ratings only when the outcome of their throw matched their intention. We argue that these findings underline the separation between the different levels of information that may contribute to body monitoring, and we provide evidence that intentions might play a central role in metacognitive motor representations.

Table Tennis Track Detection Based on Temporal Feature Multiplexing Network.

Recording the trajectory of table tennis balls in real-time enables the analysis of the opponent's attacking characteristics and weaknesses. The current analysis of the ball paths mainly relied on human viewing, which lacked certain theoretical data support. In order to solve the problem of the lack of objective data analysis in the research of table tennis competition, a target detection algorithm-based table tennis trajectory extraction network was proposed to record the trajectory of the table tennis movement in video. The network improved the feature reuse rate in order to achieve a lightweight network and enhance the detection accuracy. The core of the network was the "feature store & return" module, which could store the output of the current network layer and pass the features to the input of the network layer at the next moment to achieve efficient reuse of the features. In this module, the Transformer model was used to secondarily process the features, build the global association information, and enhance the feature richness of the feature map. According to the designed experiments, the detection accuracy of the network was 96.8% for table tennis and 89.1% for target localization. Moreover, the parameter size of the model was only 7.68 MB, and the detection frame rate could reach 634.19 FPS using the hardware for the tests. In summary, the network designed in this paper has the characteristics of both lightweight and high precision in table tennis detection, and the performance of the proposed model significantly outperforms that of the existing models.

A new thrust vector-controlled rocket based on JOA using MCDA

A thrust vector-controlled rocket model consists of army science board model responsible for computing the angles of the gravity center with a directivity of trajectory positions (x, z). A thrust vector-controlled rocket model is known to be #TVCASB for fall trajectory correction solving, but the falling issue of the trajectory correction is a hard when restricted to ASB model. In this work, we study TVCASB model, a mathematical model of TVCASB calculating which involves ASB variables in addition to position variables, and refer to this question: Why falling issue of the trajectory and correction of TVC is a hard on ASB model? Depend on MCDM results from JOA cases, we show that four MCDM methods can choose the better case for a two pieces of fractional order proportional integral derivative controller when set tuning process for TVC model on ASB with a suitable falling of trajectory correction whether be limited or specified for a high or low distance for the rocket motion. Our multi criteria decision making methods are based on mathematical calculations via three cases were proposed by JOA for rocket parameter estimation with a behavior of it, and in the following MCDM methods involving, MCDM normalization, MCDM WSM and WPM, MCDM VIKOR, are gave case 3 lower and upper bounds (0–100) is specified for a high distance and MCDM TOPSIS is gave case one lower and upper bounds (0–1) is limited for a lower distance, and pose this question: Why the MCDM methods are gave different results of JOA cases for TVCASB model? and which one between cases is better? The first reason, there is a different in mathematical calculation for weights assign. The second reason, three mentioned above methods are used for a high and specified distance for fall trajectory correct and MCDM TOPSIS is used for a limited distance, the better case between cases depends on our need whether a high (specified) or a limited (lower) distance.