Event Simulation 02

Parabolic Trajectory of a Projectile.

Case Description

This accuracy verification example demonstrates the reliability of the event simulation solver in quantifying the following three phenomena:

• Initial linear velocities
• Large free-body displacement (motion dynamics)
• Gravitational acceleration and deceleration

The model is two single-element tetrahedral bodies—one movable (Projectile) and one fixed (Stationary). The Stationary body is also defined as a rigid body, meaning that it does not deform at all. Additionally, the bodies do not interact with each other (no contact). The Stationary body is only there as a position reference. Specifically, the Stationary body is located such that its top surface exactly corresponds with the theoretical location of the bottom surface of the Projectile body at the end of the event simulation.

Two equal components of linear velocity, one vertical and one horizontal, are applied to the Projectile body as an initial condition. In other words, the resultant initial velocity is at an upward angle of 45° from the horizontal direction. This angle, in theory, results in the greatest horizontal travel distance of a projectile for any given initial velocity (ignoring wind resistance). The following results are determined from theory and compared with the Fusion event simulation results:

• The height of the projectile when its vertical velocity reaches zero and reverses (peak height along the projectile path)
• The time required to reach the peak height of the projectile
• The time required for the projectile to return to its initial elevation (back on the ground)
• The horizontal distance traversed by the projectile when it reaches the initial elevation again

Study Type and Parameters

• Event Simulation
• Total Event Duration = 0.2 seconds
• Number of Results Save Intervals = 20 (producing one results set per 10 milliseconds)
• Element Deletion Criteria: None set

Rigid Body

• The Stationary body is rigid

Model Geometry

• Description: Two identical tetrahedra, all edge lengths = 0.866 inch, all face corner angles = 60°
• The Stationary body is positioned at a horizontal distance of 7.722 inches from the initial location of the Projectile body. Additionally, the Stationary body is rotated so that its top face corresponds with the theoretical location of the bottom face of the Projectile body at the end of the event.

Mesh Parameters

• Mesh Type = Solid, Tetrahedral
• Element Order = Parabolic
• Mesh Size = 1.0 inch, absolute (resulting in a single element per part, maximizing the allowable calculation time step and minimizing the solution time)
• Adaptive Mesh Refinement: None

Material Properties

The material properties do not affect the physics of projectile motion. However, a higher density part solves more quickly than a lower density part for event simulations. Therefore, the material, Lead, was selected, which has a relatively high density.

• Modulus of Elasticity = 2.031 X 106 psi
• Density = 0.4097 lbmass/in3
• Poisson's ratio (v) = 0.43

Constraints

• Stationary body is fully fixed

Load

• Gravity: Acceleration = 32.174 ft/sec2 in the –Y direction
• Initial Linear Velocity:

  • In X direction: 3.2174 ft/s

  • In Y direction: 3.2174 ft/s

    Note: You must change the default General precision setting from 0.123 to 0.1234 to see the fourth decimal place of the initial velocity components. Find this setting in the Unit and Value Display section of the Preferences dialog.

Theoretical Results

Time required for vertical velocity to reach zero

Time1 = Initial Velocity(vertical component) / Deceleration Rate

`Time1 = (3.2174 ft/s) / (32.174 ft/s2) = 0.1 s (corresponds to step 10 of the event simulation)

Time at which projectile returns to original elevation

The time to return from the original height from the peak height is equal to the time to reach the peak height from the initial position. Therefore:

Time2 = 2 * Time1

Time2 = 2 * 0.1 s = 0.2 s (corresponds to step 20 of the event simulation)

Peak height of the projectile relative to the initial position (Y Displacement)

Peak Height = Average Y Velocity * Time1

Average Y Velocity (in/s) = (12 in/ft) * Initial Y Velocity (ft/s) / 2 = (12 in/ft) * (3.2174 ft/s) / 2 = 19.3044 in/s

Peak Height = 19.3044 in * 0.1 s = 1.93044 in

Horizontal distance projectile traverses at end of event

Without wind resistance or any other opposing forces, the horizontal velocity component remains constant. Therefore:

Horizontal Distance (in) = (12 in/ft) * Initial X Velocity (ft/s) * Time2 = (12 in/ft) * (3.2174 ft/s) * 0.2 s = 7.72166 in

Comparison of Results

Increase the precision of the legend to compare the results to the theory for more decimal places. Specifically, access the Unit and Value Display section of the Preferences dialog. Change the General precision setting from 1.123 to 1.12345. In the Unit and Value Display > Simulation and Generative Design, change the Scientific notation precision setting from 1.123E+04 to 1.12345E+04. The legend will then display five decimal places for decimal and scientific notation results alike.

Reference

Basic Newtonian physics (equations of motion) are available from numerous reference sources.