README.md

Dynamics-Simulations

This is a collection of problem sets and my simulated results from my Olin 2021 Dynamics Class.

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Problem Set 1

1. Golf Ball Simulation (Air Resistance)

Setup

For this problem, a golf ball, with diameter D and mass m, is launched from point p with an initial velocity vector V₀=v_x0i + v_y0j + v_z0k. The wind is blowing with air velocity V_w=W_xi+W_yj+W_zk resulting in a drag force, F_D, acting on te ball. The drag force acts in the direction opposite its velocity and it proportional to the square of the velocity.

Results

For a golf ball with the following characteristics and initial conditions

• Mass: 0.0459 kg
• Diameter: 4.27 cm
• Initial velocity: 130 mph at 45 degrees with respect to ground

and following drag properties

• Air density: 1.29 kg/m³
• Drag coefficient: 0.25

2. Two Mass Hopper Simulation

Setup

For this problem, two masses, m_upper and m_lower, are connected by a linear spring with an unstreched length, L₀, and a spring constant, K. The upper mass is given an initial downward displacement and is then released, resulting in the hopper jumping.

Results

For a two mass hopper with the following characteristics and inital conditions

• Upper mass: 1.0 g
• Lower mass: 0.5 g
• Spring length: 2.0 cm
• Spring constant: 40 N/m
• Initial spring compression: 1.0 cm

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Problem Set 2

1. Marble in a Bowl Simulation

Setup

For this problem, a small marble is released at the rim of a hemispherical bowl of radius R = 200 mm with an initial velocity of 100 mm/s in the circumferential direction. The marble slides on the inside surface of the bowl without friction. Gravity acts in the downward vertical direction.

Results

2. Rollercoaster Simulation

Setup

For this problem, a roller coaster cart is released along a roller coaster track. the track is defined by the function, y=h(1-B*xL)cos²(3πx/(2L)) where 0<B<1 controls the height of the second peak of the track. One design constraint is that the cart must always travel over the second peak. The cart has air resistance. The magnitude of the force is F_D=1/2ρC_DV² where ρ is the air dentisty and C_D is the drag coefficient, A is the frontal area of the car (which has mass, m), and V is the magnitude of the velocity of the car. The direction of the drag force is opposite to that of the car's velocity.

Results

For a rollercoaster with the following characteristics and initial conditions:

• Initial height of ride: 100 m
• Horizontal length of the ride: 500 m
• Mass of car: 500 kg
• Frontal area of car: 4.0 m²
• Drag coefficient: 0.1
• Air density: 1.02 kg/m³

B = 0.361 is the lower bound for how tall the second peak can be. For any value B lower than 0.361 results in a second peak that is too tall for the rollercoaster cart to go over.

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Problem Set 3

1. 3D Spring Pendulum

Setup

For this problem, a mass, m, is suspended on a massless linear spring to form a spherical pendulum. The unstreched lentgh of the spring is L₀. The spring can only chane lenth in the radial direction.

Results

For a spring pendulum with the following characteristics

• Mass: 0.1 kg
• Spring length: 20 cm
• Spring constant: 20 N/m

and initial position and velocity

• Initial spring length: 20 cm
• Initial theta: 0 rad
• Initial phi: π/2 rad
• Initial velocity in radial direction: 0.1 rad/s
• Initial velocity in theta direction: 0 rad/s
• Initial velocity in phi direction: 0 rad/s

and with the following different initial position and velocity

• Initial spring length: 30 cm
• Initial theta: 0 rad
• Initial phi: π/2 rad
• Initial velocity in radial direction: 1 rad/s
• Initial velocity in theta direction: 1 rad/s
• Initial velocity in phi direction: 0 rad/s

2. Planar Pendulum

Setup

For this problem, a plan pendulum which consists of mass, m, is suspended by a massless rod of lenth, l. Gravity acts in the downward vertical direction. The position of the mass can be described by the angle,θ, or by the cartesian coordinates (x,y).

Results

For a planar pendulum with the following characteristics and initial conditions

• Mass: 0.1 kg
• Length: 20 cm
• Initial theta: π/2
• Initial velocity in theta direction: 0 rad/s

These two simulations were calculated using different coordinate systems. The cartesian pendulum was calculated with equations of motion that were defined using a cartesion coordinate system, while the polar pendulum was calculated with equaitons of motion that were defined using a polar coordinate system. As expected, the two are identical. Both coordinate systems describe a pendulum accurately.

Adding viscious drag and aerodynamic drag to the pendulum result in the following. Viscous drag is proportional to the angular velocity of the pendulum while aerodynamic drag is proportional to the square of the angular velocity.

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Problem Set 4

Double Pendulum

Seup

For this problem, a planar double pendulum consists of masses, m₁ and m₂, suspended by massless rods of length l₁ and l₂ The position of the masses can be described by the angles, θ₁ and θ₂, with respect to vertical. The pendulum is supported at point O with a frictionless pivot. Gravity acts in the downward vertical directions.

Results

For a double pendulum with the following characteristics and initial conditions

• Mass 1: 1.0 kg
• Length 1: 0.5 m
• Theta 1: π/2
• Mass 2: 1.0 kg
• Length 2: 0.5 m
• Theta 2: π/4

Final Project

Double Pendulum on a Cart

For this problem, a planar double pendulum consisting of masses, m₂ and m₃, are suspended by massless rods of length l₁ and l₂ with a frictionless pivot upon a frictionless lateral cart with mass, m₁. The position of the cart can be described by the distance x with respect to the origin, and the positions of the pendulums can be described by the angles, θ₁ and θ₂, with respect to vertical. Gravity acts in the downward vertical directions. Furthermore, the cart and pendulum masses are subject to a drag force F_D=C_DV² where C_D is the drag coefficient and the force is acting in the oposite direction of velocity.

Results

Drag Coeficient = 0

Drag Coeficient = 0.05

Drag Coeficient = 0.5