Dynamics

Project proposal
System description with sketch (2D or 3D)

Consider a scenario where a car is driving on a hilly terrain. The car has a mass of 1000 kg. The terrain consists of hills and valleys modeled by a sinusoidal function given by: y = 5sin(x), where x represents the horizontal distance and y represents the elevation. Find the gravitational force, normal force, frictional force, spring force, and aerodynamic drag.

List of the acting forces

The car is subject to various forces as it navigates the terrain:
Gravitational Force: This force acts vertically downward and is equal to the product of the car's mass and gravitational acceleration.
Normal Force: The normal force exerted by the terrain on the car. This force acts perpendicular to the surface of the terrain and prevents the car from sinking into the ground or flying off.
Frictional Force: The frictional force between the tires of the car and the terrain. This force opposes the motion of the car and depends on the coefficient of friction between the tires and the terrain. The coefficient of friction varies across different parts of the terrain due to changes in surface conditions (e.g., mud, gravel).
Spring Force: The car is equipped with a suspension system consisting of springs. The springs provide damping as the car encounters bumps and dips in the terrain. The spring force depends on the stiffness of the springs and the displacement of the suspension system from its equilibrium position.
Aerodynamic Drag: If the car is moving at a sufficient speed, aerodynamic drag will also be present, opposing the direction of motion. This force arises due to the resistance encountered by the car as it moves through the air. It is dependent on the car's speed and aerodynamic properties.

Surface or path equation if needed

y = 5sin(x)

Project: Particle dynamics with Matlab application

Each group should submit a proposal describing the problem that they intend to formulate and simulate using Matlab. The problem should be a 2D or 3D particle motion subject to at least 4 different types of forces including friction and spring forces (force components are not counted). A clear sketch of the problem should be provided with the proper coordinate system. The problem shouldn't have an analytical solution. This means Matlab must be used to realize the possible motion for various possible scenarios.

Note:
The proposal should not be a repeated one from the previous semesters.
Unacceptable proposals will get no grade of the 20% and the instructor will provide a proposal for the students.
Must use the attached word templates for your report parts. Save each report part file as PDF and then upload it in the assignment before the due date.
To make it easy for you, the below is a checklist that you need to confirm before considering any problem as a project proposal. You need to ask yourself the following questions.
Q1: is it 2D/3D particle problem? (only 2D and 3D are accepted and a particle not a rigid body) Q2: Do we have 4 different forces or more acting all the time on the particle?
Q3: Are the forces related to the particle kinematics? (Normal and friction forces are related to particle motion. But any external force that is unknown, is not accepted).
Q4: Are the forces of different types? (should be 4 different forces)
Q5: Is it difficult to solve analytically? (this means you can't easily solve it by hand calculation)
Q6: Is it your own idea? (any repeated idea from the last term or taken from someone else, e.g., over the net, is not acceptable)
If the answer is No for any of the above questions, then the proposal is not acceptable. All answers must be YES. Once the proposal is accepted by you, then you can use the report format and fill required sections.

Derive the equations of motion
Once the proposal is accepted by the instructor (or improved/corrected), each group should work independently to derive the equations of motion and all required supporting equations and constraints (if applicable). For this you need to do/provide the following:
Establish a reference point on the path to measure the particle position from. This is the s = 0 point. You may need to use two coordinate systems to relate the path equation to the position along the curve.
Assume an arbitrary position in the positive sense for your particle along the path.
Prepare the FBD and KD at that arbitrary position.
Write the equations of motion.

Write any required auxiliary constraint equations you may need to check for if the particle stops momentarily or the speed changes directions.

Matlab Simulation
Each group should write a Matlab script to simulate the particle motion for a number of possible motion scenarios (5 cases).
Verify or correct your equations as per the instructor's feedback in part 2.
Write the required Matlab script (code).
Simulate the system response for a given initial condition (plot the position and velocity of the particle versus time)
The simulation should be done for 5 different cases (various combinations of friction coefficient, µk, and spring stiffness, k, see the report format)
Explain the results as per your expectation of the real system response.