Mec2401 Dynamics I S2 20141 School Of Mechanical And Electri

Mec2401 に Dynamics I S2 20141school Of Mechanical And Electrical Eng

Determine the specific physical and mathematical procedures to solve various dynamics-related problems involving rolling motion, link kinematics, ground reactions, rotational dynamics, and collisions, based on provided figures and data. The tasks include calculating final velocities and accelerations, analyzing energy, and assessing safety and impact severity under specific conditions.

Paper For Above instruction

The assignment requires comprehensive analysis of multiple problems in dynamics, focusing on motion of rolling and sliding objects, linkage kinematics, rotational behavior of a disk, and impact collision in a railway scenario. Each problem demands application of fundamental principles such as conservation of energy, Newton’s laws, rotational dynamics, and impact physics.

Problem 1: Motion of a Rolling Ramp and Sliding Object

The first problem involves a smooth ramp with a mass of 90 kg and a crate of 60 kg sliding down from rest, covering a distance of 6 meters. To determine the ramp's speed when the crate reaches the bottom, energy conservation principles are applied. Since both ramp and crate are considered smooth with no friction, gravitational potential energy transforms entirely into kinetic energy of the system.

Specifically, at the starting point, the crate possesses potential energy relative to its position at B, and as it slides down, this energy converts to kinetic energy of both the crate and the ramp. By equating initial potential energy to the sum of kinetic energies at B, one can derive the speed of the ramp. The combined kinetic energy includes the translation of both the ramp and the crate, and their masses are summed to compute the total kinetic energy. The calculation yields the velocity of the ramp at B.

Next, the problem asks for the distance the ramp travels when the crate reaches B, assuming no friction, which involves integrating the velocity over the time taken for the motion or using kinematic relationships. Since the initial velocity is zero and the acceleration is known from the energy considerations, the displacement can be obtained through kinematic equations.

The third part of the problem introduces a coefficient of sliding friction (0.58) and the incline angle, which affects the motion by introducing a resistive force. The presence of friction changes the energy balance, requiring accounting for work done against friction. The motion becomes decelerated, and the analysis necessitates calculating the work done by friction and how it influences the velocity and travel distance, ultimately describing the motion of the ramp in a more realistic scenario.

Problem 2: Kinematics of a Four-Bar Linkage

This problem involves a linkage system with multiple bars, where the angular velocity and acceleration of link AB are given. Using the principles of relative motion and angular kinematics, the angular velocities and accelerations of other links (BC and CD) are derived. Since each link is treated as a uniform slender bar, moments of inertia are calculated to evaluate rotational energies.

The given data, including angular velocity (6 rad/s) and angular acceleration (2 rad/s²), serve as initial conditions. Employing vector loop equations and differentiating position equations, the angular velocities and accelerations of other links are determined. These calculations use the relationships between linear and angular quantities, considering the geometry of the linkage, which involves link lengths and positional constraints.

The kinetic energies of links AB, BC, and CD are then computed using their respective moments of inertia and angular velocities. As the links are considered as slender bars, their moments of inertia are calculated via standard formulas. Kinetic energy contributions from each link are then summed to analyze the energy distribution within the system.

Finally, the components of linear acceleration at point C are determined by differentiating the velocity expressions, considering both tangential and centripetal accelerations. These components are essential for understanding the dynamic response of the linkage and for potential stress analysis at point C.

Problem 3: Reactions and Safety Analysis of a Front-End Loader

The third problem involves a loader moving under load with given mass and load parameters, where the reactions exerted by the ground on the wheels are calculated under dynamic conditions. It incorporates static equilibrium equations, considering the high mass of the loader and the additional load, with their respective centers of mass and heights.

The reactions are derived through static analysis using equilibrium equations of moments and force balances, considering acceleration of the loader at 1.6 m/s². The positional data (x and h) of the load's center of mass influence the torque calculations about the wheels' contact points. These calculations provide the normal forces exerted at each wheel, which are critical for assessing stability and loading capacity.

The second part assesses whether the loader can stop safely without tipping, given a certain coefficient of static friction. The analysis involves calculating the maximum braking force based on the friction coefficient and evaluating the resulting moments about the wheels. If the braking torque exceeds the tipping moment, the loader remains stable; otherwise, there's a risk of rollover. Comparing the braking force with the torque due to the load's weight at a specific height helps determine tipping safety.

Problem 4: Rotational Dynamics of a Disk

This problem involves a uniform disk subjected to a constant torque, leading to angular acceleration. Deriving the expression for angular acceleration involves applying Newton's second law for rotation, where torque equals the moment of inertia times angular acceleration.

Assuming the disk's moment of inertia (I) is known for a uniform disk (I = (1/2)MR²), the angular acceleration is computed using the given torque T and mass M. The instantaneous angular acceleration at a specific time is calculated based on the relation η = T/I.

Next, the rotational speed (in rpm) after a given time interval is obtained by integrating angular acceleration over time, converting radians per second to revolutions per minute. The kinetic energy associated with the disk's rotation is calculated using KE = (1/2)Iω², providing insight into the energy stored in the rotating system. The work done by the torque during the acceleration phase is obtained as the product of torque and angular displacement.

Lastly, the radial force exerted on the shaft by the disk is determined. This force results from the centripetal acceleration when the disk spins and acts radially inward, with its magnitude computed as F = M r ω², where r is the radius of the disk. The free-body diagram illustrates this force's direction along the radius line, perpendicular to the shaft, and in the plane of rotation.

Problem 5: Collision Analysis of a Locomotive and Oil Tanker

This problem examines collisions involving high-speed vehicles, employing principles of conservation of momentum and restitution. The initial velocities are provided in km/hr, requiring conversion into m/s for calculations. The first scenario assumes a perfectly inelastic collision, where the locomotive and tanker move together afterward, allowing calculation of their common velocity based on mass and initial velocities.

The subsequent analysis introduces a coefficient of restitution (e), which modifies the post-collision velocities by accounting for elasticity. Applying the restitution equation alongside conservation of momentum yields the velocities of both vehicles after impact, depending on e.

The energy loss calculations involve comparing initial kinetic energies with post-collision energies, with values of e = 0.2 and 0.8. This comparison illustrates how elastic or inelastic the collision is, with higher e indicating less energy lost. Discussions revolve around the severity of the impacts, impact energy dissipation, and safety implications based on the calculated energy and velocity outcomes.

Conclusion

The detailed analysis of these problems demonstrates core dynamics principles, including energy conservation, kinematic relationships, rotational motion, and impact physics. Mastery of these concepts enables accurate prediction of motion behaviors, stability assessments, and safety evaluations for mechanical systems and vehicles under various conditions, highlighting their practical applications in engineering design and safety management.

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