TUTO : déterminer le moment d'inertie équivalent avec mouvement de translation

Summary

In this video, we explore the process of determining the equivalent moment of inertia for a power transmission system with translation motion. The system consists of three components: a rotating shaft (piece 1), a second rotating shaft (piece 2), and a translating piece (piece 3). The video covers key concepts such as the relationships between input and output velocities, the calculation of kinetic energy for each component, and how to calculate the system's total energy and equivalent moment of inertia. The session concludes with applying the work-energy principle to establish relationships between the motor's torque and system parameters.

Takeaways

• 😀 The task is to determine the equivalent moment of inertia for a power transospector system involving both rotational and translational motion.
• 😀 The system consists of three components: Piece 1 (rotating shaft), Piece 2 (rotated by Piece 1 and drives Piece 3), and Piece 3 (translating table).
• 😀 The input consists of a couple (Cm) applied to Piece 1, and a constant force (Fx) applied to Piece 3.
• 😀 The first kinematic relationship is between the rotational speeds of Piece 1 and Piece 2, governed by the ratio of their diameters (D1/D2).
• 😀 The second kinematic relationship relates the translational speed of Piece 3 to the rotational speed of Piece 1 using a factor of D1/D2.
• 😀 The total kinetic energy of the system is the sum of the rotational energies of Pieces 1 and 2, and the translational energy of Piece 3.
• 😀 The energy of Piece 1 and Piece 2 is rotational, while the energy of Piece 3 is translational, with a velocity linked to the rotational speed of Piece 1.
• 😀 The system's total kinetic energy is expressed in terms of the angular velocity of Piece 1 and the respective moments of inertia (J1 and J2).
• 😀 The power input into the system includes the motor power (Cm * omega1), friction power (Cf * omega1), and the constant force power applied to Piece 3.
• 😀 The work-energy theorem is applied to relate the change in kinetic energy of the system to the sum of the applied external powers, linking it to the motor torque and system parameters.

Q & A

What is the primary goal of the exercise described in the video?

The primary goal is to determine the equivalent moment of inertia of a system that combines both rotational and translational motions, specifically for a power transospecteur with a translation movement.

What is the role of the input shaft (arbre 1) in the system?

The input shaft (arbre 1) is driven by the motor and is responsible for initiating the rotational motion, which, through various mechanical linkages, drives the translational motion of the table mobile (pièce 3).

How is the relationship between the rotational velocity of the input shaft and the translational velocity of the table piece (pièce 3) determined?

The relationship is determined by first finding the rotational speed of the intermediate shaft (arbre 2), which is related to the input shaft via a pulley system. The rotational speed of arbre 2 is then used to determine the translational velocity of pièce 3 through a conversion formula based on the diameter ratios and angular velocities.

What is the significance of the diameters D1 and D2 in the system?

The diameters D1 and D2 are crucial for establishing the speed ratio between the input shaft (arbre 1) and the intermediate shaft (arbre 2). They are used to relate the rotational speeds of the shafts through the pulley system.

How is the kinetic energy of each component in the system calculated?

The kinetic energy of each component is calculated using the general formula for kinetic energy. For rotating components, the formula is 1/2 * J * ω², where J is the moment of inertia and ω is the angular velocity. For the translational component, the kinetic energy is calculated as 1/2 * m * v², where m is the mass and v is the velocity of translation.

Why is it necessary to express all energies in terms of the rotational motion of the input shaft?

It is necessary to express all energies in terms of the rotational motion of the input shaft (arbre 1) to maintain consistency in the system's energy calculations and to ultimately derive the equivalent moment of inertia of the entire system relative to the input shaft.

How does the friction in the system affect the power calculations?

The friction in the system, particularly at the input shaft (arbre 1), causes a power loss, which is accounted for as a frictional power term (CF * ω₁) in the energy and power balance equations.

What is the power dissipation due to the external constant force on piece 3?

The power dissipation due to the external constant force applied to piece 3 is calculated as Mo * F * v₃, where Mo is the constant force and v₃ is the translational velocity of piece 3.

How is the total power balance of the system established?

The total power balance is established by summing the powers applied to the system: the motor power (torque * angular velocity), the frictional power loss (CF * ω₁), and the external power applied to piece 3. These are balanced to determine the overall energy flow in the system.

What role does the theorem of kinetic energy play in the analysis?

The theorem of kinetic energy states that the time derivative of the total kinetic energy of the system equals the sum of the applied external powers. This allows us to relate the motor torque and other system parameters, helping to establish the equivalent inertia of the system and finalize the calculations.