The differential equation modeling of mechanical systems. There are two types of mechanical systems based on the type of motion.
Mass
Mass is the property of a body, which stores kinetic energy. If a force is applied on a body having mass M, then it is opposed by an opposing force due to mass. This opposing force is proportional to the acceleration of the body. Assume elasticity and friction are negligible.
Fm ∝ a
Where,
Spring
Spring is an element, which stores potential energy. If a force is applied on spring K, then it is opposed by an opposing force due to elasticity of spring. This opposing force is proportional to the displacement of the spring. Assume mass and friction are negligible.
F ∝ x
⇒ Fk = Kx
F = Fk = Kx
Where,
Dashpot
If a force is applied on dashpot B, then it is opposed by an opposing force due to friction of the dashpot. This opposing force is proportional to the velocity of the body. Assume mass and elasticity are negligible.
Fb ∝ ν
Where,
Moment of Inertia
In translational mechanical system, mass stores kinetic energy. Similarly, in rotational mechanical system, moment of inertia stores kinetic energy. If a torque is applied on a body having moment of inertia J, then it is opposed by an opposing torque due to the moment of inertia. This opposing torque is proportional to angular acceleration of the body. Assume elasticity and friction are negligible.
Tj ∝ α
Where,
Torsional Spring
In translational mechanical system, spring stores potential energy. Similarly, in rotational mechanical system, torsional spring stores potential energy. If a torque is applied on torsional spring K, then it is opposed by an opposing torque due to the elasticity of torsional spring. This opposing torque is proportional to the angular displacement of the torsional spring. Assume that the moment of inertia and friction are negligible.
Tk ∝ θ
⇒ Tk = Kθ
T = Tk = Kθ
Where,
Dashpot
If a torque is applied on dashpot B, then it is opposed by an opposing torque due to the rotational friction of the dashpot. This opposing torque is proportional to the angular velocity of the body. Assume the moment of inertia and elasticity are negligible.
Tb ∝ ω
Where,
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