1. IC 8451&CONTROL SYSTEMS
Department of Electrical and Electronics Engineering
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3. MECHANICAL ROTATIONAL SYSTEMS
The model of rotational mechanical systems can be obtained by
using three elements, moment of inertia [J] of mass, dash-pot with
rotational frictional coefficient [B] and torsional spring with
stiffness [K].
The weight of the rotational mechanical system is represented by the
moment of inertia of the mass.
The moment of inertia of the system or body is considered to be
concentrated at the centre of gravity of the body.
The elastic deformation of the body can be represented by a spring
(torsional spring).
The friction existing in rotational mechanical system can be
represented by the dash-pot.
The dash-pot is a piston rotating inside a cylinder filled with
viscous fluid.
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4. When a torque is applied to a rotational mechanical system, it
is opposed by opposing torques due to moment of inertia,
friction and elasticity of the system.
The torques acting on a rotational mechanical body are
governed by Newton s second law of motion for rotational
systems.
It states that the sum of torques acting on a body is zero (or
Newton’s law states that the sum of applied torques is equal to
the sum of opposing torques on a body).
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List of symbols used in mechanical rotational
system
θ = Angular displacement, rad
dθ /dt =Angular velocity, rad /s
d2θ / dt2 =Angular acceleration, rad /s2
T =Applied torque, N-m
J =Moment of inertia, kg-m2 / rad
B =Rotational frictional coefficient, N-m / (rad /s)
K =Stiffness of the spring, N –m / rad
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Torque balance equations of idealized elements
Consider an ideal mass element shown in fig. which has
negligible friction and elasticity.
The opposing torque due to moment of inertia is proportional to
the angular acceleration.
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Consider an ideal frictional element dash-pot shown in fig. which has
negligible moment of inertia and elasticity.
Let a torque be applied on it. The dashpot will be offer an opposing
torque which is proportional to the angular velocity of the body.
8. When the dashpot has angular displacement at both the ends
as shown in fig, the opposing torque is proportional to
differential angular velocity.
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9. Consider an ideal elastic element torsional spring shown in fig
,which has negligible moment of inertia and friction.
Let a torque be applied on it. The spring will offer an opposing
torque which is proportional to angular displacement of the
body.
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10. When the spring has angular displacement at both the ends as
shown in figure, the opposing torque is proportional to
differential angular displacement.
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11. EXAMPLE PROBLEM
Write the differential equations governing the mechanical system and
determine the transfer function
The system has two nodes and they are masses with moment of inertia J1
and J2.
The differential equations governing the system are given by torque
balance equations at these nodes.
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12. Let the angular displacement of mass with moment of inertia
J1 is θ1.The opposing torques acting on J1 marked as TJ1 and Tk
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13. The free body diagram of mass with moment of inertia J2 as
shown in fig. The opposing torques acting on J2 are marked as
Tj2,Tb and Tk
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15. • HOME WORK
• Write the differential equations governing the mechanical system and determine the
transfer function
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