This document discusses rotational mechanical systems and their modeling. It describes how rotational systems can be modeled using three elements: moment of inertia to represent mass, a dashpot to represent friction, and a torsional spring to represent elasticity. It provides the torque balance equations for idealized elements of mass, friction, and elasticity. As an example, it analyzes a two-mass rotational system and derives the differential equations governing its motion and determines its transfer function.

1. IC 8451&CONTROL SYSTEMS
Department of Electrical and Electronics Engineering
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2. Prepared by
Mrs.K.Kalpana., M.E.,(Ph.D).,
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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