Understanding Concepts in Machine Element Dynamics

 
UNIT – I:
Concepts in machine
element dynamics
 
CO 1
 
Student will be able to explain
D’Alembert principle, precession and
gyroscopic couple and 
determine
 the
direction and effect of gyroscopic
couple by analyzing various
applications.
 
The inertia force is an imaginary force, which
when acts upon a rigid body, brings it in an
equilibrium position. It is numerically equal to
the accelerating force in magnitude, but
opposite 
in direction.
Mathematically, Inertia force = – Accelerating
force = – 
m.a
Where, 
m 
= Mass of the body, and 
a 
= Linear
acceleration of the centre of gravity of the
body.
 
Similarly, the inertia torque is an imaginary
torque, which when applied upon the rigid
body, brings it in equilibrium position. It is
equal to the accelerating couple in magnitude
but 
opposite 
in direction.
 
Dynamic Forces
:
 Dynamic forces are
associated with accelerating masses. As all
machines have some accelerating parts,
dynamic forces are always present when the
machine operate. In case of rotors which
rotates at speed more than 80,000 rpm, even
slightest eccentricity of centre of mass from
axis of rotation induces very high dynamic
forces. This may lead to vibration, wear, noise
or even failure.
 
D-Alembert’s Principle
 
 
The vector sum of all external forces and
inertia forces acting upon a system of rigid
bodies is zero. The vector sum of all external
moments and inertia torques acting upon a
system of rigid bodies is also separately zero.
Inertia is a property of matter by virtue of which
a body resists any change in velocity.
Inertia Force, F
i
= -m.a        (m: mass of body; a:
acceleration of centre of mass of the body)
Negative sign indicates that the force acts in the
opposite direction to that of acceleration. The
force acts through centre of mass of the body.
 
Gyroscopic Torque (Couple)
 
Problem :
  An aeroplane makes a complete half circle of
50 metres radius, towards left, when flying at
200 km per hr. The rotary engine and the
propeller of the plane have a mass of 400 kg
and a radius of gyration of 0.3 m. The engine
rotates at 2400 r.p.m. clockwise when viewed
from the rear. Find the gyroscopic couple on
the aircraft and state its effect on it.
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This unit covers the concepts of D'Alembert principle, precession, gyroscopic couple, inertia force, torque, and dynamic forces associated with accelerating masses in machine elements. Students will learn how these principles affect the equilibrium and behavior of rigid bodies in rotational motion. Practical applications and calculations related to gyroscopic effects on aircraft are also explored.


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  1. UNIT I:Concepts in machine element dynamics

  2. CO 1 Student will be able to explain D Alembert principle, precession and gyroscopic couple and determine the direction and effect of gyroscopic couple by analyzing applications. various

  3. The inertia force is an imaginary force, which when acts upon a rigid body, brings it in an equilibrium position. It is numerically equal to the accelerating force in magnitude, but opposite in direction. Mathematically, Inertia force = Accelerating force = m.a Where, m = Mass of the body, and a = Linear acceleration of the centre of gravity of the body.

  4. Similarly, the inertia torque is an imaginary torque, which when applied upon the rigid body, brings it in equilibrium position. It is equal to the accelerating couple in magnitude but opposite in direction.

  5. Dynamic associated with accelerating masses. As all machines have some dynamic forces are always present when the machine operate. In case of rotors which rotates at speed more than 80,000 rpm, even slightest eccentricity of centre of mass from axis of rotation induces very high dynamic forces. This may lead to vibration, wear, noise or even failure. Forces: Dynamic forces are accelerating parts,

  6. D-Alemberts Principle The vector sum of all external forces and inertia forces acting upon a system of rigid bodies is zero. The vector sum of all external moments and inertia torques acting upon a system of rigid bodies is also separately zero. Inertia is a property of matter by virtue of which a body resists any change in velocity. Inertia Force, Fi= -m.a acceleration of centre of mass of the body) Negative sign indicates that the force acts in the opposite direction to that of acceleration. The force acts through centre of mass of the body. (m: mass of body; a:

  7. Gyroscopic Torque (Couple)

  8. Problem : An aeroplane makes a complete half circle of 50 metres radius, towards left, when flying at 200 km per hr. The rotary engine and the propeller of the plane have a mass of 400 kg and a radius of gyration of 0.3 m. The engine rotates at 2400 r.p.m. clockwise when viewed from the rear. Find the gyroscopic couple on the aircraft and state its effect on it.

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