Energy Transfer, Gear Ratios, and Gyroscopic Motion in Mechanics

This assignment focuses on the principles of energy transfer in mechanical systems, covering energy conservation, holding capacity, channeling, and transformation. It delves into the effects of these principles in systems with uniform acceleration, considering translational, rotational, oscillatory, and circulatory motions. The solution includes calculations for a motorized winch, determining power input and output, gear ratios, and load being raised. Additionally, the assignment addresses gyroscopic effects, including calculations for precession and torque, supported by free body diagrams and relevant formulas. References to academic papers are included, and Desklib offers more solved assignments and study resources for students.

Energy Transfer in Mechanical Systems
The effect of energy transfer in mechanical systems mainly depends on the following energy
transfer systems
 Energy conversation
 Energy holding capacity
 Energy channelling
 Energy transformers
Energy conservation:
The conversion of kinetic energy into potential energy without loss in total energy is the
basic principle of the law of conservation of energy. The total mechanical energy is termed as the
sum of kinetic and potential energy. This total mechanical energy is called the prime mover.
They can be measured in terms of kilowatt or horsepower (Xu, Y. et.al 2001).
Energy holding capacity
Energy can be stored either in one form or can be transformed into other forms. Conversion
of energy entirely depends based on the usage.
Energy Channeling
It is quite rare to employ energy where it is generated thereby the generated energy should be
transmitted as a result. This energy is channelled from the converter to the generator.
Energy transformers
These are capable of retaining the nature of energy without transmitting the power. They are
capable of altering the flux or the potential.

To determine the behaviour of energy transfer in which acceleration is present we must consider
the dynamic mechanical systems.
It is important to understand the motion of the object whether it is a kinetic object or a kinematic
object.
The motions that are needed to be considered for energy transfer are
 Translational motion
 Rotational motion
 Oscillatory motion and
 Circulatory motion
Mechanics is considered based on whether it is static or dynamic. Static is concerned with forces
that act on a body at equilibrium or at rest.
The mass moment of inertia is also needed to be considered. The mass moment of inertia can be
defined using the radius of gyration.

The main laws that govern the energy transfer in mechanical systems are Newton's laws of
motion.
Friction is something that plays an important role in energy transfer. The amount of resistance
provided can significantly cause variations in energy transfer.
Task 1 :
Solution:
Output Power
Input power P= 2 πNT
60

P= 2(3.14)(1800)(12)/60
Pinput= 2261 Watts
Overall efficiency = POUTPUT/PINPUT
(0.82) 2261 =POUTPUT
The output power is calculated as: POUTPUT = 1854 W
Load being raised
POUTPUT = 2 πNT
60
1854 = 2(3.14)(90)T/60
T=196.715 N-m
T= Load ×Radius
Radius of drum = 0.2/2 = 0.1m
Load = 196.715/(0.1)
Load = 1967.5 N
Number of teeth in GEAR A
The number of teeth in Gear A can be calculated by using the gear ratio relation:
N A
NF
= T B T D T F
T A T C T E
1800
90 = 50(30)(100)
(TA)(75)(20)
The number of teeth is found as T A=5
Gyroscope Couple:
When a body is allowed to move with uniform linear velocity along a curved path, a force in the
direction of the centripetal acceleration is to be applied. This external force applied is called active force.
Similarly, if the object is subjected to centrifugal force the force applied is termed as the passive force.
Precessional angular motion

The angular velocity of the axis of spin is called the angular velocity of precession The axis
about which it turns is called the axis of precession. The angular motion about the axis of precession is
termed as precessional angular motion.
Task 3:
Solution:
Precession
The angular velocity of the axis of spin is called the angular velocity of precession The axis about which
it turns is called the axis of precession. The angular motion about the axis of precession is termed as
precessional angular motion.
Free Body Diagram:

Moment of inertia:
I=mk2
Gyroscopic couple:
C = m× g ×r
C = (0.01 Kg) (9.81) (0.020)
C= 0.001962 N-m
Moment of inertia I= 0.01 (20)2
I = 4 E-6 kg- m2
C= I× ω ×ω p (Assumeω=31.4 rad /s)
Rate of precession ω p = 0.001962 / (4e-6)(31.4)
Rate of precessionω p = 15.62 rad/s
If the distance between the them is 25mm
F =C/x
F= 0.001962 / (0.025) = 0.078 N
Torque = Force x Perpendicular distance
= (0.078) (0.025)

T = 1.95 e-3 N-m
References:
Xu, Y., Heidemann, J. and Estrin, D., 2001, July. Geography-informed energy conservation for ad hoc
routing. In Proceedings of the 7th annual international conference on Mobile computing and networking
(pp. 70-84). ACM.
Fang, J. and Ren, Y., 2011. High-precision control for a single-gimbal magnetically suspended control
moment gyro based on inverse system method. IEEE Transactions on Industrial Electronics, 58(9),
pp.4331-4342.