Derivation Reports Samples For Students

Introduction

When it comes to control systems, the inverted pendulum system is something that stands out from amongst the rest. Linear control concepts such as stabilization of unstable systems can be demonstrated using the inverted pendulum. Since the system is intrinsically non linear, many non linear concepts can also be demonstrated using the inverted pendulum. This system has basically two equilibrium states, one stable and the other unstable. In the stable equilibrium state, the pendulum points downwards and in the absence of any controlling forces, it returns to its original state. In the unstable equilibrium state, the pendulum points upwards. ...

- Analysis

actual mass.
Fig.1-1 Schematic Diagram : Spring-Mass-Pulley system

In this derivation, the following variables are defined and used, according to Fig.1-1:

k : Spring stiffness

Ms : Mass of the Spring

M : Mass of the Body

Ms : Mass of the Pulley, which includes Mass of the spring

g : Gravity Acceleration

T : Tensile Force in the Rope

1.1 The natural frequency of free vibration of the assembled system :

Equilibrium at the body:

(1)

Equilibrium at the Pulley:

(2)

Meanwhile, the body moves twice as fast as the pulley, which means:

(3)

Substituting (3) into (1) :

(4)

Substituting (4) into (2) :

(5)

At this stage, the natural frequency of this system is assumed as follows:

: Angular Velocity of the Vibration

Then the motion of the pulley is described as follows:

(6)

General solution becomes,

(7)

When mass of the pulley is included in mass of the spring, or when mass of the pulley is

neglected,
(8)
1.2 When a body is simply suspended from a spring:

Eq (2) becomes,

(9)

And eq (3) becomes,

(10)

Substituting (10) into (1) :

(11)

Substituting (11) into (9) :

(12)

Then the motion of the pulley is described as follows:

(13)

General solution becomes,

(14)

In this case, because there is no pulleys,

(15)
1.3 Equivalent Concentrated Mass of Spring

When one end of a spring is stationary and the other end moves, the equivalent mass of

the spring concentrated at the moving end is one-third of the actual spring mass.

Show that the effective mass of a spring is 1/3 its actual mass

Body with mass (M) is suspended by a spring with ...

Arthur Murray

This is a case that was affirmed by the Chief Justice within the influence of section 18 of the Judiciary Act following five awaiting appeals resulting from resolutions of a Board of Review within the the stipulations of the Income Tax and Social Services Contribution Assessment Act 1936 as revised. The income years wherein the appeals relay are the ones concluded on the 30th June within the years of 1954 through to 1957. Of the appeals four are by the taxpayer company and concern revised assessments regarding each of the original three of the years and a novel appraisal regarding the fourth. ...