- 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 ...