- 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 ...