- 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 ...