Mechanical Engineering Programme of Study
(BSc/4 Years/System of Credits)
Experiment No. 5
The Characteristics of Coulomb Damped Free Vibration of a System of
Spring-Mass
Objective
To identify the components of a Coulomb damped free vibration, study the
characteristics of the resulting vibration, and determine the natural frequency, the
stiffness, and the friction coefficient of the resulting vibration.
Fig. 1
Theory
Dry friction or Coulomb damping is used to damp vibrations because of its
simplicity and convenience. The direction of the friction force is opposite to the
direction of the velocity.
Case A: The displacement (x) increases in the positive direction and ( x ) is
positive
The equation of motion for the mass-spring system of mass m under Coulomb
damping is given as
mx  kx  N  0
(1)
where m is the mass, k is the stiffness of the spring, μ is the coefficient of friction,
and N is the normal force (N = mg). The solution of the differential equation (1) is
given as
x  A1 cos( n t )  A2 sin(  n t ) 
where n 
k
, A1 and A2 are constants.
m
N
k
(2)
Mechanical Engineering Programme of Study
Case B: The displacement (x) decreases in the positive direction and ( x ) is
negative
The equation of motion for the mass-spring system of mass m under Coulomb
damping is given as
mx  kx  N  0
(1)
The solution of the differential equation (1) is given as
x  A3 cos( n t )  A4 sin(  n t ) 
N
k
(2)
where A3 and A4 are constants.
Applying the initial conditions of
x (t  0)  0
x(t  0)  xo ,
The solution for the first half of the cycle will be
N 
N

x(t )   xo 
 cos( n t ) 
k 
k

(3)
Applying the initial conditions of
 
2N 

x    xo 
,
k 

 n 
 
x   0
 n 
for the second half of the cycle, the solution of the differential equation will be
3N 
N

x   xo 
(4)
 cos n t  
k 
k

4 N
Which means that the amplitude of the vibration has decreased by
in one
k
cycle.
Mechanical Engineering Programme of Study
Procedures
1. Add weights to the system of Encoder  1 so that you will have the total
weight of 4 pieces.
2. Record the total mass of the added weights.
3. Turn on the controller. Enter the Command menu, go to Trajectory and
select step, set-up. Select open loop step and input a step size of zero, a
duration of 300 ms and 1 repetition. Exit to the background screen by
consecutively selecting ok.
4. Go to set up Data Acquisition in the Data menu and select Encoder  1
data to acquire and specify data sampling every 2 (two) servo cycles.
Select OK to exit. Select Zero Position from the Utility menu to zero the
encoder positions.
5. Select Execute from the Command menu. Prepare to manually displace
the first mass carriage approximately 2.5 cm. With the first mass
displaced, select Run from the Execute box and release the mass
approximately 1 second later. The mass will oscillate and attenuate while
encoder data is collected to record this response. Select OK after data is
uploaded.
6. Select Set-up plot from the Plotting menu and choose Encoder  1
Position then select plot Data from the Plotting menu.
7. Print the resulted damped vibration
8. Measure x1 and x2.
9. Determine the natural frequency of the system, n.
10. Determine the stiffness of the system, k.
x
t
Fig. 2
Mechanical Engineering Programme of Study
11. Determine the value of the normal force, N.
12. Determine the friction coefficient from the following relation
4 N
x1  x 2 
k
Report
Prepare a report that includes the following:
A)
Objective
B)
Theory
C)
Procedure
D)
Results & Calculations
E)
Discussion (compare your results with theory)
(5)