Free Vibration of a Spring Mass System

1.6 Experimentation

This experiment is for the free vibration analysis of a spring-mass system without any external damper. It involves a spring, a mass, a sensor, an acquisition system and a computer with a signal processing software as shown in Fig.1.4.

Accelerometer is used as a sensor to sense the motion of vibration and transfers the data to the computer through a data acquisition system. Data acquisition system takes vibration signal from the accelerometer and encodes it into digital form. Computer acts as a data storage and analysis system, and it takes encoded data from data acquisition system and processes the data using a signal processing software.

Experiments were done by taking two springs and two masses. So, total four experiments were done by choosing one spring and one mass at a time. Any two masses can be distinguished by their weights. Similarly, springs can be distinguished by their well known property called stiffness.


1.6.1 Spring Stiffness


a) Theoretical Calculation:

The stiffness of a spring is,


where G is the Torsion Modulus,

d is the wire diameter,

D is the Mean coil diameter and D= -d; and

N is the number of active coils.


Table 1: Properties of different springs used in the experiment



Torsion modulus (G) in M Psi

Wire diameter (d) in mm

Outer coil diameter ( ) mm

Number of active coils













Fig.1.4: Experimental Setup for the free vibration of a Spring-Mass System

Fig.1.5: (a) Spring A and (b) Spring B



b) Experimental Calculation:

The slope of the Load-Displacement plot of a system indicates its Stiffness.


Fig. 1.6: Load-Displacement Plot


1.6.2 Corrective Mass

Let us consider a mass less spring with stiffness k and has an effective mass, m eff , at the free end. Hence, the natural frequency of the spring-mass system without considering mass of accelerometer and mass of stand can be written as,




Considering the mass of accelerometer and the mass of stand used to hang weights into account, the effective mass would be,



Hence, the Natural frequency of the system considering the corrective mass will be,



1.6.3 Sample Calculations

Calculate the Natural Frequency of a spring-mass system with spring ‘A’ and a weight of 5N.



Stiffness of spring ‘A’ can be obtained by using the data provided in Table 1, using Eq.(1.16)

  = 256.7 N/m


Using Eq. (1.17), corrective mass, M = (5/9.81) + 0.0182 + 0.1012 = 0.629 Kg.

Hence, the Natural Frequency of the system is,


= 20.2 rad/sec.


1.6.4 Experimental Setup

The experimental setup of a spring-mass system contains a spring which is fixed to the support at one end by using a clamper and free at the other end as shown in Fig. 1.7. At the free end a stand is attached in order to place weights. Care must be taken while giving displacements to the mass in order to have motion only along the axis of the spring.



Fig. 1.7: Experimental setup of a spring-mass system

Fig. 1.8: Accelerometer


An accelerometer (Fig.1.8) is a time-dependent vibration measuring device. It is a transducer, which converts the acceleration of vibration into equivalent voltage signal, and sends it to data acquisition system. It is a contacting type of transducer.




Fig.1.9: Data acquisition system


Data acquisition system shown in Fig.1.9 receives voltage signal from accelerometer and calibrate the data into equivalent accelerometer scale and send it to computer where by using a software these data can be analyzed as time history (time-displacement) and frequency domain (i.e., using FFT).


PULSE: Vibration measurement software shown in Fig. 1.10 uses the calibrated data from the data acquisition system for plotting the response.




Fig. 1.10: Time response from PULSE software



1.6.5 Experimental Procedure

   1.   Select a spring (either A or B) and a Mass (5N or 10N).


  2.   Clamp one end of the spring to the support and hang a stand to the other end (free end).


  3.   Place an accelerometer (with magnetic base) on the mass to measure the free vibration response (acceleration) and connect the accelerometer cord to the Data Acquisition System.


  4.   Connect the data acquisition system to the computer installed with PULSE software.


  5.   Give a small amount of displacement to the hanging mass and allow it to oscillate on its own.


  6.   Record the data obtained from the transducer in the form of graph (obtained from the PULSE software).


  7.   Repeat the procedure 5 to 10 times to check the reliability of the experiment.


  8.   Repeat the whole experiment by selecting different combinations of springs and masses.


  9.   Record the whole set of data for different spring and mass combinations into a data base.



1.6.6  Precautions during Experimentation


  1.   Should check whether the spring is fixed tightly or not.


  2.   Care must be taken in selecting the mass to a particular spring.


  3.   Small amounts of displacement should be given to the mass as it is assumed that the mass has restricted motion, only along the axis of the spring.


  4.   Position of the sensor should not create unbalance in the system.


  5.   User is suggested to use sensors and other measuring instruments with high sensitivity and minimize the noise in measuring data. By ensuring these it minimizes the error and improves the result.


  6.   It is also suggested to the user, to repeat the experiment several times in order to confirm the consistency of the results.