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Assignment 1-Vibration Analysis of SDOF system

1. Apparatus

The system consists of the following components: one supporting system, one steel strip, one bronze alloy strip, one aluminum strip, four concentrated masses (two small and two large), three accelerometers, three charge amplifiers, and one data acquisition system.

2. Procedure for the experiment: SDOF system

a) Clamp the steel strip to the supporting system and fix two concentrated masses using one of the three combinations* at the top as illustrated in Fig. 1;

b) Install an  accelerometer on  the  concentrated  mass  and  connect  it  to  the  data acquisition system through a charge amplifier;

c) Apply an initial displacement to the top of the beam and then release it abruptly;

d) Record the vibration of the beam using the data acquisition system with a sampling frequency of 500 Hz;

e) Replace the steel strip with the bronze alloy strip. Once again, fix two concentrated masses using one of the three combinations, and repeat steps c) and d).

f) Subsequently, substitute the bronze alloy strip with the aluminum strip. Fix two concentrated masses using one of the three combinations and repeat steps c) and d).

* The three combinations of masses are: 1) one small concentrated mass and one large concentrated mass; 2) two small concentrated masses; and 3) two large concentrated masses.

Fig. 1 SDOF system

3. Theoretical solution

The moment of inertia for the rectangular section (I) is given as:

where b is the width of the cross section in m, h is the thickness of the cross section in m.

The stiffness of a cantilever beam (k) is given as:

where E is the Young’s Modulus of the cantilever in  N ∙ m2,  I  is the moment of inertia of the cross-section of the cantilever in m4, L is the length of the cantilever in m.

The natural frequency (幼n, in rad/s) of SDOF system is calculated as:

where k is the stiffness of the cantilever beam in N/m, m is the mass of the system in kg.

4. Requirement

a) Please plot time history and frequency spectrum figures;

b) Please analyze the experimental measured natural frequencies;

c) Please compare measured and theoretically calculated natural frequencies; d) Please discuss the relationship between frequency and mass;

e) Please discuss the relationship between frequency and stiffness.

Appendix: Material properties for theoretical calculation

In  the  theoretical  calculation  (or  called  numerical  calculation),  some  material properties should be known in the beginning.

For bronze, aluminum, and steel cantilever beams, they share the same geometrical properties as below:

The length L = 630 mm

The width b = 50 mm

The thickness h = 2.5 mm (for aluminum strip and steel strip), h = 3.15 mm (for bronze strip)

The areas of cross-section A = b × h

For concentrated masses, their masses are as below:

The large concentrated mass m1=635.5 g

The small concentrated mass m2=316.5 g

The Young’s Modulus of beam (E) is constant. Here lists the Young’s Modulus for bronze, aluminum, and steel:

The density of beam () in constant. Here lists the density for bronze, aluminum and steel: