48660 dynamics and control project draw the free body

Question

Dynamics and Control Project - Modelling of a 2 DOF System, Vibration Analysis and Experimental Verification

The Shake Table rig (Figure 1) was designed to model the behaviour of a building during an earthquake scenario. The rigs have been developed as two-storey structures that emulate vibrations in a single direction with 2 degrees of freedom. The rigs were designed to help students to break down and understand the complex dynamics of such a system.

Students will model the 2 DOF system (using the simplified diagram in Figure 2), perform a vibration analysis and verify the results experimentally both in MATLAB and using the remote lab rigs.

You are required to read the document "Shake_Table_User_Guide_V2-1.pdf" for detailed information about the shake table rigs. Variance data on each of the rigs can be found in "Shake_Table_Variance_Data_2014-08-20.pdf".

Students are required to produce a report detailing the three parts to the project, along with an insightful discussion and reflection relating to the tasks completed.

Part 1: Modelling of a 2DOF System

1. Draw the free body diagrams for the two masses in the system based on the simplified diagram in Figure 2.  

2. Derive the differential equations of motion for the system. The displacements x₁(t) of M₁ and x₂(t) of x₂ are measured from the rest positions of the masses.

Figure 2 shows a shear building with base motion. This building is modelled as a 2 DOF dynamic system where:

M1 = M2 = M = 1.178 kg

k₁ = k₂ = k = 370.374

C₁ = C₂ = C = 0.05 N.S/m

y(t) =  y₀ sin(ωt)

Where k and c are the total values of stiffness and damping for each of the levels of the structure.

Part 2: Vibration Analysis

1. By assuming un-damped free vibration, calculate the natural frequencies of the system: ω₁ and ω₂.

2.  Calculate the normal modes of vibration corresponding to ω₁ and ω₂, and draw their modal shapes:

3. Obtain the transfer functions for each of the masses, based on the differential equations of motion from Part 1 (Damped, Forced Vibration):

X₁(S)/Y(S)

X₂(S)/Y(S)

4. Using MATLAB/Simulink, analyse the responses x₁(t) and x₂(t) due to the following inputs:

Unit step base movement:       y(t) = 1

Harmonic motion of the base: y(t) = 0.7 sin(ω₁t)

Harmonic motion of the base: y(t) = 0.7 sin(ω₂t)

Part 3: Experimental Verification

Record and plot the actual responses (i.e. produce two graphs) from the UTS remote vibration (Shaker Table) laboratory by setting the frequencies of the base movement to  f1 = ω1/2π and f2 = ω2/2π. Students should set the total system damping (% on the remote lab interface, given C₁ = C₂ = 0.05) using the information below.

Note: The relationship between the damping interface control value  (%)  and damping coefficient (N.s/m) is shown in Figure 3 and is given by the following equation:

y = 1.669x²

Provide an insightful, clear, relevant but brief discussion and reflection on the tasks performed in this report. Draw some conclusions about why modelling such a system might be useful in real life engineering practice. Also discuss how the simulated results compare with the real life experimental (shaker table) results.

Attachment:- Assignment Readings.rar