48660 Dynamics and Control

Project 2 – Modelling of a 2 DOF System (Part 1), Vibration Analysis (Part 2) and Experimental Verification (Part 3)

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”.

Figure 1- 2 Degrees of Freedom Shake Table

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 𝑀2 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 𝑁/𝑚 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.

Figure 2 - A 2 Degree of Freedom Vibration System

Part 2: Vibration Analysis

1. By assuming undamped 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:

(1)

𝑋?

(1) = {𝑋1(1)}

𝑋2

(2) = {𝑋1((22))}

𝑋?

𝑋2

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 = 𝜔2𝜋1 and 𝑓2 = 𝜔2𝜋2. Students should set the total system damping (% on the remote lab interface, given 𝑐1 = 𝑐2 = 0.05 𝑁. 𝑠/𝑚) 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

Figure 3 - Force/Velocity Coefficient (N.s/m) vs. Interface Control Value (%)

Discussion and Reflection

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.

Additional Notes

• Students must include units for all quantities measured or derived.

• Students should consult the marking guide provided on UTSOnline.

• The main body of the reports must be limited to 8 pages or less (i.e. not including title page, table of contents etc.).

• Scanned hand written reports are acceptable, so long as they are neat and easily readable. Any hand written work should be written on blank, unruled paper and scanned in only. Digital photographs of hand written work will attract deductions for the report presentation. Graphs and figures should be made using software and annotated appropriately (with legends, titles, captions, axis labels).

• Submission is via UTS Online under the Assignments tab; follow the instructions provided.

Project 2 – Modelling of a 2 DOF System (Part 1), Vibration Analysis (Part 2) and Experimental Verification (Part 3)

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”.

Figure 1- 2 Degrees of Freedom Shake Table

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 𝑀2 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 𝑁/𝑚 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.

Figure 2 - A 2 Degree of Freedom Vibration System

Part 2: Vibration Analysis

1. By assuming undamped 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:

(1)

𝑋?

(1) = {𝑋1(1)}

𝑋2

(2) = {𝑋1((22))}

𝑋?

𝑋2

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 = 𝜔2𝜋1 and 𝑓2 = 𝜔2𝜋2. Students should set the total system damping (% on the remote lab interface, given 𝑐1 = 𝑐2 = 0.05 𝑁. 𝑠/𝑚) 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

Figure 3 - Force/Velocity Coefficient (N.s/m) vs. Interface Control Value (%)

Discussion and Reflection

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.

Additional Notes

• Students must include units for all quantities measured or derived.

• Students should consult the marking guide provided on UTSOnline.

• The main body of the reports must be limited to 8 pages or less (i.e. not including title page, table of contents etc.).

• Scanned hand written reports are acceptable, so long as they are neat and easily readable. Any hand written work should be written on blank, unruled paper and scanned in only. Digital photographs of hand written work will attract deductions for the report presentation. Graphs and figures should be made using software and annotated appropriately (with legends, titles, captions, axis labels).

• Submission is via UTS Online under the Assignments tab; follow the instructions provided.

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