MECH 375 LAB 2 Experiment 2 Forced Harmonic Response Of Single Degree-Of-Freedom System 2025 2026 Co

MECH 375 LAB 2 Experiment 2: Forced Harmonic Response Of Single Degree-Of-Freedom System 2025|2026 Concordia University Experiment 2: Forced Harmonic Response Of Single Degree-Of-Freedom System

Submitted by

Lab Section EI – X Summer 2025 Professor Subhash Rakheja

Concordia University Montreal, QC, Canada

Table of Contents Table of Contents .............................................................................................................................2 Objective ......................................................................................................................................... 3 Introduction ..................................................................................................................................... 3 Procedure......................................................................................................................................... 5 Results ............................................................................................... 6 Table 1: 0K Damping Theoretical and Experimental Results .................................. 6 Table 2: 1K Damping Theoretical and Experimental Results .................................. 7 Table 3: 5K Damping Theoretical and Experimental Results .................................. 8 Table 4: 10K Damping Theoretical and Experimental Results ................................. 9 Sample Calculations............................................................................ 20 Discussion ......................................................................................... 23 Sources of Error.................................................... Error! Bookmark not defined.

Conclusion......................................................................................... 24 References ........................................................................................ 24

List of Figures Figure 1: 0K Damping Amplitude ................................................................ 10 Figure 2: 1K Damping Amplitude ................................................................ 11 Figure 3: 5K Damping Amplitude ................................................................ 12 Figure 4: 10K Damping Amplitude ............................................................... 13 Figure 5: Phase Angle 1K ......................................................................... 14 Figure 6: Phase Angle 5K ......................................................................... 15 Figure 7: Phase Angle 10K ........................................................................ 16 Figure 8: 0K Damping Test Results ............................................................... 17 Figure 9: 1K Damping Test Results ............................................................... 18 Figure 10: 5K Damping Test Results ............................................................. 19 Figure 11: 10K Damping Test Resultsw .......................................................... 20

Objective The objective of this experiment is to induce the forced harmonic response of a single degree-offreedom (SDOF) torsional vibration system and observe its reaction [1]. The experiment also aims to observe the reaction with differing dampers to outline damping importance within a mechanical system [1].

Introduction In a SDOF torsion vibration system, an input will incur a harmonic response at the inputs frequency and eventually dissipate back to the system's natural frequency due to other conflicting forces [2]. A forced harmonic response applies a repeated force input, often in the form of a sinusoidal function, to change the transient response of the system from its natural frequency to the inputs frequency [2]. The system response is of utmost importance for a wide range of inputs to determine the resonance frequency as well as the amplitude at this frequency. This data is crucial when designing systems that undergo harmonic loads as it helps one to avoid resonance through proper design. It can also help ensure that the system can withstand repeated loads at the resonance frequency if needed.

Theoretical values for the stiffness, natural/damped frequencies, damping ratio, and resonance frequency can be calculated using the same equations as seen in experiment 1. The steady state amplitude of the system can be found through the following equation: 𝜃

𝑠𝑠 | |=

𝜃𝑖

(2.1) [1]

1

√(1−[

2 𝜔 2 𝜔𝑛

𝜔

2

] ) +(2𝜁 𝜔 ) 𝑛

where θss is the steady state amplitude and θi is the excitation amplitude [1]. The phase angle response may be calculated with the following equation: 𝜙 = tan−1 |

2𝜁

𝜔

𝜔𝑛 𝜔

2

| (2.2) [1]

1−(𝜔 ) 𝑛

where φ is the phase angle between the response and the excitation [1]. Equation 2.1 gives the dynamic amplification ratio of the response and equation 2.2 gives the angle in which the

response lags the excitation [1]. The phase angle helps determine whether a harmonic input helps or hinders the harmonic motion [3].

Figure 1: Schematic of damped SDOF system [1]

Figure 1 represents the system used for the experiment with varying dampers. In theory, when plotting the amplitude ratio vs the frequency ratio, the data should look as seen in figure 2.

Figure 2: Response of a harmonically excited SDOF system --- Magnification factor vs frequency factor [1]

Figure 3 shows the theoretical plot for phase angle vs frequency ratio.

Figure 3: Response of a harmonically excited SDOF system --- Phase angle vs frequency ratio [1]

Procedure Set up experiment as seen in figure 2 but without the inertia disc. Ensure the function generator and oscilloscope are properly connected and have the proper settings as indicated by the lab manual.

Starting at a frequency of 1 hz, increase by 0.5 hz up to 10 hz and record the peak-to-peak values of both channels 1 and 2 as well as the phase angle as denoted by φ on the oscilloscope for each step. Repeat the procedure with the three provided torsional dampers of 1k, 5k, and 10k.

Results Table 1: 0K Damping Theoretical and Experimental Results 0K

FREQ 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

CH1 108 152 144 152 40 96 96 100 104 108 108 108 108 114 108 118 126 126 130

CH2 92 156 204 372 1640 300 140 100 76 60 48 44 38 30 28 26 24 24 20

PHASE 1-2 1.525 8.25 10.23 12.44 130.1 175.3 172.8 178.1 177.2 170.8 178.6 172.8 174.8 167.1 152.7 158.3 140.4 167 164.1

CH2/CH1 0.851851852 1.026315789 1.416666667 2.447368421 41 3.125 1.458333333 1 0.730769231 0.555555556 0.444444444 0.407407407 0.351851852 0.263157895 0.259259259 0.220338983 0.19047619 0.19047619 0.153846154

Theoretical Amplitude 1.024049893 1.05578942 1.103680052 1.172033055 1.268014861 1.403887509 1.601951208 1.906842533 2.422052138 3.453320523 6.470935527 128.9421293 6.632669138 3.114918496 1.987893535 1.435138315 1.108289419 0.893233282 0.741560364

Theoretical Phase Angle 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Table 2: 1K Damping Theoretical and Experimental Results

1K

FREQ 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

CH1 130 156 148 144 138 136 112 120 116 120 120 128 132 132 132 132 140 144 144

CH2 70 122 96 138 154 424 240 153 124 104 88 72 64 52 44 44 36 36 32

PHASE 1-2 53.65 35.79 63.55 38.25 51.64 71.37 126.8 132.2 141.3 155.3 158.5 163.4 164 170.7 171.2 172.3 153.8 165.5 147.5

CH2/CH1 0.538461538 0.782051282 0.648648649 0.958333333 1.115942029 3.117647059 2.142857143 1.275 1.068965517 0.866666667 0.733333333 0.5625 0.484848485 0.393939394 0.333333333 0.333333333 0.257142857 0.25 0.222222222

Theoretical Amplitude 1.108719924 1.280221805 1.620060662 2.346089117 3.458549735 2.39665627 1.340658376 0.860820722 0.609885296 0.460088849 0.362189152 0.294015896 0.24428507 0.206700094 0.177491705 0.154275861 0.135476581 0.120013678 0.107124119

Theoretical Phase Angle 10.11717796 5.918553986 3.530417581 1.881076143 0.604421372 179.541906 178.6138447 177.7758537 177.0012772 176.2731567 175.5802016 174.914614 174.2708432 173.6448375 173.0335765 172.4347668 171.8466409 171.2678179 170.6972067

Table 3: 5K Damping Theoretical and Experimental Results

5K

FREQ 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

CH1 120 148 144 136 128 126 126 122 122 128 128 130 130 130 130 132 140 142 142

CH2 14.4 15.4 15.2 15.2 15.2 15.2 14.4 15.2 13.6 15.2 15.2 15.2 15.2 15.2 16 16 17.6 33.6 32

PHASE 1-2 5.92 12.1 15.08 14.79 24.28 22.71 21.6 17.38 22.6 22.56 20.55 16.65 29.03 12.47 9.939 10.59 14.43 95.91 100.7

CH2/CH1 0.12 0.104054 0.105556 0.111765 0.11875 0.120635 0.114286 0.12459 0.111475 0.11875 0.11875 0.116923 0.116923 0.116923 0.123077 0.121212 0.125714 0.23662 0.225352

Theoretical Amplitude 1.10181311 1.262046451 1.582644169 2.334005628 5.031030355 5.077760619 1.938332636 1.1046917 0.743402506 0.545533367 0.422261657 0.338951272 0.279394668 0.235028698 0.200918759 0.17402953 0.15239653 0.134695388 0.120002875

Theoretical Phase Angle 22.12296303 13.28866079 8.000990698 4.279148614 1.376815573 178.9564355 176.8445544 174.9434368 173.1940747 171.5588894 170.0129163 168.538994 167.1249867 165.7621057 164.4438543 163.1653415 161.9228241 160.7133921 159.5347488

Table 4: 10K Damping Theoretical and Experimental Results

10K

FREQ 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

CH1 128 152 144 132 130 124 124 124 120 124 124 126 130 130 130 134 136 140 142

CH2 20 60 16 17.6 17.6 17.6 17.6 16.8 18.4 18.4 18.4 19.2 18.4 19.2 27.2 32 32.8 33.6 33.6

PHASE 12 4.835 75.07 14.91 12.6 15.55 19.84 19.35 15.52 8.637 10.25 19.49 21.15 23.23 8.793 30.46 50.66 68.42 76.58 83.08

CH2/CH1 0.15625 0.394737 0.111111 0.133333 0.135385 0.141935 0.141935 0.135484 0.153333 0.148387 0.148387 0.152381 0.141538 0.147692 0.209231 0.238806 0.241176 0.24 0.23662

Theoretical Amplitude 1.102319706 1.263761716 1.588679857 2.364676005 5.530390216 5.821079882 1.98399756 1.115170994 0.747320583 0.547401699 0.423291264 0.33957569 0.279800021 0.235305578 0.20111551 0.174173836 0.152505151 0.134778939 0.120068334

Theoretical Phase Angle 29.88216 18.46085 11.23699 6.03736 1.945816 178.5252 175.5445 172.8713 170.4248 168.1536 166.023 164.0094 162.0957 160.2695 158.5215 156.8446 155.2331 153.6824 152.1887

Figure 1: 0K Damping Amplitude

Experimental Vs Theoretical Amplitude Ratio over Frequency 1K 4 3.5

Amplitude Ratio

3 2.5 2

1K

1.5

1K

1 0.5 0 0

2

4

6

Frequency Hz Figure 2: 1K Damping Amplitude

8

10

12

Experimental Vs Theoretical Amplitude Ratio over Frequency 5K 6

Amplitude Ratio

5 4 3

5K 5K

2 1 0 0

2

4

6

Frequency Hz Figure 3: 5K Damping Amplitude

8

10

12

Experimental Vs Theoretical Amplitude Ratio over Frequency 10K 7

Amplitude Ratio

6 5 4 10K

3

10K

2 1 0 0

2

4

6

Frequency Hz Figure 4: 10K Damping Amplitude

8

10

12

Experimental Vs Theoretical Phase Angle over Excitation Frequency 1K 250

Phase Angle (Deg)

200 150 100 50 0 0 50

0.5

1

1.5

2

Frequency Theortical Phase Angle

Figure 5: Phase Angle 1K

2.5

PHASE 1-2

3

3.5

Experimental Vs Theoretical Phase Angle over Excitation Frequency 5K 250

Phase Angle (Deg)

200 150 100 50 0 0 50

0.5

1

1.5

2

Frequency Theortical Phase Angle

Figure c: Phase Angle 5K

2.5

PHASE 1-2

3

3.5

Experimental Vs Theoretical Phase Angle over Excitation Frequency 10K 250

Phase Angle (Deg)

200 150 100 50 0 0

0.5

50

1

1.5

2

Frequency Theortical Phase Angle

Figure 7: Phase Angle 10K

2.5

PHASE 1-2

3

3.5

Figure 8: 0K Damping Test Results

Figure S: 1K Damping Test Results

Figure 10: 5K Damping Test Results

Figure 11: 10K Damping Test Results

Sample Calculations Known Variables d1 = 0.00635m,

J = 0.0377 kg*m2

L1 = 0.80m,

G = 80x109 Pa

Equations 𝐾= 𝑓 = 𝑛

𝜋𝐺𝑑4

,

32𝐿 𝐾𝑒𝚐

√4𝜋2𝐽

,

𝑛

𝐾

√𝐽

,

𝜁=

)

√1−(ω/𝜔𝑛)2+(2∗𝜁∗ω/𝜔𝑛)2

φ = tan−1 |

2𝜁ω 𝜔𝑛 2 1−( ω 𝜔 )

𝑛

𝐶 2√𝐾⋅𝐽

1

|𝜃𝑠𝑠 | = ( 𝜃𝑖

𝜔 =

|

,

𝜃2

2⋅𝜋⋅𝜁

𝜃0

√1−𝜁2

ln ( ) = − (

),

Theoretical Results Given the shaft dimensions, the theoretical value for the spring constant of the vibrating system can be found using 𝜋(80𝑥109)(0.006354) 𝑁 𝜋𝐺𝑑4 = = 15.96 𝑘= 32𝑙 32 𝑥 0.8 𝑚 The spring fundamental natural frequency is given by

𝑓 = 𝑛

𝐾

𝑒𝚐 15.96 = = 3.27 𝐻𝑧 √ 4𝜋2𝐽 √ 4𝜋2(0.0377)

Using 𝑓𝑛 , the natural frequency can be calculated with 𝜔𝑛 = 2𝜋𝑓𝑛 = 2𝜋(3.27) = 20.58 𝑟𝑎𝑑/𝑠

The sample calculation for ω = 20.58 rad/s and 𝜁 = 0, the amplitude ratio for 3 Hz can be found with the following equation: 1

|𝜃𝑠𝑠 | = ( 𝜃𝑖

)

√1−(ω/𝜔𝑛)2+(2∗𝜁∗ω/𝜔𝑛)2

=(

1

)

√1−(ω/𝜔𝑛)2

=(

1 √1−(6𝜋/20.58)2

=6.22

)

0K: ζ = 0 1K: C = 0.22532 (given) 5K: C = -0.02104 (calculated previously lab #1) 10K: C = -0.09890 (calculated previously lab #1)

0K: 𝜃𝑠𝑠

= 1.024049893 , 𝜁 =

𝐶

=0

2√𝐾⋅𝐽

𝜃𝑖

φ = 10.11717796 1K: 𝜃𝑠𝑠

= 1.10871992

𝜁=

𝐶

= 0.145228328

2√𝐾⋅𝐽

𝜃𝑖

φ = 10.11717796 5K: 𝜃𝑠𝑠

= 1.10181311

𝐶

𝜁=

= 0.063745258

2√𝐾⋅𝐽

𝜃𝑖

φ = 22.12296303 10K: 𝜃𝑠𝑠

= 1.102319706

𝜃𝑖

φ = 29.88216007

𝜁=

𝐶 2√𝐾⋅𝐽

= 0.045098642

Discussion The calculation of spring constant (K), theoretical natural frequency (ωₙ) and experimental natural frequency, and amplitude ratios at their respective resonance frequencies for undamped, 1K, 5K, and 10K damped systems, we now compare results based on the amplitude ratios between the various configurations.

The experimental results demonstrate significant deviations from theoretical predictions across all damping configurations. For the undamped system (0K), the experimental amplitude ratio 1.024, at 3 Hz was 83.5% lower than the theoretical value of 6.22, while the phase angle error reached 430%, the experimental was φ = 53.65 and theoretical was φ = 10.12. The 52.5% error in frequency ratio further confirms that experimental resonance behavior diverged from theory. For the damped systems, the amplitude ratios (1.109 for 1K, 1.102 for 5K, and 1.102 for 10K) are all close in value, but fail to exhibit the expected resonance trends. This is possibly due to the data collected as readings change rapidly with big jumps in values. The damping ratios derived from Experiment #1, were likely compromised by errors from prior measurements, as evidenced through the unexpected phase angles. For instance, the 1K damper showed identical phase lag (10.12°) to the undamped case, contradicting theory which predicts increased lag with damping. Similarly, the 10K damper yielded a phase angle of 29.88° versus the theoretical 22.12° for 5K. From the plots, the peak response amplitude located at the damped natural frequency for 0k, 1k, 5k, 10k are 3 Hz, 3.5 Hz, 10 Hz and 9 Hz, respectively. The undamped system aligns closely to the computed value.

There were significant sources of error in this experiment. Most importantly, the damping ratio itself had uncertainty because it was calculated from experimental data based on Experiment 1, meaning any radical errors in the earlier calculations carried over to this experiment. The measured errors observed can be caused by a range of possible reasons like sensor miscalculation on either channels 1 and 2, torsional vibration of the transferred equipment propagating to the surroundings, and improper disc tightening and shaft alignment or adjustment.

Geometric errors like length, diameter, or shaft sizing imperfections also impact calculations for mass moment of inertia and therefore the natural frequencies. Furthermore, users' errors during

the experiment, installation of the disk, or collecting data can lead to inaccuracy, particularly since prolonged vibration risks preventing accurate readings and generating false data. % 𝐸𝑟𝑟𝑜𝑟 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑅𝑎𝑡𝑖𝑜, 0𝐾 =

(1.109 − 0.538) (Fr. thr − Fr. exp) ⋅ 100% = ⋅ 100% Fr. thr 1.109

= 52.5% % 𝐸𝑟𝑟𝑜𝑟 𝑃ℎ𝑎𝑠𝑒 𝐴𝑛𝑔𝑙𝑒, 0𝐾 = |

(𝑃𝑎. 𝑡ℎ𝑟 − 𝑃𝑎. 𝑒𝑥𝑝) 𝑃𝑎. 𝑡ℎ𝑟

| ⋅ 100% = |

(10.117 − 53.65) 10.117

| ⋅ 100%

= 430%

Conclusion This experiment incorporated the use of a single setup and three varied dampers. These tests were utilized to study and explore the effects of various dampers on a one degree-offreedom forced motion system. It should be taken into account that the said sources of error, such as potential user mistake or the equipment utilized across the lab, could have potentially influenced the results obtained.

References 1. Department of Mechanical, Industrial, and Aerospace Engineering, MECH 375 Lab Manual, Concordia University, Montreal, Summer 2025.

2. “Forced Harmonic Oscillators Explained.” Accessed: Jul. 11, 2025. [Online]. Available: https://resources.pcb.cadence.com/blog/2021-forced-harmonic-oscillators-explained 3. “Understanding Phase Angle in Simple Harmonic Motion: Key Concepts and Applications.”

Accessed: Jul. 11, 2025. [Online]. Available: https://allen.in/jee/physics/phase-angle