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-of-
freedom (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
| |=
1 (2.1) [1]
2
𝜃𝑖 𝜔 2 𝜔 2
√(1−[ ] ) +(2𝜁 𝜔 )
𝜔𝑛 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:
𝜔
2𝜁
𝜙= tan−1 | 𝜔𝑛 | (2.2) [1]
𝜔 2
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
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-of-
freedom (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
| |=
1 (2.1) [1]
2
𝜃𝑖 𝜔 2 𝜔 2
√(1−[ ] ) +(2𝜁 𝜔 )
𝜔𝑛 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:
𝜔
2𝜁
𝜙= tan−1 | 𝜔𝑛 | (2.2) [1]
𝜔 2
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
