MECH 375 Lab -1 (Experiment 1: Characteristics Of A Single Degree-Of-Freedom
System (Torsional Vibration) 2025-2026 Concordia University
Experiment 1:
Characteristics Of A Single Degree-Of-Freedom System
(Torsional Vibration)
Submitted by
Lab Section EI – X
Summer 2025
Professor Subhash Rakheja
Concordia University
Montreal, QC, Canada
1
,Table of Contents
Table of Contents .............................................................................................. 2
Objective .................................................................................................... 3
Introduction ................................................................................................. 3
Procedure .................................................................................................... 6
Part 1: Parallel and Serial Stiffness ...................................................................... 6
Part 2 - Measurement of Viscous Damping Coefficient ............................................... 7
Results ....................................................................................................... 8
Part 1: Parallel and Serial Stiffness ...................................................................... 8
Part 2: Measurement of Viscous Damping Coefficient ............................................... 11
Sample Calculations ..................................................................................... 14
Theoretical ............................................................................................ 14
Experimental .......................................................................................... 14
Discussion ................................................................................................... 16
Part 1: Parallel and Serial Stiffness ..................................................................... 16
Part 2: Measurement of Viscous Damping Coefficient ............................................... 16
Conclusion .................................................................................................. 17
References .................................................................................................. 17
List of Figures
Figure 1: Free body diagram of torsional shaft in parallel ................................................ 4
Figure 2: Free body diagram of torsional shaft in series .................................................. 4
Figure 3: Step Response of SDoF under-damped system ................................................ 5
Figure 4: Frequency vs θ2 / θ1 for parallel shaft configuration ................................................................ 9
Figure 5: Frequency vs θ2 / θ1 for series shaft configuration................................................................. 10
Figure 6: No damping test results ......................................................................... 11
Figure 7: 5K damping test results .......................................................................... 12
Figure 8: 10K damping test results ........................................................................ 13
List of Tables
Table 1: Angular Displacement of Shafts in Parallel ...................................................... 8
Table 2: Angular Displacement of Shafts in Series ........................................................ 8
2
, Objective
The first aim of the experiment is to find the equivalent stiffness coefficient (𝐾𝑒𝑞) of a
mechanical system made up of two torsion shafts with varying diameter and length. For the
second part, the objective is to find the viscous damping coefficient of a torsion damper.
Introduction
Vibration is inherent in mechanical systems and is an important component that explains how
systems respond to external forces. Every system possesses a natural frequency and certain
vibratory characteristics to provide stability and avoid failure. If a system is excited at or close to
its natural frequency, then the system can become resonant, resulting in large oscillations. This
laboratory is focused on the vibratory performance of single-degree-of-freedom (SDOF)
systems, with specific attention paid to the influences of stiffness and damping. The resistance to
the system displacement is caused by stiffness, while the rate of energy dissipation is influenced
by damping.
Part 1: Parallel and Serial Stiffness
The torsional stiffness K, of a shaft is defined as the ratio of applied torque T, to the angular
deflection θ. It is calculated using:
𝑇 𝜋𝐺𝑑4
𝐾= =
𝜃 32𝐿
Where:
• T: Applied torque (N·m)
• θ: Angular deflection (radians)
• G: Modulus of rigidity (G=80GPa)
• d: Shaft diameter (m)
• L: Shaft length (m)
When two shafts are arranged in parallel, 𝑇1 and 𝑇2 are calculated separately and then combined.
𝑇1 = 𝐾1𝜃
𝑇2 = 𝐾2𝜃
3
System (Torsional Vibration) 2025-2026 Concordia University
Experiment 1:
Characteristics Of A Single Degree-Of-Freedom System
(Torsional Vibration)
Submitted by
Lab Section EI – X
Summer 2025
Professor Subhash Rakheja
Concordia University
Montreal, QC, Canada
1
,Table of Contents
Table of Contents .............................................................................................. 2
Objective .................................................................................................... 3
Introduction ................................................................................................. 3
Procedure .................................................................................................... 6
Part 1: Parallel and Serial Stiffness ...................................................................... 6
Part 2 - Measurement of Viscous Damping Coefficient ............................................... 7
Results ....................................................................................................... 8
Part 1: Parallel and Serial Stiffness ...................................................................... 8
Part 2: Measurement of Viscous Damping Coefficient ............................................... 11
Sample Calculations ..................................................................................... 14
Theoretical ............................................................................................ 14
Experimental .......................................................................................... 14
Discussion ................................................................................................... 16
Part 1: Parallel and Serial Stiffness ..................................................................... 16
Part 2: Measurement of Viscous Damping Coefficient ............................................... 16
Conclusion .................................................................................................. 17
References .................................................................................................. 17
List of Figures
Figure 1: Free body diagram of torsional shaft in parallel ................................................ 4
Figure 2: Free body diagram of torsional shaft in series .................................................. 4
Figure 3: Step Response of SDoF under-damped system ................................................ 5
Figure 4: Frequency vs θ2 / θ1 for parallel shaft configuration ................................................................ 9
Figure 5: Frequency vs θ2 / θ1 for series shaft configuration................................................................. 10
Figure 6: No damping test results ......................................................................... 11
Figure 7: 5K damping test results .......................................................................... 12
Figure 8: 10K damping test results ........................................................................ 13
List of Tables
Table 1: Angular Displacement of Shafts in Parallel ...................................................... 8
Table 2: Angular Displacement of Shafts in Series ........................................................ 8
2
, Objective
The first aim of the experiment is to find the equivalent stiffness coefficient (𝐾𝑒𝑞) of a
mechanical system made up of two torsion shafts with varying diameter and length. For the
second part, the objective is to find the viscous damping coefficient of a torsion damper.
Introduction
Vibration is inherent in mechanical systems and is an important component that explains how
systems respond to external forces. Every system possesses a natural frequency and certain
vibratory characteristics to provide stability and avoid failure. If a system is excited at or close to
its natural frequency, then the system can become resonant, resulting in large oscillations. This
laboratory is focused on the vibratory performance of single-degree-of-freedom (SDOF)
systems, with specific attention paid to the influences of stiffness and damping. The resistance to
the system displacement is caused by stiffness, while the rate of energy dissipation is influenced
by damping.
Part 1: Parallel and Serial Stiffness
The torsional stiffness K, of a shaft is defined as the ratio of applied torque T, to the angular
deflection θ. It is calculated using:
𝑇 𝜋𝐺𝑑4
𝐾= =
𝜃 32𝐿
Where:
• T: Applied torque (N·m)
• θ: Angular deflection (radians)
• G: Modulus of rigidity (G=80GPa)
• d: Shaft diameter (m)
• L: Shaft length (m)
When two shafts are arranged in parallel, 𝑇1 and 𝑇2 are calculated separately and then combined.
𝑇1 = 𝐾1𝜃
𝑇2 = 𝐾2𝜃
3
