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Grand Valley State University
Padnos School of Engineering
EGR 345: Dynamics Systems Modeling and Control
Dr. Hugh Jack
Lab 7: Oscillation of a Torsional Spring
Dale Slotman
Partners:
Brad VanderVeen
Joel Knibbe
Eric VanderZee
Ryan Jeffries
Leanne Newman
Purpose:
The objective of this laboratory exercise was to study torsional oscillation using
both theoretical models and actual data collected using Labview.
Theory:
Figure 1: General Torsional Spring Model
Given a torsional spring as shown in figure 1, a general response can be
calculated using the equations:
M
T J
M
and
T
J
A
G
L
where T is the torque applied to the mass,
Jm is the mass moment of the mass,
JA is the area moment of the spring,
L is the length of the spring,
and G is the shear modulus of the spring.
Combining these equations yields the differential equation:
J
M
J
A
G
L
Figure 2: Torsional Spring Setup
Calculations of natural frequency:
Area Moment of the Torsional Spring:
D 0.25 in
J A
2
D
2
4
J A 1.596 10
10 m
4
Spring Coefficient:
L 30.75in
G
Steel
9 Pa
K
s
J
A
G
Steel
L
K
s 15.328N m
Mass Moment of Inertia for Square Tubing:
Outside_Depth 2.016 in Inside_Depth 1.535 in W .526 m M 4.18 kg
Cross_Sectional_Area Outside_Depth
2 Inside_Depth
2
Volume Cross_Sectional_Area W Density
M
Volume
Mass_Solid Outside_Depth
2 W Density Mass_Inner Inside_Depth
2 W Density
J M
1
12
Mass_Solid Outside_Depth
2 W
2
1
12
Mass_Inner Inside_Depth
2 W
2
J M 0.098kg m
2
The natural frequency is:
The natural frequency is then:
f
J
A
G
L J
M
f 1.992Hz
2. A Labview program was set up to record a voltage input from a potentiometer.
The program is shown in figure 3.
Figure 3: Labview Program for Accepting Potentiometer Voltage Input
3. The potentiometer was calibrated using multiple readings of angle turned and
voltage output. The relationship between angle turned and voltage output is:
Output Voltage = 0.081*Radians Turned + 0.
4. The data collection rate of Labview was calculated to be 150.3 samples per
second.
5. The torsional spring was mounted onto an overhead beam using a c-clamp. The
potentiometer was connected to the mass by setting it on an adjustable wood base
and applying hot glue to the knob when inserted into the hole on the bottom of the
mass.
Discussion:
The theoretical and measured spring constants are 15.328 Nm and 16.131Nm,
respectively. The theoretical and measured frequencies are 1.992 Hz and 2.067 Hz,
respectively. These values are similar and the small error can be accounted for by
inaccuracies in the properties of the materials. Replacing the shear modulus of the
spring material with 80Mpa yields a theoretical frequency of 2.06 Hz, which matches
the measured yield. Since the steel used in the lab was thin rod, possibly an extruded
rod, the shear modulus may have been increased by work hardening.
The actual response of the system matches the theoretical response almost
exactly. The theoretical and measured responses are graphed simultaneously in figure
Measured and Theoretical Oscillations
Time (sec)
Radial Displacement
Measured
Theoretical
Figure 5: Measured and Theoretical Oscillations
Conclusions:
An undamped theoretical torsional spring system will accurately model a physical
torsional spring. The laboratory exercise showed that both spring constants and
natural frequencies can be calculated using an undamped theoretical system.
