Bifilar trifilar suspension apparatus

Instruction Manual &
Experiment Guide
Intelligent
System
Corporation
BIFILAR / TRIFILAR SUSPENSION
APPARATUS

Bifilar / Trifilar Suspension Apparatus
INTELLIGENT SYSTEM CORPORATION
NOTE:
Every effort has been made to ensure that the information contained in this manual is accurate;
however no liability is accepted for errors. Should an error be discovered please inform the
company in writing, giving full details. Any experimental results given are for guidance only and
are not guaranteed as exact answers that can be obtained for a given apparatus; due to the
complex variables applicable to most experiments.
The basic principles set out in the following make no claim to completeness. For further
theoretical explanations, refer to the specialist literature.
The selection of experiments makes no claims of completeness but is intended to be used as a
stimulus for your own experiments.

Table of Contents
Page
1. INTRODUCTION...........................................................................................................1
2. GENERAL DESCRIPTION..........................................................................................2
2.1 Unit Assembly ............................................................................................................2
3. SUMMARY OF THEORY ............................................................................................3
3.1 General........................................................................................................................3
3.2 Performance of experiments .......................................................................................3
4. EXPERIMENT ...............................................................................................................9
Appendix A Experimental Data Sheet

INTELLIGENT SYSTEM CORPORATION 1
1 INTRODUCTION:
The BIFILAR / TRIFILAR SUSPENSION APPARATUS unit consists of frame
made up of Bar, Hollow cylinders, Mounting Plate, Locking wheel and Base plate
Apparatus is capable for determining of mass moments of inertia.

2 GENERAL DESCRIPTION:
2.1 Unit Assembly:
Figure: Parts Identification of BIFILAR / TRIFILAR SUSPENSION APPARATUS
1. Bar 2. Cylinder
3. Hollow Cylinder 4. Mounting Plate
5. Locking Wheel 6. Base Plate

3 SUMMARY OF THEORY:
3.1 General:
For investigation of pendulums with bifilar or trifilar suspension, a cylinder
(2), a hollow cylinder (3) or a bar (1) can be suspended from the mounting
plate (4) an caused to oscillate. The pendulum length can be altered by
adjusting the thread with the locking wheels (5). The bar is attached to two
threads (bifilar). The cylinder and hollow cylinder have three suspension
points (trifilar). The mounting plate is attached to a base plate (6) for wall
mounting.
Commissioning
• Fix unit to a suitable wall at a height of approx. 1.5 m using the screws and
dowels provided (drill dia. Ø 10 mm) or bolt to the universal frame TM
090 available as an accessory.
• Suspend desired pendulum for threads. The thread length can be set with
clamping screws and should always be the same for the respective bodies.
Important: In the case of wall mounting, ensure sufficient load bearing
capacity of the wall and tighten screws firmly.
If it were to fall, the unit could cause injury and/or be destroyed.
3.2 Performance of Experiment:
Theoretical principles:
3.2.1 Pendulum with bifilar suspension:
The pendulum with bifilar suspension, i.e. suspension from two
threads, corresponds to the ideal mathematical pendulum in the
equation of motion. As the mass only exhibits translatory motion
without rotation, it has the effect of a concentrated mass.

Fig. 3.2 Pendulum with bifilar suspension
Solving, this equation of motion permits determination of the period of
oscillation of the pendulum. The pendulum is deflected by the angle φ.
This raises the centre of gravity of the concentrated mass by the
amount h. on releasing the pendulum, the restoring force FR – as a
component of the force due to weight – will attempt to return the
pendulum to its initial position.
Fig. 3.3 Parallelogram of forces of mathematical pendulum
The centre of gravity theorem in x-direction, together with the
acceleration of the centre of gravity x and the restoring force FR = FG
· sin φ = m · g · sin φ, yields
m · m · g · sinx ϕ= (3.1)
The angular acceleration is substituted for the acceleration x
,x L x Lϕ ϕ=⋅ =⋅  (3.2)
And the equation expressed in canonical form
sin 0
g
L
ϕ ϕ+ = (3.3)
This non-linear differential equation can be linearized for small
deflections
sin φ = φ , φ < < π
The equation of motion for the mathematical pendulum is thus
0
g
L
ϕ ϕ+ = (3.4)

Fig. 3.4 Harmonic Oscillation
The solution is a harmonic oscillation in the form ˆ( ) sin .t tϕ ϕ ω= ,
where ω is the frequency of the oscillation and ˆϕ the initial deflection.
Differentiating twice and inserting this initial approximation into the
equation of motion gives.
2
ˆ ˆsin sin 0
g
t t
L
ϕω ω ϕ ω− + =(3.5)
Resolution for the unknown frequency yields
g
L
ϕ = (3.6)
2
2
L
T
g
π
π
ω
= = (3.7)
This is the natural frequency and period of oscillation or periodic time
of the pendulum. It becomes apparent that the only governing factors
are the length L and the gravitational constant g. The mass and hence
the shape and material of the pendulum have no influence on the
natural frequency and period of oscillation of the system.

3.2.2 Pendulum with trifilar suspension:
The pendulum with trifilar suspension is used for experimental
determination of mass moments of inertia. For this purpose the body to
be investigated is subjected to torsional oscillation. The period of
oscillation T can be used to establish the mass moment of inertia J.
When subjected to torsional oscillation, the body executes rotary
movement about its axis of rotation with the angle of rotation α; the
suspension thread moves through the angle φ. In this process the body
is raised by the height h. the force due to weight FG produces a
restoring force FR, which acts on every mass point. The following
applies:
Fig. 3.5 Pendulum with trifilar suspension
L
R
α ϕ= ⋅ (3.8)
FR = FG sin φ = m · g sin φ (3.9)
Where; L = Thread length
R = Radius of rotation

Fig. 3.6 Parallelogram of forces
The rotary movement is produced exclusively by the horizontally
acting component FH of the restoring force, to which the following
applies:
The rotary movement is produced exclusively by the horizontally
acting component FH of the restoring force, to which the following
applies:
FH = FR cos φ = m · g cos φ (3.10)
The equilibrium of moments about the axis of rotation can now be set
up.
0HJ F Rα + ⋅ = (3.11)
sin cos 0J m g Rα ϕ ϕ+ ⋅ ⋅ = (3.12)
sin cos 0
L
J m g R
R
ϕ ϕ ϕ⋅ + ⋅ ⋅ = (3.13)
For very small deflections φ << π the equation can be linearised (it is
therefore appropriate to choose the greatest possible thread length L).
the following then applies:
sin φ = φ and cos φ = 1 (3.15)
This yields:
0
L
J m g R
R
ϕ ϕ⋅ + ⋅ ⋅ ⋅ = (3.16)

or
2
0
m g R
J L
ϕ ϕ
⋅ ⋅
+ ⋅ =
⋅
 (3.17)
The initial approximation for the differential equation of a harmonic
oscillation (φ (t) = ˆϕ sin ω t, c.f. Section 3.1.1) results in
2
2
ˆ ˆsin sin 0
m g R
t t
J L
ϕω ω ϕ ω
⋅ ⋅
− + =
⋅
(3.18)
Hence the following applies to the natural frequency ω:
2
m g R
J L
ω
⋅ ⋅
=
⋅
(3.19)
or the period of oscillation T:
2
2
2
J L
T
m g R
π
π
ω
⋅
= = ⋅
⋅ ⋅
(3.20)
and the mass moment of inertia J:
2
2
2
4
m g R
J T
Lπ
⋅ ⋅
= ⋅
⋅ ⋅
(3.21)

4 Experiment:
The numerical values are specimen experimentation results.
Bifilar suspension with L = 500 mm
The bar is suspended with a thread length of L = 500 mm. This produces a period of
oscillation as per equation (3.7) of
2
2 1.419 ( 9.81 ).
L m
T s where g
g s
π= = =
Table 3.1
Measurement i Time Ti20
1 27.9s
2 28.1s
3 28.1s
Sum ΣTi20 84.1s
Mean valve
20
20
i
i
T
T
i
∑
=
28.033s
20
20
iT
T∗
= 1.4017s
This value is to be checked experimentally. For this purpose, the pendulum is
deflected by a small angle φ and a stopwatch used to measure the time taken for 20
oscillations. This process is repeated three to five times and the mean value calculated
from the readings.
The result is
T*
= 1.4017s

Bifilar suspension as second pendulum:

A second pendulum has a period of oscillation of T = 2 s (the point of rest is crossed
once per second). How long must the thread be to maintain this period of oscillation?
Rearranging equation (3.7) for L gives
2
2
0.995
4
T g
L m
π
⋅
= ≈
Table 3.2
1 39.0s
2 38.8s
3 39.0s
Sum ΣTi20 116.8s
Mean valve
20
20
i
i
T
T
i
∑
=
38.933s
20
20
iT
T∗
= 1.947s
As a check, the thread is set to the calculated length and three to five times recorded
for 20 oscillations each.
This result in
T*
= 1.947s

Torsional oscillation of a cylinder:
If the cylinder is suspended from three threads, it can exhibit torsional oscillation. If
the mass moment of inertia J of the body is known, the period of oscillation can be
calculated using equation (3.20).
The formula for the mass moment of inertia of a cylinder can be found in literature.
2
2
m
J r= ⋅ (3.22)
Give m = 3 kg and r = 80 mm the result obtained is:
J = 0.0096 kg m2
For R = 65mm and L = 995 mm this yields a theoretical value for the period of
oscillation of
T = 1.74 s

Table 3.3
1 35.5s
2 35.0s
3 34.9s
Sum ΣTi20 105.4s
Mean valve
20
20
i
i
T
T
i
∑
=
35.133s
20
20
iT
T∗
= 1.756s
In the experiment, the times required for 20 oscillations with small deflection angles φ
are again measured three to five times. The thread length is set accordingly to 995
mm.
This results in
T*
= 1.756s
Determination of mass moment of inertia of a hollow cylinder:
Use can be made of the trifilar suspension to establish the mass moment of inertia of a
body. In this case the hollow cylinder is to be investigated. The hollow cylinder is a
body easily described in geometrical terms, though the method is also suited to
complex bodies where the mass moment of inertia is difficult to calculate but the mass
is relatively easy to weigh.
Table 3.4
1 40.8s
2 41.2s
3 40.9s
Sum ΣTi20 112.9s
Mean valve
20
20
i
i
T
T
i
∑
=
40.967s
20
20
iT
T∗
= 2.048s
The hollow cylinder is suspended accordingly from three threads. The thread length L
is again 995 mm. the mass of the body is 4kg.
For determination purposes, the times for 20 oscillations are recorded three to five
times. T*
enables the mass moment of inertia to be calculated as per equation (3.21),
with the following result:

2
2 2
2
0.0177
4
m g R
J T kg m
Lπ
⋅ ⋅
= ⋅=
⋅ ⋅
The value determined experimentally can be compared to a calculated value. The
formula for the mass moment of inertia of a hollow cylinder is found in literature.
2 2
1 2( )
2
m
J r r= ⋅ + (3.23)
Given m = 4 kg, r1 = 80 mm and r2 = 50 mm the result is:
J = 0.0178 kg m2

APPENDIX A
Experiment Data Sheets

EXPERIMENT
Table 4.1 Bifilar suspension with L = 500 mm
1
2
3
4
5
Sum ΣTi20
Mean valve
20
20
i
i
T
T
i
∑
=
20
20
iT
T∗
=
Table 4.2 Bifilar suspension as second pendulum
1
2
3
4
5
Sum ΣTi20
Mean valve
20
20
i
i
T
T
i
∑
=
20
20
iT
T∗
=

Table 4.3 Torsional oscillation of a cylinder
1
2
3
4
5
Sum ΣTi20
Mean valve
20
20
i
i
T
T
i
∑
=
20
20
iT
T∗
=
Table 4.4 Determination of mass moment of inertia of a hollow cylinder
1
2
3
4
5
Sum ΣTi20
Mean valve
20
20
i
i
T
T
i
∑
=
20
20
iT
T∗
=

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