INSTRUCTIONAL MANUAL
OF
REDUNDANT JOINT
APPARATUS
By:
ENGINEERING MODELS & EQUIPMENT
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2
REDUNDANT JOINT APPARATUS
CONTENTS:
Page No.
1.0
Theory
03
2.0
Objective
04
3.0
Apparatus
05
4.0
Suggested experimental work
05
5.0
Results & Discussions
05
6.0
Sample Data Sheet
05
7.0
Precautions
05
3
REDUNDANT JOINT APPARATUS
1.0
THEORY:
If v and h are the vertical and horizontal movement of the point d under load W,
the extension of various members can be calculated as follows:
a
b
C
y
B
A
x
T3
T 1L1
c
T2 L2
D
v
D'
W
h
Three Bar Suspension System
If the length of the member AD
Length of the member CD
The length of the member BD
= L1
= L2 and
=c
If change of orientation of various members is neglected, then
Extension of BD
=v
Extension of AD
= v sin x - h cos x
Extension of CD
= v sin y + h cos y
c
c
Also, sin x 
, sin y 
L2
L1
a
b
cos x 
,
cos y 
L2
L1
Where, AB = a and BC = b
Therefore, if the force in the member AD =T1
Force in the member CD =T2
Force in the member BD = T3
Tc
T
Then extension of BD = v  3  3
A3 E 3 K 3
(cv  ah) T1 L1 T1
Extension of AD
=


L1
A1 E1 K 1
4
Extension of DC
=
T
(cv  bh) T2 L2

 2
L2
A2 E 2
K2
Therefore, from above equations
(cv  ah) K1
L1
(cv  ah) K 2

L2
cv K 3

= K3v
c
T1 
(1)
T2
(2)
T3
Now equating the vertical and horizontal forces at point D, we have
T1 sin x  T3  T2 sin y  W
T1 cos x T2 cos y  0
Now we have five equations for five unknown T1, T2, T3, v and h:
Hence by solving we get
( N1 a  N 2 b)
W
h
c N1 N 2 (a  b) 2  N 3 ( N1 a 2  N 2 b 2 )
and v 
( N1a 2  N 2 b 2 )
W
c 2 N 1 N 2 ( a  b) 2  N 3 ( N 1 a 2  N 2 b 2 )
(3)
(4)
(5)
(6)
(7)
Where,
AE
1 K
N 1  1 1  2  21
L1
L1 L1
AE
1 K
N 2  2 2  2  22
L2
L 2 L2
AE
K
1
N 3  3 3  2  23
c
c
c
AE
Here,
is a property of the member, called stiffness of the member and is
L
defined as load per unit extension. The stiffness K, can be found for each spring
separately by taking it out and by suspending from it load 1,2,3,4,5kg in succession
and measuring the extension in each case. A graph is plotted for load against
extension, which should be a straight line with in elastic range and the value of K
is obtained from it.
After knowing h and v from equations (6) and (7), the tension T1 T2 and T3 can be
calculated from equations (1), (2) and (3) respectively and compared with the
measured values in spring balance of each member.
2.0
OBJECTIVE:
Comparison of experimental and theoretical results of forces in the members and
the component displacement of the loaded joint D of a three bar suspension system
for vertical loads.
5
3.0
APPARATUS:
Apparatus consists of three suspension members (spring balances) of different
stiffness, which are jointed at a point to form the redundant joint. The upper end of
the suspension members being tied in a position to a vertical wooden board.
Arrangement is provided to apply a vertical load at the joint and to measure its
horizontal and vertical displacement on a paper and also elongations and forces in
the suspension members by the help of dial gauges. Dial gauges with magnetic
base are provided with the apparatus.
4.0
SUGGESTED EXPERIMENTAL WORK:
Step1: Measure the distances a, b, c, L1 and L2 with the help of scale.
Step2: Fix the dial gauges at the point D to measure the horizontal and vertical
displacement simultaneously for each addition of vertical load.
Step3: To remove the initial slackness of the members, the initial position may be taken as
that with a 0.5kg load suspended at joint D and treat the deflected position as the
datum.
Step4: Note down the deflection h & v and forces T1, T2 and T3 for different sets of load.
5.0
RESULTS AN DISCUSSIONS:
Compare the experimental and analytical values of T1, T2 and T3.
6.0
SAMPLE DATA SHEET:
Value of v, h, T1 T2 and T3
Value
of
v (mm)
Observed
7.0


h (mm)
Calculated
Observed
Calculated
T1 (kg)
Observed
Calculated
T2 (kg)
Observed
Calculated
T3 (kg)
Observed
PRECAUTIONS:
Distance a, b and lengths L1, L2 and L3 should also be measured accurately.
Tap the dial gauges before taking a reading for vertical and horizontal
displacements.
Calculated