the two peg test
TRANSCRIPT
The Two Peg TestThe Two Peg Test
Reciprocal MethodReciprocal Method
To Determine the collimation error of a Level InstrumentTo Determine the collimation error of a Level Instrument
A few reminders: see levelling presentation
We will use convention that error is the result of the observed value minus the true value
It follows that the correction to the observed value then must have the opposite sign as the error
RR = Rod Reading
BS = Backsight RR
FS = Foresight RR
EL = ElevationEL Elevation
DEL = difference in elevation = BS – FS ( one setup) = ΣBS – ΣFS (involving a number of turns)
DEL = final EL – initial EL
H H i t l Di tH = Horizontal Distance
Σ = summation sum of e.g. ΣBS = BS1 + BS2 + BS3 . . . + BSLast
The Level Instrument 2-Peg TestThe Instrument set up andset up and levelled
Horizontal Line
A
B
Neglecting curvature and refraction
in 60 m c+r =0.0002m
d
B
dL
Reciprocal MethodSet up as near to the A position as focus will allow
1st setup near AA RR=aed = RR due to collimation error = RR with error =ad+ed
Set up as near to the A position as focus will allow
ed = Collimation error over distance daed
Read and book BS on A
ad What proper reading should beed = Collimation error over distance d
A
B
Place 2 pegs A & B about 60m is a reasonable Horizontal distance apart
d
B
dMinimum focus distance
L
1st setup near AA RR=aedAB
RR aed
aed-be(d+L)
=(ad+ed)-(b(d+L)+ed+L)
RR=be(d+L) =RR with collimation error
R d d b k
ed+LCollimation error over distance d+L =
(ad ed) (b(d+L) ed+L)
be(d+L)
Read and book FS on B
b(d+L)What proper reading should be >
A
B
d
B
dL
1st setup near AA RR=aedAB
RR aed
RR=be(d+L)
aed-be(d+L)
=RR with collimation error
=(ad+ed)-(b(d+L)+ed+L)
be(d+L)
ed+L
(ad ed) (b(d+L) ed+L)=ad-bd+L+ed-ed+L = DELappAB1 = (DELAB)-eL
eLaed
b(d+L)eded Horizontal Linesad
A
BDELAB=EL diff. A to B = ELB-ELA
( ) (b )DELAB
d
B=(ad) - (bd+L)
Elevation B
Elevation A
dL
Datum(0)
Elevation B
1st setup near A 2nd setup Near Peg BA RR=aedAB
RR aed
RR=be(d+L)
aed-be(d+L)
=(ad+ed)-(b(d+L)+ed+L)
RR=bed
For second setup do so as near to B as
( ) ( ( ) )
bed
bded
near to B as focus will allowRead and book
A
BB
L d
1st setup near A 2nd setup Near Peg BA RR=aed RR=ae(d+L)AB
RR aed
RR=be(d+L)
aed-be(d+L)
=(ad+ed)-(b(d+L)+ed+L)
RR=bed
RR=ae(d+L)
ae(d+L) - bed
=(a(d+L)+ed+L)-(bd+ed)
Read and book
( ) ( ( ) ) ( ( ) ) ( )
ae(d+L)
a(d+L)
ed+L
Read and book
A
BB
L d
1st setup near A 2nd setup Near Peg BA RR=aed RR=ae(d+L)AB
RR aed
RR=be(d+L)
aed-be(d+L)
=(ad+ed)-(b(d+L)+ed+L)
RR=bed
RR ae(d+L)
ae(d+L) - bed
=(a(d+L)+ed+L)-(bd+ed)
Read and book
( ) ( ( ) )=a(d+L)-bd+ed+L-ed = DELappAB2 = (DELAB)+eL
( ( ) ) ( )
ae(d+L)
a(d+L)
ed+L
Read and book
eded
eL
bd
bed
A
B
eded
DELAB=EL diff. A to B = ELB-ELA
=(a(d+L))-(bd)
bd
B
L d
aed-be(d+L) =(ad+ed)-(b(d+L)+ed+L) =ad-bd+L+ed-ed+L = DELappAB1 = (DELAB)-eL1st
2nd ae(d+L) - bed =(a(d+L)+ed+L)-(bd+ed) =a(d+L)-bd+ed+L-ed = DELappAB2 = (DELAB)+eL
Adding the 1st result to 2nd leaves 2*(DELAB)
Therefore dividing by 2 leaves DELABTherefore dividing by 2 leaves DELAB
Or Subtracting the 1st result from the 2nd leaves 2* eL
Therefore dividing by 2 leaves eL The collimation error over dist. L
Note: Even if d in the 1st setup and the 2nd setup were not equal and then ed for the 1st setup would not equal ed for the 2nd setup, they would still cancel for each setup
A
BDELAB
B
Elevation B
Elevation A
L
Elevation B
Datum(0)
eL = collimation error in Horiz. Dist. L \ eL/L= collimation error per unit length = e1
Then –e1= correction to rod reading per unit of Horiz. Dist. = c
ed+L
RR including Collim. Errorbed+L
eLed
aed =RR with error
bd+LWhat rod reading should bead =what RR should be
AThus we could correct a RR for any Horiz. Dist. (H)
H*c = correction to RR
d
BThe error over d+L (ed+l) = error over d (ed) + error over L (eL)
(ed+L)=(ed)+(eL)dL
(ed+L)=(ed)+(eL)d + L
Reciprocal Method Example L=60m, d=2mSet up as near to the A position as focus will allowSet up as near to the A position as focus will allow
1st setup near ARR=1.900 = ad + edA
aed=1.900ed = Collimation error over distance d
Read and book BS on A
ed
ad What proper reading should be= Collimation error over distance d
A
B
d
B
dL
Minimum focus distance
1st setup near ARR=1 900 = a + e = a +eA
R d d b k
RR=1.900 = ad + ed = ad +ed
RR=2.456= bd+L + ed+L = bd+L + ed + eL
1.900-2.456= -0.556
AB
= ad+ed-bd+L-ed-eL = DELapp1 = DEL-eL
be(d+L)=2.456
ed+L
Read and book FS on B
eLaed
b(d+L)eded Horizontal Linesad
A
B
2m
B
2m60m
Set up as near to the B position as focus will allow2nd setup near B
ARR=1.800 = bd + edB
A
bed=1.800bded
Read and book
A
BB
60m 2m
2nd setup near BRR=1.256= a(d+L) + ed+L = ad+L + ed + eL
RR=1.800= bd + ed = bd + ed
1.256-1.800= -0.544
AB
= ad+L + ed + eL - bd - ed = DELapp2 = DEL+eL
Read and bookae(d+L)=1.256
a(d+L)
ed+L
Read and book
eded
eL
bd
bed=1.800
A
B
eded
DELAB=EL diff. A to B = ELB-ELA
=(a(d+L))-(bd)
bd
B
60m 2m
aed-be(d+L)=1.900-2.456= -0.556 =(ad+ed)-(b(d+L)+ed+L) =ad-b(d+L)+ed-ed+L =DELappAB1=(DELAB)-eL1st
2nd ae(d+L)-bed=1 256-1 800= -0 544 =(a(d+L)+ed+L)-(bd+ed) =a(d L) bd+ed L ed =DEL AB2=(DELAB)+eL2 d ae(d+L)-bed=1.256-1.800= -0.544 =(a(d+L)+ed+L)-(bd+ed) =a(d+L)-bd+ed+L-ed =DELappAB2=(DELAB)+eL
:Adding the 1st result to 2nd leaves 2*(DELAB)
:Therefore then dividing by 2 leaves DELAB
-1.100
-0.550
:Or Subtracting the 1st result from the 2nd leaves 2* eL
:Therefore then dividing by 2 leaves eL
Th lli ti di t L 60
0.012
0.006 The collimation error over dist. L=60m
Note: Even if d in the 1st setup and the 2nd setup were not equal and then ed for the 1st setup would not equal ed for the 2nd setup, they would still cancel for each setup
0.006
A
BDELAB
B
Elevation B
Elevation A
60m
Elevation B
Datum(0)
eL = collimation error in Horiz. Dist. L = 0.006m\ eL/L= collimation error per unit length = e1=0.006m/60m=0.0001m/mTh ti t d di it f H i Di t 0 0001 /Then –e1= correction to rod reading per unit of Horiz. Dist. = c = -0.0001m/m
Thus we could correct a RR for any Horiz. Dist. (H)
H*c = correction to RR
RR including Collim Error
H*c = correction to RR
ae(d+L)
a(d+L)
ed+L
eded
eL
bd
RR with error = bed
Collim. Error
Wh t dA
B
eded bdWhat rod reading should be
What rod reading should be
B
The error over d+L:(ed+l) = error over d:(ed) + error over L:(eL)(ed+L)=(ed)+(eL)
L d
(ed+L)=(ed)+(eL)
c = -0.0001m/m = Collimation Factor = c factor
H*c = correction to RR
Ex. For previous 2nd setup reading to A : H=L+d=62m
H*c = correction to RR = 62*(-.0001) = -0.0062ma ( )=1 256 1 256+( 0 006) = 1 250 = a( )
RR including Collim Error
ae(d+L)=1.256 1.256+(-0.006) = 1.250 = a(d+L)
What the correct RR should have been
If adjusting instrument this is the
ae(d+L)
a(d+L)
ed+L
eded
eL
bd
RR with error = bed
Collim. Error
Wh t d
reading you would adjust +hair to
A
B
eded bdWhat rod reading should be
What rod reading should be
B
The error over d+L:(ed+l) = error over d:(ed) + error over L:(eL)(ed+L)=(ed)+(eL)
60m 2m(ed+L)=(ed)+(eL)
A
B
d
B
dL
A
B
d
B
dL
Corrections involved in 1 setup
HF*c=cHF
HG*c=cHG
Corrected RR @ F = f = fe + cHF = fe+HF*cCorrected RR @ F f fe cHF fe HF cCorrected RR @ G = g = ge + cHG = ge+HG*c
DEL from F to G = f - g = fe - ge + (HF – HG)*cRemember DEL = BS - FS
f g
g g ( )
BSFS
Correction @ HF=cHF f
fe
g
ge
Correction @
F@ HG=cHG
G
HF
HHF+HG
HG
DEL from F to G = f – g = fe – ge + (HF – HG)*c
Then in general For any setup
DEL = BScorrected – FScorrected = BSobserved – FSobserved + (Hto BS – Hto FS)*c
DEL = BScorrected – FScorrected = BSobserved – FSobserved + (imbalance)*c
For any section involving a number of turns (since linear)
f gBS
FS
For any section involving a number of turns (since linear)
DEL = ΣBScorrected–ΣFScorrected=ΣBSobserved–ΣFSobserved+(ΣHto BS – ΣHto FS)*c
DEL = ΣBScorrected–ΣFScorrected=ΣBSobserved –ΣFSobserved+(total imbalance)*c
Correction @ HF=cHF
Correction @
f
fe
g
ge
F@ HG=cHG
G
HF
HHF+HG
HG
Correction @ HF=cHF
Correction @
f
F@ HG=cHG
G
HF
HHF+HG
HG
Example HC=30.000m HD=70.000m
C
D
HC
HHC+HD
HD
A
B
H1+H2