The Two Peg Test
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The Two Peg Test Balance Method
To 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 = Elevation DEL = difference in elevation = BS – FS ( one setup) = ΣBS – ΣFS (involving a number of turns) DEL = final EL – initial EL H
=H Horizontal i t l Di Distance t
Σ
= summation
sum of
e.g. ΣBS = BS1 + BS2 + BS3 . . . + BSLast
Neglecting Curvature and refraction
in 60m c+r c r = 0.0002m
Thee Level eve Instrument st u e t 2-Peg eg Test est aeL/2=RR due to Collimation error
eL/2= error in RR in Horizontal distance L/2 aL/2= what RR should be A B
aeL/2
1st setup @ Mid point RR=aeL/2
eL/2 aL/2
Remember RR means Rod Reading 2nd setup Near Peg A
Setup Level Instrument @ Mid Point of two pegs (A&B) that you have set Approximately 60m is sometimes used for L
Horizontal Line
A
B At Mid Point
L/2
L/2 L
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2
2nd setup Near Peg A
aeL/2-beL/2
beL/2 eL/2 bL/2
A
B L/2
L/2 L
Move the instrument as close to the A Peg as minimum focus will allow ( usually at most a couple of paces, sometimes less)
A B
L
Move the instrument as close to the A Peg as minimum focus will allow ( usually at most a couple of paces, sometimes less) Lets call this distance d
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2
2nd setup Near Peg A RR=aed
aeL/2-beL/2 aed =RR due to collimation error over horizontal distance d ed Error in RR over horizontal distance d ad =what RR should be
A B
d L
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2
2nd setup Near Peg A RR=aed
aeL/2-beL/2
A B
d L
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2 aeL/2-beL/2
2nd setup Near Peg A RR=aed RR=bed+L aed-bed+L
bed+L RR including Collim. Error
Collimation error in Horiz dist d+L
ed+L
bd+L What rod reading should be
A B
d L
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2
= =
aL/2+eL/2 bL/2+eL/2
aeL/2-beL/2
=
aL/2-bL/2 \ corrected Elevation difference =DEL The error cancelled
aeL/2
beL/2
eL/2
eL/2
aL/2
bL/2
A
B L/2
L/2 L
2nd setup Near Peg A A B
RR=aed = ad + ed = ad +ed RR=bed+L= bd+L + ed+L = bd+L + ed + eL ad-bd+L-eL = DELapp = DEL-eL
aed-bed+L
bed+L RR including Collim. Error
aed =RR with error ed
eL
ad =what RR should be
What rod reading should be
bd+L
DEL-DELapp= eL = collimation error in Hor. Dist L
A
i.e. first setup (mid point) results – 2nd setup results =eL The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d
ed+L
d+L
(ed+L)=(ed)+(eL)
L
B
DEL-DELapp= eL = collimation error in Hor. Dist L i.e. first setup (mid point) results (RR@A–RR@B) – 2nd setup (Near A) results (RR@A-RR@B) =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 bed+L RR including Collim. Error
aed =RR with error ed
eL
ad =what RR should be
A
What rod reading should be
ed+L
bd+L
Thus we could correct a RR for any Horiz. Dist. (H) H*c = correction to RR
B The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d
d+L
(ed+L)=(ed)+(eL)
L
Example L = 60m A B
1st setup @ Mid point RR=0.700 = aL/2+eL/2 RR=1.250 = bL/2+eL/2
0.700-1.250 =
aL/2+eL/2-bL/2-eL/2 \ corrected EL diff = -0.550m
=DEL from A to B The error cancelled 0 700 0.700
1 250 1.250
eL/2
eL/2
aL/2
bL/2
A DEL
= -0.550m
B L/2
L/2 L=60
Example d=2m
2nd setup Near Peg A A B
RR=0.900 = ad + ed = ad +ed RR=1.456= bd+L + ed+L = bd+L + ed + eL 0.900-1.456= -0.556
= ad+ed-bd+L-ed-eL = DELapp = DEL-eL 1.456 RR including Collim. Error
0.900 =RR with error ed
eL
ad =what RR should be
What rod reading should be
bd+L
DEL-DELapp= eL = collimation error in Hor. Dist L
A
i.e. first setup (mid point) results – 2nd setup results =eL
d=2
ed+L
The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d+L
(ed+L)=(ed)+(eL)
L=60
B
DEL-DELapp = (-0.550)-(-0.556) = eL = collim. err. in L = 0.006m i.e. first setup (mid point) results (RR@A–RR@B) – 2nd setup (Near A) results (RR@A-RR@B) =eL = collimation RR error in Horiz. Dist. L \ eL/L= collimation error per unit length = e1 = 0.006/60 = 0.0001m/m Then –e1= correction to rod reading per unit of Horiz. Dist. = c = -0.0001m/m bed+L RR including Collim. Error aed =RR with error ed
eL
ad =what RR should be
A
What rod reading should be
ed+L
bd+L
Thus we could correct a RR for any H H*c = correction to RR
B d=2
The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d+L
(ed+L)=(ed)+(eL)
L=60
Example
c = -0.0001m/m
Correction to RR = H*c
2*(-0.0001)= -0.0002 = Cor. To RR aed 0 0062 0.0062
ad = 0.900 + (-0.0002) = 0.8998
bed+L=1.456 RR including Collim. Error
aed=0.900 =RR with error ed=0.0002
0.006=eL
ad =what RR should be = 0.8998
ed+L
What rod reading should be 1.4498 = bd+L
62*( 0 0001)= -0.0062 62*(-0.0001)= 0 0062 = Cor. Cor To RR bed+L bd+L=1.456 + (-0.0062) = 1.4498
A
B d=2
The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d+L
(ed+L)=(ed)+(eL)
L=60
Example
Corrected results A ad = 0.8998 B bd+L= 1.4498 DEL
0 0062 0.0062
= -0.5500 = ad-bd+L = corrected Elevation Difference bed+L=1.456 RR including Collim. Error
aed=0.900 =RR with error ed=0.0002
0.006=eL
ad =what RR should be = 0.8998
ed+L
What rod reading should be 1.4498 = bd+L If adjusting x-hair (to this value)
A DEL
= -0.550
B d=2
The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d+L
(ed+L)=(ed)+(eL)
L=60
Corrections involved in 1 setup
HF*c=cHF HG*c=cHG Corrected RR @ F = f = fe + cHF = fe+H HF*cc Corrected RR @ G = g = ge + cHG = ge+HG*c Remember DEL = BS - FS DEL from F to G = f - g = fe - ge + ((HF – HG))*c FS
BS
fe
Correction @ HF=cHF
ge g
f
Correction @ HG=cHG
F G HF HG HF+HG
Corrections involved in 1 setup
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)
DEL = ΣBScorrected–ΣFScorrected=ΣBSobserved–ΣFSobserved+(ΣHto BS – ΣHto FS)*c DEL = ΣBScorrected–ΣFScorrected=ΣBSobserved –ΣFSobserved+(total imbalance)*c fe
BS
Correction @ HF=cHF
FS ge g
f
Correction @ HG=cHG
F G HF HG HF+HG
Correction @ HF=cHF
f
Correction @ HG=cHG
F G HF HG HF+HG
Example HC=30.000m HD=70.000m
C D HC HD HC+HD
A B
H1+H2
To 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 = Elevation DEL = difference in elevation = BS – FS ( one setup) = ΣBS – ΣFS (involving a number of turns) DEL = final EL – initial EL H
=H Horizontal i t l Di Distance t
Σ
= summation
sum of
e.g. ΣBS = BS1 + BS2 + BS3 . . . + BSLast
Neglecting Curvature and refraction
in 60m c+r c r = 0.0002m
Thee Level eve Instrument st u e t 2-Peg eg Test est aeL/2=RR due to Collimation error
eL/2= error in RR in Horizontal distance L/2 aL/2= what RR should be A B
aeL/2
1st setup @ Mid point RR=aeL/2
eL/2 aL/2
Remember RR means Rod Reading 2nd setup Near Peg A
Setup Level Instrument @ Mid Point of two pegs (A&B) that you have set Approximately 60m is sometimes used for L
Horizontal Line
A
B At Mid Point
L/2
L/2 L
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2
2nd setup Near Peg A
aeL/2-beL/2
beL/2 eL/2 bL/2
A
B L/2
L/2 L
Move the instrument as close to the A Peg as minimum focus will allow ( usually at most a couple of paces, sometimes less)
A B
L
Move the instrument as close to the A Peg as minimum focus will allow ( usually at most a couple of paces, sometimes less) Lets call this distance d
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2
2nd setup Near Peg A RR=aed
aeL/2-beL/2 aed =RR due to collimation error over horizontal distance d ed Error in RR over horizontal distance d ad =what RR should be
A B
d L
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2
2nd setup Near Peg A RR=aed
aeL/2-beL/2
A B
d L
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2 aeL/2-beL/2
2nd setup Near Peg A RR=aed RR=bed+L aed-bed+L
bed+L RR including Collim. Error
Collimation error in Horiz dist d+L
ed+L
bd+L What rod reading should be
A B
d L
A B
1st setup @ Mid point RR=aeL/2 RR=beL/2
= =
aL/2+eL/2 bL/2+eL/2
aeL/2-beL/2
=
aL/2-bL/2 \ corrected Elevation difference =DEL The error cancelled
aeL/2
beL/2
eL/2
eL/2
aL/2
bL/2
A
B L/2
L/2 L
2nd setup Near Peg A A B
RR=aed = ad + ed = ad +ed RR=bed+L= bd+L + ed+L = bd+L + ed + eL ad-bd+L-eL = DELapp = DEL-eL
aed-bed+L
bed+L RR including Collim. Error
aed =RR with error ed
eL
ad =what RR should be
What rod reading should be
bd+L
DEL-DELapp= eL = collimation error in Hor. Dist L
A
i.e. first setup (mid point) results – 2nd setup results =eL The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d
ed+L
d+L
(ed+L)=(ed)+(eL)
L
B
DEL-DELapp= eL = collimation error in Hor. Dist L i.e. first setup (mid point) results (RR@A–RR@B) – 2nd setup (Near A) results (RR@A-RR@B) =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 bed+L RR including Collim. Error
aed =RR with error ed
eL
ad =what RR should be
A
What rod reading should be
ed+L
bd+L
Thus we could correct a RR for any Horiz. Dist. (H) H*c = correction to RR
B The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d
d+L
(ed+L)=(ed)+(eL)
L
Example L = 60m A B
1st setup @ Mid point RR=0.700 = aL/2+eL/2 RR=1.250 = bL/2+eL/2
0.700-1.250 =
aL/2+eL/2-bL/2-eL/2 \ corrected EL diff = -0.550m
=DEL from A to B The error cancelled 0 700 0.700
1 250 1.250
eL/2
eL/2
aL/2
bL/2
A DEL
= -0.550m
B L/2
L/2 L=60
Example d=2m
2nd setup Near Peg A A B
RR=0.900 = ad + ed = ad +ed RR=1.456= bd+L + ed+L = bd+L + ed + eL 0.900-1.456= -0.556
= ad+ed-bd+L-ed-eL = DELapp = DEL-eL 1.456 RR including Collim. Error
0.900 =RR with error ed
eL
ad =what RR should be
What rod reading should be
bd+L
DEL-DELapp= eL = collimation error in Hor. Dist L
A
i.e. first setup (mid point) results – 2nd setup results =eL
d=2
ed+L
The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d+L
(ed+L)=(ed)+(eL)
L=60
B
DEL-DELapp = (-0.550)-(-0.556) = eL = collim. err. in L = 0.006m i.e. first setup (mid point) results (RR@A–RR@B) – 2nd setup (Near A) results (RR@A-RR@B) =eL = collimation RR error in Horiz. Dist. L \ eL/L= collimation error per unit length = e1 = 0.006/60 = 0.0001m/m Then –e1= correction to rod reading per unit of Horiz. Dist. = c = -0.0001m/m bed+L RR including Collim. Error aed =RR with error ed
eL
ad =what RR should be
A
What rod reading should be
ed+L
bd+L
Thus we could correct a RR for any H H*c = correction to RR
B d=2
The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d+L
(ed+L)=(ed)+(eL)
L=60
Example
c = -0.0001m/m
Correction to RR = H*c
2*(-0.0001)= -0.0002 = Cor. To RR aed 0 0062 0.0062
ad = 0.900 + (-0.0002) = 0.8998
bed+L=1.456 RR including Collim. Error
aed=0.900 =RR with error ed=0.0002
0.006=eL
ad =what RR should be = 0.8998
ed+L
What rod reading should be 1.4498 = bd+L
62*( 0 0001)= -0.0062 62*(-0.0001)= 0 0062 = Cor. Cor To RR bed+L bd+L=1.456 + (-0.0062) = 1.4498
A
B d=2
The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d+L
(ed+L)=(ed)+(eL)
L=60
Example
Corrected results A ad = 0.8998 B bd+L= 1.4498 DEL
0 0062 0.0062
= -0.5500 = ad-bd+L = corrected Elevation Difference bed+L=1.456 RR including Collim. Error
aed=0.900 =RR with error ed=0.0002
0.006=eL
ad =what RR should be = 0.8998
ed+L
What rod reading should be 1.4498 = bd+L If adjusting x-hair (to this value)
A DEL
= -0.550
B d=2
The error over d+L (ed+l) = error over d (ed) + error over L (eL)
d+L
(ed+L)=(ed)+(eL)
L=60
Corrections involved in 1 setup
HF*c=cHF HG*c=cHG Corrected RR @ F = f = fe + cHF = fe+H HF*cc Corrected RR @ G = g = ge + cHG = ge+HG*c Remember DEL = BS - FS DEL from F to G = f - g = fe - ge + ((HF – HG))*c FS
BS
fe
Correction @ HF=cHF
ge g
f
Correction @ HG=cHG
F G HF HG HF+HG
Corrections involved in 1 setup
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)
DEL = ΣBScorrected–ΣFScorrected=ΣBSobserved–ΣFSobserved+(ΣHto BS – ΣHto FS)*c DEL = ΣBScorrected–ΣFScorrected=ΣBSobserved –ΣFSobserved+(total imbalance)*c fe
BS
Correction @ HF=cHF
FS ge g
f
Correction @ HG=cHG
F G HF HG HF+HG
Correction @ HF=cHF
f
Correction @ HG=cHG
F G HF HG HF+HG
Example HC=30.000m HD=70.000m
C D HC HD HC+HD
A B
H1+H2
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