Pantograph
PANTOGRAPH
A pantograph is a four-bar linkage used to produce paths exactly similar to the ones traced out by a point on the linkage. The paths so produced are, usually, on an enlarged or reduced scale and may be straight or curved ones.
A pantograph is a geometrical instrument used in drawing offices for reproducing given geometrical figures or plane areas of any shape on an enlarged or reduced scale. It is also used for guiding cutting tools. Its mechanism is utilized as an indicator rig for reproducing the displacement of the crosshead of a reciprocating engine, which, in effect, gives the position of displacement.
Fig. (a) shows the line diagram of a pantograph in which AB = CD, BC = AD and ABCD is located/jointed in such a way that it remains in a parallelogram position. OQP is a straight line and P describes a path similar to that described by Q.
Proof: To prove that the path described by P is similar to that described by Q, let us consider similar triangles ΔOAQ and ΔOBP in which ∠BOP is common.
As AQ || BP, ∠OQA = ∠OPB
So, OA/OB = OQ/OP = AQ/BP …(3.1)
All links are rigid, form the displaced position shown in Fig. (b)
B1O = BO
D1A1 = DA
A1O = AO
and P1B1 = PB
B1A1 = BA
A1Q1 = AQ
OA1/OB1 = A1Q1/B1P1
As A1 B1 C1 D1 is also a parallelogram in displaced position,
A1D1 || B1C1 i.e. A1Q1 || B1P1
OQ1P1 is again a straight line so that ΔOA1Q1 and ΔOB1P1 is similar.
OA1/OB1 = OQ1/OP1 …(3.2)
From Equation 3.1 and 3.2, we get
OQ/OP = OQ1/OP1 because OA = OA1 and OB = OB1
Hence, OQ1 is similar to OP1 or they are parallel. Thus, path traced by P is similar to that of Q.
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