Geometrical Precision of Mechanism - KTS · Geometrical Precision of Mechanism Influence of...
Transcript of Geometrical Precision of Mechanism - KTS · Geometrical Precision of Mechanism Influence of...
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Geometrical Precision ofMechanism
Mechanism Design
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Geometrical Precision ofMechanism
Influence of dimension deviation ofmechanism to driven member position
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Transfer function
If the system has one possibility to move relatively(angle ϕ) , the relative positions of all other bodies(like angle ψ) depend on this variable. The transferof motion can be expressed with geometricalfunctions, which are called (kinematic) transferfunctions.
Geometrical Precision of Mechanism
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Geometrical Precision of Mechanism
When we have constant dimension of mechanism q1, q2, … , qn, angles ϕ and ψ determine positions of driving and driven members,
we can write transfer function in the explicit or implicit form.
explicit equation
implicit equation
In mathematics, an implicit equation is a relation of the form G(q1,..., qn) = 0, where G is a function of several variables
In mathematics, an explicit equation is a function that is written in terms of an independent variable. Example: crank mechanism
Example: four bar mechanism
Transfer function
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Geometrical Precision of Mechanism
Transfer function of 4bar mechanismFreudenstein equation
Known parameters of mechanism: a, b, c, d, ϕ, Ao
Coordinates of points A, B
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Geometrical Precision of Mechanism
Transfer function of 4bar mechanismFreudenstein equation
For calculating of equation G(ϕ, ψ )=0 we can write
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Geometrical Precision of Mechanism
Transfer function of 4bar mechanismFreudenstein equation
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Geometrical Precision of Mechanism
Transfer function of 4bar mechanismFreudenstein equation
We can write Freudenstein´s equation in the form
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Geometrical Precision of Mechanism
The transfer function with a dimension deviation is
Where Δq1, Δq2, … , Δqn – the deviations of dimensionsΔ ψ - deviation of driven member position
explicit equation
We use Taylor´s formula (we are not considering a higher orders ofa partial derivations) and we write
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Geometrical Precision of Mechanism
explicit equation
Partial derivation of a function F with respect to the variable qk
equation for calculating of deviation of driven member position(for the explicit form of transfer function)
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Geometrical Precision of Mechanism
partial derivation of a function F with respect to the variable qk
divided bypartial derivation of a function F with respect to the variable ψ
equation for calculating of deviation of driven member position(for the implicit form of transfer function)
implicit equation
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Geometrical Precision of Mechanism
The solutions of a driven member deviation of 4bar mechanism
We can use Freudenstein´s equation in the form
The solutions of a driven memberdeviation Δ ψ from influence of deviationsof dimension Δa, Δb, Δc, Δd
equation for calculating of deviation ofdriven member position
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Geometrical Precision of Mechanism
The solutions of a driven member deviation of 4bar mechanism
We can use Freudenstein´s equation in the form
deviations of dimension Δb
𝜕𝐺
𝜕𝑏= −
𝑏
𝑎𝑐
𝜕𝐺
𝜕ψ= −
𝑑
𝑎𝑠𝑖𝑛 ψ - sin(ϕ - ψ)
∆ψb = −
𝜕𝐺𝜕𝑏𝜕𝐺𝜕ψ
∆𝑏
∆ψb = −𝑏
𝑐𝑑 𝑠𝑖𝑛ψ − 𝑎𝑐 𝑠𝑖𝑛(ϕ − ψ)∆𝑏
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Geometrical Precision of Mechanism
The solutions of a driven member deviation of 4bar mechanism
We can use Freudenstein´s equation in the form
deviations of dimension Δd
𝜕𝐺
𝜕𝑑=
𝑑
𝑎𝑐+𝑐𝑜𝑠ϕ
𝑐−cosψ
𝑎
𝜕𝐺
𝜕ψ= −
𝑑
𝑎𝑠𝑖𝑛 ψ - sin(ϕ - ψ)
∆ψd = −
𝜕𝐺𝜕𝑏𝜕𝐺𝜕ψ
∆𝑑
∆ψd =𝑐 𝑐𝑜𝑠 ψ − d − a cosϕ
𝑐𝑑 𝑠𝑖𝑛ψ − 𝑎𝑐 𝑠𝑖𝑛(ϕ − ψ)∆𝑑
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Geometrical Precision of Mechanism
Transfer function of crank mechanism
explicit form of transfer function
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Geometrical Precision of Mechanism
after differentiation
The solutions of a driven member deviation of crank mechanism
Position of driven member
For small values we can write