Geometrical Precision ofMechanism
Mechanism Design
Geometrical Precision ofMechanism
Influence of dimension deviation ofmechanism to driven member position
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
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
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
Geometrical Precision of Mechanism
Transfer function of 4bar mechanismFreudenstein equation
For calculating of equation G(ϕ, ψ )=0 we can write
Geometrical Precision of Mechanism
Transfer function of 4bar mechanismFreudenstein equation
Geometrical Precision of Mechanism
Transfer function of 4bar mechanismFreudenstein equation
We can write Freudenstein´s equation in the form
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
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)
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
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
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
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
Geometrical Precision of Mechanism
Transfer function of crank mechanism
explicit form of transfer function
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
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