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