Computerized tolerance analysis of disk cam mechanisms with a roller follower

Abstract

The tolerance analysis is one of the key elements of design and manufacturing of cam mechanisms. This paper presents a computerized method for analyzing the tolerances in disk cam mechanisms with a roller follower. By employing the concept of simulated higher-pair contact analysis, the kinematic deviation of the follower motion arising from the tolerance amount of each design parameter can be determined numerically. The offset translating roller follower and the oscillating roller follower cases are given to illustrate the proposed method. The tolerance analysis results show that, owing to the combined effects of various design parameters, the position accuracy of the follower motion may degrade considerably. But, the tolerance amounts of the design parameters may also have a compensating effect to enhance the relative accuracy of the follower motion between its high and low dwell positions. Hence, it is possible to find the optimal tolerance combination for balancing the kinematic accuracy and the tolerance amounts of disk cam mechanisms with a roller follower.

Appendix

In reference to Fig. 5, the steps of implementing the computation process for the simulated contact analysis method are described as follows.

• Step 1 Read data of design parameters and their corresponding errors (or tolerances) of the cam mechanism and also the program control parameters Δθ, ε₁ and ε₂. For the translating follower case, the values of r _b , r _f and e should be read, and for the oscillating follower case, those of r _b , r _f , f and l should be read; the correspondingly specified error (or tolerance) amounts (Δr, Δr _f , Δe) or (Δr, Δr _f , Δf, Δl) are also read. The parameter Δθ is a specified increment of the cam rotation angle, and the parameters ε₁ and ε₂ are small specified and positive numbers. Then, let θ = 0°.

• Step 2 Calculate the theoretical displacement of the follower ψ(θ). In this step, the value of S(θ) is specified or calculated first. Then, for the translating follower case, ψ(θ) = L(θ), whose value can be calculated by using Eq. 5. Also, for the oscillating follower case, ψ(θ) = ξ(θ), whose value can be calculated by using Eq. 11.

• Step 3 Input estimated values of parameters θ*, ψ* and υ*, the three unknowns to be solved. For the translating follower case, the initial guesses are suggested by θ* = θ, $\psi^{*} = \psi (\theta ) = L(\theta )$ and υ* = ϕ(θ), in which, ϕ(θ) can be calculated by using Eq. 4. For the oscillating follower case, the initial guesses are suggested by θ* = θ, $\psi^{*} = \psi (\theta ) = \xi (\theta )$ and υ* = ϕ(θ), in which, ϕ(θ) can be calculated by using Eq. 12.

• Step 4 Calculate R _T (θ*) by using Eq. 1 for the translating follower case or by using Eq. 7 for the oscillating follower case. Also, calculate R′ _T (θ*) by using a numerical differentiation algorithm. Then, calculate R _R (θ*) and n _R (θ*) by using Eqs. 14 and 15, respectively.

• Step 5 Let R _C (θ, θ*) = R _R (θ*) and n _C (θ, θ*) = n _R (θ*) as those shown in Eqs. 23 and 24, respectively. Then, for the translating follower case, calculate R _F (θ, ψ*, υ*) and n _F (θ, ψ*, υ*) by using Eqs. 35 and 36, respectively. Also, for the oscillating follower case, calculate R _F (θ, ψ*, υ*) and n _F (θ, ψ*, υ*) by using Eqs. 40 and 41, respectively.

• Step 6 Substitute the values of R _C (θ, θ*), n _C (θ, θ*), R _F (θ, ψ*, υ*) and n _F (θ, ψ*, υ*) into Eqs. 29–31. Then, let F be the vector of the calculated function values, that is,

  $$ {\mathbf{F}} = \left\{ {\begin{array}{*{20}l} {F_{1} } \\ {F_{2} } \\ {F_{3} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}l} {R_{Cx} (\theta , \theta^{ * } ) - R_{Fx} (\theta , \psi^{ * } , \upsilon^{ * } )} \\ {R_{Cy} (\theta , \theta^{ * } ) - R_{Fy} (\theta , \psi^{ * } , \upsilon^{ * } )} \\ {n_{Cx} (\theta , \theta^{ * } )n_{Fy} (\theta , \psi^{ * } , \upsilon^{ * } ) - n_{Cy} (\theta , \theta^{ * } )n_{Fx} (\theta , \psi^{ * } , \upsilon^{ * } )} \\ \end{array} } \right\} . $$

  (48)

  Check whether ‖F‖ ≤ ɛ₁ or not. If so, go to step 7, or else, go to step 8.

• Step 7 Calculate the displacement error of the follower ΔS(θ) by using Eq. 37 for the translating follower case or by using Eq. 42 for the oscillating follower case. Then, print ΔS(θ) and update the cam rotation angle by θ = θ + Δθ. Check whether θ > 360^∘ or not. If so, stop the computation process. Else, go to step 2.

• Step 8 Calculate the Jacobian matrix of vector F by

  $$ {\mathbf{J}} = \left[ {\begin{array}{*{20}c} {\partial F_{1} /\partial \theta^{ * } } & {\partial F_{1} /\partial \psi^{ * } } & {\partial F_{1} /\partial \upsilon^{ * } } \\ {\partial F_{2} /\partial \theta^{ * } } & {\partial F_{2} /\partial \psi^{ * } } & {\partial F_{2} /\partial \upsilon^{ * } } \\ {\partial F_{3} /\partial \theta^{ * } } & {\partial F_{3} /\partial \psi^{ * } } & {\partial F_{3} /\partial \upsilon^{ * } } \\ \end{array} } \right], $$

  (49)

  in which, employing a numerical differentiation algorithm for calculating the partial differential terms is suggested.

• Step 9 Calculate the vector of updating values for the unknowns by

  $$ {\varvec{\Updelta}} = \left\{ {\begin{array}{*{20}c} {\Updelta \theta^{ * } } \\ {\Updelta \psi^{ * } } \\ {\Updelta \upsilon^{ * } } \\ \end{array} } \right\} = - {\mathbf{J}}^{ - 1} {\mathbf{F}}. $$

  (50)

  Check whether $\left\| {\varvec{\Updelta}}\right\| \le \varepsilon_{2}$ or not. If so, go to step 3 for inputting new estimated values of parameters θ*, ψ* and υ*. Else, update the parameters by

  $$ \left\{ {\begin{array}{*{20}c} {\theta^{ * } } \\ {\psi^{ * } } \\ {\upsilon^{ * } } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\theta^{ * } + \Updelta \theta^{ * } } \\ {\psi^{ * } + \Updelta \psi^{ * } } \\ {\upsilon^{ * } + \Updelta \upsilon^{ * } } \\ \end{array} } \right\}, $$

  (51)

  and then go to step 4.

The above steps demonstrate the use of Newton–Raphson method to solve the equations of tangency, and also to determine functions θ*(θ), ψ*(θ) and υ*(θ) numerically. After the actual follower displacement function ψ*(θ) is determined, the follower displacement error function ΔS(θ) can be correspondingly obtained. After the computation process being programmed, the follower motion deviations caused by different tolerance degrees can all be simulated.