Derivation of dynamic equation of viscoelastic manipulator with revolute–prismatic joint using recursive Gibbs–Appell formulation

Abstract

In this paper, the motion analysis of a viscoelastic manipulator with N-flexible revolute–prismatic joints is being studied with the help of a systematic algorithm. The presence of prismatic joints, along with revolute ones, makes the derivation of the equations complicated. The link’s axial motions cause variation of its flexible parts with respect to time. In order to modify the associated mode shapes concerning an instant link length, dynamic interaction between the rotary reciprocating motion and transverse vibration of the flexible arm is evaluated. The links are modeled on the assumed mode method using the Timoshenko beam theory (TBT). Dynamic equations are derived from the recursive Gibbs–Appell formulation. The formulation involves fewer mathematical calculations but shows efficient computational performance when compared to other formulations. The dynamic model of each joint shows flexibility, damping, backlash and frictions resulting in accuracy of the formulations. Applying recursive algorithm based on the $3\times 3$ rotational matrix instead of the $4\times 4$ one causes the computational performance to fall by separating the rotating matrix. Furthermore, motion equations are obtained symbolically and systematically. Links linear motion causes TBT mode shapes changes with respect to time. This is implemented in a non-dimensional form to avoid computing for each step. Finally, the following dynamic equations are solved numerically by MATLAB software for a spatial two-armed manipulator. The outcome of the simulations represents the ability of the proposed algorithm to derive and solve the equations of motion. Moreover, the data are compared with the rigid and elastic links, modeled by the Euler–Bernoulli beam theory.

Appendices

Appendix A

1.1 Non-dimensional TBT mode shapes calculations

Han et al. [35] studied mode shapes extraction of TBT at different boundary conditions in dimensionless forms. With the help of Eqs (52 and 53), the following expressions include the generations approach for coefficients represented in these Equations:

$$\begin{aligned}&\left[ {{\begin{array}{*{20}c} {D_{1} } \\ {\begin{array}{l} D_{2} \\ D_{3} \\ D_{4} \\ \end{array}} \\ \end{array} }} \right] =\left[ {{\begin{array}{*{20}c} 0 &{} \alpha &{} 0 &{} 0 \\ {-\alpha } &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \beta \\ 0 &{} 0 &{} \beta &{} 0 \\ \end{array} }} \right] \left[ {{\begin{array}{*{20}c} {C_{1} } \\ {\begin{array}{l} C_{2} \\ C_{3} \\ C_{4} \\ \end{array}} \\ \end{array} }} \right] ;\nonumber \\&\alpha =-\frac{a^{2}+\gamma ^{2}b^{2}}{a(1+\gamma ^{2})};\,\beta =\frac{b^{2}+\gamma ^{2}a^{2}}{b(1+\gamma ^{2})};\nonumber \\&\gamma =\sqrt{\frac{2(1+\nu )}{{k}'}} \end{aligned}$$

(A.1)

$$\begin{aligned}&a=\sqrt{(I^{*}+\frac{1}{{k}'G^{*}})\frac{\rho ^{*}\omega ^{*{2}}}{2}+\sqrt{(I^{*}+\frac{1}{{k}'G^{*}})^{2}\frac{\rho ^{*^{2} }\omega ^{*{4}}}{2}+\rho ^{*}A^{*}\omega ^{*{2}}} } ;\, \nonumber \\&b=\sqrt{-(I^{*}+\frac{1}{{k}'G^{*}})\frac{\rho ^{*}\omega ^{*{2}}}{2}+\sqrt{(I^{*}+\frac{1}{{k}'G^{*}})^{2}\frac{\rho ^{*{2}}\omega ^{*{4}}}{2}+\rho ^{*}A^{*}\omega ^{*{2}}} }\nonumber \\ \end{aligned}$$

(A.2)

where a and b are wave numbers. Dimensionless variables are shown by asterisk. Dimensionless variables introduced below, respectively:

$$\begin{aligned} L^{*}= & {} L/L=1;\,\,\,\,\,\nu ^{*}=\nu /L;\,\,\,x^{*}=x/L\nonumber \\ A^{*}= & {} A/L^{2};\,\,\,\rho ^{*}=\rho /\left( \frac{EI}{L^{6}}\right) ;\,\,\,\,t^{*}=t/(1/\omega _{1} ) \nonumber \\ I^{*}= & {} I/L^{4};\,\,\,G^{*}=G/\left( \frac{EI}{L^{4}}\right) ; \end{aligned}$$

(A.3)

For a cantilever beam, boundary conditions are represented as:

$$\begin{aligned} \left[ {{\begin{array}{*{20}c} {y(0)} \\ {\theta _{z} (0)} \\ \end{array} }} \right] \,=\left[ {{\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} }} \right] ;\,\,\,\,\left[ {{\begin{array}{*{20}c} {\frac{\partial \theta _{z} (1)}{\partial \eta }} \\ {\frac{\partial y(1)}{\partial \eta }-\theta _{z} (1)} \\ \end{array} }} \right] \,=\left[ {{\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} }} \right] ,\nonumber \\ \end{aligned}$$

(A.4)

The boundary conditions were applied on Eqs. (52 and 53). Thus, the obtained frequency equations represent as:

$$\begin{aligned}&(a^{2}-b^{2})\sin a\,\sinh b-ab\,\cos a\,\cosh b\nonumber \\&\quad \times \frac{a^{4}+\,a^{4}\gamma ^{4}+4a^{2}b^{2}\gamma ^{2}+b^{4}+\,b^{4}\gamma ^{4}}{(a^{2}+b^{2}\gamma ^{2})(b^{2}+a^{2}\gamma ^{2})}-2ab=0,\nonumber \\ \end{aligned}$$

(A.5)

And mode shapes coefficients are calculated as:

$$\begin{aligned} \left[ {{\begin{array}{*{20}c} {C_{1} } \\ {\begin{array}{l} C_{2} \\ C_{3} \\ C_{4} \\ \end{array}} \\ \end{array} }} \right] =\left[ {{\begin{array}{*{20}c} 1 \\ {-C_{4} } \\ {\frac{\alpha }{\beta }} \\ {\alpha \frac{2a\,\sin \,a\mathrm{e}^{b}+b\,\mathrm{e}^{a}-b}{2\alpha a\,\cos \,a\mathrm{e}^{b}-\beta b\mathrm{e}^{2b}-b\beta }} \\ \end{array} }} \right] \end{aligned}$$

(A.6)

Appendix B

Dependent coefficients ${\mathbf{C}}$ expressions:

$$\begin{aligned} {\mathbf{C}}_{1ij}= & {} \int _{L_{i} }^{L_{i} +\,l_{i} } {\mu _{i} \,{\mathbf{r}}_{ij} d\eta _{i} } ; \end{aligned}$$

(B.1)

$$\begin{aligned} {{\tilde{\mathbf{C}}}}_{1ij}= & {} \int _{L_{i} }^{L_{i} +\,l_{i} } {\mu _{i} \,{{\tilde{\mathbf{r}}}}_{ij} d\eta _{i} } ; \end{aligned}$$

(B.2)

$$\begin{aligned} C_{2i}= & {} \int _{l_{i} }^{\,L_{i} +l_{i} } {\mu _{i} \,\eta _{i} {}^{i}{{\tilde{\mathbf{x}}}}_{i} d\eta _{i} } \end{aligned}$$

(B.3)

$$\begin{aligned} C_{3ij}= & {} \int _{L_{i} }^{L_{i} +\,l_{i} } {\mu _{i} \,{}^{i}{\mathbf{x}}_{i}^{\mathrm{T}} \cdot {\mathbf{r}}_{ij} d\eta _{i} } ; \end{aligned}$$

(B.4)

$$\begin{aligned} C_{4ijk}= & {} \int _{L_{i} }^{\,L_{i} +l_{i} } {\mu _{i} \,{\mathbf{r}}_{ij}^{\mathrm{T}} \cdot \,{\mathbf{r}}_{ik} d\eta _{i} } ; \end{aligned}$$

(B.5)

$$\begin{aligned} {\mathbf{C}}_{5ij}= & {} \int _{L_{i} }^{\,L_{i} +l_{i} } {\mu _{i} {}^{i}{{\tilde{\mathbf{x}}}}_{i} \,{\mathbf{r}}_{ij} d\eta _{i} } \end{aligned}$$

(B.6)

$$\begin{aligned} {\mathbf{{C}'}}_{5ij}= & {} \int _{L_{i} }^{\,L_{i} +l_{i} } {\mu _{i} \eta _{i} {}^{i}{{\tilde{\mathbf{x}}}}_{i} \,{\mathbf{r}}_{ij} d\eta _{i} }; \end{aligned}$$

(B.7)

$$\begin{aligned} {\mathbf{C}}_{6ijk}= & {} \int _{L_{i} }^{L_{i} +\,l_{i} } {\mu _{i} \,{{\tilde{\mathbf{r}}}}_{ij} {\mathbf{r}}_{ik} d\eta _{i} } ; \end{aligned}$$

(B.8)

$$\begin{aligned} C_{7i}= & {} \int _{l_{i} }^{\,L_{i} +l_{i} } {\mu _{i} \,\eta _{i} {}^{i}{{\tilde{\mathbf{x}}}}_{i}^{\mathrm{T}} \,{}^{i}{{\tilde{\mathbf{x}}}}_{i} d\eta _{i} } \end{aligned}$$

(B.9)

$$\begin{aligned} {C}'_{7i}= & {} \int _{l_{i} }^{\,L_{i} +l_{i} } {\mu _{i} \,\eta _{i}^{2} {}^{i}{{\tilde{\mathbf{x}}}}_{i}^{\mathrm{T}} \,{}^{i}{{\tilde{\mathbf{x}}}}_{i} d\eta _{i} } ; \end{aligned}$$

(B.10)

$$\begin{aligned} C_{8ij}= & {} \int _{L_{i} }^{\,L_{i} +l_{i} } {\mu _{i} {}^{i}{{\tilde{\mathbf{x}}}}_{i}^{\mathrm{T}} \,\,{{\tilde{\mathbf{r}}}}_{ij} d\eta _{i} } ; \end{aligned}$$

(B.11)

$$\begin{aligned} {C}'_{8ij}= & {} \int _{L_{i} }^{\,L_{i} +l_{i} } {\mu _{i} \eta _{i} {}^{i}{{\tilde{\mathbf{x}}}}_{i}^{\mathrm{T}} \,\,{{\tilde{\mathbf{r}}}}_{ij} d\eta _{i} } \end{aligned}$$

(B.12)

$$\begin{aligned} C_{9ijk}= & {} \int _{L_{i} }^{L_{i} +\,l_{i} } {\mu _{i} \,{{\tilde{\mathbf{r}}}}_{ij}^{\mathrm{T}} {{\tilde{\mathbf{r}}}}_{ik} d\eta _{i} } ; \end{aligned}$$

(B.13)

$$\begin{aligned} C_{10ijk}= & {} \int _{L_{i} }^{L_{i} +l_{i} } {{\varvec{\uptheta }}_{ij}^{\mathrm{T}} \cdot \,J_{i} \,{\varvec{\uptheta }}_{ik} d\eta }; \end{aligned}$$

(B.14)