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\begin{document}

\linenumbers

\title{Synthesis of PR/RP-chain based compliant mechanisms - 
Design of applications exploiting fibre reinforced material characteristics}

\Author[1]{Uwe}{Hanke}
\Author[2]{Erwin-Christian}{Lovasz}
\Author[1]{Marco}{Zichner}
\Author[1]{Niels}{Modler}
\Author[1]{Andrei}{Comsa}
\Author[1]{Karl-Heinz}{Modler}
%\authorrunning{P. Flores and F. Viadero}
% Use \authorrunning{Short Title} for an abbreviated version of
% your contribution title if the original one is too long
\affil[1]{TU Dresden, Germany} \\
\affil[2]{Politehnica University Timi\c{s}oara, Romania}





\runningtitle{Synthesis of PR/RP-chain based compliant mechanisms} 


\runningauthor{Hanke, U.}

\correspondence{Hanke (uwe.hanke@tu-dresden.de)}




\received{}
\pubdiscuss{} %% only important for two-stage journals
\revised{}
\accepted{}
\published{}

%% These dates will be inserted by Copernicus Publications during the typesetting process.


\firstpage{1}

\maketitle



\begin{abstract}
Compliant mechanisms have several advantages, especially smaller
number of elements and therefore less movable joints. The flexural members
furthermore allow an integration of special functions like balancing or
locking. Synthesis methods based on the rigid body model
(\cite{howell,Soenmez:2008}) or topology optimization (\cite{Zhou:13}) bring
the advantages of compliant elements into applications. A
robust synthesis tool is the geometrical approach (\cite{Hanke:MeTrApp2013})
concentrating on solving guidance tasks by RR-chain\footnote{one link with two
rotational joints (\textbf{R})} based compliant linkages with maximum design
freedom within the installation space. This step-by-step synthesis procedure is
extended to PR\footnote{one link with one frame fixed prismatic joint
(\textbf{P}) and one moving rotational joint (\textbf{R})}/RP\footnote{one link
with one frame fixed rotational joint (\textbf{R}) and one moving prismatic
joint (\textbf{P})}-chain based compliant linkages. The necessary analysis of
the compliant beam element can be done by numerical analysis as well as through
experiments and the elastic similitude. The synthesis method is presented and
discussed by using an application for a cup holder mechanism made of fiber
reinforced material and comparing this results to the result given in
(\cite{Hanke:MeTrApp2013}).

\end{abstract}

\keywords{Compliant mechanisms, large deflections, synthesis}


\introduction  %% \introduction[modified heading if necessary]
Classical linkage structures need additional elements (i.e. springs) for
balancing or locking, which gives rise to the structure complexity. These
features can be directly implemented by using compliant linkages. This
function integration used in a compliant cup holder mechanism
(\cite{Hanke:MeTrApp2013}) lead to a simple two part assembly
(RR-chain coupled to a beam element (B): RRB-mechanism, s. Fig.~\ref{fig:CH}).
The integrated compliant section allows the implementation of locking, so that additional
springs are not required. The applied geometric based synthesis method, allows a
direct use in the design process providing maximum design freedom regarding
position and size of the mechanism.

This paper will extend the advances of this geometrical method to RP- and
PR-chains and discuss differences to the RR-chain based linkages.

\ifpdf
\begin{figure}[ht]
\center
\includegraphics[width=12cm]{fig_CH.pdf} % links unten rechts oben
\caption{Compliant cup holder mechanism (\cite{Hanke:MeTrApp2013}): a) closed
position b) open position c) kinematic scheme of the RRB-mechanism d) RRB/PRB
and RPB-mechanism}
\label{fig:CH}
\end{figure}
\fi



\section{Synthesis}

The compliant beam section $B_0B$ of the mechanisms shown in
Fig.~\ref{fig:CH}c) is integrated in the guidance link $AB$ and acts
like a spring element. For the synthesis model the large deflections of this
beam section are introduced by a force/moment load, defined through the
coupled joint pair. This can be a RR-, PR- or RP-joint pair. Its direct
implementation into a graphical synthesis method is problematic. By using the
elastic similitude of the beam a graphical implementation is possible. The
result is a synthesis method with two serial coupled synthesis
steps (\cite{Hanke:MeTrApp2013}):
\begin{itemize}
  \item synthesis of the RR/PR/RP-chain and
  \item analysis and similiarity transformation of the compliant beam element.
\end{itemize} 
The synthesis of the RR/PR/RP-chain define the load configuration of the beam
element. For the kinematic solution of the beam element two parameter can be
used to define the load case. These parameter are direct coupled to the scaling
and orientation of the beam element.

\subsection{Graphical 2-pose synthesis of PR- and RP-chains}
RR-, PR- and RP-chains are link systems with three links $a, b, f$. The further
example will deal with $f$ as frame link, $a$ as guidance link and $b$ as
intermediate link. The graphical algorithm solving the two pose task
is the construction of the relative pole triangle, for a given vertex of this
triangle (\cite{Modler:1990,McCharty:2000}). In the case of a RR-chain three
parameter can be arbitrarily chosen. The triangle of the PR- and RP-chain
degenerate to a triangle with two parallel sides and one vertex at infinity.
Therefore the synthesis of this special chains reduces to two parameters.

By introducing the rules refering to the Appendix the two pose synthesis of
the PR- and RP-chain can be done by the synthesis steps listet in Table
\ref{tab:synthesis-steps} refering Fig.~\ref{fig:PR-RP-chainsynthesis}.

\begin{table}
\caption{Synthesis of the RP- and PR-chain}
\label{tab:synthesis-steps}
\centering
\begin{tabular}{p{0.5cm}p{5.5cm}p{5.5cm}}
\svhline\noalign{\smallskip}
step & PR-chain & RP-chain  \\
\hline\noalign{\smallskip}
1 & set moving joint $A^{ab}_{1}$ & set rotatory ground joint $A^{bf}_{0}$\\
2 & draw line $a_1$ defined by joint $A^{ab}_{1}$ and pole $P^{af}_{12}$ & draw
line $f_1$ defined by joint $A^{bf}_{0}$ and pole $P^{af}_{12}$\\
3 & draw line $f_1$ defined by $P^{af}_{12},\alpha^{af}_{12}$ and line $a_1$ &
draw line $a_1$ defined by $P^{af}_{12},\alpha^{af}_{12}$ and line $f_1$\\
4 & draw line $b_1$ defined by $A^{ab}_{1},\alpha^{ab}_{12}=\alpha^{af}_{12}$
and line $a_1$ & draw line $b_1$ defined by
$A^{bf}_{0},\alpha^{bf}_{12}=\alpha^{af}_{12}$ and line $f_1$ \\
5 & the direction of translation of the prismatic pair $ab$ is perpenticular to
line $b_1$ & the direction of translation of the prismatic pair $bf$ is
perpenticular to line $b_1$\\ \svhline\noalign{\smallskip}
\end{tabular}
\end{table}

\begin{figure}
\center
\includegraphics[width=10cm]{fig_PR-RP-synthesis.pdf}
\caption{Two pose synthesis of the: a) PR-chain
b) RP-chain}\label{fig:PR-RP-chainsynthesis}
\end{figure}

The scheme in Fig.~\ref{fig:PR-RP-chainsynthesis} is statically balanced in
position $1$, when the force action line of the between link $a$ and $f$ is
equal to link line $b_1$. This fact is of fundamental
importance for the implementation of the compliant beam element. The shortest
distance to $b_1$ measured from $P^{af}_{12}$ is given by vector $\overrightarrow{r}$. 

\subsection{Synthesis of the compliant beam element by using the elastic
similitude}

The deflection of the compliant beam depends on its material parameter (Young's
modulus $E$), geometrical parameters (the cross section with the area moment of
inertia $I$ and beam length $L$) and the load conditions.

\ifpdf
\begin{figure}[ht]
\center
\includegraphics[width=10cm]{fig_beam.pdf}\\
a)\hspace{5cm}b)\\ % links unten rechts oben
\caption{Cantilevered compliant beam under external load: a) beam model b)
internal moment $M$}
\label{fig:beam}
\end{figure}
\fi

The mathematical description of large-deflection behavior of cantilevered beams is subject to 
several scientific publications. The following statements and equations are
based upon the considerations of \textsc{Howell}
(\cite{Bisshopp:1943,howell,Guimin:13,Campanile:2008}).

The Bernoulli-Euler equation is the vital element when dealing with large-deflection analysis. 
It says that for constant material parameter $E,I$ the curvature $\kappa$ is
proportial to the internal beam moment:

\begin{equation}
\kappa = \frac{d \vartheta}{ds}  = \frac{M}{EI} .
\label{eq:bernoulli}
\end{equation}

A constant internal moment means a constant curvature with radius
$r=EI/M$ (\cite{howell,mse}). This case is according to Fig.~\ref{fig:beam}
equal to $d\rightarrow\infty$. For all other cases the internal moment reads $M=-F\cdot
d+F(\sin\Phi x+\cos\Phi y)$. Hence it makes sense to bring Eq.
(\ref{eq:bernoulli}) in the second order form (\cite{Holst:2011}) and use the
energy method (\cite{Zelenika:12})to establish the elliptic integral

\begin{equation}
\frac{1}{\sqrt{2}}\sqrt{\frac{F}{EI}}L  =
\int\limits_{\vartheta_1=0}^{\vartheta_2}\frac{d
\vartheta}{\sqrt{\cos(\Phi+\vartheta)+C}} ,
\label{eq:integral}
\end{equation}

where $L$ is the beam length and $C$ a integration constant. The constant $C$
is directly coupled to the intial conditions ont the clamped side of the beam:
$C=Fd^2/(2EI)-\cos\Phi$. For a set of known parameters $F,d,E,I$ parameters the
integral (\ref{eq:integral}) can be resolved numericaly (s.
\cite{howell,Campanile:2008,Kimball:2002,Venanzi:2005}).

Using the similitude theory (\cite{Kerle:BWT2006,Weber:1930}) a more general
way of handling compliant beam elements can drawn. The idea behind the synthesis
is using the elastic similitude theory to implement a precalculated beam
element. The elastic similitude is defined by the \textsc{Hooke}-number:

\begin{equation}
	\textrm{Ho}=\frac{F}{EL^2}\quad .
\label{eq:Ho}
\end{equation} 

In this case a scaling of dimensions affects all dimensions and therefore the
section modulus $I$ as well. For the compliant beam element, bending is the
major load case. So it is possible to use a modified \textsc{Hooke}-number

\begin{equation}
	\textrm{mHo}=\frac{FL^2}{EI}\quad ,
\label{eq:mHo}
\end{equation}

where the section modulus $I$ can be independently scaled. This number can be
directly seen in Eq. (\ref{eq:integral}) (s.
$\sqrt{\frac{F}{EI}}L=\sqrt{\textrm{mHo}}$). A set of different integration
constants $C$ lead to a set of deflected model beams with different modified
Hooke numbers $\textrm{mHo}$. These model beams are the base to start the
synthesis.

The two positions of a undeflected and the deflected beam end define the pole
of rotation $P^{af}_{12}$. This pole is coupled to the load conditions
of the beam. This load conditions are defined by two parameters: the load
direction $\Phi$ and the distance $d$, defining the moment introduced at the
beam end. These parameter define the line $f_F$ of action of the force $F^{af}$.
The shortest distance between the pole $P^{af}_{12}$ and $f_F$ is given by the
vector $\overrightarrow{r}$ (s. Fig.~\ref{fig:beam}a and
Fig.~\ref{fig:beam_deflection}a,b). This vector is used to find the dimensions
of the compliant beam element by using a similarity transformation to scale and
orient the beam according to the derived PR/RP-chain.

\ifpdf
\begin{figure}[ht]
\center
\includegraphics[width=7cm]{fig_beam_calc.pdf} 
\includegraphics[width=6cm]{fig_beam_model.pdf} \\
\begin{tabular}{p{5.5cm}p{4cm}}
a) & b)
\end{tabular}% links unten rechts oben
\caption{Deflection analysis of a cantilevered beam: a) calculation for
different load sets b) measurement using a simple test bench }
\label{fig:beam_deflection} 
\end{figure}
\fi

The elastic similitude makes no difference if the deflections are calculated
(s. Fig.~\ref{fig:beam_deflection}a) or found by experiment (s.
Fig.~\ref{fig:beam_deflection}b). The rough measurement with the simple test
bench still give results usefull doing a further synthesis. In this case a more
advanced tool kit (\cite{Limaye}) could bring more exact results. Doing
measurements at a beam model is a fast and easy way to get to reliable
solutions. In this special case the implementation is done by
use of the \textsc{Matlab} progamming environment. The geometric algorithms are
based object  oriented programming with three objects: points, lines and
circles with their related methods and the beam is implemented by solving the
Eq. (\ref{eq:bernoulli}) numerically as initial value problem for a normalized
beam model.

\section{Application}

The described synthesis method will be used to derive a cup holder mechanism as
we can find in mobile vehicles (\cite{Corves:12}). This special application is feature
implemented in luxure cars to fix a cup during driving. The loacation of this
feature has to be in reaching distance to the driver. While not using this the
cup holder shouldn't be visible hiding behind the board. The aim is the
reduction of parts via integration of all locking functions within the
compliance of the mechanism structure and discuss practical differences to the
RR-chain based mechanism shown in (\cite{Hanke:MeTrApp2013}).

The synthesis task is a two pose task refering to (\cite{Hanke:MeTrApp2013}, s.
Table \ref{tab:sol}). In the initial position $a_1$ the cupholder should be
hiding behind the front board. The second position $a_2$ offers a panel with a
circular hole to place the cup. When the holder is not in use the frontend
should flush with the surface of the board. All joints should be located behind
the board.

\ifpdf
\begin{figure}[ht]
\center
\includegraphics[width=5.5cm]{PR_sol.pdf} 
\includegraphics[width=5.5cm]{RP_sol.pdf} \\
\begin{tabular}{p{5.5cm}p{4cm}}
a) & b)
\end{tabular}% links unten rechts oben
\caption{Task solution for compliant: a) PRB-linkage b)
RPB-linkage}
\label{fig:PR-RP-solution} 
\end{figure}
\fi

\begin{table}
\caption{Synthesis parameter and results for the RPB- and PRB-mechanism}
\label{tab:sol}
\centering
\begin{tabular}{lrrrrrrrr}
\svhline\noalign{\smallskip}
 & \makebox[1cm][r]{$a_1$} & \makebox[1cm][r]{$a_2$} &
 \makebox[1cm][r]{$P^{af}_{12}$} & \makebox[1cm][r]{PR:$A^{ab}_{1}$} &
 \makebox[1cm][r]{PR:$A^{ab}_{2}$} & \makebox[1cm][r]{$B_0$} &
 \makebox[1cm][r]{$B_1$} & \makebox[1cm][r]{$B_2$}  \\
  &  &  &  &  &   \makebox[1cm][r]{RP:$A^{bf}_{0}$} &  &  & \\ 
  \hline\noalign{\smallskip} x & 0.0 & 20.0  & 10.0 & -30.0 & 50.0 & -46.2 &
  113.5 & 30.0\\ y & 10.0 & 10.0 & 20.0 & -20.0 & -20.0 & -0.0 & -0.0   &
  123.5 \\ $\varphi$ & 0.0 & 90.0 & & & & & & \\ \svhline\noalign{\smallskip}
\end{tabular}
\end{table}

The derived PRB-linkage (\textbf{P}rismatic joint, \textbf{R}otatory joint and
compliant \textbf{B}eam element) given in Fig.~\ref{fig:PR-RP-solution}a) is
characterised by a small distance of the moving joint $A^{ab}$ to the front end
of the board. The beam element is relavitely large, compared to the RR-chain
solution given in (\cite{Hanke:MeTrApp2013}). Extending this distance will
enlarge the beam and therefore lead to difficulties in the linkage design and its
stability. Thermoplastic fibre reinforced material (FRM) enable the adjustment
their mechanical properties. Material with high strength, coupled with a low
Young modulus $E$ is extremly qualified for the use in compliant structures. A
very compact design shown in Fig.~\ref{fig:sol} is possible. 

\begin{figure}
\center
\includegraphics[width=12cm]{fig_sol.pdf}
\caption{Cup holder linkage: a) PRB-linkage model b) PRB-linkage transparent top
view a) RPB-linkage model b) RPB-linkage transparent top
view}\label{fig:sol}
\end{figure}

The use of thermoplastic matrix material enables a easy post processing by
thermobending. Using preprocessed FRM preforms (s.Fig.~\ref{fig:sol_assembly}a),
the final processing in manufacturing this linkage is a three step folding
processes (s. Fig.~\ref{fig:sol_assembly}b,c,d) with the final assembly of one
bolt.

\begin{figure}
\center
\includegraphics[width=10cm]{fig_sol_assembly.pdf}
\caption{Processing and assembly of a PRB-linkage: a) preform b) 1 folding
step c) second folding step d) bolt assembly}\label{fig:sol_assembly}
\end{figure}

As seen in Fig.~\ref{fig:PR-RP-solution}b) and Fig.~\ref{fig:sol}c,d) the
RPB-linkage is similiar to the discused PRB-linkage. Differences between these
solutions are in design and kinematic behavior. Both linkages are less complex than
the RR-chain based solution given in (\cite{Hanke:MeTrApp2013}). Instead of two
bolts and one link arm, the PRB/RPB-linkage only use one bolt to complete the
linkage.




\conclusions  %% \conclusions[modified heading if necessary]
Two parameters defining the PR/RP-chain and one the size of the beam element,
give three free parameter that allow the geometrical synthesis for
PR/RP-chain based mechanisms with a compliant beam element. Finally, it has been
shown that the extension of the synthesis scheme given in
(\cite{Hanke:MeTrApp2013}) to PR/RP-chain based compliant linkages is possible.
These structures are less complex and characterised by a simplified design.






\appendix
\section{Index rules}
Using graphical synthesis methods it is useful to name links with small letters
and joints/ poles as big letters. Motions are defined by link positions and
the reference link from where this movement is measured. Handling
definite positions we have to deal with link pairs and position pairs. Therefore
all kinematic symbols are marked with indices $P^{af}_{12}$ ($af:$ link
index $\to$ $a$ moves in reference to $b$; $12:$ pose index $\to$ position $1$
moves to position $2$). This nomenclature leads to the rules given in
Fig.~\ref{fig:rules}.
%\newpage
%$ P^{af \quad \to \textrm{link index:} a \textrm{ moves in reference to }
%f}_{12 \quad	\to \textrm{pose index: position } 1 \textrm{ moves to position }
%2}$

\ifpdf
\begin{figure}[ht]
\center
\includegraphics[width=8cm]{fig_RPD.pdf}
\caption{Rules and nomenclature for the graphical synthesis demonstrated at a
two pose RR-chain example}
\label{fig:rules} 
\end{figure}
\fi




\begin{acknowledgement}
The authors would like to express their gratitude towards the Deutsche Forschungsgemeinschaft 
(DFG), which supports this research within the scope of the subproject D2 of the Collaborative 
Research Centre SFB 639 "Textile-Reinforced Composite Components in Function-Integrating 
Multi-Material Design for Complex Lightweight Applications".
\end{acknowledgement}



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% Use the following syntax and markup for your references

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