TECHNICAL TRANSACTIONS 8/2017

CZASOPISMO TECHNICZNE 8/2017

ARCHITECTURE
DOI: 10.4467/2353737XCT.17.124.6875

Anita Pawlak-Jakubowska (anita.pawlak-jakubowska@polsl.pl)

Krystyna Romaniak
Geometry and Engineering Graphics Centre, Silesian University of Technology

Class ii mechanisms in the construction of retractable roofs

Mechanizmy klasy ii w budowie ruchomych

przekryć dachowych

Abstract
This work presents retractable roofs of new designs based on class II mechanisms. Firstly, roofs that are
currently in existence were analysed in terms of the mechanisms involved in their construction. Then, the
possibility of constructing a roof based on the variety of quadruples in which all links are joined with roof
panels was investigated. The boundary conditions that have to be fulfilled by such a  construction were
specified. Ultimately, research models were constructed with the use of construction solutions similar to
those applied in buildings currently in existence.
Keywords: membrane roof, retractable roof, mechanisms class II

Streszczenie
W artykule przedstawiono propozycję nowych rozwiązań ruchomych dachów w oparciu o mechanizmy klasy
II. W  pierwszej kolejności przeanalizowano istniejące zadaszenia pod kątem występujących w  ich budowie
mechanizmów. W kolejnym etapie sprawdzono możliwość skonstruowania zadaszenia opartego na różnego
rodzaju czworobokach, których wszystkie człony połączone są z  panelami dachowymi. Określono warunki
brzegowe, które musi spełniać tego typu konstrukcja. Ostatecznie zbudowano modele badawcze, stosując
rozwiązania konstrukcyjne analogiczne do rozwiązań używanych w istniejących obiektach.
Słowa kluczowe: dachy membranowe, ruchome dachy, mechanizmy klasy II

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 1.  Introduction

One of the biggest challenges that 21st century designers are faced with is mobile
architecture, sometimes also described as the architecture in motion, kinetic or dynamic
architecture. Moving walls and ceilings, roofs that can retract horizontally (in and out) and
vertically (up and down), rotating facades and segments of a  construction are examples of
this type of architecture. The element of building construction that recently has undergone
the most rapid development is the retractable roof. New possibilities in terms of materials
and manufacturing technologies bring about the construction of retractable roofs that are
increasingly larger in size and retract much faster. The changing shape of the retractable cover
of a construction constitutes a key factor in its visual and esthetic perception. It is also expected
to be in harmony with the surroundings and enhance the attractiveness of the entire building.
Retractable roofs with their construction based on class II mechanisms were singled out
for the analysis. It included retractable roofs on constructions such as the Reliant Stadium in
Houston (USA), Marlins Park Stadium in Florida (USA) and Wimbledon Centre Court (UK).
The analysis of building structures with retractable roofs inspired a quest for new solutions
regarding the structure of mechanisms. With reference to already existing solutions, the class
II mechanisms were subject to thorough examination. Preliminary analysis provided all
possible solutions containing rotary and prismatic kinematic pairs for two assembly options.
The research covered constructions with roof panels directly connected with the mechanism
coupler or mounted on the coupler plane. This preliminary analysis resulted in determining
the boundary conditions following a number of conditions restricting the area of research.
The results obtained from the analysis constitute the main topic discussed in this paper.

2.  Second class mechanisms in the construction of roofs for sports stadiums

At the first stage, the retractable roofs were analysed in terms of the movement that they
performed. Two methods of movement were specified i.e. rotary and translation/forward
movement (Fig. 1). In general, roof panels are directly attached to drives, of which movement
is imposed by the shape of railings. If it is circular, the independent variable appearing in
translation equations is an angle, whereas in the case of line-shaped railings, it is a line quality.

Fig. 1. Retractable roofs with a four-bar linkage: a) Reliant Stadium in Houston, USA (source: http://
www.uni-systems.com/en/projects/featured-projects/reliant-stadium, access: 29.12.2015),
b) Marlins Park in Florida, USA (source: www.uni-systems.com, access: 29.12.2015)

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 Fig. 2. Roofs with movement: a) rotary along railings – Fukuoka Dome stadium, Japan (source:
https://upload.wikimedia.org/wikipedia/commons/8/84/Fukuoka_Seaside_Momochi_Aerial_
Shoot.jpg, access: 29.12.2015), b) translating along lines – PGE National Stadium in Warsaw
(source: https://en.wikipedia.org/wiki/National_Stadium,_Warsaw#/media/File:POL_Stadion_
Narodowy_Warszawa_09.jpg, access: 29.12.2015)

The roofs of the Reliant Stadium in Houston (USA) and Marlins Park in Florida (USA) are
constructed with the use of two four-bar-linkages that can accommodate the impact of sudden gusts
of wind (Fig. 2). Hence, each panel is able to perform movement that is transversal to the direction
in which the roof is progressing. The range of movement of the Reliant Stadium roof panels is
21.5 inches with respect to the base.

Fig. 3. The roof of Wimbledon Centre Court (UK) with 2nd class mechanisms:
a) four-bar linkage connected to the roof panels (source: https://commons.wikimedia.org/wiki/
File:Flickr_-_Carine06_-_The_roof_is_closing.jpg, access: 29.12.2015),
b) the mechanism determining the maximum distance between the panels (source: http://www.
moog.com/literature/ICD/Moog-Wimbledon_Roof_Electromechanical_Actuators-article-en.pdf,
access: 29.12.2015), c) diagram of the mechanisms (by A. Pawlak-Jakubowska)

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 Two class II mechanisms were specified in the roof of the Wimbledon Centre Court (UK).
The roof panels covered with technical materials are attached to one of the mechanisms (Fig.
3). The other mechanism maintains constant distances between trusses with membranes.
One link of this mechanism is called a stabilizing arm with its length defining the maximum
distance between girders (Fig. 3b).
Located between two drives, the two roof panels (links 2  and 3  in Fig. 3c) are inter-
connected to each other and to the lattices in a rotary manner (kinematic pairs B, C, D). The
movement of the roof panels is triggered by the motion of the first drive (link 1 in Fig. 3c)
while the other drive is blocked. This first drive is a driving device for both the roof panels
and the mechanism determining the maximum opening of the roof segment (links 4 and 5).
The stabilising arm 5 is connected with the lattice girder and link 4 (kinematic pairs F and G)
in a rotary manner. Link 4 performs a translation along the lattice girder (kinematic pair E).
Fig. 4 presents the translatory movement of the roof links located between two drives. The
movement of the driving device 1 causes links 2 and 3 to move the roof to the position, which
ensures the minimum gradient required for water drainage.

Fig. 4. Roof panels located between two driven trucks in Wimbledon Centre Court: a) open , 
b) movement, c) retracted roof (all by A. Pawlak-Jakubowska)

The analysis of construction solutions used in the retractable roofs such as Reliant
Stadium in Houston (USA), Marlins Park in Florida (USA) and Wimbledon Centre Court
(UK) inspired a search for new roof forms regarding the structure of mechanisms.

3.  Construction of research roof models based on class II mechanisms

In order to specify the possible designs for retractable membrane roofs that are linked
with class II mechanisms, a number of conditions have been considered, which have to be
fulfilled by such mechanisms. Correct functioning of the mechanisms depends on the length
of the moving links and the range of movement of the driving link. The ratio of the lengths of
individual links decides on the category the mechanism belongs to. There are three categories,
namely: double-crank, crank-rocker, drag-link and double-rocker mechanisms1.

1
This division is connected witch Grashof condition [4, p. 40-49], [5, p. 46-53].

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 In order to specify the movement performed by the roof joined with the mechanism,
all its theoretical solutions that contained translation and rotary kinematic pairs have been
considered [6, p. 95-101]. The extreme (turning) as well as idle positions were determined.
In addition, the issue of the roof - coupler joining point was covered in the research, as in the
following two cases:
▶▶ a roof panel directly joined with the coupler – then its movement defines the trajectory
of the roof movement.
▶▶ The connecting point of the roof panel is located on the so-called coupler plane – then
the roof trajectory depends on where the point is on the coupler plane.
Having done the analysis and considered that a vast area of research needs to be covered,
the discussion on the issue of the roof connection with the use of a coupler plane was omitted.
Hence, very detailed research has been carried out on roof panels directly joined to the
links of the mechanism. Exploiting a wide range of possibilities of parametric modelling in
AutoCAD, the translation of the mechanism links was determined. The obtained data was
used to define viable solutions that fulfil the following imposed boundary conditions:
1) The inclination angles of panels such as, for example, roof slopes should take values
between 10° and 85°. This assumption brings about proper water drainage and
simultaneously sorts out the issue of roofing over an enclosed area, such as a courtyard.
Criteria adopted regarding the inclination of panels (links) enable a  collision-free
movement of the roof.
2) All panels connected to links of the mechanism must function as roof slopes. Hence, if,
during a movement, one panel covers the other from the top (Fig. 5a), such a position
is eliminated from the movement range. This ondition restricted the movement range
of individual links.
3) The solutions requiring the installation of a circular gutter were eliminated. They are
cases in which the adjacent links form a V-shaped basket while moving, thus making
rainwater gather between the panels (links). The task of draining water while the
panels move horizontally is relatively straightforward, whereas doing it while the
panels move along an arch (Fig. 5b) becomes a bit of a challenge, among others, in
terms of installing an arch-shaped gutter.
4) Translation kinematic pairs are located exclusively at the base. This condition results
from the analysis of constructions with retractable roofs in which the prismatic
kinematic pairs often connect the roof with the lower link of the construction.

The obtained results were analysed in terms of the adopted boundary conditions, whereby
several mechanisms were selected for further analysis. A variety of possibilities for covering
an area with the use of roofings that harnesses two mechanisms of class II were taken into
consideration. Symmetrical and non-symmetrical mechanism connections were examined
considering the same and different types of mechanisms. Consequently, the roofs constructed
on the basis of a  symmetrical connection of two identical mechanisms were selected for
a detailed analysis due to their viable and technologically less challenging construction (two
driving devices of the same type, the links in the joined mechanisms of identical lengths and

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 Fig. 5. Class II mechanisms with a rotary driving link: a) mechanism diagram (by K. Romaniak),
b) roof panel 2 covers panel 1 from the top (by A. Pawlak-Jakubowska), c) an arch-shaped gutter
(by A. Pawlak-Jakubowska), d) restricted movement range (by A. Pawlak-Jakubowska)

the gutters can be installed at the same level). A mixed combination of class II mechanisms
imposes the use of different drives (rotary and translatory), panels of different dimensions
and, in some cases, the gutters must be installed at different levels.
A detailed analysis has been carried out for four models constructed on the basis of two class
II mechanisms that were joined together. The research method was illustrated on the example of
the mechanisms with a rotary drives, three rotary kinematic pairs and one prismatic kinematic
pair (Fig. 5a). In the two symmetrically joined mechanisms, the drives are located on the
outer edges of the roof and the prismatic kinematic pairs can be found in the central link of the
construction (Fig. 6a). Thus, the roof retracts starting from the outer edges towards the centre
of the covered area. Symmetrically located roof panels can perform a synchronised movement
or each link of the roof can move independently. The percentage rate of the covered area was
assessed following a dimensionless analysis, which mainly aimed at establishing proportional
relationships between the lengths of individual links. The following letter symbols have been
adopted: a, b, c – lengths of the moving links, d – the length of the base (Fig. 6b).
A constant angle between the base and link 3 (a shape angle) has been noted as κ.
A detailed analysis exploited the possibilities of parametric modelling available through
the AutoCAD software. For model number 1 the following assumptions were adopted:
▶▶ constant length of a driven link (coupler) b,
▶▶ the lengths of links 1 and 2 proportional to b:
▶▶ a = (i + 0.1)b, c = (i + 0.1)b, where i = 0, 0.1, …, 0.9,
▶▶ different values of a shape angle κ (85°, 60°, 45°, 30°, 15°),
▶▶ movement range of link 1 (10° - 85°).

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 Fig. 6. Model with a rotary driving link: a) open and retracted roof respectively, b) notation of the
lengths of individual links of mechanism, c) roof axonometry (all by A. Pawlak-Jakubowska)

As a result of the carried out research, the percentage values of the open area (noted as
O in Table 1) and the d values were assessed. The adopted assumption was that the satisfying
result would be the opening of 70% or more of the covered area. In addition, it was significant
to obtain as large as possible span d, which defines the area of covered space. It was agreed
that a satisfactory result would be a value greater than 1.8b. Table 1 presents a fragment of the
results obtained; in bold the data fulfilling adopted criteria. Empty spaces mean that driving
link 1 does not reach values within the full range of movement (10°-85°).

Table 1. Percentage values of the open roof space and d lengths in relation to selected lengths a and
c proportional to a given b and adopted values of angle ĸ
Angle κ = 85° Angle κ = 60° Angle κ = 45° Angle κ = 30° Angle κ = 15°
O d O d O d O d O d
[%] [%] [%] [%] [%]
c=0 56 1,68b 56 1,68b 56 1,68b 56 1,68b 56 1,68b
c=0,1b 60 1,67b 58 1,71b 56 1,74b 54 1,76b 53 1,77b

a=0,7b c=0,2b 69 1,65b 63 1,74b 59 1,79b 55 1,83b 51 1,86b

c=0,3b 90 1,63b 72 1,76b 63 1,84b 56 1,91b 50 1,95b
c=0,4b – – – – 72 1,88b 58 1,98b 50 2,05b
c=0,5b – – – – – – 61 2,05b 49 2,14b

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 Angle κ = 85° Angle κ = 60° Angle κ = 45° Angle κ = 30° Angle κ = 15°
O d O d O d O d O d
[%] [%] [%] [%] [%]
c=0 63 1,78b 63 1,78b 63 1,78b 63 1,78b 63 1,78b
c=0,1b 71 1,76b 68 1,8b 66 1,83b 63 1,85b 61 1,87b

a=0,8b c=0,2b 90 1,74b 78 1,83b 71 1,88b 65 1,93b 60 1,96b

c=0,3b – – – – – – 68 2b 59 2,05b
c=0,4b – – – – – – 77 2,07b 59 2,14b
c=0,5b – – – – – – – – 59 2,23b
c=0 73 1,87b 73 1,87b 73 1,87b 73 1,87b 73 1,87b
c=0,1b 91 1,86b 84 1,9b 80 1,93b 75 1,95b 72 1,96b
a=0,9b
c=0,2b – – – – – – 84 2,02b 72 2,05b
c=0,3b – – – – – – – – 73 2,14b

Table 2 presents selected examples of roofs constructed with the use of data highlighted in
bold in Table 1. For consecutive values of angle ĸ, the diagrams of mechanisms constituting the
base of a roof construction were presented with given lengths of individual links determined
proportionally to the b length. Horizontal projection of the roof illustrates the open space
percentage, whereas a vertical projection shows translations of individual roof panels.

Table 2. Examples of roofs fulfilling boundary conditions

Angle Values of a, b, c and d specified
No. Horizontal and vertical roof projections
κ for one mechanism

1 85°

2 60°

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 Angle Values of a, b, c and d specified
No. Horizontal and vertical roof projections
κ for one mechanism

3 45°

4 30°

For the model where a = 0,8b, c = 0,2b and angle κ = 60°, as much as 78% of the roof
open space can be obtained (Fig. 7). When specified, the translations of individual panels
determine a working space of the whole roof. By analogy, a similar analysis was carried out for
the remaining solutions presented in Table 2.

Fig. 7. Movements of: a) links 1,2 and 3, b) driving link 1, c) link 2, d) link 3, e) roof working space (all
by A. Pawlak-Jakubowska)

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 To drain water from the panels of a model with a rotary driving link, the installation of gutters
and downpipes is required. From the driving panels, the rainwater runs down to a gutter located
at the base level and then into a downpipe (Fig. 8). As the angle between panels 2 and 3 is variable,
a flexible gutter can be installed, which will take the water to a gutter that is linked to a downpipe.
This type of gutter system makes it possible for water drainage to occur in every position of the
roof, open or closed, and even when the roof stops at any moment of its movement.

Fig. 8. Water drainage from the roof (by A. Pawlak-Jakubowska)

Finally, the last stage of the research focused on digital models made with the use
of the Autodesk Inventor Professional 2016 software. This particular software provides tools
for making a model and then performing kinematic simulation.

Fig. 9. Components of the research model (by A. Pawlak-Jakubowska)

Exploiting information regarding retractable roof construction technologies [2], the

solutions for the construction of individual elements of research models were proposed (Fig. 9).

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4.  Conclusions

The quest for new designs of retractable roofs went through several stages. Firstly, all
theoretically possible solutions for class II mechanisms with rotary and translation kinematic
pairs were determined. The set of solutions was subject to numerous selections that considered
a variety of conditions. Thus, boundary conditions were specified, which should be fulfilled
by roofs joined with selected mechanisms. Finally, several solutions were selected and used to
build research models. A research model is a set of roofs built on the basis of one mechanism of
a specific structure. For each such a model, a detailed investigation was carried out in terms of
its kinematics, for example, its movement range and working area, as well as the water drainage
method. The solutions, which fulfilled adopted assumptions i.e. the percentage of the open
space over 70%, were selected. Ultimately, construction solutions for individual elements of
the models were proposed. The solutions were modelled on real roofs currently in existence.
The research carried out and the results obtained indicate the possibility of determining
new designs of retractable roofs based on structural and kinematic research of mechanisms.
In currently existing retractable roofs built on the basis of class II mechanisms, two panels are
joined with the driving links of mechanism at most. In the solutions proposed in this work,
a driving link is also joined with roof panels and functions as a roof segment.

References

[1] Gerfen K., Wimbledon Centre Court Retractable roof, The Journal of the American
Institute of Architects ARCHITECKT, 2009, http://www.architectmagazine.com/
technology/detail/wimbledon-centre-court-retractable-roof_o (access: 18.12.2015).
[2] Ishii K., Structural Design of Retractable Roof Structures, WIT Press, Southampton,
Boston, 2000.
[3] MACHINE Design by Engiers for Engineers.
[4] http://www.moog.com/literature/ICD/Moog-Wimbledon_Roof_Electromechanical_
Actuators-article-en.pdf (access: 18.12.2015).
[5] Młynarski T., Listwan A. i  Pazderski E., Teoria mechanizmów i  maszyn. Cz. III Analiza
kinematyczna mechanizmów, Wydawnictwo Politechniki Krakowskiej, Kraków 1999, 40–49.
[6] Norton R.L., Kinematics Design of Machinery: An Introduction to the Synthesis and Analysis
of Mechanisms and Machine, McGraw Hill, 1999, 46–53.
[7] Pawlak-Jakubowska A., Romaniak K., Modelowanie geometrii przekryć ruchomych,
Modelling in Engineering, 25(56)/2015, 95–101.
[8] Uni–System, www.uni-systems.com (access: 18.12.2015).

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