*Corr. Author’s Address: Faculty of International Training, Thai Nguyen University of Technology, Vietnam, ngocnd@tnut.edu.vn 299
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307 Received for review: 2023-01-29
© 2023 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2023-05-15
DOI:10.5545/sv-jme.2023.539 Original Scientific Paper Accepted for publication: 2023-06-06
Applying Parametric Analysis in Enhancing Performance  
for Double-Layer Scissor Lifts
Dang, A.T. – Nguyen, D.-V . – Nguyen, D.-N.
Anh-Tuan Dang – Dang-Viet Nguyen – Dinh-Ngoc Nguyen
*
Thai Nguyen University of Technology, Vietnam
This study presents a calculation procedure of applying the statistical method in determining design parameters for double-layer scissor 
lifts to improve their operation (e.g., lifting height, loading, stability). By parameterizing the mounting orientation of cylinders, the working 
of the mechanism is summarized as functions and evaluated. To verify the accuracy of obtained equations, a 2D model of the mechanism 
was constructed and simulated using Working Model software. The results obtained from the simulation indicate that by adjusting the 
mounting positions of cylinders, the platform’s position and reactions on the device can be determined, enabling improvement of the system’s 
performance. The results obtained from the study show the practical significance of applying parametric methods to the calculating process 
and optimizing the structure of multiple-layer scissor lifts.
Keywords: double-layer scissor lift, cylinder’s orientation, kinematic analysis, parametric method
Highlights
• 	 Applying parametric analysis for a distinctive model of a double-layer scissor lift to obtain the reactions of each joint in the 
whole mechanism.
• 	 The proposed method can assist the calculation process without the need to construct 3D models for complex simulations.
• 	 Based on the acquired reactions and position information, the structure of the system can be further optimized, reducing the 
production cost and the calculation time.
• 	 From the initial design parameters, hydraulic cylinders can be easily selected to meet the designer’s requirements.
0  INTRODUCTION
Scissor lifts are popular aerial platform lifts that 
use a scissor-shaped mechanism to lift objects or 
people over short distances in height. Although the 
first mechanism was used in Sweden in the early 
1900s, it was not until the 1970s that scissor lifts 
gained popularity as commonly manufactured lifting 
structures. To select the appropriate lift for the job, 
lifting height, and loading are factors to be considered 
when choosing the equipment.
Based on the above requirements, several studies 
have been conducted on devices to test their functional 
ability. Solmazyiğit et al. [1] proposed the construction 
of and successfully manufactured a scissor lift device 
that enables lifting loads of up to 25 tons. Pan et al. [2] 
conducted a falling arrest test on a multi-layer scissor 
lift to check the structural stability, thereby ensuring 
tip-over safety for the system during the dropping 
process. Another study by Dong et al. [3] on an actual 
model also shows that the tip-over potential of the 
scissor lift system depends on the cylinder’s speed. 
The study also indicates that the system’s stability 
reduces when connecting joints are severely worn, or 
the structure is damaged. 
In addition to manufacturing and testing on 
real devices, other studies also focus on theoretical 
models to evaluate the performance of lifting 
systems to select the optimized layouts. Spackman 
[4] applied mathematical techniques to analyse the 
mechanism of n-layer scissor lifts. The study not only 
analysed the reactions in the scissor members but 
also presented some design issues related to actuator 
placement, member strength, and rigidity. Kosucki 
et al. [5] suggested using a volumetric controller to 
regulate the speed of the cylinders. With significant 
advantages such as a simple structure and low cost, 
the paper presents the possibility of extending the 
use of volumetric control to operate low-power 
driving systems. Karagülle et al. [6] employed finite 
element analysis (FEA) on a 3D model created in 
Solidworks to ascertain the internal loads acting on 
each component of a single-layer system. Dang et al. 
[7] proposed using the parametric method to model 
single-layer scissor lifts and construct mathematical 
equations to determine cylinder loads and reactions 
at the joints. Using the same approach, Todorović et 
al. [8] applied Harris Hawks optimization (HHO) to 
reduce mass in the mechanism frames. By solving 
static equations for a single-layer structure, Čuchor 
et al. [9] suggested an accurate method for selecting 
a hydraulic cylinder capable of lifting loads up to 3.5 
tons. 
These studies demonstrate the significant 
influence of the cylinder on the operation of 
the lifts. It can be observed that for 3D models 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307
300 Dang, A.T. – Nguyen, D.-V. – Nguyen, D.-N.
constructed using graphic software, the calculation 
process is only performed for limited positions, thus 
making it difficult to evaluate the overall impacts 
of the cylinder’s operation. In contrast, employing 
mathematical equations as a solving approach can 
greatly reduce the calculation and testing time, 
providing a more efficient method.
In this study, a specific structure of a double-layer 
scissor lift consisting of two cylinders is selected for 
analysis (Fig. 1). By determining the mathematical 
relationship between the movements of the cylinders 
and the platform, designers can choose the suitable 
dimensional arrangement for the cylinders that is 
beneficial for the required lifting.
Fig. 1.  Structure of the studied system
1  KINETIC ANALYSIS
From the 3D model of the system, a 2D model is 
represented, as shown in Fig. 2. The design problem 
can be described as follows: for a double-layer scissor 
mechanism with a fixed frame length a, designers 
must arrange two given hydraulic cylinders to utilize 
the maximum potential of the structure. Assuming the 
cylinders operate between the zero-stroke (l
Cyl.min
) and 
full-stroke positions (l
Cyl.max
) with a maximum thrust 
force F
Cyl
, and the platform has a raising requirement 
of Dh.
The problem then becomes one of determining 
the mounting position of joints P and Q in the two 
connected frames (AF and FC) while ensuring the 
stability of the lift and limiting the loading on the 
cylinders. Analysis of the lift construction shows that 
when the cylinder operates (the distance between P and 
Q changes), platform 5 moves to the corresponding 
height h. Although the length of the frames (a) and the 
arrangement dimensions (FQ, FP) of the cylinder are 
fixed, the stability of the mechanism is altered (angles 
γ at joints between frames E and F, and the distance l 
between supports A and B). Since there are too many 
parameter dimensions involved in the operation, 
designers may have difficulty choosing the correct 
layout to represent the movement of the platform.
a)                                                 b)
Fig. 2.  Schematic diagram of the mechanism in:  
a) lowest position, and b) highest position
Assign parameters for assembling dimensions  
QF =  β
1
·a, PF =  β
2
·a, and PQ = λ·a (with 0 < β
1
, β
2
 < 1, 
λ
max
 > λ > λ
min
 > 0 is the length ratio of the cylinder 
compared to one frame); the system is parameterized 
as shown in Fig. 3.
Fig. 3.  Parameterizing the device in Fig. 2
When cylinder extends, platform height h can be 
calculated using the following equation:
 ha a 

2
2
2
1
2
sin
cos
.

 (1)
The angle γ between frames is determined based 
on the relationship of the triangle MPQ:
 cos. 




1
2
2
22
12
2
 (2)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307
301 Applying Parametric Analysis in Enhancing Performance for Double-Layer Scissor Lifts 
Substitute Eq. (2) into Eq. (1); the height of the 
platform can be determined by the layout dimensions 
of the structure:
 ha 
   

2
12
2
12
. (3)
The lifting efficiency of the system can be 
evaluated by the lifting ratio, which is determined by 
the lifting height of the platform when the cylinder 
extends its length from zero-stroke to full-stroke:
 k
hh
a
f
h




maxm in
maxm in
,, ,. 
12
 (4)
It should be noted that when the platform is 
raised, γ between the frames increases and the loading 
distance between supports l decreases, which is also a 
factor affecting the stability of lift. This distance can 
be calculated using the equation:
 la a 
  
cos.

 
 24
12
2
2
12
 (5)
2  FORCE ANALYSIS
As described in the modelling step, due to the 
symmetry of the 3D structure, it is possible to 
investigate the reactions on the lift by analysing 
force on one side of the system. If the loading is 2P’, 
the reactions at the bearings and cylinders in Fig. 
2 can be correspondingly determined in terms of  
P
G
 = (P’ + W
p
)/2 (where W
p
 represents the weight of 
the platform, and G is the centroid of the total load 
P
G
). In practice, the load on the system, specifically 
the lifting load P
G
 on the platform, may continuously 
change during the operation of the device. This occurs 
when workers walk on the platform to perform repair 
and maintenance tasks while the lift is still in motion. 
This situation not only exerts forces on joints but 
also causes deformation of the component frames, 
including bending, compression, or tensile stresses. 
Since these frames can be designed with rigid and 
sustainable materials, the impact of forces on them 
can be ignored, allowing the problem to be focused on 
revolution joints where bearings are assembled. 
In the scope of this study, this paper is only 
concerned with analysing the static influence of forces 
on the structure (it considers that the load P
G 
is fixed 
on the platform during the movement of the cylinder, 
and the frame weight W of each frame is at its centre). 
This means that the distance between point G and 
support A is constant when lifting persons or objects 
from height h to h’ (see Fig. 4). However, since the 
support distance l changes during the operation of 
cylinders, the magnitude of loads acting on joints A 
and B also changes, influencing the balance of the 
device and reactions on joints.
Fig. 4.  Location of load PG when the platform raises from the 
lowest to the highest position
Assuming that the platform is raised slowly 
enough that the effect of acceleration can be ignored. 
Given the load on the system and the cylinder’s 
weights are much smaller than the weight of the 
frames (W), the calculation of reactions on the 
structure can be outlined in the following steps:
First, the scissor structure is separated from 
moving platform 5 and ground platform 1 (see Fig. 
5). Based on this figure, the reactions at supports A 
and B can be determined using moment equilibrium 
equations:
 PP
P
BB
GG
  
l
l
, (6)
 PPPPP
P
AAGBG
GG
  
l
l
. (7)
Similarly, the reactions at supports C and D can 
be calculated as:
 PP
P
P
CC
GG
GG
  


lW
l
l
l
l
W
4
2
2. (8)
 PPP
P
W.
DDG
GG
   
l
l
2 (9)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307
302 Dang, A.T. – Nguyen, D.-V. – Nguyen, D.-N.
Second, release the connection at joints M 
and N, and separate these reactions into directional 
components Fx and Fy. (see Fig. 6):
Balancing the moment in DEB frame:
 M
E 
 0, →
 
PP FF
WW F
DB MN
NM
 
 
 


a
a
a
xx
yy
coss in
cos.


22 2
2
22
0 (10)
Third, continuing to release connections at E, F, 
and apply the moment equilibrium equation at point E 
for BE frame (see Fig.7);
 
MF W
PF



  
E
N
BN
y
x
a
a
a
22
22 2
0
cos
coss in .


 (11)
Draw F
Nx
 and F
Mx
 from Eq. (10):
Fig. 5.  Separating component frames to determine reactions in the mechanism
Fig. 6.  Release the connection at joints M and N

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307
303 Applying Parametric Analysis in Enhancing Performance for Double-Layer Scissor Lifts 
 FF
W
NM
tan
xx


2
. (12)
Draw F
Ny
 from Eq. (12):
 FP W
P
W
Ny B
l
l
   






222
GG
. (13)
Apply the moment equilibrium equation for AF 
frame:
 
MP FW
FF

  


F
AN
NQ
a
a
a
a
y
x
cosc os
sins in .



22 2
22
0
1
 (14)
Substitute Eqs. (7), (12) and (13) into Eq. (14) 
and draw the cylinder’s thrust force F
Q
:
  F
PW PW
Q

 

 
 




GG
2
2
2
2
12 12
2
12
2




   sin
. (15)
It can be observed that the coefficient l
G
 in Eq. 
(15) has been omitted, implying that the position of 
loading P
G
 does not impact the magnitude of thrust 
force in cylinders. However, as this parameter remains 
present in the reaction at other joints, their magnitudes 
will vary and influence the stability of the system. 
By applying the equilibrium equation for frame AF, 
directional reactions at joint F can be computed as:
 FF F
FQ N xx







 cos,


2
 (16)
 FF FP W.
FQ MA yy
 






   sin 

2
 (17)
Subtitle β
1
, β
2
 and λ into these equations:
F
PW
W
F
G
x

   









  
 
2
2
21
12
12
2
2
2
12
2


 
 
, (18)
 
FP
P
W
W
FG
GG
G
y
l
a
P

  

   
4
3
2
2
12
12
2
2
12
12

 


. (19)
The total reactions in the remaining joints 
can be determined by combining their directional 
components:
 FF F
EE E

xy
22
, (20)
  
Fig. 7.  Release the connection at joints E and F

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307
304 Dang, A.T. – Nguyen, D.-V. – Nguyen, D.-N.
 FF F
FF F

xy
22
, (21)
 FF FF
MN NN
 
xy
22
. (22)
3  RESULTS
To validate the accuracy of the calculation method, 
five 2D physical models were created using Working 
P
G
A
B
G
D C
E
F
N
M
Cylinder
Reaction at point M
Reaction at point E
Reaction at point N
Thrush force on cylinder 
Reaction at point F
Fig. 8.  Export reactions at joints using Working Model 
Table 1.  Compare result of forces between calculation method and 2D model simulation
a = 1 m; W = 75 N; P
G
 = 5000 N
l
G
 = 1 m l
G
 = 5 m
F
1
 [N] F
2
 [N] D [%] F
1
 [N] F
2
 [N] D [%]
β
1
 = 0.20
β
2
 = 0.65
l = 0.46
FCyl 68,865.00 68,876.76 0.02 68,866.50 68,876.76 0.01
FM 807.30 807.18 (0.01 2,657.36 2,657.27 0.00
FE 67,138.51 67,150.27 0.02 66,835.78 6,846.14 0.02
FF 1,288.87 1,287.93 0.07 5,224.77 5,224.64 0.00
β
1
 = 0.20
β
2
 = 0.65
l = 0.74
FCyl 17,991.60 17,992.64 0.01 17,990.90 17,992.64 0.01
FM 938.33 938.66 0.04 4,385.43 4,386.01 0.01
FE 12,454.58 12,455.55 0.01 9,637.26 9,638.13 0.01
FF 1,874.53 1,875.04 0.03 8,770.51 8,771.54 0.01
β
1
 = 0.20
β
2
 = 0.80
l = 0.62
FCyl 50,998.50 51,102.72 0.20 51,006.00 51,102.72 0.19
FM 693.59 695.64 0.30 2,647.89 2,650.97 0.12
FE 49,127.06 49,227.89 0.20 48,696.98 48,839.48 0.29
FF 1,225.13 1,228.80 0.30 5,257.09 5,261.63 0.09
β
1
 = 0.10
β
2
 = 0.30
l = 0.22
FCyl 71,199.00 71,372.29 0.24 71,226.80 71,372.29 0.20
FM 652.18 653.41 0.19 2,678.20 2,681.35 0.12
FE 68,427.42 68,545.39 0.17 67,549.09 67,706.62 0.23
FF 1,215.76 1,218.03 0.19 5,334.71 5,341.76 0.13
β
1
 = 0.25
β
2
 = 0.50
l = 0.30
FCyl 26,339.80 26,351.61 0.04 26,339.80 26,351.61 0.04
FM 665.36 666.31 0.14 2,664.44 2,664.92 0.02
FE 23,158.51 23,169.49 0.05 22,373.90 22,385.29 0.05
FF 1,217.95 1,219.00 0.09 5,301.81 5,302.57 0.01
with F
1
 is reaction measured from Working Model;  
F
2
 is reaction obtained by calculation; and [ ]
12
2
100 %
FF
F
−
D= × is the difference between the two methods.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307
305 Applying Parametric Analysis in Enhancing Performance for Double-Layer Scissor Lifts 
Model software to measure the reactions at the joints 
(refer to Fig. 8). The obtained results from this step 
are then compared with the results calculated using 
equations, as presented in Table 1.
The first two models represent the results of 
the same structure with cylinder lengths of 0.46 m 
and 0.74 m, respectively. The remaining models 
depict structures with random cylinder orientations. 
The differences of less than 0.3 %, demonstrate the 
accuracy of the calculation model in determining the 
reactions in the mechanism. This indicates that we can 
accurately determine the design parameters without 
the need to construct or simulate the system in 3D 
virtual models, which can be time-consuming.
For detailed analyses, constraints such as the 
range of frame angular movement are applied  
( γ
min
 = 10° and γ
max
 = 120°), the operational length of 
the cylinder (50 mm for zero-stroke and 80 mm for 
full-stroke, corresponding to λ
min
 = 0.5 and λ
max
 = 0.8 
for the given model with a = 1 m), and the maximum 
thrust force F
Cyl
 = 16 kN. Based on this information, 
two graphs of lifting ratio k
h
 and the maximum loading 
for the cylinder at zero-stroke (lowest position) are 
constructed, as presented in Fig. 9.
The graph also indicates that to achieve the 
highest efficiency for the cylinder (maximum lifting 
ratio), the cylinder orientation must be selected as 
β
1 
= 0.31 and β
2 
= 0.65; Additionally, the loads in the 
other joints of this system can also be calculated, as 
shown in Fig. 10.
Eq. (3) shows that the position of the platform is 
determined by the displacement of the cylinder l
Cyl
 and 
the arrangement coefficients β
1
, β
2
. This implies that 
Fig. 9.  Graphs constructed from the given data ( β
1
, β
2
); a) lifting ratio k
h
, and b) maximum thrust force F
Cyl
Fig. 10.  Structure of the new system: a) highest and lowest positions of the lift; and  
b) reactions on joints corresponding the movement of platform

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307
306 Dang, A.T. – Nguyen, D.-V. – Nguyen, D.-N.
The results from Fig. 11 show that when the 
cylinder extend or raises the platform at a slower 
rate, there is a smaller difference in velocity during 
the movement. Conversely, faster movement of the 
cylinder leads to system instability, as varying velocity 
not only causes vibration but also creates acceleration 
and inertia forces.
Another application of the parametric method is 
selecting the appropriate cylinder in case the position 
of the cylinder has been given. For example, if the 
dimensions for the cylinders’ arrangement are β
1 
= 
0.3 and β
2 
= 0.6, the structure of the cylinder (length, 
stroke and force) can be calculated and summarized as 
shown in Fig. 12.
According to the requirement for structural 
stability, the angle γ between frames should be adjusted 
between 10 degrees and 120 degrees, meaning the 
cylinder length should be selected between 0.31 
meters and 0.8 meters, according to Eq. (2). 
Suppose the platform is required to be raised 
0.5 meters. In that case, we can either select the 
cylinder with a 32 cm initial length and 12 cm stroke 
the lifting velocity of the platform can be determined 
by taking the derivative of Eq. (3):
 


h
A

 





  
12
2
12
2
, (23)
where 

λ is the extend rate of the cylinder. 
The equation also indicates that by controlling the 
oil pump system, designers have the ability to modify 
the system’s operation, particularly the movement of 
the platform. This feature can be achieved by using a 
time-control unit for the pumping system, allowing for 
complex platform movements. such as acquiring the 
complex movement of the platform by using a time-
control unit for the pumping system. Fig. 11 illustrates 
the platform movement for a specific system selected 
from Fig. 10, in which the cylinder moves at different 
constant speeds of 1 m/s, 2 m/s, and 5 m/s. This 
corresponds to operation times for cylinder extension 
from zero stroke to full stroke length of 60 s, 150 s, 
and 300 s, respectively.
0
5
10
15
 -
 0.25
 0.50
 0.75
 1.00
0.5 0.55 0.6 0.65 0.7 0.75 0.8
Cylinder's length [m]
Platform's displacement Platform's velocity 1
Platform's velocity 2 Platform's velocity 3
Platform's displacement
h [m]
Platform's velocity
[mm/s]
λ3 = 1 [m/s]
λ1 = 5 [m/s]
λ2 = 2 [m/s]
.
.
.
Fig. 11.  Movement of platform for selected structure with cylinder moving extends with different speeds
0
15
30
45
60
0
0.5
1
1.5
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cylinder’s length  [m]
Maximum thrush Platform's height
h [m] F
cyl
[kN]
γ<10 ° γ>120 °
1.25 m
1.75 m
λ
2 
= 0.61÷0.80
13.91 kN
0.76 m
0.26 m
λ
1
=0.32÷0.44
34.88 kN
Fig. 12.  Data of cylinder corresponding to the raising height and loading requirement (a = 1 m; β
1 
= 0,3; β
2 
= 0,6; P
G 
= 5 kN; W = 75 N)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)7-8, 299-307
307 Applying Parametric Analysis in Enhancing Performance for Double-Layer Scissor Lifts 
(corresponding to h
min
 = 0.26 m and h
max
 = 0.76 m) 
with a maximum capacity of P
max 
= 34.88 kN or 
select another one with a 61 cm initial length and 19 
cm stroke (h
min
 =1.25 m, and h
max 
= 1.75 m) with a 
maximum capacity P
max
 = 13.91 kN (having a longer 
stroke but a smaller thrust force requirement).
4  CONCLUSIONS
This study analyses the applicability of the parametric 
method in designing double-layer scissor lifts. The 
conclusions drawn from the study are presented as 
follows:
• By using simple dimensional parameters, it is 
possible to determine the important information 
of double-layer scissor lifts, such as the platform’s 
height, cylinder’s thrush force, and loading in 
revolution joints. This enables easy component 
selection without the need for complex 3D 
modelling or experimental tests.
• The parametric method allows for efficient 
selection of the appropriate cylinder based on 
design parameters and the given requirements.
• By analysing the reactions and position of the 
lift, it is possible to optimize the structure of the 
system, leading to cost reduction and shorter 
calculation time.
5  ACKNOWLEDGEMENTS
The authors wish to thank Thai Nguyen University of 
Technology for supporting this work.
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