12572


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 22, No 4, 2024, pp. 633 - 642   

https://doi.org/10.22190/FUME240314022Z 

© 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

GECKO-INSPIRED FRACTAL BUFFER  

FOR PASSENGER ELEVATOR 

Yu-Ting Zuo1,2, Fu-Fang Luo3, Shi-Long Zeng3 

1College of Mechanical Engineering, Baoji University of Arts and Sciences, Baoji, China 
2Shaanxi Key Laboratory of Advanced Manufacturing and Evaluation of Robot Key 

Components, Baoji University of Arts and Sciences, Baoji, China 
3Gansu Province Special Equipment Inspection and Testing Institute, Lanzhou, China 

ORCID iDs: Yu-Ting Zuo   https://orcid.org/0000-0001-5260-6926 
 Fu-Fang Luo   https://orcid.org/0009-0001-6030-3279    
 Shi-Long Zeng   https://orcid.org/0009-0000-6718-2150    

Abstract. Elevator accidents are common occurrences and can cause serious injuries 

and property damages. If an elevator falls from a tall building and hits the ground at 

high speed, passengers in the car have little chance of survival. We designed a fractal 

buffer with hierarchical structure, which is inspired by the gecko’s pad system, to 

minimize the damage. A fractal-fractional oscillator is established to show the frequency-

amplitude relationship of the fractal buffer using He’s frequency formulation. The 

fractal-structured vibration-absorbing metamaterial in low frequency contributes to the 

elevator safety. This paper opens a new window for designing safe and reliable buffers, 

and provides entrepreneurs with new ideas for the next generation of elevators. 

Key words: Metamaterials, Passenger elevator, Elevator crash, Energy absorption, 

Fractal theory, Hierarchical structure, Duffing oscillator, He’s frequency 

formulation 

1. INTRODUCTION 

The first safety elevator in history was exhibited at the 1854 New York World's Fair by 

an American mechanical engineer Elisha Graves Otis. Subsequently, as the technology 

developing, the traction drive passenger elevator was used. Due to the defects during 

elevator construction, or lacking regular maintenance or faulty operation, accidents happen 

easily. Elevator accidents include failure of the elevator’s door which may trap passengers, 

failure of the brake system which may lead to elevator crash such as rushing to the top or 

falling to the ground. Among all of these, elevator crash is the most serious case. Assuming 

that an elevator car falls from the 10th floor (35 meters) and if other systems fail, it will 

reach the speed of 94.29 km/h when it hits the ground. In a traction drive elevator system, 

 
Received: March 14, 2024 / Accepted May 10, 2024 

Corresponding author: Yu-Ting Zuo 
Baoji University of Arts and Sciences, Baoji 721016, China  

E-mail: zuomath@163.com 



634 Y.T. ZUO, F.F. LUO, S.L. ZENG 

a buffer is installed at the bottom of the elevator shaft. There are two types of buffers, one 

is for energy accumulation, such as polyurethane buffers and steel spring buffers. The other 

is for energy dissipation, such as oil buffers [1]. 

A buffer is used to absorb the enormous kinetic energy to prevent the car from falling. 

Usually, the buffer is made of polyurethane, alloy or other materials, which have the 

advantage of high elasticity and effective energy absorption. 

To avoid elevator accidents, Yao et al. proposed an elevator fault monitoring system 

based on the Internet of Things technology [2]. Tome et al. designed an automatic speed 

measurement system applied to elevator overspeed governors [3]. Lozzi et al. investigated 

the failure models of an inclined elevator [4]. Zheng et al. recommended a soft-landing 

system for a full-load elevator [5]. 

However, there are still many things need to be improved in an elevator buffer system. 

At present, elevator buffers are mainly made of a single material with simple structure, 

which cannot fully absorb the energy within a short period of time. If the kinetic energy of 

the falling car cannot be fully absorbed by the buffer, passengers will get injured easily. 

In this paper, inspired by the unique structure of the gecko’s pad [6,7], we propose a 

fractal buffer with a hierarchical structure which can drastically absorb the kinetic energy 

brought by the elevator car in a short distance, so that passengers are better protected and 

have greater chance of survival. The two-scale fractal theory [8,9,10] is used in the design 

of the elevator buffer. This study provides a theoretical insight into the innovative elevator 

design and sheds light on advanced elevator manufacturing. 

2. GECKO’S PAD AND ITS FRACTAL STRUCTURE 

Fractal geometry is a valuable mathematical tool for the description of surface 
microstructures [11,12] and the quantification of complex networks [13,14]. The surface 
fractality and the network fractality provide a straightforward yet effective approach to 
predicting the complexity, simplifying theoretical analysis and enhancing its reliability. 

The gecko is famous for its ability of fast movement, strong adhesiveness, and smooth 
landing. When a gecko falls from a high place to the ground, it hardly gets hurt. Many 
attempts were made to discover the mechanisms of the gecko's pad and many gecko-
inspired systems appeared. For example, gecko-like dry adhesive surfaces [15], gecko-
inspired adhesives [16,17,18], gecko-inspired gripping pads [19], switchable dry/wet 
adhesives [20], soft robotic grippers [21] and other bio-inspired materials [22].  

Mathematically, the microstructure of a gecko’s pad can be approximated by fractal 
geometry, which is characterized by self-similarity. The microstructure of a gecko’s pad is 
shown in Fig. 1. 

There is a hierarchical structure in the gecko’s pad system. When a gecko falls onto the 
ground, the kinetic energy can be completely absorbed by the pad in a very short time, 
therefore the structure of the gecko’s pad has a super energy absorption capability. 

To understand better the relationship between the structure of a gecko’s pad and the 
energy absorption ability, we use fractal theory as a mathematical tool. Fractal theory is 
good at describing unsmooth boundaries, porous media, discontinuous problems, etc. In 
the 1960s, Mandelbrot raised a question about measuring the length of the coastline of 
Britain, and he found that as the measurement scale decreased to a very small value, the 
length of the coastline tended to infinite large. This work, published in Science, ushered in 
the era of fractals [23]. 



 Gecko-inspired Fractal Buffer for Passenger Elevator 635 

 

Fig. 1 Gecko’s pad system 

Currently, Hausdorff fractal dimension is widely used in engineering, and it requires 

self-similarity at all scales. A hierarchical structure with three or four cascades is often 

used, but an additional cascade with self-similarity requires high cost. Therefore, in this 

paper we allow the two-scale fractal dimensions to mimic the fractal nature of the gecko’s 

pad system.   

A gecko’s pad has multiple cascades from centimeters to nanometers, a soft landing 

induces only the first nanoscale cascade, and a hard landing puts the whole pad system into 

work in a very short time. This property is very useful for designing a fractal buffer. 

3. FRACTAL BUFFER SYSTEM 

Inspired by the gecko’s pad, a fractal buffer can be made of some metamaterials with a 

hierarchical structure. The main idea is to build a gecko-inspired system that can 

completely absorb the kinetic energy of the fast-falling elevator car, so that the car ends up 

with soft landing in a short time. The speed of the elevator car should reduce gradually so 

as to avoid any injury to the passengers. 

Xu et al. proposed a fractal cushioning using the Hilbert fractal structure and investigated 

its quasi-static property [24]. Traditionally, the buffer is made of alloy or polyurethane, which 

can absorb energy to some extent. However, in an elevator crash, it is of little help to the 

passengers. To improve the property of the buffer, we start by changing its material. By using 

metamaterials [25], we can obtain a wider range of elastic coefficients.  

Fractal metamaterials have been employed extensively in noise and sound absorption 

[26] and wave absorption [27]. They exhibit distinctive mechanical properties [28] and 



636 Y.T. ZUO, F.F. LUO, S.L. ZENG 

damping characteristics [29]. A fractal buffer made of metamaterials is proposed, as shown 

in Fig. 2. The top of the metamaterials has the smallest elastic coefficient, while the bottom 

has the largest elastic coefficient. Several layers of different materials are placed according 

to different elasticity coefficients. By taking into account the weight of the elevator car 

(including passengers) and the fall distance, the choice of buffer metamaterials and the 

corresponding thickness can be determined. Metamaterials can be fabricated by 3D 

printing technology [30,31]. Generally, 3D printed mechanical metamaterials have 

negative Poisson's ratio, steady-state structures, and origami structures [32], these 

properties offer a total new road to design smart programmable mechanical materials, e.g., 

metamaterial bones [33] and smart fabrics [34]. 

 

Fig. 2 Hierarchical metamaterial with different properties at different cascades  

The gecko's pad is made of proteins, a type of soft and high molecular weight material. 

Similarly, man-made soft materials such as metamaterials [25] and auxetic materials 

[35,36] with hierarchical structure can be applied to the design of fractal buffers.  

The best candidate for metamaterials or auxetic materials with hierarchical structure is 

the fractal-like porous structure, such as Menger's sponge. Different porosity leads to 

different property, and the fractal dimensions are the key factor influencing the property. 

The different porosities for different cascades in a fractal buffer result in different elastic 

coefficients and can be adjusted during the manufacturing process. 

4. THEORETICAL ANALYSIS 

The fractal buffer system consists of a cascade of rigidities. The top layer has the lowest 

stiffness, while the bottom layer has the highest stiffness. The equivalent coefficient of 

elasticity can be calculated as 

 
0 0 0

0 1

1
...

n

n

N aN a N

k k k k
= + + + , (1) 

where k is the equivalent elastic coefficient, ki (i = 0 ~ n) is the elastic coefficient of the 

buffer in the i-th layer, and N0 is number of the first layer, a is the self-similarity index, 

generally, we choose a=2 or a=3. The equivalent elastic coefficient is a function of the 

displacement of the fractal buffer: 

 ( ) exp( )k k h b ch= = , (2) 



 Gecko-inspired Fractal Buffer for Passenger Elevator 637 

where b and c are positive constants, h is the displacement. Eq. (2) implies that when the 

fractal buffer is hit, it can adjust its elastic coefficient according to the displacement, so 

that the elevator car can land smoothly and passengers can be prevented from injury in 

elevator accidents. 

If we view the elevator car as a mass point, its total potential energy is 

 E mgH= , (3) 

where m is the mass of the car, g the gravity acceleration, H is height. The initial velocity is: 

 2u gH= . (4) 

The governing equation of the fractal buffer can be expressed as 

 
2

3

2
0, (0) 0, (0) 2

d u du
m ku u u gH

dt dt 
+ + = = = , (5) 

where εu3 is the nonlinear term, and ε can be negative for metamaterials, du/dtα is the two-

scale fractal derivative [10], α is the two-scale fractal dimension, which can be calculated 

by He-Liu fractal dimension formulation [8]. Furthermore, the genetic programming 

methodology [13] offers an alternative approach to the accurate determination of fractality. 

As demonstrated in Ref. [13], the accuracy of this approach is as high as 98%. 

The approximate solution of Eq. (5) reads [37] 

 sinu A t= , (6) 

where A is the amplitude and ω is the frequency. According to He’s frequency formulation 

[38], we have  

 
2

/2

( )

u A

df u

du


=

= . (7) 

Here f(u) is given as  

 
3

( )
ku u

f u
m

+
= . (8) 

So we obtain  

 
2

2 3

4

k A

m m


 = + . (9) 

According to Eq. (6), we have  

 cos
du

A t
dt




 = . (10) 

That means 

 (0)
du

A
dt

= . (11) 

According to the initial conditions, we, therefore, have  

 2A gH = . (12) 

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638 Y.T. ZUO, F.F. LUO, S.L. ZENG 

Combining Eqs. (9) and (12), we have   

 
2

2

3

2

k gH

m m





= + . (13) 

Solving ω from Eq. (13) yields  

 

22 4 24

4

k k mgH

m




+ +
= . (14) 

From Eq. (12), the amplitude can be solved as  

 
2

8

2 4 24

mgH
A

k k mgH
=

+ +
, (15) 

and the period reads 

 
2

2 4
2

2 4 24

m
T

k k mgH




 
= =

+ +
. (16) 

We assume that the total energy is absorbed by the fractal buffer system within one quarter 

period, and the landing velocity is zero. The falling time is  

 t
v


= , (17) 

where v is average velocity of the falling car, Δ is the maximal amplitude. It requires that  

 A  . (18) 

According to Eqs. (2) and (15), the amplitude can be solved from the following 

transcendental equation 

 
2

8

2 exp( ) 4 exp(2 ) 24

mgH
A

b cA b cA mgH
=

+ +
. (19) 

The average velocity can be calculated as  

 
2

2

gH
v = . (20) 

So the falling time is  

 
2

t
v gH

 
= = . (21) 

It should meet the following requirement 

 
4

T
t  . (22) 

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 Gecko-inspired Fractal Buffer for Passenger Elevator 639 

That is  

 
2

2

2 4 24

m

gH k k mgH







+ +
. (23) 

In view of Eq. (2), we write Eq. (23) in the form  

 
2

2

2

2 exp( ) 4 exp(2 ) 24

m

gH b c b c mgH








 +  +
. (24) 

Eq. (24) is the criterion for designing the needed fractal buffer.  

5. FRACTAL BUFFER’S MAXIMAL DISPLACEMENT 

We have obtained the criterion given in Eq. (24) for smooth landing during elevator 

accidents, but it is an inexplicit form. An explicit form for the maximal displacement of the 

fractal buffer’s motion is much needed. We consider the following equation 

 
2

2

2
( ) 0

2 exp( ) 4 exp(2 ) 24

m
R

gH b c b c mgH






 = − =

 +  +
. (25) 

In order to solve Δ from Eq. (25), we apply Chun-Hui He’s iteration method [39-41]. 

During experiment, its value can be approximately estimated, for example,  

 
1 / 2 = , (26) 

 
2 / 4 = , (27) 

where λ is the height of the fractal buffer. According to Ref. [41], we have  

 
2

1
1

2

1 1

2

2 exp( ) 4 exp(2 ) 24

m
R

gH b c b c mgH






= −

 +  +
, (28) 

and 

 

2

2
2

2

2 2

2

2 exp( ) 4 exp(2 ) 24

m
R

gH b c b c mgH






= −

 +  +
. (29) 

So an explicit form for the maximal displacement of the fractal buffer’s motion is obtained 

 1 2 2 1

1 2

R R

R R

 − 
 =

−
. (30) 

The iteration can be continued for a better explicit form. When the buffer reaches its 

maximum displacement, a much longer time is needed, which can be guaranteed by the 

fractal vibration system. Fig. 3 shows the dynamic property of the fractal buffer for 

different fractal dimensions. It is obvious that a smaller value of the fractal dimension leads 

to a larger half period. When the half period tends to infinity, the fractal buffer becomes 

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640 Y.T. ZUO, F.F. LUO, S.L. ZENG 

extremely stable and the passengers inside the car can be protected. This extended period 

property is also called the low frequency property [42,43]. 

 

Fig. 3 Dynamical property of the fractal buffer with different fractal dimensions 

6. CONCLUSION 

In order to prevent injury to passengers in an elevator crash, a fractal buffer system, 

which can completely absorb the kinetic energy of the falling elevator, is proposed, and the 

criterion for designing the needed fractal buffer is discussed. The system exhibits a low 

equivalent frequency, which can be adjusted by the fractality, and the maximal 

displacement should be at the quarter period. These two properties guarantee the system’s 

safety and reliability during an accident. 

The metamaterials or auxetic materials with special property, as given by Eq. (2), can be 

fabricated with ease by 3D printing technology [32,33] and by spinning technology [44,45]. 

The 3D-printed metamaterials exhibit a negative Poisson's ratio, which can be adjusted during 

the printing process. Furthermore, the fractality can be precisely determined, allowing for the 

smart fabrication and AI optimization of the fractal buffer system. 

The amplitude-frequency of the fractal buffer’s vibration can be readily and 

efficaciously revealed by He’s frequency formulation [38,46]. In a forthcoming article, we 

will discuss the effect of the negative Poisson's ratio on the periodic property of the fractal 

buffer system. 

Acknowledgement: The paper is supported by Science and Technology Program of Gansu Administration 

for Market Regulation (Grant No. SSCJG-TS-B202202) and National Natural Science Foundation of 

China. 



 Gecko-inspired Fractal Buffer for Passenger Elevator 641 

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