Department of Mechanical & Nuclear Engineering
0408491 Senior Design Project 1
A B.S. REPORT
PREPARED IN PARTIAL FULFILLMENT OF THE
REQUIREMENT FOR THE DEGREE OF
BACHELOR OF SCIENCE
IN
MECHANICAL ENGINEERING
Fall 2024/2025
Seated Exercise Machine for The Legs of The Elderly
Khalid Khalid Abuamra U21100691
Omar Hani Shaheen U21105502
Omar Wael Abukhalil U20101302
Supervisor: Naser Khaled Hasan Nawayseh
ABSTRACT
This project aims to design a safe, and user-friendly seated exercise machine for the Elderly to
enhance leg mobility and strength. The machine incorporates two motions, namely a walking-like
movement using a rack-and-pinion system with synchronized sliders for smooth, controlled linear
motion, and a vertical flexing movement powered by linear actuators to target calf muscles.
Designed for ease of use, the compact machine allows users to exercise while engaging in daily
activities like reading or watching TV. Safety features, such as a stable base and non-slip footrests,
ensure confidence and prevent accidents. The development process involved research, design,
analysis, and validation to create an effective and reliable solution for elderly users.
ACKNOWLEDGMENTS
We would like to extend our heartfelt thanks to Prof. Naser Nawayseh for his exceptional guidance,
expertise, and steadfast support, which were crucial to the success of this project. His mentorship
and valuable insights had a profound impact on the direction and results of our work. Additionally,
we would like to express our appreciation to the University of Sharjah for giving us the opportunity
to pursue this project and explore these design topics.
Table of Contents
INTRODUCTION .......................................................................................................................... 1
1.1
Motivation ....................................................................................................................... 1
1.2
Literature Review............................................................................................................ 2
1.2.1 Previous Solutions to leg exercise machine for the elderly ........................................................................................... 2
1.2.2 Mechanisms for the leg movement.................................................................................................................................... 3
1.3
Objectives........................................................................................................................ 5
THEORETICAL BACKGROUND ................................................................................................ 6
2.1 Kinematics and Dynamics .................................................................................................... 6
2.2 Movement ............................................................................................................................. 6
2.3 Position, Velocity, and Acceleration .................................................................................... 7
2.4 Design of Pinion & Rack ...................................................................................................... 8
2.5 Von Mises Theory and Factor of Safety ............................................................................. 10
2.6 Linear actuators................................................................................................................... 11
2.7 Electric Motors.................................................................................................................... 12
2.8 3D printing material ............................................................................................................ 12
2.9 Gears materials.................................................................................................................... 13
2.10 Critical components material ............................................................................................ 13
METHODOLOGY........................................................................................................................ 14
3.1 Mechanism design................................................................................................................... 14
3.1.1 First move ment ....................................................................................................................................................................14
3.1.2 Second movement and selection of linear actuators ....................................................................................................15
3.2 Components ........................................................................................................................ 17
3.3 Material selection ................................................................................................................ 23
3.3 Force and stress analysis ..................................................................................................... 26
3.4 Motion analysis ................................................................................................................... 30
3.5 Motor selection ................................................................................................................... 35
3.6 Design of a Pinion and Rack Mechanism ........................................................................... 38
MANAGMENT ............................................................................................................................ 51
4.1 Breakdown of Work into Specific Tasks ............................................................................ 51
4.2 Breakdown of Work into Specific Tasks Flow Chart ......................................................... 52
4.3 Gantt Chart for the Organization of Work and Timeline.................................................... 53
4.4 Breakdown of Responsibilities Among Team Members .................................................... 54
4.5 Budget Management ........................................................................................................... 55
CONCLUSIONS........................................................................................................................... 56
References ..................................................................................................................................... 57
APPENDICES .............................................................................................................................. 59
List of Figures
Figure 1 Importance of Leg exercise for elderly ………………………………………………….1
Figure 2 Leg exercise assembly…………………………………………………………………...2
Figure 3 exercise machine for elderly people …………………………………………………….5
Figure 4 first movement walking-like motion ……………………………………………………..5
Figure 5 second movement up and down…………………………………………………………..5
Figure 6 rack & pinion mechanism…..……………………………………………………………6
Figure 7 signal of a sinusoidal motion…………………………………………………………….7
Figure 8 rack and pinion ……………………………………………………………………..........8
Figure 9 linear actuator ……………………………………………………………………..........11
Figure 10 Electric motor ………………………………...………...……………………………..12
Figure 11 Stress Strain graph for3D printing material………………………………………...…13
Figure 12 first movement ………………………………………………….………...…………..14
Figure 13 second movement ………………………………………………….……...…………..14
Figure 14 Base ……………………………………………………………………….…………..17
Figure 15 Bracket ………………………………………………………………………………..17
Figure 16 Rails …..………………………………………………………………………………17
Figure 17 Slider ……………………………………………………………………....………….18
Figure 18 Rack & pinion …………………………………………………………………....……19
Figure 19 Support platform ………………………………….…………………………………..19
Figure 20 Bracket for leg pad ………………………………………...………………….………20
Figure 21 Pin ……………………………………………………………..……………….……..20
Figure 22 Rotating part ………………………………………………………………………... 21
Figure 23 Foot pad ……………………………………………………………………….………21
Figure 24 Linear actuator …………………………………………………………………..……22
Figure 25 Rod ……………………………………………………………………………..……..22
Figure 26 Seated Exercise Machine for The Legs of The Elderly ………………………..……..23
Figure 27 Fixing the base ………………………………………………………………….……..26
Figure 28 Applying force on the pad …………………………………………………….………26
Figure 29 Deformation …………………………………………………………………….…….27
Figure 30 Pin …………………………………………………………………………………….27
Figure 31 Rod …………………………………………...……………………………….………27
Figure 32 Position plot on MATLAB ………………..…………………………………….……31
Figure 33 Velocity plot on MATLAB …………………………………………………….……..32
Figure 34 Acceleration plot on MATLAB ………………………………………………………32
Figure 35 Angular Displacement plot on MATLAB …………………………...……………….33
Figure 36 Angular Velocity plot on MATLAB ………………………………………….………34
Figure 37 Angular Acceleration plot on MATLAB ……………………………………….…….35
Figure 38 Force plot on MATLAB ………………………………………………...……………36
Figure 39 RPM plot on MATLAB ………………………………………………...……….……37
Figure 40 Torque plot on MATLAB ……………………………………………………….……38
Figure 41 Time vs Actual maximum power……………………………………………….……..38
Figure 42 Gantt chart……………………..………………………………………………………54
Figure 43 Comparison Between different material ………………………………………………60
Figure 44 Technical data sheet for ABS 1 ………………………………………………………61
Figure 45 Technical data sheet for ABS 2……………………………………………………….62
List of Tables
Table 1 Length & Weight………………………………………………………………………..15
Table 2 Force, Speed & Current data ……………………………………………………………16
Table 3 Material type for each component ………………………………………………...……24
Table 4 Force analysis results ……………………………...……………………………………28
Table 5 Factor of Safety results for the rod ……………………………………………………..28
Table 6 Factor of Safety results for the pin ……………………………………………………..29
Table 7 Allowable contact stress number for iron and bronze gears ……………………………46
Table 8 Allowable bending stress number for iron and bronze gears …………………………..46
Table 9 Gantt table………………………………………………………………………………53
Table 10 Budget Management …………………………..………………………………………………………………..…55
Chapter One
INTRODUCTION
1.1 Motivation
As people age, their bodies undergo various changes that can affect strength, balance, and mobility.
For many elderly individuals, these changes can lead to difficulties with daily activities, such as
walking, climbing stairs, or even standing up from a chair. This is where leg exercises become
essential. Regular leg exercises are one of the most effective ways to address common aging
challenges and maintain physical function. Regular leg exercises are crucial for elderly individuals
as they address several key aspects of aging and significantly improve overall health. They help
maintain muscle strength and endurance, which counteracts the natural muscle loss that occurs
with age. Leg exercises also enhance balance and stability by strengthening the muscles that
support proper posture and movement, reducing the risk of falling which is one of the leading
causes of injury for elderly. These exercises help make the joints more flexible and easier to move,
reducing pain from conditions like arthritis and improving movement in the hips, knees, and
ankles. Regular exercise also improves blood flow and heart health, helping to reduce the chances
of problems like high blood pressure. Strengthening their legs helps elderly people stay
independent, allowing them to do daily tasks on their own, which improves their overall quality of
life. leg exercises help prevent falls and injuries by boosting muscle strength, joint health, heart
health, and overall independence. Also, it helps their body to move rather than just sit for a long
time which could result in mobility problems and back pain. It is because of the long period of
time they spend just sitting or lying down without moving or exercising their body that could lead
to adverse health problems. The Benefits of exercising for the elderly can be shown in Figure 1
[1].
Figure 1: Importance of Leg exercise for elderly
1
1.2 Literature Review
1.2.1 Previous Solutions to leg exercise machine for the elderly
This literature review examines the design and utility of seated exercise assemblies, focusing on
their potential to improve health outcomes for sitting users and their relevance to enhancing
mobility and circulation, particularly for the elderly. The reviewed patents (US8550963B1) [2]
and (US8894551) [3] present a solution by offering a leg exercise assembly specifically designed
to facilitate seated exercise. This assembly bridges the gap by providing low-cost, accessible, and
effective leg exercise options, making it especially suitable for those who spend long standing
hours seated,
The patented assembly introduces a system structured to promote leg movement and exercise,
particularly in a seated orientation. It features two support members (or pedals) that move in a
linear, oppositely directed, reciprocal manner. This motion encourages the feet and legs to mimic
a walking-like movement, providing a form of low-impact exercise.
The design shown in Figure 2 features a compact base housing an electrically powered motor that
drives reciprocating pedals. These pedals move in opposite directions to simulate natural leg
motions, such as walking or pedaling, providing seated leg exercise. Each pedal has a treaded
surface for stability and comfort, with a compressible layer to reduce pressure on the feet. The
motor connects to the pedals via a drive linkage system with cam assemblies and bearings, ensuring
smooth and efficient movement. The compact, lightweight design allows for easy placement under
a desk or table, making it ideal for office or small space use.
Figure 2: Leg exercise assembly
2
The current leg exercise assembly primarily focuses on general leg movement but does not
effectively target muscles like the calves. To address this, the design will be modified to include a
vertical, up-and-down motion in addition to the existing linear reciprocal motion, specifically
targeting the calf muscles. Additionally, incorporating adjustable resistance will enhance the
device's versatility, making it suitable for users with varying fitness levels and needs
1.2.2 Mechanisms for the leg movement
1. Walking-Like Motion
Walking is a low-impact exercise that offers numerous benefits for elderly individuals [4],
particularly in improving cardiovascular health, joint mobility, and muscle strength. Regular
walking enhances circulation, reduces blood pressure, and maintains bone density, which is crucial
for aging adults. It also strengthens muscles in the lower body, such as the quadriceps, hamstrings,
calves, and glutes, which are essential for mobility. Additionally, walking improves balance and
coordination, helping to prevent falls, a common issue for seniors. Machines that simulate walking,
like elliptical trainers or those with a rack-and-pinion system, offer these benefits without the high
impact of outdoor walking [5].
2. Pinion and Rack Mechanism
The rack-and-pinion system is widely used in exercise equipment because it converts rotational
motion into smooth, linear motion. This mechanism ensures consistent, controlled movement,
which is crucial for elderly exercise machines. It is commonly found in devices that simulate
walking or cycling, offering low-impact, precise motion. For seniors, the rack-and-pinion system
reduces the risk of jerky movements that could cause discomfort or injury, making it an excellent
choice for machines that help improve mobility and strength. The system is also durable and low
maintenance, making it suitable for long-term use in exercise equipment for older adults [6].
3. Targeting The Calf Muscle
Strengthening the calf muscles is important for elderly individuals as it helps with mobility,
balance, and reduces the risk of falls. Calf raises are an effective exercise to target these muscles,
but many seniors find it difficult to perform them due to joint pain or limited mobility. Exercise
machines that simulate calf raises using controlled vertical movements can provide an easy way
for seniors to strengthen their calves without putting strain on their joints [7].
3
4. Linear Actuators in Exercise Machines
Linear actuators are used in exercise machines to provide smooth, controlled movement by
converting rotational motion into linear motion. This is ideal for machines like calf-raisers, where
precise, adjustable movement is needed. For elderly individuals with limited mobility or joint
issues, linear actuators reduce effort and strain during exercises. They allow for customizable range
and speed, ensuring safe and effective workouts. This makes them perfect for seniors who need
low-impact, controlled exercises to improve strength and mobility without risking injury [8].
4
1.3 Objectives
The objective of our product is to create a comfortable and easy-to-use seated exercise machine
for elderly people, ensuring it can facilitate movement for both legs of the user as shown in Figure
3. The first motion will incorporate two distinct positions. To achieve a walking-like motion, the
two sliders within the machine will operate in a synchronized manner, moving in opposite
directions relative to each other shown in Figure 4. This reciprocal movement will simulate the
natural motion of walking, enabling users to engage both legs effectively. The second motion will
target the calf muscles, allowing users to perform a vertical flexing movement shown in Figure 5.
The product should be lightweight and compact, making it easy for users to lift and move around
their home or to other places. Also, one of the advantages of this product is that elderly users will
be exercising their legs while, for example, watching the TV or reading the newspaper. The design
must also prioritize safety, with features like a stable base, non-slip footrests, and smooth
movement to prevent accidents. These safety elements will help elderly users feel secure and
confident while exercising. Overall, this machine is designed to offer an easy and effective way
for elderly individuals to remain active and improve their mobility, allowing them to use it
wherever and whenever they want.
Figure 3: exercise machine for elderly people
Figure 4: First movement walking-like motion
Figure 5: Second movement vertical movement up and down
5
Chapter Two
THEORETICAL BACKGROUND
2.1 Kinematics and Dynamics
Kinematics studies motion, focusing on position, velocity, and acceleration without considering
the forces behind it. It answers "how" objects move. Dynamics, however, explores the forces that
cause or affect motion, answering "why" things move.
2.2 Movement
The back-and-forth motion for the legs, achieved using a pinion and rack mechanism, which
converts the pinion’s circular motion into translational motion, moving the pads along the rails.
Additionally, the first movement follows a sinusoidal pattern, creating a smooth, repetitive
oscillation like ocean waves. [9].
Figure 6: pinion and rack mechanism
The sinusoidal motion can be described by these equations:
Position: x(t) = Asin (2πft)
(eq. 4)
Velocity: v(t) = (2πf)(A)cos(2πft)
(eq. 5)
Acceleration: a(t) = -(2πf)^2(A)sin(2πft)
(eq. 6)
Where:
A = amplitude of the rack motion, maximum displacement (m)
f = assumed frequency (Hz)
t = time (s)
6
This graph shows how the position changes over time in a sinusoidal pattern:
Figure 7: signal of a sinusoidal motion
2.3 Position, Velocity, and Acceleration
When designing a mechanism, analyzing angular position, angular velocity, and angular
acceleration is essential. Angular position describes the orientation of parts, angular velocity
indicates how fast they are rotating and in which direction, and angular acceleration reveals how
quickly their rotational speed changes. Studying these factors ensures smooth motion, optimal
performance, and helps prevent mechanical issues or failure.
The following equations were used to analyze the angular Position, Velocity, and Acceleration:
𝑥
Position: (𝜗 = )
(eq. 1)
𝑅
•
•
•
x = horizontal displacement of the rack
R = pitch circle radius of the pinion
ϑ = angular displacement of the pinion
𝑉
Velocity: (𝜔 = 𝑅 )
•
•
(eq. 2)
V = linear velocity of the rack
ω = angular velocity of the pinion
𝑎
Acceleration: (𝛼 = 𝑅 )
•
•
(eq. 3)
a = linear acceleration of the rack
α = angular acceleration of the pinion
7
2.4 Design of Pinion & Rack
Pinion and rack are two parts that work together to convert rotary motion into straight-line motion
shown in figure 8. The pinion is a small round gear that engages with the rack, which is a flat bar
with teeth. When the pinion turns, its teeth push against the rack’s teeth, causing the rack to move
back and forth in a straight line. This system is often used in steering mechanisms, like in cars, and
in machines that need to convert rotational motion into linear motion [10].
Figure 8: rack and pinion
The AGMA (American Gear Manufacturers Association) methodology is critical in the design
and analysis of rack and pinion systems because it ensures the reliability, durability, and efficiency
of the gear mechanism, two fundamental stress equations are used in the AGMA methodology,
one for bending stress and another for pitting resistance (contact stress) [11].
The fundamental equation for Bending stress:
(eq. 7)
where for U.S. customary units (SI units):
Wt is the tangential transmitted load, lbf (N)
Ko is the overload factor
Kv is the dynamic factor
Ks is the size factor
Pd is the transverse diametral pitch
F (b) is the face width of the narrower member, in (mm)
Km (KH) is the load-distribution factor
KB is the rim-thickness factor
J(YJ) is the geometry factor for bending strength (which includes root fillet
stress-concentration factor Kf)
(mt) is the transverse metric module
8
The fundamental equation for pitting resistance (contact stress):
(eq. 8)
where Wt, Ko, Kv, Ks, Km, F, and b are the same terms as in the bending stress, For U.S.
customary units (SI units), the additional terms are:
Cp (ZE) Is an elastic coefficient, √lbf/in2 (√N/mm2 )
Cf (ZR) is the surface condition factor
dP (dw1) is the pitch diameter of the pinion, in (mm)
I (ZI) is the geometry factor for pitting resistance
AGMA Strength Equations
The equation for the allowable bending stress is:
(eq. 9)
where for U.S. customary units (SI units)
St is the allowable bending stress, lbf/in2(N/mm2)
YN is the stress cycle factor for bending stress
KT (Yθ) are the temperature factors
KR(YZ)are the reliability factors
SF is the AGMA bending factor of safety
9
The equation for the allowable contact stress σc,all is
(eq. 10)
Sc is the allowable contact stress, lbf/in2 (N/mm2)
ZN is the stress cycle life factor
CH(ZW) are the hardness ratio factors for pitting resistance
KT (Yθ) are the temperature factors
KR(YZ) are the reliability factors
SH is the AGMA contact wear factor of safety
2.5 Von Mises Theory and Factor of Safety
Von Mises Stress Theory, also known as the Distortion Energy Theory, is widely used in
engineering to determine the strength of materials under complex loading conditions. It is
particularly effective for ductile materials, which are prone to yielding rather than brittle failure
[12].
Key Concept of Von Mises Theory
Yield Criterion: The theory states that yielding begins when the distortion energy per unit
volume in a material reaches the same level as the distortion energy at yielding under uniaxial
tension.
For principal stresses:
(eq. 11)
Where σ1, σ2, σ3 are the principal stresses
10
For plane stress:
(eq. 12)
Here σx, σy are normal stresses and τxy is the shear stress
Factor of Safety (FoS) using Von Mises Stress
The Factor of Safety quantifies the margin between the maximum stress a material can withstand
and the stress it experiences. It is calculated as:
(eq. 13)
Where:
•
Sy Yield strength of the material.
•
σ́ Calculated von Mises stress
2.6 Linear actuators
Linear actuators shown in figure 9 convert electrical energy into straight-line motion, they extend
or retract a rod or shaft to move components along a straight path. Key specifications, such as
speed, load capacity, and stroke length, are important for performance. In our design, linear
actuators move the pads vertically, powered by a battery or voltage source, to achieve precise
movement [13].
Figure 9: Linear actuator
11
2.7 Electric Motors
Choosing the right motor is crucial for ensuring efficiency and performance in a given application.
DC motors shown in Figure 10 are easy to control with adjustable speed but may wear out over
time. AC motors are durable and efficient but have fixed speeds. Servo motors offer high precision
for tasks needing exact positioning, though they require a controller and are more costly. St epper
motors provide precise movement in small steps, ideal for accurate positioning but may struggle
under heavy loads. Selecting the right motor ensures optimal performance and longevity for the
task at hand [14].
Figure 10: Electric Motor
2.8 3D printing material
When comparing ABS, ABA, and polycarbonate shown in Figure 11 for 3D printed
parts, ABS often emerges as the best choice for many projects due to its excellent balance of
strength, durability, and ease of use. ABS is more affordable and easier to print with than both
polycarbonate and ABA, requiring only a heated bed for optimal results and exhibiting less
warping during the printing process. While polycarbonate offers outstanding heat resistance and
superior mechanical strength, making it ideal for high-performance applications exposed to
elevated temperatures, it is significantly more expensive and requires higher printing temperatures,
making it more challenging to work with. Additionally, polycarbonate can be prone to cracking if
not printed under controlled conditions. ABA, a modified version of ABS, provides increased
flexibility and impact resistance, but it can be trickier to print with and still doesn't offer the same
level of heat resistance as polycarbonate. Overall, ABS is the more cost-effective, user-friendly
material, making it a better choice for most general-purpose 3D printing projects, while
polycarbonate is better suited for specialized applications requiring extreme heat resistance and
strength for more information about 3D printing material see Appendix A.
12
Figure 11: Stress vs Strain graph for 3D printed materials
2.9 Gears materials
Steel, cast iron, and bronze each offer unique advantages for gears based on the application. Steel
is the strongest and most durable, ideal for high-load, high-performance uses, but it is more
expensive and requires advanced manufacturing. Cast iron is cost-effective, offers good wear
resistance and vibration damping, making it suitable for moderate-load and noise-sensitive
applications, though it lacks steel’s tensile strength and can be brittle. Bronze excels in corrosion
resistance and low friction, making it ideal for environments with moisture or corrosive elements,
but its softness limits its use in high-load applications.
2.10 Critical components material
Aluminum is often better than steel or ABS for critical components due to its excellent strengthto-weight ratio, making it ideal for applications where reducing weight without compromising
strength is essential, such as in aerospace and automotive industries. Unlike ABS, aluminum is
much stronger and more rigid, providing greater durability for high-stress applications.
Additionally, aluminum is naturally corrosion-resistant, making it more reliable in harsh
environments, whereas ABS is more prone to degradation over time. Overall, aluminum's
combination of strength, lightweight properties, and corrosion resistance makes it a superior choice
for critical components.
13
Chapter three
METHODOLOGY
This project aims to design a Seated Exercise Machine for the Legs of the Elderly, focusing on
mechanism design, components, materials, force and stress analysis, motion analysis, motor
selection, and rack-and-pinion design. The first motion mimics walking with two sliders moving
in opposite directions, while the second targets the calf muscles with a vertical flexing movement.
3.1 Mechanism design
3.1.1 First movement
For the first movement shown in figure 12, the mechanism includes two racks, each measuring
500 mm in length. These racks are connected to two sliders positioned on four parallel rails. The
rack-and-pinion system converts rotational motion into linear motion. A motor drives the pinion,
which in turn moves the racks in a straight line. As the racks move, they propel the sliders along
the rails.
Once the racks travel their full 250 mm, dragging the sliders with them, the pinion reverses its
rotation, moving the racks in the opposite direction and creating a sinusoidal motion. The four
parallel rails constrain this movement, ensuring the sliders move only along the intended axis
without sideways or rotational deviation. This setup guarantees smooth, precise, and stable
motion along the rails.
The racks are designed to move in opposite directions simultaneously, synchronized with the
pinion's rotation. Each rack contributes to the linear motion of the sliders, enabling controlled
and accurate travel over a fixed range.
Figure 12: first movement
12: First movement
14
3.1.2 Second movement and selection of linear actuators
For the second movement shown in Figure 13 of our leg exercise machine, we chose a micro
linear actuator over a cam and follower system for its lighter weight, smaller size, and smoother
motion. The actuator provides precise, controlled up-and-down movement. In contrast, the cam
and follower system is bulkier, heavier, and requires more maintenance, making it less spaceefficient and harder to control with the same precision. Thus, the micro linear actuator offers a
more compact, efficient, and safer solution for our design.
The system features four linear actuators in total, with two positioned on each side of the
machine. These actuators are mounted on a supporting pad and connected via a rod. At each end
of the rod, a cap-like structure secures the actuators’ top heads, ensuring stability and precise
alignment during operation.
Figure 13: Second movement
After evaluating electric, pneumatic, and hydraulic options, we selected electric linear actuators
for their precise control, reliability, and seamless integration with electronic systems. Among the
available choices, the PA-MC1 micro linear actuator stood out for its compact size, IP65
ingress protection rating, and durable construction, making it highly suitable for our design
requirements [15].
Table 1 Length & Weight data
15
The selection process began by defining key performance specifications. Each actuator provides a
dynamic force of 17 lbs or 7.7 kg, with two actuators working together to generate 34 lbs or 15.4
kg, enough to support the average human leg (30 lbs or 13.6 kg). A static force of 48 lbs or 21.7
kg ensures the leg remains securely in place, preventing unwanted movement. The chosen 4-inch
extended length offers optimal calf muscle engagement without excessive motion. Actuators
operate at 0.71 inches per second or 18 mm/s without load and 0.51 inches per second 13 mm/s
under full load, balancing effective muscle engagement and user comfort. Powered by 12 VDC,
they integrate seamlessly into the system’s controls. Compact and lightweight, the actuators have
a retracted length of 6.76 inches or 171 mm and an extended length of 10.76 inches or 273 mm,
fitting within the machine’s design without adding significant weight (0.21 lbs each or 0.095 kg).
Table 2 Force, Speed & Current data
Using trigonometry, we can calculate the angle by which the foot pad will be raised. The linear
actuator has an extended length of 4 inches (equivalent to 101.6 mm), and the distance between
the linear actuator and the rotation point of the pad is 302.80 mm ,we found below that the angle
is 18.5 degrees.
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
tan θ = 𝑎𝑑𝑗𝑒𝑐𝑒𝑛𝑡
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
θ = 𝑡𝑎𝑛−1 (𝑎𝑑𝑗𝑒𝑐𝑒𝑛𝑡 )
101.6
θ = 𝑡𝑎𝑛−1 (302.8) = 18.5°
16
3.2 Components
In this section, we will provide a detailed description of the components that make up our machine.
Each component has been carefully designed or selected to fulfill specific functions, ensuring the
machine operates effectively and meets its intended purpose.
Part 1 Base: The base shown in Figure 14 serves as the foundation for the machine, providing
structural stability and support
Figure 14: Base
Part 2 Bracket: The bracket is essential for securely supporting the rails at both ends, ensuring
stability and alignment. Its primary role is to prevent deflection or movement that could affect
the mechanism's performance shown in figure 15
Figure 15: Bracket
17
Part 3 Rails: The rails serve as the guiding structure for the slider's movement, ensuring smooth
and precise motion shown in figure 16
Figure 16: Rails
Part 4 slider: The slider is a moving component designed to travel along the rails, providing
smooth and controlled motion. It is securely attached to the rack, allowing it to transfer the motion
generated by the rack-and-pinion mechanism shown in figure 17
Figure 17: Slider
18
Part 5 rack and pinion shown in Figure 18
Figure 18: rack and pinion
Part 6 support platform: the support platform is positioned on top of the slider and acts as a
stable base for both the leg pad and the linear actuators. Its primary function is to ensure proper
positioning and secure attachment of these components during operation. As the slider moves
along the rails, the platform remains level and provides a solid foundation shown in figure 19
Figure 19: Support platform
19
Part 7 Brackets for the leg pad: Two brackets will be mounted on the support platform, as
shown in Figure 20, to securely hold the leg pad in place. These brackets are designed to provide
stability and ensure the leg pad remains fixed during use.
Figure 20: Bracket for the leg pad
Part 8 pin: The pin shown in figure 21 will be placed between the two brackets
Figure 21: Pin
20
Part 9 rotating part: The rotating part shown in Figure 22 will be placed on the pin and is
designed to rotate around it. This part facilitates movement within the system by allowing
controlled rotation while remaining securely attached to the pin
Figure 22: rotating part
Part 10 foot pad: this part is where the user reset his or her foot on shown in figure 23
Figure 23: foot pad
21
Part 11 linear actuators: the device that will perform the vertical movement shown in figure 24
Figure 24: Linear actuators
Part 12 Rod: The rod is shown in Figure 25 a crucial component designed to support the foot
pad, providing stability during vertical movement, it is connected to the linear actuator, which
enables the rod to move vertically
Figure 25: Rod
22
After assembling all the parts, we have obtained the final product, as shown in Figure 26. This
complete assembly integrates all the individual components, ensuring they work together
seamlessly. The design and structure are now fully functional, with each part contributing to the
overall performance of the mechanism.
Figure 26: Seated Exercise Machine for The Legs of The Elderly
3.3 Material selection
Table 3 Material type for each component
Material
type
Table 1: Material
selection
ASTM A48 Gray
Cast Iron class 20
Name of the component Image of the component
Rack & Pinion
23
Slider
Support platform
ABS (Acrylonitrile
Butadiene Styrene)
Brackets for the leg pad
Pin
Rotating part
Leg pad
Base
24
Rod
Aluminum(1060
Alloy)
Bracket
Rails
ASTM A48 Gray Cast Iron Class 20 is an excellent choice for rack and pinion mechanisms due to
its cost-effectiveness, ease of machining, and good strength and durability. It is well-suited for
moderate load applications, making it ideal for this system. ABS (Acrylonitrile Butadiene Styrene)
is a popular material for 3D printing, offering strength, durability, and lightweight properties see
Appendix B for more information about ABS. It has good impact resistance, is easy to mold,
making it a reliable choice for parts exposed to stress. Aluminum, known for its lightweight and
durable nature, is commonly used in mechanical systems. It reduces overall weight, resists rust
and corrosion, and is easy to machine, making it versatile for different applications.
25
3.3 Force and stress analysis
For the force and stress analysis we are going to use SolidWorks Simulation to analyze our design
accurately, so for the setup of the simulation we started by fixing the base to imitate the ground in
real life as shown in figure 27
Figure 27: fixing the base
After fixing the base we applied two forces shown in figure 28 that act on the two-foot pads
these forces were determent by getting the average weight of human legs which is 27.2 [16] kg
we then used this formula to calculate the force:
The wight of one leg is:
𝑚=
27.2 𝐾𝑔
= 13.6 𝑘𝑔
2
Calculating the Force:
𝐹 =𝑚 ∗𝑔
𝐹 = 13.6 ∗ 9.81 ≈ 134 𝑁
Figure 28: Appling force on the pad
26
After running the simulation, we found the reaction forces shown in table 4 below also we can
clearly see that from the figure 29 (*Note that the deformation is scale up in the figure) that we
have two critical parts that may fail the first part shown in figure 30 is the rod that sits on the two
linear actuator and its purpose is to act as a support for the pad and move the pad in a vertical
motion when the liner actuators are switched on, The second part is the pin that will also supports
the pad shown in figure 31.
Figure 29: Deformation
Figure 31: pin
Figure 30: rod
27
Table 4: force analysis results
Fixture
name
Fixture Image
Fixture Details
Entities:
Type:
Fixed-1
1 face(s)
Fixed
Geometry
Resultant Forces
Components
Reaction force(N)
Reaction
Moment(N.m)
X
o
Y
267.969
Z
o
Resultant
267.969
0
0
0
0
After knowing these parts can fail, we need to in investigate them and find the factor of safety for
both parts by using Von Mises Stress, For the first part as shown in table 2 we found the minimum
factor of safety to be 1.85 in our application it is satisfactory as in our application we don’t have a
large consequence when that particular part fails, it won’t cause any harm to the person using the
machine and it is easily replaceable if it fails, for the second part the minimum factor of safety
came out to be 6.1 as shown in table 3 which is more than enough to.
Table 5: Factor of Safety results for the rod
Name
Factor of
Safety1
Type
Von Mises Stress
Min
Max
1.850e+00
Node: 165879
5.706e+02
Node: 166315
Assem1-1-Factor of Safety-Factor of Safety1
28
Table 6: Factor of Safety results for the pin
Name
Factor of Safety2
Name
Factor of Safety1
Type
Von Mises Stress
Type
Max Normal Stress
Min
Max
6.163e+00
Node:Min
235467
1.104e+00
Node: 165879
2.529e+03
Node:
Max
234796
5.706e+02
Node: 166315
Assem1-1-Factor of Safety-Factor of Safety1
Assem1-1-Factor of Safety-Factor of Safety2
Table 2
29
3.4 Motion analysis
For sinusoidal motion, the position, velocity, acceleration, and angular parameters (θ, ω, α) are
calculated based on the principles of simple harmonic motion. These calculations are crucial for
designing the rack and pinion mechanism to ensure it meets performance requirements.
Assumptions and Initial Parameters:
1. Frequency (f = 1 Hz): A frequency of 1 Hz was selected to ensure smooth and
controlled motion suitable for elderly users, prioritizing comfort and simplifying system
analysis.
2. Pitch Radius of Pinion (rp = 0.06 m): Chosen to match the dimensions of the leg
exercise machine while providing effective motion transfer.
3. Amplitude (A = 0.125 m): Determined based on the rack length of 0.5 m. The motion
begins at the center of the rack, making the amplitude half the distance from the center to
the rack's end.
The maximum position, velocity, and acceleration for the mechanism are calculated using
sinusoidal equations. MATLAB is utilized to plot and validate these results, ensuring accuracy:
1-calculating the maximum position:
𝑋 (𝑡) = 𝐴 × sin(2 × π × f × t) (𝑚)
A=0.125 m
f=1 Hz
t=0.25 s (maximum position occurs at this time)
𝑋𝑚𝑎𝑥 = 0.125 × sin(2 × 𝜋 × 1 × 0.25) = 0.125 𝑚
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 32:
Figure 32: Position plot on MATLAB
30
2-Calculating the maximum velocity:
𝑉(𝑡) = (2 × 𝜋 × 𝑓) × 𝐴 × cos(2 × 𝜋 × 𝑓 × 𝑡)
𝑚
( )
𝑠
A=0.125 m
f=1 Hz
t=1 s (maximum velocity occurs at this time)
𝑚
𝑉𝑚𝑎𝑥 = (2 × 𝜋 × 1) × 0.125 × cos(2 × 𝜋 × 1 × 1) = 0.785 ( )
𝑠
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 33:
Figure 33: Velocity plot on MATLAB
3- maximum acceleration calculations:
𝑎 (𝑡) = −(2 × 𝜋 × 𝑓)2 × 𝐴 × sin (2 × 𝜋 × 𝑓 × 𝑡)
𝑚
( 2)
𝑠
A=0.125 m
F=1 HZ
t=0.25 s (maximum acceleration occurs at this time)
𝑎𝑚𝑎𝑥 = (2 × 𝜋 × 1)2 × 𝐴 × sin(2 × 𝜋 × 1 × 0.25) = 4.934
𝑚
( 2)
𝑠
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 34:
Graph 1: Time vs Position
B
Figure 33:
34: Acceleration plot on MATLAB
31
Angular displacement, angular velocity, and angular acceleration of the pinion will be
determined using kinematics. These values are essential for calculating the force, torque, and
power required by the mechanism:
1-calculating maximum angular displacement:
𝜃=
𝑋𝑚𝑎𝑥
𝑟𝑝
(𝑟𝑎𝑑)
𝑋𝑚𝑎𝑥 = 0.125 𝑚
𝑟𝑝 = 0.06 𝑚
𝜃=
𝑋𝑚𝑎𝑥 0.125
=
= 2.083
𝑟𝑝
0.06
(𝑟𝑎𝑑)
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 35:
Figure 35: Angular Displacement plot on MATLAB
32
2-calculating maximum angular velocity:
𝜔=
𝑉𝑚𝑎𝑥 = 0.785
𝑉𝑚𝑎𝑥
𝑟𝑝
(
𝑟𝑎𝑑
)
𝑠
𝑚
𝑠
𝑟𝑝 = 0.06 𝑚
𝜔=
𝑉𝑚𝑎𝑥 0.785
𝑟𝑎𝑑
=
= 13.08 (
)
𝑟𝑝
0.06
𝑠
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 36:
Figure 36: Angular Velocity plot on MATLAB
3- calculating maximum angular acceleration:
𝛼𝑚𝑎𝑥 =
𝑎𝑚𝑎𝑥 = 4.934
𝑎𝑚𝑎𝑥
𝑟𝑝
(
𝑟𝑎𝑑
)
𝑠2
𝑚
𝑠2
𝑟𝑝 = 0.06 𝑚
𝛼𝑚𝑎𝑥 =
𝑎𝑚𝑎𝑥 4.934
𝑟𝑎𝑑
=
= 82 ( 2 )
𝑟𝑝
0.06
𝑠
33
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 37:
Figure 37: Angular Acceleration plot on MATLAB
After performing the hand calculations and plotting the equations in MATLAB to generate the
corresponding graphs, the exact maximum values for displacement (X), velocity (V), acceleration
(a), angular velocity (ω), angular displacement (θ), and angular acceleration (α) were determined.
These values were extracted at the points where each respective variable reached its peak value
during the motion.
34
3.5 Motor selection
The methodology for motor calculations begins by determining the force required to move the
rack in the pinion and rack mechanism. This includes accounting for the load mass, acceleration,
and frictional resistance in the system. The calculated force is then used to determine the torque
at the pinion, which is derived based on the radius of the pinion gear. Additionally, the power
requirement of the motor is calculated to ensure it can deliver sufficient energy to meet the
operational demands. These calculations are essential to ensure the motor selection aligns with
the mechanical and functional requirements of the system.
The total mass being moved consists of the following:
1. Mass of both legs that will be on top of the pad is 𝑀𝐿 = 27.2 𝐾𝑔
2. Mass of the components on both sides is 𝑀𝑐 = (5 × 2) + (0.01 × 4) = 10.04 𝐾𝑔
3. Mass of the pinion is 𝑀𝑝 = 2.94 𝐾𝑔
1- calculating maximum Force:
𝐹𝑚𝑎𝑥 = 𝑀𝑡𝑜𝑡𝑎𝑙 × 𝑎𝑚𝑎𝑥 = (𝑀𝐿 + 𝑀𝑐 ) × 𝑎𝑚𝑎𝑥
(𝑁)
𝑀𝑡𝑜𝑡𝑎𝑙 = 𝑀𝐿 + 𝑀𝑐 = 27.2 + 10.04 = 37.24 𝐾𝑔
𝑎𝑚𝑎𝑥 = 4.934
𝑚
𝑠2
𝐹𝑚𝑎𝑥 = 𝑀𝑡𝑜𝑡𝑎𝑙 × 𝑎𝑚𝑎𝑥 = (𝑀𝐿 + 𝑀𝑐 ) × 𝑎𝑚𝑎𝑥 = 37.24 × 4.934 = 183.74 (𝑁)
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 38:
Figure 38: Force plot on MATLAB
35
2- Calculating maximum n (RPM):
𝑛𝑚𝑎𝑥 =
𝜔𝑚𝑎𝑥 = 13.08
𝜔𝑚𝑎𝑥 × 60
2×𝜋
(𝑅𝑃𝑀)
𝑟𝑎𝑑
𝑠2
𝑛𝑚𝑎𝑥 =
𝜔𝑚𝑎𝑥 × 60 13.08 × 60
=
= 124.9
2×𝜋
2×𝜋
(𝑅𝑃𝑀)
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 39:
Figure 39: RPM plot on MATLAB
3- Calculating maximum torque:
𝜏𝑚𝑎𝑥 = (𝐹𝑚𝑎𝑥 × 𝑟𝑝 ) + (𝐼𝐺 × 𝛼)
(𝑁. 𝑚)
𝐹𝑚𝑎𝑥 = 183.74 𝑁
𝑟𝑝 = 0.06 𝑚
𝐼𝐺 = 0.5 × 𝑀𝑝 × 𝑟𝑝2 = 0.5 × 2.94 × 0.062 = 5.292 × 10−3 𝐾𝑔. 𝑚2
𝜏𝑚𝑎𝑥 = (𝐹𝑚𝑎𝑥 × 𝑟𝑝 ) + (𝐼𝐺 × 𝛼) = (183.74 × 0.06) + (5.292 × 10−3 × 82)
= 11.45 (𝑁. 𝑚)
36
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 40:
Figure 40: Torque plot on MATLAB
4- calculating the maximum actual power:
𝑃𝑚𝑎𝑥 = 𝜏𝑚𝑎𝑥 × 𝜔𝑚𝑎𝑥 (𝑊)
𝜏𝑚𝑎𝑥 = 11.45 𝑁. 𝑚
𝑟𝑎𝑑
𝜔𝑚𝑎𝑥 = 13.08
𝑠
𝑃𝑚𝑎𝑥 = 𝜏𝑚𝑎𝑥 × 𝜔𝑚𝑎𝑥 = 11.45 × 13.08 = 149.76 (𝑊)
𝑃𝑎𝑐𝑡𝑢𝑎𝑙 =
𝑃𝑚𝑎𝑥
(𝑊 )
𝜂
𝑃 = 149.76 (𝑊)
𝜂 = 95%, 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑎𝑐𝑘 𝑎𝑛𝑑 𝑝𝑖𝑛𝑖𝑜𝑛 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑠𝑚 𝑑𝑢𝑒 𝑡𝑜 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛
𝑃𝑎𝑐𝑡𝑢𝑎𝑙 =
𝑃𝑚𝑎𝑥
149.76
=
= 157.64 (𝑊 )
𝜂
0.95
Plotting the above equation in MATLAB from 0 to 10 seconds produces the following graph
shown in Figure 41:
Figure 41: Time vs Actual maximum power
37
After performing the hand calculations and plotting the equations in MATLAB, the maximum
values for RPM, force, torque, and actual power were determined. These values were extracted at
the points where each variable reached its peak during the motion. Also, the kind of motor we will
choose based on these calculations and our application will be reversible DC motor.
3.6 Design of a Pinion and Rack Mechanism
Designing a pinion and rack mechanism involves both a priori and design decisions. A priori
decisions define the system’s function, including load capacity, speed, reliability, and lifespan,
while considering safety factors to mitigate risks. Key parameters such as the tooth system
(pressure angle, addendum, dedendum, root fillet radius), gear ratio, and quality number are set to
ensure smooth operation and precise manufacturing. Design decisions begin with selecting a trial
diametral pitch, which affects tooth size, load capacity, and spatial constraints. Materials and
hardness for the pinion and rack are chosen to balance strength, wear resistance, and compatibility,
and the face width is estimated and adjusted to meet safety factors. Detailed analyses, including
bending analysis to assess cyclic stresses and wear analysis to verify surface hardness and
durability, are performed. If the safety factors or performance criteria are inadequate, parameters
are revised and the process is iterated until all requirements are met.
Priori decisions:
1-Motor specifications that were calculated: Power is 0.212 HP; speed is 125 rpm.
2-Function: reliability as 99%, life of 107 cycles.
3-design factor: 𝑛𝑑 = 2.
4-pressure angle 𝜙 = 20∘ .
5-Rack ratio= 4.16 to 1.
6-quality number: 𝑄𝑣 = 6.
Design decisions:
1-module: module is 3 mm/teeth, converted to diametral pitch for calculations 𝑃𝑑 =
8.46 𝑡𝑒𝑒𝑡ℎ/𝑖𝑛𝑐ℎ.
2-dimensions: pitch diameter 𝑑𝑝 = 120 𝑚𝑚 = 4.72 𝑖𝑛, length of the rack 𝐿 = 500 𝑚𝑚 =
19.6 𝑖𝑛 .
2-material: ASTM A48 Gray Cast Iron class 20
3-face width: recommended value of Face width should be 3p<F<5p, 𝑝 = 𝜋 × 𝑚, F=30
mm=1.1811 in.
4- Backup ratio: assume backup ratio 𝑚 𝐵 ≥ 1.2 so 𝐾𝐵 = 1
38
For a walking exercise machine operating at 125 RPM for 1 hour per day, the rack and pinion
mechanism would undergo approximately 7,500 cycles per day. Over 4 years, this totals around
10.95 million cycles (2,737,500 cycles per year), which is very close to the target of 10 million
cycles, making it a reasonable estimate for the mechanism's expected lifespan. These values align
with widely accepted mechanical design standards, such as a 99% reliability for AGMA gear mesh
design. A safety factor of 2, a 20° pressure angle, and a quality number of 6 are standard choices
for balancing performance, manufacturability, and cost. ASTM A48 Gray Cast Iron class 20 is
selected for its strength and wear resistance, while face width and module are based on established
guidelines to ensure proper load distribution and compatibility. Following the a priori and design
decisions, calculations will adhere to the methodology outlined in ANSI/AGMA 2001-D04,
covering key aspects like load analysis, material selection, safety factors, and wear and bending
strength calculations.
Tooth count calculations:
𝑁𝑝 =
𝑑𝑝 120
=
= 40 𝑡𝑒𝑒𝑡ℎ 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑝𝑖𝑛𝑖𝑜𝑛
𝑚
3
𝑁𝑟 =
𝐿
500
=
= 167 𝑡𝑒𝑒𝑡ℎ 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑟𝑎𝑐𝑘
𝑚
3
Interference check for the pinion:
𝑁𝑝 =
2×𝐾
sin (𝜙)2
K=0.8 for stub tooth
𝑁𝑝 =
2×𝐾
2 × 0.8
=
= 14 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑒𝑡ℎ 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑝𝑖𝑛𝑖𝑜𝑛
2
sin (𝜙)
sin (20)2
39
Lewis form factor Y:
𝑌𝑝 = 0.389 (𝑏𝑦 𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛)
𝑌𝑟 = 0.485
Geometry factor J (figure 14-6 in the book):
𝐽𝑝 = 0.43
40
𝐽𝑟 = 0.47
Transmitted load:
𝑉=
𝜋 × 𝑑𝑝 × 𝑛 𝜋 × 4.72 × 125
𝑓𝑡
=
= 155 (
)
12
12
min
𝑊𝑡 =
33000 × 𝐻 33000 × 0.212
=
= 45.13 𝑙𝑏𝑓
𝑉
155
Dynamic factor:
𝐵
𝐴 + √𝑉
𝐾𝑣 = (
)
𝐴
𝐵 = 0.25(12 − 𝑄𝑣 )2/3 = 0.25(12 − 6)2/3 = 0.825
𝐴 = 50 + 56(1 − 𝐵 ) = 50 + 56(1 − 0.825) = 59.8
𝐵
0.825
𝐴 + √𝑉
59.8 + √ 155
𝐾𝑣 = (
) =(
)
𝐴
59.8
= 1.168
41
Reliability factor:
𝑠𝑖𝑛𝑐𝑒 𝑅 = 0.99 → 𝐾𝑅 = 1
Stress cycle factors 𝑌𝑁 𝑎𝑛𝑑 𝑍𝑁 :
𝑠𝑖𝑛𝑐𝑒 𝑁 = 107 𝑐𝑦𝑐𝑙𝑒𝑠 → 𝑌𝑁𝑃 = 1 𝑎𝑛𝑑 𝑌𝑁𝑟 = 1
42
𝑠𝑖𝑛𝑐𝑒 𝑁 = 107 𝑐𝑦𝑐𝑙𝑒𝑠 → 𝑍𝑁𝑃 = 1 𝑎𝑛𝑑 𝑍𝑁𝑟 = 1
Size factor 𝐾𝑠 :
0.0535
𝐹 × √𝑌𝑝
𝐾𝑠 = 1.192 (
)
𝑃
0.0535
1.1811 × √ 0.3892
= 1.192 (
)
8.466
= 1.046
Load correction factor:
𝐶𝑚𝑐 = 1 𝑓𝑜𝑟 𝑢𝑛𝑐𝑟𝑜𝑤𝑛𝑒𝑑 𝑡𝑒𝑒𝑡ℎ
43
Pinion proportion factor (
𝐹
10×𝑑
< 0.05 𝑎𝑛𝑑 1 < 𝐹 < 17 𝑖𝑛):
Since F/10d<0.05, we will consider F/10d=0.05
𝐶𝑝𝑓 =
𝐹
− 0.0375 + 0.0125 × 𝐹
10 × 𝑑
𝐶𝑝𝑓 = 0.05 − 0.0375 + 0.0125 × 1.1811 = 0.0272
Pinion proportion modifier:
𝐶𝑝𝑚 = 1 𝑓𝑜𝑟 𝑠𝑡𝑟𝑎𝑑𝑑𝑙𝑒 𝑚𝑜𝑢𝑛𝑡𝑒𝑑 𝑝𝑖𝑛𝑖𝑜𝑛 𝑤𝑖𝑡ℎ
𝑆1
< 0.175
𝑆
Mesh alignment factor (for commercial enclosed units):
𝐶𝑚𝑎 = 𝐴 + 𝐵 × 𝐹 + 𝐶 × 𝐹 2
𝐶𝑚𝑎 = 0.127 + 0.0158 × 1.1811 + (−0.930 × 10−4 × 1.18112 ) = 0.1455
44
Mesh alignment correction factor (for all other conditions):
𝐶𝑒 = 1
Load distribution factor:
𝐾𝑚 = 1 + 𝐶𝑚𝑐 (𝐶𝑝𝑓 × 𝐶𝑝𝑚 + 𝐶𝑚𝑎 × 𝐶𝑒 )
𝐾𝑚 = 1 + 1(0.0297 × 1 + 0.1455 × 1) = 1.1752
Geometry factor of pitting resistance (external gears):
𝑚 𝑛 = 1 𝑓𝑜𝑟 𝑠𝑝𝑢𝑟 𝑔𝑒𝑎𝑟𝑠
𝐼=
𝐼=
cos (𝜙) + sin (𝜙)
𝑚𝑟
×
2𝑚 𝑛
𝑚𝑟 + 1
cos (20) + sin (20)
4.166
×
= 0.1295
2×1
4.166 + 1
Temperature factor:
𝐾𝑇 = 1 → 𝑎𝑠𝑠𝑢𝑚𝑖𝑛𝑔 𝑡𝑒𝑚𝑝𝑟𝑒𝑡𝑢𝑟𝑒 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 120 𝑑𝑒𝑔𝑟𝑒𝑒 𝑐𝑒𝑙𝑠𝑖𝑢𝑠
45
Applied bending strength and contact strength (ASTM A48 Gray Cast Iron class 20):
Table 7 Allowable contact stress number for iron and bronze gears
𝑆𝑐 = 55000 𝑃𝑠𝑖
Table 8 Allowable bending stress number for iron and bronze gears
𝑆𝑡 = 5000 𝑃𝑠𝑖
46
Elastic coefficient (Cast Iron for pinion and rack):
𝐶𝑝 = 2100 √𝑃𝑠𝑖
Surface condition factor:
Standard surface conditions for gear teeth have not yet been established. When a detrimental
surface finish effect is known to exist, AGMA specifies a value of Cf greater than unity.
𝐴𝑠𝑠𝑢𝑚𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 𝐶𝑓 = 1
Hardness ratio factor:
𝐶𝐻 = 1 → 𝑠𝑖𝑛𝑐𝑒 𝐻𝐵𝑟 = 𝐻𝐵𝑃 (𝑠𝑎𝑚𝑒 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙)
47
Minimum face width for bending and wear:
𝐹𝑏𝑒𝑛𝑑 = 𝑛𝑑 × 𝑤𝑡 × 𝑘𝑜 × 𝑘𝑣 × 𝑘𝑠 × 𝑃𝑑 ×
𝐹𝑏𝑒𝑛𝑑 = 2 × 45.13 × 1 × 1.168 × 1.046 × 8.46 ×
𝑘𝑚 × 𝑘𝐵 𝑘 𝑇 × 𝑘𝑅
×
𝐽𝑝
𝑆𝑡 × 𝑌𝑁𝑃
1.1752 × 1
1×1
×
= 0.510𝑖𝑛
0.43
5000 × 1
𝑘𝑚 × 𝐶𝑓
𝐶𝑝 × 𝑘𝑅 × 𝐾𝑇 2
)
𝐹𝑤𝑒𝑎𝑟 = 𝑛𝑑 × 𝑤𝑡 × 𝑘𝑜 × 𝑘𝑣 × 𝑘𝑠 ×
×(
𝑑𝑝 × 𝐼
𝑆𝑐 × 𝑍𝑁𝑃
𝐹𝑤𝑒𝑎𝑟 = 2 × 45.13 × 1 × 1.168 × 1.046 ×
1.1752 × 1
2100 × 1 × 1 2
) = 0.309 𝑖𝑛
×(
4.72 × 0.1295
55000 × 1
Our face width is 1.1811 in or 30 mm which is more than the minimum value of face width for
bending and face width for wear.
Factors of safety for Pinion bending and Pinion wear:
𝜎𝑝 = 𝑤𝑡 × 𝑘𝑜 × 𝑘𝑣 × 𝑘𝑠 ×
𝜎𝑝 = 45.13 × 1 × 1.168 × 1.046 ×
𝑃𝑑 𝑘𝑚 × 𝑘𝐵
×
𝐹
𝐽𝑝
8.46
1.1752 × 1
×
= 1079.36 𝑃𝑠𝑖
1.1811
0.43
𝑆𝑡 × 𝑌𝑁𝑃
5000 × 1
𝐾𝑇 × 𝐾𝑅
𝑆𝐹𝑝 =
= 1 × 1 = 4.63
𝜎𝑝
1079.36
1/2
𝑘 × 𝐶𝑓
𝜎𝑐𝑝 = (𝑤𝑡 × 𝑘𝑜 × 𝑘𝑣 × 𝑘𝑠 × 𝑚
)
𝑑𝑝 × 𝐹 × 𝐼
𝜎𝑐𝑝 = (45.13 × 1 × 1.168 × 1.046 ×
× 𝐶𝑝
1/2
1.1752 × 1
) × 2100 = 19895.08 𝑃𝑠𝑖
4.72 × 1.1811 × 0.1295
𝑆𝑐 × 𝑍𝑁𝑃
55000 × 1
𝐾𝑇 × 𝐾𝑅
1×1
𝑆𝐻𝑝 =
=
= 2.76
𝜎𝑐𝑝
19895.08
48
Factors of safety for Rack bending and Rack wear:
𝐽
0.43
𝜎𝑅 = 𝜎𝑝 × 𝑝 = 1079.36 ×
= 987.49 𝑃𝑠𝑖
𝐽𝑅
0.47
𝑆𝑡 × 𝑌𝑁𝑅
5000 × 1
𝐾𝑇 × 𝐾𝑅
𝑆𝐹𝑅 =
= 1 × 1 = 5.06
𝜎𝑅
987.49
𝐻𝑎𝑟𝑑𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑟𝑎𝑐𝑘 𝑎𝑛𝑑 𝑝𝑖𝑛𝑖𝑜𝑛 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑠𝑜 𝑆𝐻𝑅 = 𝑆𝐻𝑝 = 2.76
Comparing factors of safety to see whether bending or wear is the threat:
𝑆𝐹𝑝 = 4.63
2
𝑆𝐻𝑝
= 2.762 = 7.61
So, from this comparison we can see that for the Pinion, bending is the main threat.
𝑆𝐹𝑅 = 5.06
2
𝑆𝐻𝑅
= 2.762 = 7.61
So, from this comparison we can see that for the Rack bending is the main threat.
49
Minimum Rim thickness below the tooth:
ℎ𝑡 =
1 1.25
1
1.25
+
=
+
= 0.2657 𝑖𝑛 (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑡𝑜𝑜𝑡ℎ)
𝑃𝑑
𝑃𝑑
8.466 8.466
𝑡𝑅 ≥ 𝑚 𝐵 × ℎ𝑡 ≥ 1.2 × 0.2657 ≥ 0.3188 𝑖𝑛 (𝑟𝑖𝑚 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑠ℎ𝑜𝑢𝑙𝑑 𝑒𝑥𝑐𝑒𝑒𝑑 0.3188 𝑖𝑛)
After performing the rack and pinion calculations to determine the factors of safety for both wear
and bending using AGMA standards, we can conclude that our factors of safety are adequate. The
lowest factor of safety, which is for wear, was 2.76. This value is not excessively high, indicating
that the design is not overly conservative, and it is well above the minimum required value of 1,
ensuring the component's reliability under operating conditions. According to AGMA, the typical
range for the factor of safety for wear is between 1.5 and 3, which aligns with our calculated value,
further confirming the design's robustness and durability under the expected loads and conditions.
50
Chapter four
MANAGMENT
4.1 Breakdown of Work into Specific Tasks
The project was divided into four phases: Research, Design, Analysis, and Finalization.
In the Research Phase, we focused on gathering knowledge and exploring methods to achieve the
project objectives, including evaluating different mechanical systems like rack and pinion
mechanisms and selecting appropriate materials based on strength, durability, and cost.
During the Design Phase, we translated our research into detailed models using SolidWorks, with
special attention to the rack and pinion mechanism. We carefully integrated each component to
ensure functionality, ergonomics, and safety.
In the Analysis Phase, we validated the design through force analysis, motor calculations, and
motion analysis to ensure that the system could handle operational loads and operate smoothly.
Finally, in the Finalization Phase, we compiled the project report and prepared presentation slides
to summarize key design considerations, analysis results, and conclusions.
51
4.2 Breakdown of Work into Specific Tasks Flow Chart
52
4.3 Gantt Chart for the Organization of Work and Timeline
Table 9: Gantt table
Table 9: force analysis results
Figure 42 : Gantt chart
53
4.4 Breakdown of Responsibilities Among Team Members
Khalid Khalid Abuamra:
•
•
•
•
•
•
•
•
•
Conducted researched for the two movements
Design the leg exercise machine in SolidWorks
Selected the material for the components and selected the 3D material
Done the SolidWorks simulation and found the factor of safety
Selected the appropriate linear actuators for our application
Worked on the calculations of finding the suitable motor
Worked on the calculations of rack and pinion calculations
Worked on the calculations of the motion analysis
Contributed to the project report, presentation slides, and project
Omar Hani Shaheen:
•
•
•
•
•
•
•
•
•
Conducted research for the two types of movements.
Assisted in developing design concepts using SolidWorks.
Researched and selected suitable materials for 3D printing.
Performed rack and pinion analysis.
Conducted force analysis to ensure structural integrity and functionality.
Completed motion analysis for both movements.
Calculated motor requirements to meet operational demands.
Selected appropriate linear actuators for the application.
Contributed to drafting the project report and creating the presentation.
Omar Wael:
•
•
•
•
•
•
•
Conducted research for the two types of movements.
Assisted in designing our mechanism.
Researched for the appropriate linear actuator for our mechanism.
Researched for the right motor that meets the specifications of our mechanism.
Used MATLAB software for calculations and analysis of position, velocity, acceleration,
force, etc.
Assisted in choosing the right parts for our mechanism.
Contributed to writing the report and presentation.
54
4.5 Budget Management
Table 10 Budget Management
Expense Category
Linear actuators
ABS Filaments
Power sources
Motor
assembly
Quantity
4
4
3
1
1
Total cost
Amount (AED)
1000
390
950
200
1600
4140
55
Chapter Five
CONCLUSIONS
In conclusion, this project creates a seated exercise machine for the elderly, combining safety,
comfort, and ease of use. It features a walking-like motion driven by a rack-and-pinion mechanism
and a vertical calf-flexing motion powered by linear actuators, designed to engage the legs while
minimizing joint strain. The compact design, stable base, and non-slip footrests make it ideal for
home use, promoting regular exercise and improving mobility. The development followed a
structured process, from research and design to analysis and finalization, resulting in a safe and
practical solution to help elderly users maintain physical activity and improve their quality of life.
56
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58
APPENDICES
Appendix A
Figure 43: Comparison Between different material
59
Appendix B
Figure 44: Technical data sheet for ABS 1
60
Figure 45: Technical data sheet for ABS 2
61