Final Dynamics Report

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Dynamics of a Commercial Elevator

ES 2503 Introduction to Dynamic Systems

Professor M. S. Fofana, PhD, [email protected]

Mechanical Engineering Department

Ryan Coran - Civil Engineering - [email protected]

Thai Dinh - Mechanical Engineering - [email protected]

Patrick Fitzgerald - Mechanical Engineering and Physics - [email protected]

Nour Krayem - Biomedical Engineering - [email protected]

Worcester Polytechnic Institute


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1 Table of Contents:1 Table of Contents: ................................................................................................................. 1

2 Abstract ................................................................................................................................. 3

3 Research and First Steps ...................................................................................................... 4 3.1 Project Objectives and Goals .......................................................................................... 4

3.2 Elevator Components and Overview of Function ............................................................ 4

3.3 Range of Speeds and Accelerations and Other Related Variables ................................. 6

3.4 Engineering and Technological Development of the Elevator ......................................... 8

3.5 Power Sys tems and Components Important to the Elevators Operation ........................ 9

4 The Elevator System ............................................................................................................12

4.1 Components of the System............................................................................................12

4.1 Defining Variables ......................................................................................................14 4.1.1 Types of Variables ...............................................................................................14

4.1.2 Subscript Meanings .............................................................................................16

4.1.3 Additional Parameters .........................................................................................17

4.1.4 Numerical Assignments .......................................................................................19

4.1.4.1 Fixed Values .................................................................................................20

4.1.4.2 Variable Values and Their Ranges ................................................................20

4.2 Defining the Kinematics: ................................................................................................21

4.2.1 Cables Constraint ...................................................................................................21

4.2.2 Defining the Path of Motion .....................................................................................22

4.2.2.1 Long Elevator Trips ...........................................................................................23

4.2.2.2 Short Elevator Trips ..........................................................................................25

4.2.2.3 Trip Displacement Cutoff Distance ....................................................................27

4.2.2.4 A Note ..............................................................................................................27

4.3 Determining the Equations Relating Forces ...................................................................27

4.3.1 Free Body Diagrams ...............................................................................................28

4.3.1.1 Subsystem 1: Passenger Car ...........................................................................29

4.3.1.2 Subsystem 2: Wire 1 .........................................................................................30

4.3.1.3 Subsystem 8: Counterweight ............................................................................31

4.3.1.4 Subsystem 7: Wire 3 .........................................................................................32

4.3.1.5 Subsystem 6: Pulley 2 ......................................................................................33


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4.3.1.6 Subsystem 5: Wire 2 .........................................................................................34

4.3.1.7 Subsystem 3: Pulley 1 ......................................................................................35

4.3.1.8 Subsystem 4: Motor ..........................................................................................36

4.3.2 Additional Parameters .............................................................................................37

4.3.2.1 System Power Consumption .............................................................................37

4.3.2.2 Change in Force Felt by Passenger ..................................................................37

5 Visualizing the Mathematical Relationships ..........................................................................39

6 Conclusion ...........................................................................................................................52

7 Appendices ..........................................................................................................................54

7.1 Appendix A: Deriving Motion ..........................................................................................54

7.1.1 Long Trip Motion .....................................................................................................54

7.1.2 Short Trip Motion .....................................................................................................55

7.1.3 Long-Short Boundary Condition ..............................................................................56

7.2 Appendix B: Force Equations ........................................................................................56

7.2.1 Equations for Pulley 2 .............................................................................................56

7.2.2 Equations for Pulley 1 .............................................................................................57

7.2.3 Equations for Motor .................................................................................................58

8 References...........................................................................................................................59


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2 Abstract

In this project, we will examine the kinematics and dynamics of a gearless elevator

powered by a DC motor installed in a tall building. Our objectives are to investigate the

components of a commercial elevator, to understand how the individual components work

together to achieve the overall function of the elevator, to research and calculate the dynamic and

kinematic properties of an elevator (distances, speeds, accelerations, forces, torques, momenta,

etc.), to research the history and development of the elevator. We are motivated by what

engineers have done in the past then we would like to analyze that to improve the functionality

and safety of commercial elevators, and through this examination we hope to gain insight into

those improvements and even think of more improvements ourselves.

The methods that we use to analyze are breaking project prompts up between members tospread out workload, deciding on a specific type of elevator and motor system to focus on,

breaking down the system into components and study their interactions, determining

/approximating the forces and torques and resultant motion between the different components,

tracing the transfer of energy through the system, interpreting the individual motions of

components to understand the overall function of the elevator. The tools that help us to analyze

are sketches/free body diagrams, kinematic equations, engineering handbook, online engineering

resources.

From this project, we hope to learn how to better analyze a complex physical system

whose function is often taken for granted. We also intend to make use of the theories of

dynamics that we will learn in class to understand a pragmatic example of an elevator. We intend

to channel this newfound experience from this project to make more informed decisions to

improve the elevators safety and function. In a broader sense, we would also like to improve our

physical understanding and awareness of problems we will face in the future.


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3 Research and First Steps

3.1 Project Objectives and Goals

The purpose of the project is to examine the components of a commercial elevator, and to

decompose the system in order to analyze the dynamic and kinematic properties. In order to

specifically define the problem, our group researched the history and development of the elevator

and decided to examine a 1:1 gearless traction elevator powered by a DC motor, since it

appeared to be the most commonly used elevator type in tall commercial buildings.[16] With this

specification, we were able to find particular components used in these settings, and we are able

to look up the necessary values related to them, which are needed to complete the mathematical

portion of our project. In the secondary phase of this examination of elevators, we will analyze

each individual component and trace the energy from one component to the next to understand

how each component interacts to achieve the overall function. Using this knowledge of

interaction and the assumed kinematic values and physical constants, we will ascertain the

unknown dynamic and kinematic properties, including forces and torques. We are also going to

analyze the safety components of the elevator to explain and ensure the safety of the elevator we

are analyzing.

3.2 Elevator Components and Overview of Function

Normally, when a building owner or contractor wants to set up an elevator in their

building, the size specifications and expected capacity loosely determine the type of elevator

they must set up. Once the type of elevator is chosen (which largely defines the type of motor

assembly and pulley system), the building owner or contractor will likely need to buy the

different components separately, or have an elevator company come in and install all their own

products. The different components are comprised of the motor assembly, pulley systems, theelevator car, the counterweight, extra safety precautions, and any electronics necessary to get

everything to run smoothly. The dynamics and kinematics of the safety systems and the inner-

workings of the electrical systems are both beyond the scope of this analysis, since they are not


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directly involved in the mechanical and physical interactions needed for normal function of the

elevator.

The primary part of the elevator system, from a functional point of view, is the motor

assembly. The motor assembly is comprised of a motor and a braking system, and may or may

not include a system of gears to manipulate the speed and torque of the system. The effect of

gears on a motor system are important to understand to discern the usefulness of a geared or

gearless motor assembly in a given situation: Many of the most efficient motors in production

produce very high rotational speeds and rather low torques, but the magnitudes of these

respective variables is very much the opposite for the application of elevators. To meet this end,

gear boxes are often employed to simultaneously slow the rotational speed and increase the

torque at the output shaft. Short elevator systems normally dont need much speed since

distances are low too, but for taller buildings, the speed is expected to be higher to maintain thetravel-time. The torque required to move it is only dependent on the system mass, not the system

height. This presents a problem for the geared systems in tall buildings, since the expected high

speeds are sacrificed for the necessary amount of torquehowever there is a solution, for a

greater price: if the motor itself is upgraded to a much more powerful motor, then the gear

system can be omitted. In this case, the new, default torque is is comparable to the amplified

torque of a lower-power motor using gears, and the default speed of the higher-power motor

meets the expectation of faster movement [16]. With respect to this examination of a commercial

elevator, the gearless motor assembly will be considered for a tall building, since many large

scale commercial buildings are tall.

Directly on this motor assembly, regardless of the geared/gearless specification, is a

braking system, which can severely reduce the allowable torque transfer and rotational speed of

the assembly. The brake effectively cuts off the transfer of mechanical power to the entire system

and also disallows further motion of any component of the elevator when a cease of motion is

necessary. This brake is enabled during both normal use and emergency use: the electronic

circuits will enable the brakes when the passenger car arrives as a floor to stabilize everything for

the riders, and in emergencies, if the electronics system recognizes something is amiss, for

example power supply to the motor is cut, this braking mechanism will bring the system to a stop

and disallow the car from free falling. The brake inside the motor assembly serves as both a


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frequently used component in the elevators standard function , and also as an emergency system

to prevent dangerous accidents.

Directly on the end of this motor assembly shaft is the sheave, which is the driving pulley

for all the vertical motion[5]. The outside of the diameter of this pulley is grooved to hold the

cables in their intended position as they are rolled along from one side of the pulley to the other.

These grooves also have a high coefficient of friction with respect to the cables to ensure against

slipping of the cable with respect to the pulleythis is is what the term traction refers to in the

name traction elevator[1]. The sh eave serves to pull the cables in the desired direction based on

the direction the motor assembly turns it. Another pulley exists in the machine room, which is

free to spin without restriction, and redirects the side of the cable going to the counterweight so

that the elevator car and the counterweight are out of each others way during their movements.

For an interactive three dimensional visual representation of the sheave, motor assembly, cables,and this secondary pulley, see [2].

This motion of the cables directly controls the height of the elevator car and the

counterweight, since they are directly connected. When the elevator car is pulled up, the

counterweight goes down, and visa-versa. The counterweight is designed to have the weight

approximately equal to the weight of the elevator car plus 40 to 50 % of the passenger capacity,

so that for the mean elevator ride, the weight of the counterweight cancels the weight of the

passenger car and passengers [14]. By doing this, the amount of work needed to raise or lower

the elevator car is minimized, and the braking force needed at each floor is also minimized.

The functionality of the motor assembly and cable system is a very interesting one, and is

often best displayed through visual representations. See [6] for a video showing the different

components operate together in the machine room, and [9] for a more academic presentation of

the inner workings of the entire elevator through 3D simulations.

3.3 Range of Speeds and Accelerations and Other Related

Variables

In order to properly break down the kinematics of the elevator for later calculations, it is

necessary to determine specifications for some given elevatorthe expected speeds,


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accelerations, jerks, and allowable heights are all important to understand the motion of the

elevator, but actual values for these variables will be needed to to attain any appreciable

numerical results. To meet this end, we have selected a specific model of elevator from a

supplier, and the specifications associated are shown in Table 1 below.

Variable 1:1 Gearless

Min Max

Acceleration Rates [ft/sec 2] 2 5

Jerk Rates [ft/sec 3] 4 10

Floor-to-Floor Time [sec] 4 7

Speed [ft/min] 400 1400

Travel [floors] 10 120

Leveling Error [in] 0 3/8

Ride Quality Very Smooth

Table 1: Dynamic and Kinematic Properties Associated with the Elevator [7]

The values stated in Table 1 will be used in the examination of the kinematic properties

in Phase 2. As far as this project is concerned, these kinematic values are assumed, and must be

acknowledged as such. In addition to these kinematic values, their dynamic counterparts must

also be assumed, in order to properly execute a number of calculations in Phase 2. However, the

properties necessary for determining the function of the elevator are not directly correlated to the

elevator itself; instead they correlate to the motor assembly which drives the system. For this

purpose we selected a brand of motor, which is compatible with our selected elevator, and we

have included Table 2 below, in order to convey the dynamic properties that are associated with

our choice of motor.

Variable Elevator Motor

Traction Ratio 2:1

Rated Capacity [kg] 800

Max Shaft Load [kg] 3000

Applicable Converter 3 phase 400V


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DC Voltage for Brake Opening [V] 200

DC Voltage for Brake Holding [V] 100

Traction Sheave Diameter [mm] 400

Table 2: Dynamic and Kinematic Properties Associated with the Motor [10] As previously stated, the above Table 2 defines our assumptions for specifications

defined by our motor assembly. These values will be heavily used in examining the forces and

torques throughout the system, and Table 1 and Table 2 will be referenced and drawn upon in

Phase 2.

3.4 Engineering and Technological Development of the Elevator

One of the first uses of the elevator was around 80 AD, where passengers were moved between floors of the Roman Colosseum by gladiators and wild animal power. Little progress

was made in the following centuries, and even during the Middle Ages elevators were powered

by means of human and animal force as a source of energy. Also during these early days of the

elevator, there were not any significant safety systems to ensure the wellbeing of the passengers

if and when the cable snapped or something went awry [15]. In 1853, Elisha Graves Otis

invented the safety equipment for the elevator by using ratchet bars and spring. He publicly

demonstrated his safe elevator design in the following year in New York City [15]. Later, the

first steam-powered elevator was brought into service in a store at Broadway. In the late 1800s,

the first hydraulic elevator was introduced and went into service, which offered increased

speeds many times than the previous types based on pulleys. After that, with the advent of

electricity, the electric motor was introduced into elevator technology [3]. Taller buildings

became feasible with the development of the direct-connected geared electric elevator in 1889,

since the height was only a problem for the buildings ease of use, and had much less to do with

any engineering limitations of the time. As buildings became even taller, the elevator design

evolved into the gearless traction electric elevator to make better use of the available motors. The

traction electric elevator has become the standard in the elevator industry because it can be built

in any building with varied heights and its speed is much faster than that of the steam-powered

one.


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3.5 Power Systems and Components Important to the Elevators

Operation

For the elevator to move there will be an electrical power source converted to mechanical

(rotational) power by means of the motor. The elevator will have brakes to ensure safety in case

the cable of the elevator brakes. The job of the elevator is to pick up and drop off passengers, and

the space inside an elevator needs to be enough to fit a safe amount of passengers. The expected

amount of passengers in the elevator defines the specifications such as capacity for the elevator.

The most important structures of the elevator are the motor assembly, the cable-pulley system

and the brakes [11].

The system of cables and pulleys is a crucial component that attaches the motor, the

elevator car, and the counterweight. The type of elevator that the team aimed to examine was thegearless elevator that is powered by a DC motor installed in a tall building, such as shown in

[Fig. 1] to the right. Cable system is firmly pulled against the stationary sheave, and it is through

this physical connection that the rotational motion of the sheave causes the motion of the cables

along their existent path, which then causes the vertical movement of the elevator car and

counterweight. This transfer of force is done through the high friction area on the sheave, which

directly constrains the motion of the cable touching the sheave to be the same motion as the outer

diameter of the sheave itself. The difference in weight between the elevator car plus passengers

and the counterweight also creates an imbalance, and in addition to the sheave needing to apply a

torque to move the system, the sheave needs to apply additional torque to keep the system in

stationary equilibrium while stopped [11].


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powered, the spring will necessarily cause the brake to engage, which also effectively serves as a

failsafe in the case of power-outages [11]. The tensioning band has a high coefficient of friction

with respect to the braking drum, to improve the stopping power of the brakes [11].


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4 The Elevator System

4.1 Components of the System

For this analysis of an elevator, it is necessary to reduce the large set of possible

elevator systems into one specific version in order to properly analyze it from an engineering

and mathematical point of view. This specificity also includes a few simplifications, in order to

make the system reasonable to find solutions for. Figure 2-A is the culmination of these

simplifications, where it has been reduced to a system of different rigid bodies held together by

flexible but unstretching cables. The following briefing on the elevator s components are done

with reference to numbered regions in Figure 2-B .

Figure 2: Whole system with components numbered: Part A is a clean view of the entire system when the passenger car is atsome intermediate position. Part B is the region layout used to point to each fundamental component in the system, where theregions are overlaid onto the diagram from part A. Part C is a more zoomed view of part A specifically on the top of the building


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containing the motor room. Since a lot of features are inside the third region, part D provides a more focused view of the subregionsof region 3, which are labeled from 3.1 to 3.4. Part D is shown from above so the motor is easily viewable.

The two rigid bodies that are entirely free to move are the passenger car and the

counterweight (region 1 and region 2, respectively, in Figure 2-B ). While these two are free to

move on any set of axes, they will only be moving vertically, due to the assumption that no

motion exists in non-vertical directions to begin with, and the applied forces only exist in thevertical direction. Combined in the subsystem of the passenger car are the passengers, where

they are assumed to move with motion identical to that of the car.

The next components to consider are found in the Motor Room, which for this

generalized elevator (and most elevators) is located at the top of the elevator shaft, near the top

of the building (Figure 2-B ) region 3, larger view in Figure 2-C . Important components in this

region are the driving pulley (also known as the sheave), the secondary pulley, the motor, and

finally the short length of cable that passes over the pulleys (parts 3.1, 3.2, 3.3, and 3.4,

respectively, in Figure 2-D ). The driving pulley transfers a torque from the motor to the cablepulled against its circumference through friction, causing a no-slip condition between the pulley

and the cables. The secondary pulley serves only to reposition the counterweight side of the

cables, so that the counterweight hangs sufficiently far away from the passenger car in order to

prevent any sort of collisions between components of the elevator. The motor applies the

aforementioned torque to the driving pulley through its shaft, and that shaft also supports the

pulley dynamically such that no motion of the driving pulley other than rotational motion exists.

This state of only-rotation exists for the secondary pulley as well, ensured similarly by its own

supporting shaft. The motor is bolted securely to the floor, so no motion other than the internalrotation exists there either. The wire held tight between the two pulleys will be referred to as the

second segment of wire, since it is the second straight region of the wire starting from the

passenger car.

Two other crucial components in this system are the cables that transfer forces

generated in the motor-sheave subsystem to do useful work down at the passenger car and the

counterweight. The length of the first, leftmost cable (region 4 in Figure 2-B ) supports the

passenger car through internal tension, and hangs from the driving pulley. The third section of

cables (region 5 in Figure 2-B ) functions similarly, but supports the counterweight and hangsfrom the secondary pulley.

One assumption that has been made, in order to make mathematical solutions more

simply available, concerns friction and damping forces: all friction forces due to the contact

between surfaces are assumed to be negligible except the contact area between the pulley and

cables, which is designed to use friction to ensure a no slip condition. All damping forces due to


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air resistance or any other similar mechanisms are assumed to also be negligible, since the

speeds of any object here is reasonably low, especially compared to their mass.

The last element of the elevator system to consider is the building in which it is situated.

Based on our research, we had concluded that the gearless motor was used in buildings from

10 floors to hundreds of floors. For ease of demonstrating the results, we have decided to gowith the lower bound of this, 10 floors, so that the 10 floors are individually recognizable in

pertinent figures. Increasing the number of floors would not function differently than these 10

floors, so 10 floors seems sufficient to exemplify the situation. Region 6 in Figure 2-B

demonstrates this parameter. Through additional research, the average height of a single floor

is approximately 10 feet, which is a value we will also assume to be the case for our elevator

[17].

4.1 Defining VariablesNow that a physical understanding of the relative positions and inter-component

relations has been established, it is necessary to break the system down further still, by means

of symbolic representations of parameters and specifications. The following is the breakdown of

our definitions for each of the variables we will be using in the following mathematical analyses.

4.1.1 Types of Variables

In this analysis, there will be extensive use of a number of different types of variables,

and while some of these variable types will only be used once in reference to a specific value,

many others will be used to refer to a trait of many different objects. Subscripts will be used to

differentiate between a given trait held by different objects, and the layout of our use of

subscripts will follow this briefing on the different traits being examined.


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Symbol Meaning Units

Tension acting in a given cross section of support cables

Weight of an object caused by gravity

Normal force on an object, resulting from pressing against a surface

Moment caused by an object Length of an object (specifically cables) Mass of an object

Height of an object with respect to a specified reference point Velocity of an object with respect to a specified reference point / Acceleration of an object with respect to a specified reference point / Angular acceleration of a pulley /

Table 3: All general variables to be used with subscripts: These variables pertain specifically to parameters that have subscriptsaffixed to them. See Section 4.1.3 for exceptions and specific or system-wide symbols.


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4.1.2 Subscript Meanings

In tandem with these symbols are the subscripts which modify and specify them, and

these subscripts are important to defining, manipulating, and understanding generalized

mathematical and engineering concepts with respect to a specific subsystem. We have

assigned this nomenclature to make the object as relatable as possible to its subscript

representation, to alleviate time spent on discerning sets of equations.

Subscript Refers to the...

passenger car counterweight first length of cable (wire)

second length of cable (wire)

3 third length of cable (wire) primary/driving pulley secondary pulley passenger load on car first shaft second shaft

, ,..,8 ( )

value of something acting on or

from the nth subsystem.*

Table 4: Subscript meanings: These subscripts will be used in the following analysis to easily refer to specific subsystems. Tobetter understand the naming scheme, the part of the meaning that contributes most to the abbreviated subscript is underlined. Ageneralized variable type is used to indicate that any of the symbols in Section 4.1.1 can easily be interchanged, as far as thesubscripts are concerned. Note that the word wire is the contributing term for the cable -related subscripts, so it is not easilyconfused with th e passenger cars .*See Figure 9 for the definition of these subsystems, and see the force body diagrams in Section 4.3.1 for these.


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4.1.3 Additional Parameters

There are, however, some variables that pertain to the elevator and its functionalit y that

do not directly relate to any single part of the system, or are more globally applicable variables.

Symbol Meaning Units

Reaction moment which arises from being attached to a fixed location Reaction force which arises from being attached to a fixed location

The maximum speed () attainable by the elevator system / The maximum acceleration () attainable by the elevator system /

The position variable with respect to the curvature/path of the cable* AKA ( ) The rate of change of versus time, AKA ( ) / The rate of change of versus time, AKA ( ) /

Linear mass density of the cables / , , 3 Time since start of elevator trip marking specific periods of time for long trip , Time since start of elevator trip marking specific periods of time for short trip

Starting position of elevator trip (corresponds to integer value floor number) Ending position of elevator trip (corresponds to integer value floor number) Standard acceleration of gravity / Angle from horizontal made by the cable between the two pulleys ( 2) Angle from horizontal made by

Angle from horizontal made by

Horizontal distance between the centers of pulley 1 and pulley 2 Radius of pulley 1 Radius of pulley 2

Mechanical power applied by the motor onto the system Table 5: Specific parameters that will not be used with dynamic subscripts: This table outlines all the variables andparameters that are fixed in their usage, and do not follow the dynamic subscript notation described previously. *This path is defined as the set of all positions that the cable could occupy in the system, which ranges from zero displacement atthe passenger car while at the lowest floor, to its maximum displacement at the lowest vertical position of the counterweight. SeeFigure Figure 3-A System path for a graphical interpretation of this path, and see Figure 3-B System path and 3-C System path for these minimum and maximum points and correlated system configurations.


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Figure 3: System path: Part A depicts the whole path, which is the path defined by the possible locations of the cable. Part Bshows the = 0, ie starting end, of the path, which is located at the lowest height of the passenger car. Part C shows theending point of the path, which is defined by the lowest position of the counterweight (occurring when the passenger car is at itsmax height). This system path will mostly be used to relate the velocities and acceleration of each object. This usage is outlined inSection 4.2.1 .


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Figure 4: Representations of heights: Part A depicts all the vertical and the horizontal variables, with respect to the top of thepassenger car while at floor 1 (this also corresponds to = 0). Part B depicts that first state, and part C depicts that second state.Note also that the maximum height with respect to that baseline is the same for both the passenger car and the counterweight.These could have each been off by a constant amount, but for ease we chose the easiest option of that constant, zero.

4.1.4 Numerical AssignmentsNow with a proper understanding of the qualitative definitions of these variables, it is

important to also have a comprehensive quantitative understanding too. For some variables asingle value is sufficient towards this end, but many too are better defined by a range, sincethey are subject to change.


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4.1.4.1 Fixed Values

Mass Variable Value Units

34

60

4

3

0.1 / 0.6015

Table 6: Fixed values of masses: Masses for the car and counterweight were determined based on the information in Table 2. Themasses of the pulleys and the linear mass density were estimated. Theta was calculated using the radii of the pulleys, and theirrelative displacements, through some complicated math not worth discussing in depth, loosely described as using a requiredtangency of the wire segment to both pulleys.

Elevator/SituationalVariable

Value Units

32.2

/

2.0 / 6.667 / 90

0.656 0.30 95 98 5

Table 7: Fixed values of situational variables: Gravity is assumed to be standard gravity on earth, and acceleration and velocity

were determined by the minimum values of the respective entries in Table 1. The maximum height is derived based on ourconstruction that the lowest height is zero, and the greatest height is 10 ( ) = 10 9 = 90 4.1.4.2 Variable Values and Their Ranges

Variable Minimum Maximum Units

0 52

0 (C @ 1st floor) 90 (C @ 10th floor) 0 (C @ 10th floor) 90 (C @ 1st floor)

Table 8: Variable ranges: Maximum passenger load weight was determined using the data in Table 1. This comes out to be about10 average people.

Another value that we estimated was shape of the pulleys themselves, and thedistribution of their masses. We chose to simplify their shape to be solid cylinders of uniform

density, which makes their moment of inertias:

= (1/2) and = (1/2)


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4.2 Defining the Kinematics:

For this project, it is the kinematics of the elevator that are predefined and follow a

specific, determinable form. The specificity arises due to constraints enforced by physical

parameters of the driving motor: the maximum rotational speed and the maximum angularacceleration. Another constraint is the fact that every element in the structure is attached by a

bundle of cables, whose length is constant. We will first consider the constraint due to the cable,

in order to understand the relative motions of each component of the elevator, and then once

that is established, the path of the elevator will be fleshed out.

4.2.1 Cables Constraint

Since the elevator system is just that, a system, we need to determine how different

components in the overall system interact. Other than the forces examined in Section 4.3.1 ,

there is still the relation of motion between each subsystem. By exploiting the seemingly

obscure variable and its time derivatives, we can relate crucial parameters of each

subsystem together.

Since we know the length of the cables is unchanging, and all the components lie on the

path, then they are all move uniformly together on that dynamic axis. This can be shown

starting from this idea: for the nth component equals the for the (n-1)th component plus

the length of the cable between them:

( ) = ( ) +( ) ( ) Differentiating this versus time once and twice shows us how tightly defined this system is:

( ) = ( ) ( ) = ( )


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These aver very powerful equations, since they indicate to us that the entire system can

be represented by a single velocity and acceleration along a given (curvilinear) axis. Relating a

positive to acceleration of particular components with respect to fixed linear axes yields the

following:

Component Acceleration variable Positive equivalence orientation

Passenger Car with positive vertical

Wire 1 with positive vertical

Pulley 1 with clockwise at radius

Wire 2 with axis rotated angle clockwise from horizontal

Pulley 2 with clockwise at radius

Wire 3 3 with negative verticalCounterweight with negative vertical

Table 9: Verbose relationships of system acceleration

An even more useful way of putting this is string of equivalences:

= = = = = = 3 = While the heights of each component are not especially important beyond knowing

the vertical position of the passenger car and the counterweight, the importance of seeps

into the entire system, and when these acceleration variables are used with respect to forces,

this string of equations will provide a much needed route to mathematical solution.

4.2.2 Defining the Path of Motion

The path of motion versus time, specifically ( ), is defined separate from the forcesthat cause it. The motion is confined by the idea that the system will accelerate at its maximum

possible acceleration until it reaches its maximum possible speed, and cannot continue to

accelerate. At this point, the system must remain in motion at maximum speed until the right

time when it must begin to slow down, with the same acceleration as it had started. Instead of

assuming that acceleration jumps on and off, one could look at the jerk rate of the system too

(which we do have a value for), but in many cases this would not change the valuessignificantly, the time frame would show almost no difference, and the mathematics would just

be uglier. With this considered, we continue on to deriving the path of the motion without

considering jerk.[18]

There is, however, one thing to consider: does scheme described above allow for all

possible situations of transferring over a given number of floors? One easy way to answer this is


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to think of an extreme case, where the elevator is moving up some very small distance, say an

inch. Using the system described above with reasonable and (see Table 7), under the

system of accelerating to maximum speed, coasting, and then coming back to a stop, the

elevator would necessarily overshoot its 1 inch target, well before it even got up to full speed.

From this investigation, we arrive at an important conclusion: we must consider situations in theway previously described when the start-to-end displacement of the elevator is longer than

some value, and if it is shorter than that cutoff value, it must be handled in a different way.

Note: In this section we refer to ( ), ( ), and () extensively. They are all values inrespect to the system, ie () = , () = , () = 4.2.2.1 Long Elevator Trips

In order to begin tackling the problem, let us first consider the case where the distance to

travel is sufficiently large. Under these conditions, the position, velocity, and acceleration would

take the form exemplified in Figure 5. The discontinuous acceleration graph determines the rest

of the velocity and position graphs, with use of the initial and final conditions, of = =0 and and are known constants for any given trip. Important to this derivation is use of atimescale to break up the piecewise function into well-defined regions.

If we say that the motion starts at = 0, and from 0 < , the system is acceleratingpositively with an acceleration of . At , the velocity reaches its maximum (this is the crux

of what means), and ( )drops back off to zero, during < . At , the system starts toslow down again, and by 3, the speed is back to zero and the total distance traveled was ( ).


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Figure 5: Kinematics of a Long Distance Elevator Trip : This system is restricted by the fact that possible values of accelerationare quantized, and are piecewise versus time.


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Following these relationships and boundary conditions, we can arrive at these piecewise

definition of acceleration, velocity, and vertical position (See Appendix A-1):

( ) = , 0 ;( ) = 0, < < ;( ) = , < 3;

( ) = , 0 ;( ) = , < < ;( ) = ( ), < 3;

( ) = 1/2 + , 0 ;( ) = (1/2 + )+ ( ), < < ;( ) = (1/2 + + ( )) 1/2( ) + ( ), < 3;

Where

= / = ( )/

3 = ( )/ + /

4.2.2.2 Short Elevator Trips

After having solved the situation of long trips explicitly, we can look at the situation

where the trip is too short to possibly reach v max . In order to properly modify this system (whose

acceleration values are fixed) we can only modify the time the system remains in each state of

acceleration, namely, shortening it. Whatever speed the car gets up to, it must also be slowed

down that same amount, which will require the same amount of time. If we work with two new

points in time, and , where = 2 from this symmetry, we have what is shown in Figure6.


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Figure 6: Kinematics of a short distance elevator trip: The two new variables are equally spaced, and this system is accelerating100% of the time. It does not reach maximum speed, and the ride is very short.


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The same procedure was done for this as for the long trip (see Appendix A-2). This is a

summary of the findings:

( ) = , for 0 < ( ) =

, for

<

( ) = , for 0 < ( ) = (2 ), for < ( ) = +1/2 , for 0 < ( ) = +1/2( ) + ( ) 1/2( ), for <

= ( )/

= 2 ( )/ 4.2.2.3 Trip Displacement Cutoff Distance

Now, weve solved for the two different cases of motion, but we need to be able to know

which is the correct one to use in a given situation. Using the fact that cannot be lower than ,

we find the following (see Appendix A-3 for derivation):

( ) > If this is true, then the y f - y 0 is considered a long distance. Otherwise, it is short distance.

4.2.2.4 A Note

It is important to point out that this derivation assumed that was positive, and thatand was in the positive direction. However, the entirety of the derivation could easily

receive a negative sign everywhere, and would still function equally. In fact, assigning that

negative value follows exactly the sign of , and everything else will work correctly in bothdirections.

4.3 Determining the Equations Relating Forces

Now that the system has been thoroughly defined and the nomenclature has been laid

down, everything is in place to perform the actual analysis of the system from a mathematical

point of view. In order to properly do this, the elevator system has been broken into eight

subsystems, identified in Figures 5.


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Figure 7: 8 subsystems: In A, the different subsystems are labeled from the point of view of the whole elevator. Subsystems 1, 2,7, and 8 are very large areas, and are easily discernable from this scale, but for subsystems 3 through 6, part B of this figure is moreappropriate to portray those subsystems.

4.3.1 Free Body Diagrams

The following is the investigation of the dynamics of each subsystem (1-8), where the

forces/moments are defined physically in the free body diagram, and are related using NewtonsSecond Law. The different subsystems are interdependent, but in a way that we still can solve

explicitly for all terms. In order to attain this ability to solve in a linear fashion, we will reorder the

analysis of the FBDs so that any given FBD is explicitly solvable for its single unknown, and all

values necessary for that explicit definition have themselves been solved in the preceding steps.


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4.3.1.1 Subsystem 1: Passenger Car

Figure 8: FBD Passenger Car: This is the free body diagram of the passenger car, where the weight of the car and the weight ofthe passenger load are directed in the negative y direction, and there is a tension pulling upward that counteracts those other forces.

Based on Figure 8, two types of forces are exerted on the car: gravity and the tension

from the cable. Applying the Newtons Second Law and the data of cars mass, load, cars range

of acceleration, we can derive the result of the tension exerting on the car.

= ( + ) Using the equivalence between and , and the fact that = , and solving

for we find that

= ( + )( +)


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4.3.1.2 Subsystem 2: Wire 1

Figure 9: FBD Wire 1: This is the free body diagram of the wire extending from the passenger car to pulley 1. Breaklines have been

drawn so that the forces are easier to view on the wire, and the center of the wire has been kept in view to show that acts at itscenter.

Figure 9 shows that three forces are acting on the wire: the pulley pulling up with tension

, gravity pulling down with , and the passenger car pulling down with tension . Since we

already have solved for in terms of knowns, it is essentially a known now too. Applying the

Newtons Second Law and considering the forces in the y -direction:

= = The acceleration of the wire is the same as that of the system and = so we can

find the relation between tensions:

= + ( + ) Using the definitions of the system variables, we can replace to yield the following:

= + ( )( +)


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4.3.1.3 Subsystem 8: Counterweight

Figure 10: FBD Counterweight: This is the force body diagram of the counterweight where the weight of the counterweight is in

the negative y direction and there is a tension pulling in the positive y-direction.

Since we need T 3 for subsystem 3, we go over to subsystem 8 and work from the other side.

Figure 10 shows that two forces are acting on the counterweight: The tension pulling

up and gravity pulling down with . Applying the Newtons Second Law and the data of the

counterweight mass and the cars acceleration, we can derive the result of the tension exerting

on the car.

= = ( ) Substituting with and that

= and simplifying, we get

= ( )


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Figure 11: FBD Wire 3: This is the free body diagram of the wire 3 extending from the secondary pulley to the counterweight .Breaklines have been drawn so that the forces are easier to view on the wire, and the center of the wire has been kept in view toshow that 3acts at its center. Tension 5 from the secondary pulley pulling upward in the positive y-axis and Tension 6 from thecounterweight is pulling downward on the wire in the negative y-axis.

According to the diagram, there are 3 forces exerted on the cable: 2 tension forces and

gravity. The tensions are exerted on both ends of the cable: one is to attach to the pulley, the

other is to hold the counterweight in place. Similar to wire 1, we can determine the tension to

hold the counterweight by using the 2nd law of motion in y-direction.

= 3 = 3 3 The acceleration of the wire upward is the negative of the system acceleration, and

3 = 3 so we can find the relation between tensions: = + 3( )

Using the physical relationships of distance, we can simplify this further into known variables:

= + ( + )( )


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4.3.1.5 Subsystem 6: Pulley 2

Figure 12: FBD Pulley 2: This is a free body diagram of the secondary pulley. There are a couple different angles defined here, aswell as all the forces being applied to the pulley in different directions . The forces being applied are tension 4 is pulling the pulleyfrom an angle theta , normal force of shaft 2 is pulling the pulley from an angle of beta, tension 5 and the weight of the pulley pullingdownward in the negative y-direction.

The x and y-components of the normal force can be derived by using the 2nd law of

motion and the tension from wire 3 can be determined by using moment equation. Angle is

the angle formed by tension from wire 2 and x-axis. Both forces and moments (positive in

clockwise direction) are used to solve for these variables, which consists of three unknowns: 4,and the x and y components of . Since this pulley is just sitting on a simple, frictionless shaft,

it applies no moment.

= 4 = = +4 = = 4 =

See Appendix B.1 for full derivations. Below are the final results.

4 = (1/2) = 4

= + 4


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Figure 13: FBD Wire 2: This is a free body diagram of wire 2 . extending from the primary /driving pulley to secondary pulley.Tension 3 from the primary pulley pulling on the wire in the positive x -axis and Tension 4 from the secondary pulley pullingdownward on the wire on the direct ion of negative x -axis.

Based on the diagram, there are tension forces and gravity force on this segment of

cable. We assume that the weight of this segment is negligible because its length is small

compared to the whole length of the wire that support the system. Therefore, from the 2nd law

of motion and substituting that = , the tensions on both ends are equal. = 4 3 =

3 = 4


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4.3.1.7 Subsystem 3: Pulley 1

Figure 14: FBD Pulley 1: This is the free body diagram of the primary pulley. There are a couple different angles defined here, as

well as all the forces being applied to the pulley in different directions . the forces applied are Tension 2 is pulling the pulleydownward in the negative y-direction ,tension 3 causing the the pulley to rotate clockwise with an angle theta , moment of the thepulley is rotating clockwise ,and the weight of the pulley is acting downward in the negative y-axis.on.

According to the diagram, 3 forces are exerted on the pulley: tension, gravity and the

normal force. The tensions are from wire 1 and wire 2. The normal force is from the shaft

exerting a force to the pulley. When we apply 2nd law of motion and with respect to our system,

the pulley does not move therefore we can assume its linear acceleration equals 0. Then we

can derive the x and y components of the normal force. By using the moment equation (positive

in the clockwise direction), we can derive the moment applied by the motor. Angle is the angle

formed by tension from wire 2 and x-axis.

= 3 = = 3 =

= 3 + = See Appendix B.2 for full derivations. Below are the final results of the simplifications and

solving:

= (12

+ 3) = 3 + + = 3


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4.3.1.8 Subsystem 4: Motor

Figure 15: FBD Motor: This is the free body diagram of the motor. There is the equal and opposite moment exerted on the

motor and there must be a reaction moment from the floor to cancel this. The forces that are acting on the motor are normal force, reaction force , and the weight of the motor acting downward.

According to the sketch, there are gravity, normal force, and reaction force acting on the

motor. The reaction force can be determined from normal forces of the shaft. The motor does

not rotate with respect to the system so the moment equation will equal 0. Therefore the

reaction moment from the shaft is equal to the moment of the motor. Angle phi is formed by the

normal force of the shaft and x-axis.

= = ( + ) = ( + ) = ()

= = See Appendix B.3 for full derivations. Below are the final results. =

= + () =

We arent too concerned with these reaction forces, but, we have solved for them to be

thorough.


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4.3.2 Additional Parameters

There are some extra variables we can learn about, that will give a bit more perspective

to the reader about this elevator system. These are some more tangible results, since much of

the tensions and forces are very internal, only useful to the engineer.

4.3.2.1 System Power Consumption

The power of the system can be determined from the moment of the motor, the velocity

of the system, and the radius of pulley 1 (the pulley at the motor) from

= Which reduces to the variables weve been using as

= / / 550 Where is in horsepower and the remaining variables are in the same units as they have been

previously used.

4.3.2.2 Change in Force Felt by Passenger

Another interesting parameter useful to the average person is what a passenger feels,

for example if they feel heavier or if they feel lighter as the elevator moves up and down. The

exact force is very particular to the persons rest mass, but if we look at just the percent change

over the trip, we can generalize it and just have 100% be when they are standing still on the

ground.

If the passengers weight is and their mass is , and their is a normal force

from the floor pushing them up, the following equation would result from Newtons Second Law,

under the assumption that they move with the elevator:

= When the person is on a non-accelerating surface, this reduces to

= = Finding the general ( / 1)100%, we find the general percent

change of what the passenger feels relative to what the normally feel on un-acceleratingsurfaces.

% = /


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Since only has three possible values, , 0, , we can solve for the three

different cases, with respect to the actual values of the variables:

% = 6.2%, 0


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5 Visualizing the Mathematical Relationships

Now that explicit equations of motion have been solidified, it would be helpful to also

have a graphical representation of the motion, forces, and torques. Plugging in numbers for the

equations, we can create graphs for each of the forces. We decided to have 5 different cases

that analyze the motion of an elevator to determine how it acts with different loads, directions of

motion, changes in travel distance, and differences in height of elevator. The cases are as

follows:

Case 1 2 3 4 5

Start floor 1 = 0 ft 10 = 90 ft 1 = 0 ft 10 = 90 ft 10 = 90 ft

End floor 2 = 10 ft 9 = 80 ft 10 = 90 ft 1 = 0 ft 1 = 0 ftPassengerMass

3 people =15.65 slugs




0 people =0 slugs

Table 10: Some example situations through which the reader can glimpse at some states of this complicated system.

For case 1, the elevator will move up from floor 1 to floor 2, a height of 0 feet to 10 feet,

with 3 people on the elevator. Using the average mass of a person, we calculated this to be a

mass of 15.65 slugs. Plugging these numbers into the equations, we can determine the values

of all of the forces in the system, graphically, for this situation. This situation shows a short

distance, small load, upward motion, and low elevator height.For case 2, the elevator will move down from floor 10 to floor 9, a height of 90 feet to 80

feet, with 5 people on the elevator. Using the average mass of a person, we calculated this to

be a mass of 26.08 slugs. Plugging these numbers into the equations, we can determine the

values of all of the forces in the system, graphically, for this situation. This situation varies from

case 1 in that it has a medium load, downward motion, and high elevator height. It is similar in

that they both have short travel distances.


with 5 people on the elevator. Using the average mass of a person, we calculated this to be amass of 26.08 slugs. Plugging these numbers into the equations, we can determine the values

of all of the forces in the system, graphically, for this situation. This situation shows how the

forces act due to long travel distances. Like case 1, it has upward motion. Like case 2, it has a

medium load. However, it starts at a very low height and ends at a high height.


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with 10 people on the elevator. Using the average mass of a person, we calculated this to be a

mass of 52.16 slugs. Plugging these numbers into the equations, we can determine the values

of all of the forces in the system, graphically, for this situation. This situation shows what

happens under the opposite circumstance of case 3. The motion is the same except in reverse.The only other difference is that there is a large load instead of a medium load.


with 0 people on the elevator. This means that there is a mass of 0 slugs. Plugging these

numbers into the equations, we can determine the values of all of the forces in the system,

graphically, for this situation. This situation is the same as case 4 in terms of motion. The

elevator has the same motion in both cases. The only difference is that case 5 has no load in

the elevator, but case 4 has the max load.

To analyze the difference in the forces for each case, we will look at the graphs for eachforce separately and analyze the graphs of that force for each case. We will go in order of T 1 to

T2.


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Figure 16: Graph of tension 1 with respect to time: the tension 1 is graphed for 5 cases

Cases 1 and 2 last for shorter time periods than cases 3, 4, and 5. Cases 3, 4, and 5

have 3 different tension values, but cases 1 and 2 only have 2 tension values. In cases 2, 4, and

5, the tension increases over time, but the tension decreases in cases 1 and 3. The values are

constant, for all of the cases, within each region.


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Figure 17: Graph of tension 2 with respect to time: the tension 2 is graphed for 5 cases Cases 1 and 2 last for shorter time periods than cases 3, 4, and 5. Cases 3, 4, and 5

have 3 different tension regions, but cases 1 and 2 only have 2 tension regions. In cases 2, 4,

and 5, the tension increases over time, but the tension decreases in cases 1 and 3. The values

change, for all of the cases, within each region.


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and 5, the tension decreases over time, but the tension increases in cases 1 and 3. The values



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have 3 different tension valuess, but cases 1 and 2 only have 2 tension values. In cases 2, 4,

and 5, the tension decreases over time, but the tension increases in cases 1 and 3. The tension,

for all of the cases, is constant within each region.


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Figure 22: Graph of normal force on shaft 1 with respect to time: the normal force on shaft 1 is graphed for 5 cases


have 3 different normal force regions, but cases 1 and 2 only have 2 normal force regions. In

cases 1, 2, and 5, the normal force decreases over time, but the normal force increases in case

4. In case 3, the normal force decreases and then increases. The values change, for all of the

cases, within each region.


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Figure 23: Graph of normal force on shaft 2 with respect to time: the normal force on shaft 2 is graphed for 5 cases


have 3 different normal force regions, but cases 1 and 2 only have 2 normal force regions. In

cases 2, 4, and 5, the normal force decreases over time, but the normal force increases in

cases 1 and 3. The values change, for all of the cases, within each region.


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Figure 24: Graph of moment of the motor with respect to time: the moment of the motor is graphed for 5 cases


have 3 different moment regions, but cases 1 and 2 only have 2 moment regions. In cases 2, 4,

and 5, the moment increases over time, but the moment decreases in cases 1 and 3. The

values change, for all of the cases, within each region.


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Figure 25: Graph of power with respect to time: the power is graphed for 5 cases


have 3 different power regions, but cases 1 and 2 only have 2 power regions. In cases 1, 2, 3,

and 5, the power increases in the first region, but the power decreases in case 4. In cases 3, 4,

and 5, the power decreases in the middle region, but cases 1 and 2 do not have a middle

region. In cases 1, 2, 3, and 4, the power increases in the last region, but the power decreases

in case 5. Cases 1 and 2 jump from positive to negative between the 2 regions.


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Figure 24: Graph of height of passenger car with respect to time: the height of passenger car is graphed for 5 cases

Cases 1 and 2 last for shorter time periods than cases 3, 4, and 5. In cases 2, 4, and 5,

the position decreases over time, but the position increases in cases 1 and 3. The values

change, for all of the cases, directly, with respect to time.


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6 Conclusion

In this project we examine the kinematic and dynamic properties of the 1:1 gearless

elevator powered by a DC motor used in a tall building. The team used free body diagrams to

investigate each component of the commercial elevator. Free body diagrams help us to analyze

and calculate the dynamic and kinematic properties of an elevator such as distances, speeds,

accelerations, forces, torque, and moments. Also through free body diagram we broke down the

components of the system to motor assembly, pulley systems, the elevator car, the

counterweight. We analyzed the overall interaction by determining the forces and torque and

resultant motion taking in consideration the energy transfer of the overall motion of the elevator.

After analyzing the movement of each component of the elevator we get interested in

analyzing the forces that the person feels while he is in the elevator. We looked at different

cases that analyzes the motion of the elevator. From the engineering free body diagram of each

part of the elevator we used dimensional analysis to predict the relationship among tension

force (N), displacement (m), mass (kg), and angular frequency (rad/s). Knowing that for each

action there will be an equal and opposite reaction (newtons second law) there will always be a

weight W=mg of a person with mass m, that is locating on the surface of earth and a normal

force that will support the person weigh t that will be exerting back towards the person. Now,

when a person stand in an elevator and both of the person and an elevator are experiencing

acceleration there will be a change in the normal force between the elevator and the person.

That change in normal force well be felt by the person on the elevator. In all of the cases of the

elevator we assumed that going up is a positive direction and going down is the negative

direction. When the elevator was going up and speeding up. Here the acceleration was positive

upward. The elevator started from rest at the lowest floor. Then as it started to accelerate the

elevators floor pushed up on the person so it made him moves upward along with the elevator.

The normal force here were greater than the true weight of the person. Here the acceleration

was negative moving downward opposite to the upward motion which caused a reduction of the

velocity. When the elevator was going down and slowing down. In this case the accelerationwere moving to the opposite of the negative direction of the velocity. Here where velocity

magnitude reduced. The elevator pushed up on the person to make him accelerating upward, in

this case the normal force increased. Finally when the elevator were going down and speeding

up. Here the elevator and the person started at a higher floor. Then the elevator were speeding


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down to the lower floor. The elevator were pushing upward to support the person weight with

less force. Therefore in this case the normal force decreased.

The elevator design has been improved a lot over these 15 years. As we analyze the

elevator, we can improve the elevator by preventing problems or accidents. For example, thecable is the most vulnerable part of the system. It is the component that holds the car and the

counterweight together. The cable endures a massive load from the car and the counterweight.

Furthermore, its size is large enough to reduce the energy to move the cable so we can save

energy. We can find a better material for the cable or replace/assess it regularly. Another

component that needs to be considered is the motor. The motor is used to rotate the cable and

it also needs regular assessments to ensure the system works properly. We can deduct 5

percent of the current max load to the max load so we can prevent the situation when the weight

of the people is approximately equal to the max load.


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7 Appendices

7.1 Appendix A: Deriving Motion

7.1.1 Long Trip MotionThe time spent from = 0 to = is the time it takes to constantly accelerate from =0 to = . This timeframe is also the same as the timeframe of the last segment of the

motion, from = to = 3. Hence, = (3 ) = / From the physical motion, we defined the acceleration:for 0 <

( ) = for < ( ) = 0 for < 3 ( ) = Integrating this, we getfor 0 < ( ) = (0)+ = for <

( ) = ()+0( ) = () =

() = () = for < 3 ( ) = () ( ) = ( ) With the velocity solidly defined, we can then integrate again and find the vertical positionfor 0 < ( ) = 1/2 + for < ( ) = ()+ ( ) = (1/2 + )+ ( ) for < 3

( ) = () 1/2( ) + ( )= (1/2 + + ( )) 1/2( ) + ( ) With that defined (even if complicated and lengthy), we can apply another boundary condition torelate more of these variables to each other. We will use the end condition (3) = = (1/2 + + ( )) 1/2(3 ) + (3 ) = +( )(3 ) = + (3 )


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And so we can start defining the time constants with respect to the initial and situationalconditions

( )/ = 3 ( )/ + / = 3 And since = 3 , then = 3

= ( )/ So in summary: = / = ( )/

3 = ( )/ + / 7.1.2 Short Trip Motion

For the short trips, where the passenger car cannot possibly reach and notovershoot its target, we tackle the problem in the different way proposed in Figure 6. If it still

accelerates with but does not reach the same max speed, then and will be lessthan t 1. Describing the motion in the same way, we find thatfor 0 < ( ) = for < ( ) =

And continuing on with integration,for 0 <

( ) = (0)+ =

for < ( ) = () ( ) = ( ) = (2 ) And yet again,for 0 < ( ) = (0)+1/2( ) = +1/2 for < ( ) = ()+ ()( ) 1/2( )= +1/2( ) + ( ) 1/2( )

Finally, we can substitute the end condition that () = , and that = yielding = +1/2( ) + ( ) 1/2( )

= ( ) And therefore we can define these times to be

( )/ = = (1/2)


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7.1.3 Long-Short Boundary ConditionIf, however, the distance ( ) is small it becomes possible that could be less than

, which physically cannot occur. Using this limiting factor, we can determine when eachcondition must occur:

Time Boundary Condition:

> Recalling the definitions

= ( )/ = /

and substituting, we get

( )/ > / And solving, we find a final result,

( ) > If this is true, then the y f - y 0 is considered a long distance. Otherwise, it is short distance.

7.2 Appendix B: Force Equations

7.2.1 Equations for Pulley 2

Based on Figure 12, two types of forces are exerted on the pulley: normal force and the

tension from the cable. Applying the Newtons Second Law and the data of pulleys range of

acceleration, we can derive the result of the normal force exerting on the pulley. First, we will

consider the total forces in the x-direction:

= 4 = Because the acceleration of the pulley is 0 with respect to the system so we can find the

x-component of the normal force:

= 4 Theta is the angle formed by tension from wire 2 and x-axis. Now, we will consider the

total forces in the y-direction:

= +4 = Again, acceleration of the pulley is 0 so we can find the y-component of the normal force:

= + 4 To find the relation between tension from wire 2 and tension from wire 3, we will

consider the moment equation:


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= 4 = I is the moment of inertia and is the angular acceleration. We also have:

= (1/2) = /

Therefore, we can determine the relation of tensions from wire 2 and wire 3:

4 = ( (1/2) ( / ))/ = (1/2)

7.2.2 Equations for Pulley 1

Based on Figure 14, two types of forces are exerted on the pulley: normal force and the

tension from the cable. Applying the Newtons Second Law and the data of pulleys range of

acceleration, we can derive the result of the normal force exerting on the pulley. First, we will

consider the total forces in the y-direction:

= 3 = Because the acceleration of the pulley is 0 with respect to the system so we can find the

y-component of the normal force:

= 3 + + Theta is the angle formed by tension from wire 2 and x-axis. Now, we will consider the

total forces in the x-direction:

= 3 = Again, acceleration of the pulley is 0 so we can find the y-component of the normal force: = 3

To find the relation between tension from wire 2 and tension from wire 1, we will

consider the moment equation:

= 3 + = I is the moment of inertia and is the angular acceleration. We a lso have:

= (1/2) = / Therefore, we can determine the relation of tensions from wire 2 and wire 1:

= 3 +(1/2) ( / ) = (12 + 3)


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7.2.3 Equations for Motor

Based on Figure 15, two types of forces are exerted on the motor: normal force, the

reaction force, and gravity. Applying the Newtons Second Law and the data of motors mass,

motors range of acceleration, we can derive the result of the reaction force exerting on the

motor. First, we will consider the forces in the x-direction:

= = ( + ) With respect to the motor, the acceleration of the shaft is 0. Therefore, the reaction force

in the x-direction can be determined:

= Phi is a constant angle formed by the tension and x-axis. Then, we will consider the

forces in the y-direction:

= ( + ) = ( + )

Again, the acceleration with respect to the motor is 0 so the reaction force in the y-

direction can be determined:

= + ( + ) We can determine the reaction moment on the motor by considering the moment equation:

= = The angular acceleration of the motor is equal 0 therefore the reaction moment is:

=


60/60

59

8 References

[1] "ABOUT ELEVATORS." Elevators. Web. 14 Nov. 2015.

[2] "Classroom On Demand." Lesson Viewer -. Web. 14 Nov. 2015.

[3] "Columbia Elevator - Elevator Cabs, Entrances and Door Systems." Elevator History. Web. 14 Nov.2015

[4] "Elevators." How Do Elevators and Lifts Work? N.p., n.d. Web. 11 Nov. 2015.

[5] "Elevator Terminology for Home Elevators." Inclinator. N.p., n.d. Web. 13 Nov. 2015.

[6] "Epic Westbrook Geared Traction Elevator with Machine Room Tour." YouTube. YouTube, n.d. Web. 13

. Nov. 2015

[7] "Elevator & Hoist Modeling, Simulation & Speed Control." Elevator & Hoist Modeling, Simulation &

. Speed Control. N.p., n.d. Web. 11 Nov. 2015.

[8] "Elevators." How Do Elevators and Lifts Work? N.p., n.d. Web. 11 Nov. 2015. [9] "Fuji Lift - How Elevators Work." YouTube.YouTube, n.d. Web. 13 Nov. 2015

[10] "Gearless Traction Motor for MR and MRLElevator." - China Manufacturer & Supplier. N.p., n.d.

. Web. 11 Nov.2015

[11] How an Elevator Works." How an Elevator Works. N.p., n.d. Web. 11 Nov. 2015.

. < http://web.mit.edu/2.972/www/reports/elevator/elevator.html >

[12] "How Does an Elevator Work? Pulley System Explained." YouTube. YouTube, n.d. Web. 11

Nov. 2015.

[13] Harris, Tom. HowStuffWorks. HowStuffWorks.com. Web. 14 Nov. 2015.

.

[14] "Otis Worldwide - Home - Elevator - Escalator." Otis Worldwide - Home - Elevator - Escalator. Web.

14 Nov. 2015.

[15] "Sustainability." Archtoolbox. N.p., n.d. Web.

[16] "Traction Elevators." Elevator Wiki. Web. 14 Nov. 2015

[17]Calculating the height of a tall building where only the number of stories is known. (n.d.). Retrieved

December 14, 2015.

[18] Model, 1St Order, and Input Data. "Physical Models of an Elevator." (n.d.): n. pag. Web.
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