{"id":1008,"date":"2020-11-17T11:29:47","date_gmt":"2020-11-17T11:29:47","guid":{"rendered":"https:\/\/wordpress.peters-research.com\/?page_id=1008"},"modified":"2021-06-15T12:45:56","modified_gmt":"2021-06-15T12:45:56","slug":"ideal-lift-kinematics","status":"publish","type":"page","link":"https:\/\/wordpress.peters-research.com\/index.php\/papers\/ideal-lift-kinematics\/","title":{"rendered":"Ideal Lift Kinematics"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"1008\" class=\"elementor elementor-1008\">\n\t\t\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-2bd44fca elementor-section-full_width elementor-section-height-default elementor-section-height-default\" data-eae-slider=\"79462\" data-id=\"2bd44fca\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"has_eae_slider 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class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-624c56e elementor-widget elementor-widget-heading\" data-id=\"624c56e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Ideal Lift Kinematics\n<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-550f1783 elementor-widget elementor-widget-text-editor\" data-id=\"550f1783\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Richard D Peters<\/p>\n<p>Brunel University<br>Arup Research &amp; Development<\/p>\n<p><em>This paper was presented at ELEVCON&nbsp;HONG KONG&nbsp;1995, The International Congress on Vertical Transportation Technologies&nbsp;and first published in the IAEE book &#8220;Elevator Technology 6&#8221;, edited by G. C. Barney.&nbsp; It is reproduced with permission from The International Association of Elevator Engineers.&nbsp; The paper was republished by Elevator World (April 1996) and Elevatori (May\/June 1996).&nbsp; At the time of writing Richard Peters was working for Ove Arup &amp; Partners, and enrolled on the Engineering Doctorate program at Brunel University.&nbsp; This web version \u00a9 Peters Research Ltd 2009.<\/em><\/p>\n<h3>Abstract<\/h3>\n<p>In this paper the author considers lift kinematics, the study of the motion of a lift car.&nbsp; Ideal lift kinematics are constrained by human comfort criteria which limit the maximum acceleration and jerk (rate of change of acceleration) that are acceptable.&nbsp; Equations are presented which allow ideal lift kinematics to be plotted as continuous functions for any value of journey distance, speed, acceleration and jerk.&nbsp; Applications include generation of motor speed reference control curves.&nbsp; Supplementary results&nbsp; include&nbsp; journey time&nbsp; and formulae for&nbsp; use in lift traffic analysis.<\/p>\n<p><strong>LIST OF SYMBOLS<\/strong><\/p>\n<table border=\"0\">\n<tbody>\n<tr>\n<td>d<\/td>\n<td>Lift journey distance (m)<\/td>\n<\/tr>\n<tr>\n<td>a<\/td>\n<td>Maximum acceleration\/deceleration (m\/s<sup>2<\/sup>)<\/td>\n<\/tr>\n<tr>\n<td>v&nbsp;<\/td>\n<td>Maximum velocity (m\/s)&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>j&nbsp;<\/td>\n<td>Maximum jerk (rate of change of acceleration\/deceleration) (m\/s<sup>3<\/sup>)&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>D(t)&nbsp;<\/td>\n<td>Distance travelled at time t (m)&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>A(t)&nbsp;<\/td>\n<td>Acceleration at time t (m\/s<sup>2<\/sup>)&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>V(t)&nbsp;<\/td>\n<td>Velocity at time t (m\/s)&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>J(t)&nbsp;<\/td>\n<td>Velocity at time t (m\/s<sup>3<\/sup>)&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>dmin<\/td>\n<td>Minimum journey distance if lift commanded to stop immediately&nbsp;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>1. Introduction&nbsp;<\/h3>\n<p>Lift kinematics is the study of the motion of a lift car in a shaft without reference to mass or force.&nbsp; The maximum acceleration and jerk (rate of change of acceleration) which can be withstood by human beings without discomfort limits this motion.&nbsp; Ideal lift kinematics are the optimum velocity, acceleration and jerk profiles that can be obtained given human constraints.<\/p>\n<p>Microprocessor controlled variable speed drives can be programmed to match reference speed profiles generated through the study of lift kinematics.&nbsp; Examples of these speed reference curves, similar to those shown in Figure 1, are sometimes presented in lift manufactures&#8217; sales literature as a demonstration of the fast, comfortable and efficient lift transportation available for a particular drive system.<\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/figure1.jpg\" alt=\"\" border=\"0\"><\/p>\n<p><strong><em>Figure 1&nbsp;&nbsp; Example lift velocity-time profile for one, two and four floor runs<\/em><\/strong><\/p>\n<h3>2. Overview of previous research &amp; author\u2019s contribution<\/h3>\n<h4>2.1&nbsp; Previous work<\/h4>\n<p>P D Day and G C Barney provide references of previous published work in this field in section 11.4 of CIBSE Guide D,&nbsp;<em>Transportation Systems in Buildings<\/em>&nbsp;(1).&nbsp; In summary:<\/p>\n<p>H D Molz presented the first major work in this area in 1986.&nbsp; In his paper,&nbsp;<em>On the ideal kinematics of lifts<\/em>&nbsp;(2) (in German) he derives equations which enable minimum travel times to be calculated, taking to account maximum values of jerk, acceleration, and speed.&nbsp; If the lift trip is too short for the lift contract speed or acceleration to be obtained, the maximum speed and acceleration attained during the trip may be calculated.&nbsp; Some other points on the ideal kinematic curves are calculated.&nbsp; This paper was edited by G C Barney and re-published (3) by Elevatori in 1991 (in English and Italian).<\/p>\n<p>N R Roschier and M J Kaakinen apply Molz\u2019 formulae to provided summary tables of results for round trip time calculations (4).<\/p>\n<p>In&nbsp;<em>Elevator Trip Profiles<\/em>&nbsp;(5), J Schroeder presented a computer program that calculates the maximum speed, and minimum journey time that a lift can achieve for given flight distances if there is no speed limit.&nbsp; This produces interesting observations such as it would take a total trip of about 17 floors for an 8 m\/s lift to reach its full speed.<\/p>\n<p>In Elevator Electric Drives (6) G C Barney and A G Loher suggest a computer program based on H D Molz&#8217; equations.&nbsp; This is reproduced in&nbsp;<em>CIBSE Guide D, Transportation Systems in Buildings<\/em>&nbsp;(1).<\/p>\n<h4>2.2&nbsp; Author\u2019s contribution<\/h4>\n<p>The author has derived equations which allow ideal lift kinematics to be plotted as continuous functions for any value of journey distance, speed, acceleration and jerk.&nbsp; Supplementary results include journey time formulae for use in lift traffic analysis.&nbsp; The remainder of this paper is a discussion for the author\u2019s research.&nbsp;<\/p>\n<h3>3. Derivation of ideal kinematics equations<\/h3>\n<h4>3.1 Approach to derivation&nbsp;<\/h4>\n<p>The derivation is divided into three major sections, corresponding to the journey conditions where: (A) the lift reaches full speed; (B) the lift reaches full acceleration, but not full speed; and (C) the lift does not reach full speed or acceleration.&nbsp; The condition where full speed is reached before full acceleration(a<sup>2<\/sup>&gt;v.j) is discarded as this would be an illogical design.&nbsp;<\/p>\n<p>Conditions A to C are represented graphically in Figure 2.&nbsp; Each of the three conditions is divided into time slices, beginning and ending at each change in jerk or change in sign of acceleration.&nbsp; Functions of jerk were written down for each time interval, then integrated to give functions of acceleration, speed and distance over time.&nbsp; The end conditions of each set of functions provided the start conditions for the next time slice.<\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/figure2.jpg\" alt=\"\" border=\"0\"><br><strong><em>Figure 2&nbsp;&nbsp; Ideal lift kinematics for: (A) lift reaches full speed; (B) lift reaches full acceleration, but not full speed; (C) lift does not reach full speed or acceleration<\/em><\/strong><\/p>\n<p>Formulae were derived to establish which of conditions (A), (B) or (C) apply given journey distance, lift speed, acceleration and jerk.&nbsp; The derivation is recorded in reference (7).&nbsp; The mathematics is relatively complex and laborious, but was aided by use of mathematical computer software (Mathcad version 4.0 from Mathsoft Inc.) which has a built in symbolic processor for equation solving.<\/p>\n<p>The complete set of ideal lift kinematic equations is given in appendicies A to C.&nbsp; These equations may be implemented in a programming language such as Basic, Fortran, Pascal C++, etc.&nbsp; An example part of the derivation for condition (A) is reproduced in section 3.2.<\/p>\n<h4>3.2 Example section of derivation for condition (A)<\/h4>\n<p>Motion during time period 0&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;<\/em><em>\u2264<\/em><em>t<\/em><sub>1<\/sub><\/p>\n<p>For motion during time period 0&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264<\/em><em>t<\/em><sub>1<\/sub>, referring to Figure 2(A) we can write down:<\/p>\n<p>J(<em>t<\/em>) =&nbsp;<em>j<\/em><\/p>\n<p>A(<em>t<\/em>) =<em>&nbsp;j\u2219t<\/em><\/p>\n<p>The velocity can be determined by integrating the acceleration:<\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image017.png\" alt=\"\" border=\"0\"><\/p>\n<p>The distance travelled can be determined by integrating the velocity:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image018.png\" alt=\"\" border=\"0\"><\/p>\n<p>Motion during time period&nbsp;&nbsp;<em>t<\/em><sub>1&nbsp;<\/sub><em>\u2264<\/em><em>t&nbsp;\u2264<\/em><em>&nbsp;t<\/em><sub>2<\/sub>, referring to figure 2(A) we can write down:<\/p>\n<p>t<sub>1<\/sub>= a\/j<\/p>\n<p>J(<em>t<\/em>) = 0<\/p>\n<p>A(<em>t<\/em>) = a<\/p>\n<p>The velocity can be found by adding the velocity at the end of the previous time slice to the current acceleration, integrated:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image021.png\" alt=\"\" border=\"0\"><\/p>\n<p>Similarly, the distance travelled can be found by adding the distance travelled at the end of the previous time slice to the current velocity, integrated:<\/p>\n<h4><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image022.png\" alt=\"\" border=\"0\"><\/h4>\n<p>3.3 Example&nbsp; results&nbsp;<\/p>\n<p>Take journey distance, d=8 m; velocity v=2.5m\/s; acceleration, a=1 m\/s<sup>2<\/sup>, and jerk, j=2m\/s<sup>2<\/sup>.&nbsp; Inputting this data into equations in appendices A to C&nbsp; gives us the results plotted in Figure 3.&nbsp; The lift reaches full speed during its journey.&nbsp; Calculated values of t<sub>n<\/sub>&nbsp;are:&nbsp; t<sub>1<\/sub>=0.5 s, t<sub>2<\/sub>=2.5 s, t<sub>3<\/sub>-3 s, t<sub>4<\/sub>=3.2 s, t<sub>5<\/sub>=3.7 s, t<sub>6<\/sub>=5.7 s, t<sub>7<\/sub>=6.2 s.<\/p>\n<p><strong><em><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/kinematicsexample.jpg\" alt=\"\" border=\"0\"><\/em><\/strong><\/p>\n<p>Figure 3&nbsp; Example plots of jerk, acceleration, velocity&nbsp; and distance<\/p>\n<h3>4. Applications<\/h3>\n<h4>4.1 Generation of motor speed reference curves<\/h4>\n<p>Motor speed reference curves are commonly held in software look up tables.&nbsp; It is envisaged that a software implementation of the equations presented in this paper will provide a fast, flexible and efficient way of generating optimum reference speed profiles, on line in lift system controllers.&nbsp;<\/p>\n<h4>4.2 Formulae for lift traffic analysis<\/h4>\n<p>To calculate the handling capacity and performance of a lift system it is necessary to know how long it takes a lift to travel given distances.&nbsp; Using the appropriate formulae taken from the previous sections, the travel time of a variable speed lift (with optimum control) can be written down as follows:<\/p>\n<p>If&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image023.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp; then&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image024.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(condition A)<\/p>\n<p>If&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image025.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;&nbsp;then&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image026.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (condition B)<\/p>\n<p>If&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image027.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;then&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image028.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image029.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(condition C)<\/p>\n<p>These equations are consistent with those presented by H D Molz[2], but are in simpler form.<\/p>\n<p>It is advisable to add an additional time component to allow for motor start up time and any deviations from the optimum speed profile.&nbsp; Depending on drive quality, Day and Barney(1) recommend that this component should be between 0.2 and 0.5 seconds.&nbsp;<\/p>\n<h4>4.3&nbsp; Other kinematic problems<br>&nbsp;<\/h4>\n<p>The equations derived have applications to other kinematic problems such as power door control.<\/p>\n<h3>5. Conclusions<\/h3>\n<p>Ideal lift kinematics provide the basis for optimum speed control of lifts, an essential component for fast, efficient and comfortable transportation.&nbsp; The equations derived and presented by the author of this paper further previous research by allowing continuous , optimum functions of jerk, acceleration, speed and distance travelled profiles to be plotted against time.&nbsp; The results have applications in motor control and lift traffic analysis.&nbsp;<\/p>\n<p><strong>ACKNOWLEDGEMENTS<\/strong><\/p>\n<p>The author would like to thank his supervisors, lecturers and colleagues at Brunel University, Ove Arup &amp; Partners and CIBSE Lift Group for sharing their knowledge and experience which are providing an excellent basis for research.&nbsp; The author acknowledges, with gratitude, financial support from the Engineering and Physical Sciences Research Council, The Ove&nbsp;Arup Partnership and the Chartered Institution of Building Services Engineers.<\/p>\n<p><strong>BIOGRAPHY<\/strong><\/p>\n<p>Richard Peters studied at Southampton University.&nbsp; He was sponsored by the Ove Arup Partnership and in 1987 was awarded a BSc Hons in Electrical Engineering.&nbsp; After graduation he joined Arup where he participated in and led the design of electrical services for a number of major, national and international construction projects.&nbsp; He has a special interest in Vertical Transportation and has published a&nbsp; number of research papers in this field.&nbsp; In October 1993 he joined the Environmental Technology Doctorate programme run jointly by Brunel and Surrey Universities.&nbsp; His project, Vertical Transportation Planning in Buildings is sponsored by Arup and CIBSE.<\/p>\n<p><strong>REFERENCES<\/strong><\/p>\n<ol>\n<li>(Various authors)&nbsp;<em>CIBSE Guide D, Transportation Systems in Buildings<\/em>&nbsp;The Chartered Institution of Building Services Engineers (1993)<\/li>\n<li>Motz H D&nbsp;&nbsp;<em>On the kinematics of the ideal motion of lifts<\/em>&nbsp; Forden und haben 26 (1) (1976) (in German)<\/li>\n<li>Motz H D&nbsp;&nbsp;<em>On the ideal kinematics of lifts<\/em>&nbsp; Elevatori 1\/91 (1991) and&nbsp; Elevatori 2\/92 (1991) (in English and Italian) (beware typographical errors in formulae)<\/li>\n<li>Roschier N R and Kaakinen M J&nbsp;<em>&nbsp;New formulae for elevator round trip time<\/em>&nbsp;<em>calculation<\/em>&nbsp; Elevator World 28 (8) (August 1980 supplement)<\/li>\n<li>Schroeder J&nbsp;<em>&nbsp;Elevator trip profiles<\/em>&nbsp; Elevator World 35 (10) (November 1987)<\/li>\n<li>Barney G C and Loher A G&nbsp;<em>Elevator Electric Drives<\/em>&nbsp;Ellis Horwood, Chichester (1990)<\/li>\n<li>Peters R D&nbsp;&nbsp;<em>Ideal Lift Kinematics: Formulae for the equations of motion of a lift<\/em>&nbsp; (Internal Brunel University\/Arup Research &amp; Development paper)(November 1993)&nbsp;<\/li>\n<\/ol>\n<div>&nbsp;<\/div>\n<h3>APPENDIX A&nbsp; Lift Reaching Full Speed During Journey<\/h3>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image030.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image031.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image032.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image033.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image034.png\" alt=\"\" border=\"0\"><\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image035.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image036.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image037.png\" alt=\"\" border=\"0\"><\/p>\n<p><strong>Motion during time period 0&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264<\/em><em>t<\/em><sub>1<\/sub><\/strong><\/p>\n<p>J(<em>t<\/em>) =<em>&nbsp;j<\/em><\/p>\n<p>A(<em>t<\/em>) =&nbsp;<em>j\u2219t<\/em><\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image038.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image039.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;<\/p>\n<p><strong>Motion during time period&nbsp; t<sub>1<\/sub>&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<\/em><\/strong><sub><strong>2<br><\/strong><\/sub><\/p>\n<p>J(<em>t<\/em>) = 0<\/p>\n<p>A(<em>t<\/em>) = a<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image040.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image041.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;<\/p>\n<p><strong>Motion during time period t<sub>2<\/sub>&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>3<\/sub><\/em><br><\/strong><\/p>\n<p>J(<em>t<\/em>) =&nbsp;\u2212<em>j<\/em><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image044.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image045.png\" alt=\"\" border=\"0\"><\/p>\n<p><strong>Motion during time period&nbsp;t<sub>3<\/sub>&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>4<\/sub><\/em><br><\/strong>&nbsp;<\/p>\n<p>J(<em>t<\/em>) =0<\/p>\n<p>A(<em>t<\/em>) =0<\/p>\n<p>V(<em>t<\/em>) = v<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image047.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;<\/p>\n<p><strong>Motion during time period t<sub>4<\/sub>&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>5<\/sub><\/em><\/strong><\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image049.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image050.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image051.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image052.png\" alt=\"\" border=\"0\"><\/p>\n<p><strong>Motion during the time period t<sub>5<\/sub>&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>6<\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image054.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image055.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image056.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image057.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;<\/p>\n<p><strong>Motion during the time period t<sub>6<\/sub>&nbsp;<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>7<\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image059.png\" alt=\"\" border=\"0\"><\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image060.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image061.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image062.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<h3>APPENDIX B&nbsp; Lift Reaching Maximum Acceleration, But Not Full Speed<\/h3>\n<p>Results apply over the range<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image063.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image064.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image065.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image066.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image067.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;<\/p><p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\"><br><\/p>\n<p><img decoding=\"async\" src=\"data:image\/png;base64,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\" alt=\"\"><br><\/p>\n<p><strong>Motion during the time period 0<em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><\/strong><em><strong>t<\/strong><sub><strong>1<br><\/strong><\/sub><\/em><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image071.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image072.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image073.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image074.png\" alt=\"\" border=\"0\"><\/p>\n<p><strong>Motion during the time period&nbsp;<em>t<\/em><sub>1<\/sub><em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>2<\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image076.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image077.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image078.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image079.png\" alt=\"\" border=\"0\"><\/p>\n<p><strong>Motion during the time period&nbsp;<em>t<sub>2<\/sub><\/em><em>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>3<\/sub><\/em><\/strong><\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image080.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image081.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image082.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image083.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><strong>Motion during the time period&nbsp;<em>t<sub>3<\/sub>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>4<\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image080.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image085.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image086.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image087.png\" alt=\"\" width=\"615\" height=\"104\" border=\"0\"><\/p>\n<p><strong>Motion during the time period&nbsp;<em>t<sub>4<\/sub>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>5<\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image054.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image055.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image089.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image090.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;&nbsp;<\/p>\n<p><strong>Motion during the time period<\/strong>&nbsp;<strong><em>t<sub>5<\/sub>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>6<\/sub><\/em><\/strong><\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image071.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image093.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image094.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image095.png\" alt=\"\" border=\"0\"><\/p>\n<h3>APPENDIX C&nbsp; Lift Not Reaching Maximum Speed or Acceleration<\/h3>\n<p>Results apply over the range&nbsp;&nbsp;&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image096.png\" alt=\"\" border=\"0\"><\/p>\n<p><strong>Values of&nbsp;<em>t<sub>n<\/sub><\/em><\/strong><\/p>\n<p>&nbsp;<img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image098.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image099.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image100.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image101.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><strong>Motion during time period<\/strong>&nbsp;<strong><em>0\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>1<\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image059.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image104.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image105.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image106.png\" alt=\"\" border=\"0\"><\/p>\n<p><strong>Motion during time period<\/strong>&nbsp;<strong><em>t<sub>1<\/sub>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>2<\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image107.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image108.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image109.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image110.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><strong>Motion during time period<\/strong>&nbsp;<strong><em>t<sub>2<\/sub>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>3<br><\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image107.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image111.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image112.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image113.png\" alt=\"\" border=\"0\"><\/p>\n<p><strong>Motion during time period<\/strong>&nbsp;<strong><em>t<sub>3<\/sub>\u2264<\/em>&nbsp;<em>t&nbsp;\u2264&nbsp;<\/em><em>t<sub>4<\/sub><\/em><\/strong><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image059.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image114.png\" alt=\"\" border=\"0\">&nbsp;<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image115.png\" alt=\"\" border=\"0\"><\/p>\n<p><img decoding=\"async\" src=\"https:\/\/wordpress.peters-research.com\/images\/stories\/papers\/idealliftkinematics\/image116.png\" alt=\"\" border=\"0\">&nbsp;&nbsp;<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"has_eae_slider elementor-section elementor-top-section elementor-element elementor-element-4703451 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-eae-slider=\"16376\" data-id=\"4703451\" data-element_type=\"section\" data-e-type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"has_eae_slider elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-021529b\" data-eae-slider=\"67075\" data-id=\"021529b\" data-element_type=\"column\" data-e-type=\"column\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-9b49e14 elementor-widget elementor-widget-text-editor\" data-id=\"9b49e14\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>\u00a0<\/p><p>\u00a0<\/p><p><a href=\"\/paper-content\/kinematics.html\" target=\"_blank\" rel=\"noopener\">See redrawn equations<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Ideal Lift Kinematics Richard D Peters Brunel UniversityArup Research &amp; Development This paper was presented at ELEVCON&nbsp;HONG KONG&nbsp;1995, The International Congress on Vertical Transportation Technologies&nbsp;and first published in the IAEE book &#8220;Elevator Technology 6&#8221;, edited by G. C. Barney.&nbsp; It is reproduced with permission from The International Association of Elevator Engineers.&nbsp; The paper was republished [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":860,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"elementor_canvas","meta":{"footnotes":""},"class_list":["post-1008","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Ideal Lift Kinematics - Peters Research<\/title>\n<meta name=\"robots\" content=\"noindex, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Ideal Lift Kinematics - Peters Research\" \/>\n<meta property=\"og:description\" content=\"Ideal Lift Kinematics Richard D Peters Brunel UniversityArup Research &amp; Development This paper was presented at ELEVCON&nbsp;HONG KONG&nbsp;1995, The International Congress on Vertical Transportation Technologies&nbsp;and first published in the IAEE book &#8220;Elevator Technology 6&#8221;, edited by G. 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