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47-412: The Klien-Lindner axle uses a double, or hollow, axle, one inside the other. It has a hollow axle ( Hohlachse ) on the outside, connected at its centre by a Cardan joint to a fixed driving axle running through it. The Cardan joint comprises two spherical elements that are interlinked - a solid one on the fixed axle and a hollow one on the outer hollow axle, each oriented at 90° to the other that transfer

94-793: A 1 cos ⁡ β 1 − sin 2 ⁡ β cos 2 ⁡ γ 1 − ω 1 2 cos ⁡ β sin 2 ⁡ β sin ⁡ 2 γ 1 ( 1 − sin 2 ⁡ β cos 2 ⁡ γ 1 ) 2 {\displaystyle a_{2}={\frac {a_{1}\cos \beta }{1-\sin ^{2}\beta \,\cos ^{2}\gamma _{1}}}-{\frac {\omega _{1}^{2}\cos \beta \,\sin ^{2}\beta \,\sin 2\gamma _{1}}{\left(1-\sin ^{2}\beta \,\cos ^{2}\gamma _{1}\right)^{2}}}} A configuration known as

141-467: A universal coupling or U-joint ) is a joint or coupling connecting rigid shafts whose axes are inclined to each other. It is commonly used in shafts that transmit rotary motion . It consists of a pair of hinges located close together, oriented at 90° to each other, connected by a cross shaft. The universal joint is not a constant-velocity joint . U-joints are also sometimes called by various eponymous names, as follows: The main concept of

188-523: A German language school in Stockholm until the age of 8 when his father died. His mother, Christina Eriksdotter Schening, who was from Vadstena in Östergötland , later remarried. Because of conflicts with his stepfather, his private tuition was no longer paid and Polhem was sent to live with his uncle in Stockholm. He took a job as a farmhand on Vansta, a property in Södertörn , Stockholm. Quickly, he rose to

235-409: A century. In 1690 Polhem was appointed to improve upon the current mining operations of Sweden. His contribution was a construction for lifting and transporting ore from mines, a process that was rather risky and inefficient at the time. The construction consisted of a track system for lifting the ore, as opposed to wires; the construction was powered entirely by a water wheel . Human labor needed

282-402: A double Cardan joint drive shaft partially overcomes the problem of jerky rotation. This configuration uses two U-joints joined by an intermediate shaft, with the second U-joint phased in relation to the first U-joint to cancel the changing angular velocity. In this configuration, the angular velocity of the driven shaft will match that of the driving shaft, provided that both the driving shaft and

329-467: A facility for the training of engineers, as well as a laboratory for testing and exhibiting his designs. It has since become the prestigious KTH Royal Institute of Technology , whose history began with Charles XI and his praise for Polhem for his mining efforts. His greatest achievement was an automated factory powered entirely by water; automation was very unusual at the time. Built in 1699 in Stjärnsund ,

376-711: A latecomer to the English language. Many early uses in the 19th century appear in translations from French or are strongly influenced by French usage. Examples include an 1868 report on the Exposition Universelle of 1867 and an article on the dynamometer translated from French in 1881. In the 20th century, Clarence W. Spicer and the Spicer Manufacturing Company , as well as the Hardy Spicer successor brand, helped further popularize universal joints in

423-923: A rotating joint will be functions of time. Differentiating the equation of motion with respect to time and using the equation of motion itself to eliminate a variable yields the relationship between the angular velocities ω 1 = d γ 1 / d t {\displaystyle \omega _{1}=d\gamma _{1}/dt} and ω 2 = d γ 2 / d t {\displaystyle \omega _{2}=d\gamma _{2}/dt} : ω 2 = ω 1 ( cos ⁡ β 1 − sin 2 ⁡ β cos 2 ⁡ γ 1 ) {\displaystyle \omega _{2}=\omega _{1}\left({\frac {\cos \beta }{1-\sin ^{2}\beta \,\cos ^{2}\gamma _{1}}}\right)} As shown in

470-464: A solution to the nonuniform rotary speed of the universal joint: a pair of Hooke's joints 90° out of phase at either end of an intermediate shaft, an arrangement that is now known as a type of constant-velocity joint. Christopher Polhem of Sweden later re-invented the universal joint, giving rise to the name Polhemsknut ("Polhem knot") in Swedish. In 1841, the English scientist Robert Willis analyzed

517-438: A tutor. In exchange for constructing a complex clock, he was given Latin lessons by a local vicar. Word of Polhem's mechanical skill spread quickly and a member of the clergy wrote to Anders Spole , professor of mathematics at Uppsala University to recommend Polhem. Anders Spole, grandfather of Anders Celsius , presented two broken clocks to Polhem and offered to let him study under him if he could repair them. Polhem repaired

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564-821: Is at angle γ 2 {\displaystyle \gamma _{2}} with respect to its beginning position along the y axis. x ^ 1 {\displaystyle {\hat {\mathbf {x} }}_{1}} is confined to the "red plane" in the diagram and is related to γ 1 {\displaystyle \gamma _{1}} by: x ^ 1 = [ cos ⁡ γ 1 , sin ⁡ γ 1 , 0 ] {\displaystyle {\hat {\mathbf {x} }}_{1}=\left[\cos \gamma _{1}\,,\,\sin \gamma _{1}\,,\,0\right]} x ^ 2 {\displaystyle {\hat {\mathbf {x} }}_{2}}

611-918: Is confined to the "blue plane" in the diagram and is the result of the unit vector on the x axis x ^ = [ 1 , 0 , 0 ] {\displaystyle {\hat {x}}=[1,0,0]} being rotated through Euler angles [ π / 2 , β , γ 2 {\displaystyle [\pi \!/2\,,\,\beta \,,\,\gamma _{2}} ]: x ^ 2 = [ − cos ⁡ β sin ⁡ γ 2 , cos ⁡ γ 2 , sin ⁡ β sin ⁡ γ 2 ] {\displaystyle {\hat {\mathbf {x} }}_{2}=\left[-\cos \beta \sin \gamma _{2}\,,\,\cos \gamma _{2}\,,\,\sin \beta \sin \gamma _{2}\right]} A constraint on

658-408: Is mentioned between the forks.) A double Cardan joint consists of two universal joints mounted back to back with a centre yoke; the centre yoke replaces the intermediate shaft. Provided that the angle between the input shaft and centre yoke is equal to the angle between the centre yoke and the output shaft, the second Cardan joint will cancel the velocity errors introduced by the first Cardan joint and

705-982: Is not unique since the arctangent function is multivalued, however it is required that the solution for γ 2 {\displaystyle \gamma _{2}} be continuous over the angles of interest. For example, the following explicit solution using the atan2 (y, x) function will be valid for − π < γ 1 < π {\displaystyle -\pi <\gamma _{1}<\pi } : γ 2 = atan2 ⁡ ( sin ⁡ γ 1 , cos ⁡ β cos ⁡ γ 1 ) {\displaystyle \gamma _{2}=\operatorname {atan2} \left(\sin \gamma _{1},\cos \beta \,\cos \gamma _{1}\right)} The angles γ 1 {\displaystyle \gamma _{1}} and γ 2 {\displaystyle \gamma _{2}} in

752-450: Is seen that the output drive is just 90 degrees out of phase with the input shaft, yielding a constant-velocity drive. NOTE: The reference for measuring angles of input and output shafts of universal joint are mutually perpendicular axes. So, in absolute sense the forks of the intermediate shaft are parallel to each other. (Since, one fork is acting as input and the other fork is acting as output for shafts and above 90 degree phase difference

799-608: The x ^ 1 {\displaystyle {\hat {\mathbf {x} }}_{1}} and x ^ 2 {\displaystyle {\hat {\mathbf {x} }}_{2}} vectors is that since they are fixed in the gimbal , they must remain at right angles to each other. This is so when their dot product equals zero: x ^ 1 ⋅ x ^ 2 = 0 {\displaystyle {\hat {\mathbf {x} }}_{1}\cdot {\hat {\mathbf {x} }}_{2}=0} Thus

846-516: The Royal Swedish Academy of Sciences in 1739, the same year the academy was founded. Polhem died of natural causes in 1751 in Stockholm, at the age of 90. According to Polhem's autobiography , the event that marked the beginning of his career was the successful repair of the unfinished medieval (16th century) astronomical clock designed by Petrus Astronomus at Uppsala Cathedral , which had remained unfinished and broken for more than

893-466: The automotive , farm equipment , heavy equipment , and industrial machinery industries. The Cardan joint suffers from one major problem: even when the input drive shaft axle rotates at a constant speed, the output drive shaft axle rotates at a variable speed, thus causing vibration and wear. The variation in the speed of the driven shaft depends on the configuration of the joint, which is specified by three variables: These variables are illustrated in

940-498: The equation of time which accounts for the tilt of the equatorial plane relative to the ecliptic is entirely analogous to the mathematical description of the universal joint. The first recorded use of the term 'universal joint' for this device was by Hooke in 1676, in his book Helioscopes . He published a description in 1678, resulting in the use of the term Hooke's joint in the English-speaking world. In 1683, Hooke proposed

987-522: The 14th century of possible Austrian descent. Polhem's father, Wolf Christoph Polhammer, born c.  1610 in Hungary to a baron, moved from Hungary to Swedish Pomerania because the Austrian catholics had the then predominantly protestant Hungarians under religious persecution. Wolf Polhammer traded with Visby . He would eventually settle down to become a skipper. Christopher Polhammar attended

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1034-410: The aligned double Cardan joint will act as a CV joint. A Thompson coupling is a refined version of the double Cardan joint. It offers slightly increased efficiency with the penalty of great increase in complexity. Christopher Polhem Christopher Polhammar (18 December 1661 – 30 August 1751) better known as Christopher Polhem ( listen ), which he took after his ennoblement in 1716,

1081-954: The angles for the input and output of the universal joint connecting the drive and the intermediate shafts respectively, and γ 3 {\displaystyle \gamma _{3}\,} and γ 4 {\displaystyle \gamma _{4}\,} are the angles for the input and output of the universal joint connecting the intermediate and the output shafts respectively, and each pair are at angle β {\displaystyle \beta \,} with respect to each other, then: tan ⁡ γ 2 = cos ⁡ β tan ⁡ γ 1 tan ⁡ γ 4 = cos ⁡ β tan ⁡ γ 3 {\displaystyle \tan \gamma _{2}=\cos \beta \,\tan \gamma _{1}\qquad \tan \gamma _{4}=\cos \beta \,\tan \gamma _{3}} If

1128-586: The axles rotate. The two axles are joined by a gimbal which is not shown. However, axle 1 attaches to the gimbal at the red points on the red plane of rotation in the diagram, and axle 2 attaches at the blue points on the blue plane. Coordinate systems fixed with respect to the rotating axles are defined as having their x-axis unit vectors ( x ^ 1 {\displaystyle {\hat {\mathbf {x} }}_{1}} and x ^ 2 {\displaystyle {\hat {\mathbf {x} }}_{2}} ) pointing from

1175-546: The clocks with no difficulty and in 1687, entered the University of Uppsala at the age of 26. Polhem married Maria Hoffman (1671–1735) on 28 December 1691. Together they had 8 children between 1692 and 1705. In 1716, he was ennobled by King Charles XII of Sweden in gratitude of his services to the nation by the king and changed his surname from Polhammar to Polheim , which he later shortened to Polhem . He and his son Gabriel Polhem (1700–1772) were both elected members of

1222-484: The construction of Göta Canal , a canal connecting the east and west coasts of Sweden. Together with Charles XII of Sweden, he planned the construction of parts of the canal, particularly the canal locks in the 18th century; it was not to be finished until 1832, long after his death. Other major contributions made by Polhem were the constructions of dry docks , dams , and canal locks, which he designed together with his assistant and friend, Emanuel Swedenborg . Polhem

1269-433: The conventional driving wheels in the centre act on drive cranks on the inner axles fixed to the frame. In spite of their relatively simple design Klien-Lindner axles were not widely used. Derailments were common when they were used as leading axles . The axles often caused uneven, jerky running as a result of the resistance forces that arise from this type of Cardan joint, and they were expensive to maintain, something which

1316-414: The diagram on the right. Also shown are a set of fixed coordinate axes with unit vectors x ^ {\displaystyle {\hat {\mathbf {x} }}} and y ^ {\displaystyle {\hat {\mathbf {y} }}} and the planes of rotation of each axle. These planes of rotation are perpendicular to the axes of rotation and do not move as

1363-485: The driven shaft are at equal angles with respect to the intermediate shaft (but not necessarily in the same plane) and that the two universal joints are 90 degrees out of phase. This assembly is commonly employed in rear wheel drive vehicles, where it is known as a drive shaft or propeller (prop) shaft. Even when the driving and driven shafts are at equal angles with respect to the intermediate shaft, if these angles are greater than zero, oscillating moments are applied to

1410-406: The driving forces from the rigid axle to the hollow one. The hollow spheroid acts as a sort of link motion. In this way the hollow axle can be turned by the fixed axle. In addition, the connecting link is shaped so that the axles can slide relative to one another, parallel to their axes, to a small extent. The degree to which the hollow axle can swivel is set by the outer diameter of the fixed axle and

1457-411: The engine and rolling mill shafts. Ephriam Shay's locomotive patent of 1881, for example, used double universal joints in the locomotive's drive shaft . Charles Amidon used a much smaller universal joint in his bit-brace patented 1884. Beauchamp Tower 's spherical, rotary, high speed steam engine used an adaptation of the universal joint c.  1885 . The term 'Cardan joint' appears to be

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1504-768: The equation of motion relating the two angular positions is given by: tan ⁡ γ 1 = cos ⁡ β tan ⁡ γ 2 {\displaystyle \tan \gamma _{1}=\cos \beta \tan \gamma _{2}\,} with a formal solution for γ 2 {\displaystyle \gamma _{2}} : γ 2 = tan − 1 ⁡ [ tan ⁡ γ 1 sec ⁡ β ] {\displaystyle \gamma _{2}=\tan ^{-1}\left[\tan \gamma _{1}\sec \beta \right]\,} The solution for γ 2 {\displaystyle \gamma _{2}}

1551-415: The factory produced a number of products including manufacture of knives, locks and clocks. Development of the factory was derived from the idea that Sweden should export fewer raw materials and instead process them within their own borders. The factory met great resistance among workers who feared they would be replaced by machinery. Eventually most of the factory was destroyed in a fire in 1734, leaving only

1598-477: The first design of the variation of padlocks common today. Economically, the factory was unfeasible, but the king at the time, Charles XII , was supportive and gave Polhem freedom from taxes to encourage his efforts. The factory of Stjärnsund was visited by one of his contemporaries, Carl Linnaeus , who wrote about the factory in his diaries as "Nothing is more optimistic than Stjärnsund" (Swedish: Intet är spekulativare än Stjärnsund ). Polhem also contributed to

1645-416: The internal diameter of the hollow one. This system is used on steam locomotives with fixed outer frames and coupled axles . Typically the conventionally driven wheels are in the centre and there are outer Klien-Lindner axles, front and rear. In this way the wheels, which are fixed to the hollow axles, are 'steered' by shafts that pivot on the frame when the locomotive is curve running. Coupling rods from

1692-484: The motion of the universal joint. By 1845, the French engineer and mathematician Jean-Victor Poncelet had analyzed the movement of the universal joint using spherical trigonometry. The term universal joint was used in the 18th century and was in common use in the 19th century. Edmund Morewood's 1844 patent for a metal coating machine called for a universal joint, by that name, to accommodate small alignment errors between

1739-440: The origin towards one of the connection points. As shown in the diagram, x ^ 1 {\displaystyle {\hat {\mathbf {x} }}_{1}} is at angle γ 1 {\displaystyle \gamma _{1}} with respect to its beginning position along the x axis and x ^ 2 {\displaystyle {\hat {\mathbf {x} }}_{2}}

1786-483: The part of the factory that produced clocks. The factory continued producing clocks, known for their high quality and low price. Although the popularity of the clocks diminished during the beginning of the 19th century, clock-making continues to this day at Stjärnsund, still producing around twenty clocks of the Polhem design per year. Another product from the factory was the "Polhem locks" (Swedish: Polhemslås ), essentially

1833-408: The plots, the angular velocities are not linearly related, but rather are periodic with a period half that of the rotating shafts. The angular velocity equation can again be differentiated to get the relation between the angular accelerations a 1 {\displaystyle a_{1}} and a 2 {\displaystyle a_{2}} : a 2 =

1880-522: The position of supervisor, being responsible for supervision and accounting, for which he was well suited by his affinity for mathematics. He worked at Vansta for ten years, during which period he constructed a workshop where he made tools, repaired and constructed simple machinery to earn money. Hungering for knowledge within his fields of interest, mathematics and mechanics, he soon realized that he would get no further without learning Latin. Self-studies were attempted, but given up; Polhem realized he needed

1927-974: The second universal joint is rotated 90 degrees with respect to the first, then γ 3 = γ 2 + π / 2 {\displaystyle \gamma _{3}=\gamma _{2}+\pi /2} . Using the fact that tan ⁡ ( γ + π / 2 ) = 1 / tan ⁡ γ {\displaystyle \tan(\gamma +\pi /2)=1/\tan \gamma } yields: tan ⁡ γ 4 = cos ⁡ β tan ⁡ γ 2 = 1 tan ⁡ γ 1 = tan ⁡ ( γ 1 + π 2 ) {\displaystyle \tan \gamma _{4}={\frac {\cos \beta }{\tan \gamma _{2}}}={\frac {1}{\tan \gamma _{1}}}=\tan \left(\gamma _{1}+{\frac {\pi }{2}}\right)\,} and it

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1974-580: The three shafts as they rotate. These tend to bend them in a direction perpendicular to the common plane of the shafts. This applies forces to the support bearings and can cause "launch shudder" in rear wheel drive vehicles. The intermediate shaft will also have a sinusoidal component to its angular velocity, which contributes to vibration and stresses. Mathematically, this can be shown as follows: If γ 1 {\displaystyle \gamma _{1}\,} and γ 2 {\displaystyle \gamma _{2}\,} are

2021-557: The universal joint is based on the design of gimbals , which have been in use since antiquity. One anticipation of the universal joint was its use by the ancient Greeks on ballistae . In Europe the universal joint is often called the Cardano joint (and a drive shaft that uses the joints, a Cardan shaft), after the 16th century Italian mathematician, Gerolamo Cardano , who was an early writer on gimbals, although his writings mentioned only gimbal mountings, not universal joints. The mechanism

2068-473: Was a Swedish scientist, inventor, and industrialist. He made significant contributions to the economic and industrial development of Sweden, particularly mining. He was ennobled by King Charles XII of Sweden for his contributions to Swedish technological development. Polhem (Polhammar) was born on the island of Gotland in the village of Tingstäde , north-east of Visby . Originally, the Polhem family came from Hungary , where they were established latest in

2115-454: Was later described in Technica curiosa sive mirabilia artis (1664) by Gaspar Schott , who mistakenly claimed that it was a constant-velocity joint . Shortly afterward, between 1667 and 1675, Robert Hooke analysed the joint and found that its speed of rotation was nonuniform, but that property could be used to track the motion of the shadow on the face of a sundial. In fact, the component of

2162-604: Was limited to loading the containers. Being new and revolutionary , word of Polhem's work reached King Charles XI of Sweden who was so impressed with the work that he assigned him to improve Sweden's main mining operation; the Falun Mine in Dalarna . Funded by the Swedish mining authority, Polhem traveled throughout Europe, studying mechanical development. He returned to Sweden in 1697 to establish laboratorium mechanicum in Stockholm ,

2209-856: Was not offset by the reduced wear and tear on wheel flanges and rails. Several locomotives of this type were ordered for the Matheran Hill Railway in India , which has curves as sharp as 18.25 m (59.88 ft), traversed at a speed of 8 km/h (5 mph). Consulting engineer Everard Calthrop designed a 0-6-0T with Klien-Lindner articulated coupled axles to provide a flexible wheelbase, and four were supplied by Orenstein & Koppel . Steam locomotives with Klien-Lindner axle are still widely used for narrow gauge sugarcane railway in Java, Indonesia . Locomotives with Klien-Lindner hollow axles (selection): Cardan joint A universal joint (also called

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