Slider Crank and Grashof 1 2 3 4 Equivalent Linkage 1 2 3 4

Slider Crank and Grashof 1 2 3 4 1 2 3 Infinite radius 4 e = l1 - l4 l1= inf l4= inf Equivalent Linkage

Slider Crank and Grashof 1 2 3 4 Schematic 1 2 3 Infinite radius 4 e = l1 - l4 l1= inf l4= inf

Grashof’s criteria for Slider Cranks Now substitute in Grashof’s criteria for RRRR linkages. Here 1 is definitely longest. Let 2 be less than 3 making it shortest Shortest + longest < sum of other two Hence l2+ l1 < l3+ l4 Or l2+ l1 - l4 < l3 Or l2+ ( l1 - l4 ) < l3 Or l2+ e < l3 Or s + e < L

Grashof ‘s criteria for RRRR linkages l + s < p + q where s (link 1) is the shortest link All other links rotate fully about s. Hence if the shortest link is grounded Both 2 and 4 rotate fully about the ground making it a Double Crank l + s < p + q where s (link 1) is the shortest link Link 3 does not rotate fully about either 2 or 4. Hence if link 3 is grounded i.e. if the shortest link is made the coupler Neither 2 nor 4 will rotate fully about the ground making it a Double Rocker l + s < p + q where s (link 1) is the shortest link If link 2 (or 4) is grounded 1 rotates fully about 2 (or 4) 3 rotates partially about 2 (or 4) making it a Crank Rocker

Grashof’s criteria for Slider Cranks Analyze using same logic as used for RRRR linkages, remembering that link 2 is the shortest here. 1 2 3 4 s+ e < L where s (link 2) is the shortest link All other links rotate fully about s (link 2) . If link 2 is grounded, i.e. shortest link is the ground, Both 1 and 3 rotate fully about 2, i.e. the ground. Since link 1 is not grounded , We have an Inverted Slider Crank (RRPR) with a fully rotating crank (link 2)and a fully rotating slider. 1 s+ e < L where s is the shortest link All other links rotate fully about s (link 2). If the shortest link is the driver or crank, 1 rotates fully about 2 and vice versa. 3 and 4 rotate partially about 2. If 3 is grounded , We have an Inverted Slider Rocker (RPRR) with a partially rotating crank (link 2) and a fixed slider. 2

Grashof’s criteria for Slider Cranks 1 2 3 4 Link 2 is the shortest. s+ e < L where s is the shortest link All other links rotate fully about s. If the shortest link is the driver, 1 rotates fully about 2. 3 and 4 rotate partially about 1. If 1 is grounded , We have a Slider Crank (RRRP) with a fully rotating crank (link 2) and a fixed slider 3 s+ e < L where s is the shortest link All other links rotate fully about s. If link 4 is grounded. Neither 1 nor 3 rotate fully about 4. We have a Slider Rocker (RRRP) with a partially rotating crank (link 3) and a fixed slider. 4 All these linkages are Grashofian.

Grashof’s criteria for Slider Cranks Consider the other possibility where 3 is shorter than 2. 2 is the shortest link, 1 is the longest. We substitute in Grashof’s criteria for RRRR linkages. Shortest + longest < sum of other two Hence l3+ l1 < l2+ l4 Or l3+ l1 - l4 < l2 Or l3+ ( l1 - l4 ) < l2 Or l3+ e < l2 Or s + e < L

Grashof’s criteria for Slider Cranks Analyze using same logic as used for RRRR linkages, considering that link 3 is the shortest here. 1 2 3 4 s+ e < L where s is the shortest link All other links rotate fully about s (link 3) . If link 3 is grounded, i.e. shortest link is the ground, Both 2 and 4 rotate fully about 3, i.e. the ground. Since link 1 is not grounded , We have an Inverted Slider Crank (RRPR) with a fully rotating crank (link 2) and a fully rotating slider. 5 s+ e < L where s is the shortest link All other links rotate fully about s (link 3) . If link 1 is grounded, i.e. shortest link is the coupler 2 does not rotate fully about 1. We have a Slider Rocker (RRRP) with a partially rotating crank (link 2) and a fixed slider. 6

Grashof’s criteria for Slider Cranks Analyze using same logic as used for RRRR linkages, considering that link 3 is the shortest here. 1 2 3 4 s+ e < L where s is the shortest link All other links rotate fully about s (link 3) . If link 2 is grounded, i.e. shortest link (link 3) is the crank, 3 rotates fully about 2, i.e. the ground. 1 rotates partially about 2, i.e. the ground. We have an Inverted Slider Rocker (RRPR) with a fully rotating crank and a partially rotating slider. 7 s+ e < L where s is the shortest link All other links rotate fully about s (link 3) . If link 4 is grounded, 3 rotates fully about 4, i.e. the ground. 2 does not rotate fully about 4. We have a Slider Crank (RRRP) with a fully rotating crank and a fixed slider. 8

Grashof’s criteria for Slider Cranks : Summary s+ e < L where s is the shortest link Shortest link is Ground Inverted Slider Crank with fully rotating crank 1 5 s+ e < L where s is the shortest link Shortest link is Crank (driver) Inverted Slider Rocker with fully rotating crank and partially rotating slider Or Slider Crank with fully rotating crank 2 7 8 3 s+ e < L where s is the shortest link Shortest link is Coupler Slider Rocker with a partially rotating crank 4 6 s+ e > L where s is the shortest link Non Grashofian Slider Rocker with a partially rotating crank and fixed slider. Or Non Grashofian Inverted Slider Rocker with a partially rotating crank and partially rotating slider.