TECHNICAL DIRECTOR-DESIGNER

BRADLEY BRANAM

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Tartuffe Automation Math

 

The math below was used to figure out the setup of the automation hardware for the full stage wagon move.  The results of the math allowed me to select the appropriate configuration of stock components to achieve the motion of the wagon.  Automating the wagon added very little to my overall budget because so many of the components were in stock at UMKC. 

 

 

 

Weight of wall wagon:                                Castor Size = 5”

                                                                  Motor = 5hp @ 1760 rpm

Wagon         = 1783#                                    Gear Red. = 50:1 w/ 80% ineff

Walls           = 2350#                                    μ_r= 0.08

Wall Details = 1641#                                     a= 0.01664 ^ft/_sec²

Total                5774#                                   Dia. Drum= 12.75”

                                                                  Circ. Drum = 40.06”

Wagon Travel= 13’-3.75”                             F to move Wagon= 387.1#   

Time for Effect= 60sec                                Cable Pull=2246.844#

 

Accel/Decel Math:

 

ΔX=((v₁∙Δt)+(1/2a_A∙Δt²))+((V₂∙Δt)+(1/2a_cv∙Δt²))+((V₂∙Δt)+(1/2a_A∙Δt²))=13.3125’

13.3125’=((0∙20sec)+( 1/2a_A∙400))+((V₂∙20sec)+( 1/2∙0∙400))+((V²∙20sec)+(1/2a_A∙400))

13.3125’=(20V₂)+(20V₂)

13.3125’=40V₂

0.3328 ^ft/_sec=V₂

 

Accel:   V₂=V₁+a_A∙Δt=

             0.3328 ^ft/_sec=0+a_A∙20sec

             0.3328 ^ft/_sec=20a_A

             0.01664 ^ft/_sec²=a_A

 

Ƭ_motor= HP=^Ƭ∙rpm/₅₂₅₂=   5=^1760Ƭ/₅₂₅₂=  14.92 ^ft/_lbs

Ƭ_reducer=14.92 ^ft/_lbs∙⁵⁰/₁∙0.8=596.818 ^ft/_lbs

Ƭ_drum=596.818 ^ft/_lbs∙³⁰/₁₅=1193.636 ^ft/_lbs

Ƭ_req’d= r∙F= 6.375”∙387.1#= 2467.7625 ⁱⁿ/_lbs ÷ 12= 205.64688 ^ft/_lbs       

 

F_accel= M∙a=^6000#/_32.2=186.335∙0.01664 ^ft/_sec²=3.1#

F_friction=^N∙μr/_rcast=^{6000 #∙0.08}/_2.5=192∙2=384#

F_{to move wagon}=384# +3.1#=387.1#

F=^Ƭ/_r=14323.632ⁱⁿ/_lbs ÷ 6.375” = 2246.844#

 

Max speed for effect:

 

1760rpm ÷ 50= 35.2rpm ÷ 2= 17.6rpm ∙ 40.6”=714.56 ⁱⁿ/_min ÷ 12= 59.54667 ^ft/_min

59.54667 ^ft/_min ÷ 60sec=0.99244 ^ft/_sec ∙ 13.3125’= 13.21192 sec= death to set

 

 

 

Effect in 30 sec:

 

Accel/Decel Math:

 

ΔX=((v₁∙Δt)+(1/2a_A∙Δt²))+((V₂∙Δt)+(1/2a_cv∙Δt²))+((V₂∙Δt)+(1/2a_A∙Δt²))=13.3125’

13.3125’=((0∙10sec)+( 1/2a_A∙100))+((V₂∙10sec)+( 1/2∙0∙100))+((V²∙10sec)+(1/2a_A∙100))

13.3125’=(10V₂)+(10V₂)

13.3125’=20V₂

0.665625 ^ft/_sec=V₂

 

Accel:   V₂=V₁+a_A∙Δt=

             0. 665625 ^ft/_sec=0+a_A∙10sec

             0. 665625 ^ft/_sec=10a_A

             0.066563 ^ft/_sec²=a_A

 

Ƭ_motor= HP=^Ƭ∙rpm/₅₂₅₂=   5=^1760Ƭ/₅₂₅₂=  14.92 ^ft/_lbs

Ƭ_reducer=14.92 ^ft/_lbs∙⁵⁰/₁∙0.8=596.818 ^ft/_lbs

Ƭ_drum=596.818 ^ft/_lbs∙³⁰/₁₅=1193.636 ^ft/_lbs

Ƭ_req’d= r∙F= 6.375”∙396.403#= 2527.06913 ⁱⁿ/_lbs ÷ 12= 210.5891 ^ft/_lbs       

 

F_accel= M∙a=^6000#/_32.2=186.335∙0.066563 ^ft/_sec²=12.403#

F_friction=^N∙μr/_rcast=^{6000 #∙0.08}/_2.5=192∙2=384#

F_{to move wagon}=384# +12.403#=396.403#

F=^Ƭ/_r=14323.632ⁱⁿ/_lbs ÷ 6.375” = 2246.844#

 

Effect in 15 sec:

 

Accel/Decel Math:

 

ΔX=((v₁∙Δt)+(1/2a_A∙Δt²))+((V₂∙Δt)+(1/2a_cv∙Δt²))+((V₂∙Δt)+(1/2a_A∙Δt²))=13.3125’

13.3125’=((0∙5sec)+( 1/2a_A∙25))+((V₂∙5sec)+( 1/2∙0∙25))+((V²∙5sec)+(1/2a_A∙25))

13.3125’=(5V₂)+(5V₂)

13.3125’=10V₂

1.33125 ^ft/_sec=V₂

 

Accel:   V₂=V₁+a_A∙Δt=

             1.33125 ^ft/_sec=0+a_A∙5sec

             1.33125 ^ft/_sec=5a_A

             0.26625 ^ft/_sec²=a_A

 

Ƭ_motor= HP=^Ƭ∙rpm/₅₂₅₂=   5=^1760Ƭ/₅₂₅₂=  14.92 ^ft/_lbs

Ƭ_reducer=14.92 ^ft/_lbs∙⁵⁰/₁∙0.8=596.818 ^ft/_lbs

Ƭ_drum=596.818 ^ft/_lbs∙³⁰/₁₅=1193.636 ^ft/_lbs

Ƭ_req’d= r∙F= 6.375”∙433.611694#= 2764.274548 ⁱⁿ/_lbs ÷ 12= 230.356212 ^ft/_lbs       

 

F_accel= M∙a=^6000#/_32.2=186.335∙0.26625 ^ft/_sec²=49.611694#

F_friction=^N∙μr/_rcast=^{6000 #∙0.08}/_2.5=192∙2=384#

F_{to move wagon}=384# +49.611694#=433.611694#

F=^Ƭ/_r=14323.632ⁱⁿ/_lbs ÷ 6.375” = 2246.844#