First-Year Engineering Program

1

Roller Coaster Design Calculations

Goal•Learn a method for calculating roller

coaster velocities

First-Year Engineering Program

2

Initial Paper Design (8% of course grade)

IMPORTANT - Read project description document in your course package carefully.

Initial Paper Design:• Produce a sketch of your design with

dimensions• Produce a comprehensive set of velocities for

all segments of your design• Analysis of your calculations and design

First-Year Engineering Program

Φ 254

508

813

1219

914

914457

203 Φ 254

Sample Initial Design Sketch Top View

Front View

Sketch by Amanda Zook, Nick Persico, Scott Stultz, Dante Henderson from ENG183 Spring ’04(Modified for instructional purposes)

Dimensions are in mm

First-Year Engineering Program

4

Excel Calculations• An Excel spreadsheet has been provided for you in

the course website .• This spreadsheet is based on the Physics of

conservation of energy as described in the labs:The sum of kinetic, potential, and energy losses must be the same at all points on the roller coaster

• Three types of energy loss coefficients are included:• Frictional losses on straight track sections• Losses associated with speed (i.e. air resistance)• Losses associated with “G” forces (centripetal

acceleration – changed normal forces)

First-Year Engineering Program

Excel Spreadsheet for Calculations

5

PE1=Potential EnergyTKE = Translational KERKE = Rotational KEE = Total PE + KE

Set the loss coefficientsto the values shown forthis lecture’s examples.

First-Year Engineering Program

6

Calculations for each segment of the Roller Coaster (without curves/loops)

1. For the start of the segment identify the height and put this value in h1

2. For the end of the segment identify the height and put this value in h2

3. The difference between s1 and s2 is the track length of the segment

4. Enter the velocity coming into the segment as v1 (zero if this is the start of the track)

5. Adjust values of v2 until the two energy values in blue are equal

First-Year Engineering Program

For segments without curves/loops

7

Height of start Height of finish

Track length

Final velocity-Adjust until blue E’s are equal

Initial velocity

First-Year Engineering Program

8

Segment 1: Start to loop entry

Let’s calculate some of the velocities for this design

No loops or curves in this segment so:h1 = .762 mh2 = .254 mv1 = 0 m/secs1 = 0 ms2 = .683 m (determined by trigonometry, or measured)

First-Year Engineering Program

9

Plugging in the values h1,h2,s1,s2,v1

NOTE: These Energy’s are not equal

First-Year Engineering Program

10

Adjust v2 until blue E’s are equal

MUST BE CLOSE TO EQUAL

Final v2 = 2.4665 m/sec

First-Year Engineering Program

11

Segment calculations with curves/loops

• In addition to the h1, h2, v1, s2-s1 values there are other entries needed when centripetal acceleration is involved

• In the lower portion of the spreadsheet is a section for horizontal and vertical curves/loops

• Enter the diameter, Dh(horizontal) or Dv(vertical) curve/loop

• Adjust values of v2 until the two energy values in blue are equal

First-Year Engineering Program

12

Loops and curves

Step 3: Adjust final velocity until energies are equal

Step 1: Enterh1,h2,s1,s2,v1

Step2: EnterDh or Dv

First-Year Engineering Program

13

Vertical loops• For vertical loops the calculation needs to be

broken into two separate segments.• First perform a calculation for the velocity from

the entry of the loop to the top.• If velocity on the loop top is less than ,

where r is the radius of vertical loop, the ball cannot go through the loop.

• Then perform a subsequent calculation for the velocity from the top of the loop to the exit

rg ⋅

First-Year Engineering Program

14

Bump Calculations

• It is important to calculate the velocity of the ball when it reaches to top of the bump

• If its velocity exceeds , where r is the radius of curvature of the bump, the ball will leave the track

• To do this calculation break the bump into two ramps, one from the start to the top, and the other from the top to the end

rg ⋅

First-Year Engineering Program

15

In-class exercise:Calculate the velocity at the top of the loop

From drawing:h1 = ?h2 = ?s1 = ?s2 = ?v1 = ?Dv = ?

Plug into spreadsheetand iterate to determine:v2 final = ???

First-Year Engineering Program

16

In-class exercise : Answer

First-Year Engineering Program

17

Is the velocity high enough to stay on the top of the loop (with some extra for insurance)?

(gr)1/2 is 1.1156 m/s – so our 1.60 looks good

Now let’s walk through the next three segments of the roller coaster design.

First-Year Engineering Program

18

Top of Loop to Bottom

First-Year Engineering Program

19

Loop Summary

The velocity into the loop was: 2.4665 m/s

The velocity out of the loop was: 2.2883 m/s

Some energy was lost.

First-Year Engineering Program

20

Straightaway

NOTE: Small Loss

First-Year Engineering Program

21

Horizontal Loop

2 ½ horizontal loops

First-Year Engineering Program

Loss Coefficients• During the early roller coaster labs you will

determine values for the loss coefficients that you can use in your initial paper design.

• These do not account for all losses in a roller coaster design. Poor supports or steep drops can produce losses in addition to those calculated by this spreadsheet.

• Initial values for your design are given in the spreadsheet.

22

First-Year Engineering Program

23

Horizontal Loop Summary

• Velocity going in: 2.274 m/s

• Velocity going out: 1.983 m/s

• What kind of bank angles will be needed? r g

vtan(2

=)φ

φ

r

Φ in 76.5º, Φ out 72.4ºDo you think this will be stable?What could this team have done differently?

First-Year Engineering Program

24

Roller Coaster Design ConsiderationsOne of the most difficult things in creating an

interesting Roller Coaster design is controlling the velocity of the ball. If you think about the most fun roller coasters that you’ve ever been on, they’ve had changes in velocity rather than everything at one speed. Controlling the velocity of the ball makes performance less dependent on small changes between runs (and small defects in the track). This and a good support structure lead to a stable, reliable coaster design.

First-Year Engineering Program

25

Initial Paper Design• The Excel spreadsheet can be used for most

of the calculations required in the initial design. There are other calculations which will use the information you learned during the labs, i.e. • banking angle for horizontal curves• distance ball will land after leaving the end of the

track• etc.

• Read the project description document to understand all of the requirements!

First-Year Engineering Program

26

Brainstorming

Use the rest of this session to brainstorm ideas for your coaster with your team. Remember to include all of the required features. Remember that you need to submit calculations using the spreadsheet shown today for your entire initial design.