Forces, centre of gravity, reactions and stability

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Forces, centre of gravity, reactions and stability Topic areas Mecanical engineering: Centre of gravity

Matematics: Angles Trigonometric identities Prerequisites It may be useful to look at te resource Centre

Problem statement Many engineered objects, suc as buses, boats, diggers, puscairs, furniture, etc.

1 Forces, centre of gravity, reactions and stability Topic areas Mecanical engineering: Centre of gravity Forces Moments Reactions Resolving forces on an inclined plane. Matematics: Angles Trigonometric identities Prerequisites It may be useful to look at te resource Centre of gravity of composite bodies to introduce te concepts of te centre of gravity and tipping. Problem statement Many engineered objects, suc as buses, boats, diggers, puscairs, furniture, etc. are required to be resist to tipping over under normal use conditions. Ho can an engineer determine en an object ill tip over? Forces, centre of gravity, reactions and stability

acceleration due to gravity, usually taken to be 9.8 ms 2 at te eart s surface.

2 Background Te eigt of an object is te force due to gravity acting beteen te eart and te object, and is measured in netons (N). As te scale of te eart is large compared it many everyday objects, te eigt force acting on an object, F, can be related to its mass, m, troug te approximating expression F = mg, ere g is te acceleration due to gravity, usually taken to be 9.8 ms 2 at te eart s surface. Wen a force acts on a body te body ill accelerate in te direction of te force unless tere is a balancing force to oppose it. For an object sitting on a plane, tis balancing force is te normal reaction, R, ic acts normally and upards (or perpendicular) to te plane. If tere is friction beteen te object and te plane ic acts against any sliding tendencies, a frictional force acts upard along te plane. Activity Discussion Look at te folloing stationary block on a level plane (left) and an inclined plane (rigt). Figure Stationary blocks Discuss at forces are acting and dra force arros soing ere tey are acting. Wat assumptions do you make? 2 Forces, centre of gravity, reactions and stability

3 Activity 2 Stability Te resource.raeng.org.uk/tipping-, son in Figure 2, allos you to tilt an object it a uniform mass distribution and a mass of kg. It is assumed tat te friction beteen te object and te plane is large enoug to prevent te object from sliding. Figure 2 Screen sot of resource Te object can be tilted by dragging te slider, or clicking/tapping to buttons for fine control. Additionally, te object can be canged for a one of te same mass but different dimensions (te outline of te original square and original centre of gravity are also son). Investigate te angle at ic sape becomes unstable and ants to tip over. Wy does it tip over? 2 Repeat te above it sapes 2 and 3. Ho does te sape affect te position of te centre of gravity and at does tis do to te stability? Forces, centre of gravity, reactions and stability 3

Background Tipping angle Many engineered objects, suc as buses and puscairs, require a pysical tilt test to demonstrate adequate stability.

During te design pase, engineers ill ave a good idea as to te expected stability by considering te location of te centre of gravity and ere te limiting line of action lies.

Figure 3 An object on a roug plane at te tipping point Te diagram in Figure 3 sos a uniform block of idt and eigt, so tat te centre of gravity is at te centre, at te point of tipping.

4 Background Tipping angle Many engineered objects, suc as buses and puscairs, require a pysical tilt test to demonstrate adequate stability. Hoever, it ouldn t be ise to build someting as complex and expensive as a bus and ten ope it passes suc a test! During te design pase, engineers ill ave a good idea as to te expected stability by considering te location of te centre of gravity and ere te limiting line of action lies. Indeed, many CAD packages ill calculate te location of te centre of gravity. Figure 3 An object on a roug plane at te tipping point Te diagram in Figure 3 sos a uniform block of idt and eigt, so tat te centre of gravity is at te centre, at te point of tipping. Note, te plane is roug so te object does not slide. Te tipping point is reaced en te eigt force acts troug te corner of te object. At tis point, angles θ and ϕ sum to 90. θ + ϕ = 90 Te angle θ is te angle of te plane at ic te object tips, i.e. θ = 90 ϕ Te angle ϕ is te angle beteen te bottom edge and te line joining te corner and te centre of gravity, as son in Figure 0 in te Appendix (page ). Figure 4 Using trigonometry 2 = = 2 so tat te tipping angle is = 90. It can also be son tat (see Appendix) = =. 4 Centre of gravity of composite bodies

Activity 3 Finding te tipping angle Calculate te tipping angle for te tree objects considered in Activity 2 and compare te results it te predictions of te resource.raeng.org.uk/tipping-.

5 Activity 3 Finding te tipping angle Calculate te tipping angle for te tree objects considered in Activity 2 and compare te results it te predictions of te resource.raeng.org.uk/tipping-. Te dimensions of te objects are given in Table. Object Widt, (cm) Heigt, (cm) = Calculated tipping angle Tipping angle from interactive resource θ θ Table Stretc and callenge activity Te resource.grallator.co.uk/bus/bus.tml, son in Figure 5, allos you to tilt a bus tat can be configured as A double decker (it and itout upstairs passengers). A single decker Just te cassis Figure 5 Screen sot of resource Te up and don arros control te tilt angle of te bus. Te left arro at te bottom-rigt of te screen pulls out an options menu. Te large yello arro sos te eigt force acting from te centre of gravity of te bus. Pull out te options menu and turn on te options for soing te tipping line, as son in Figure 6. Centre of gravity of composite bodies 5

Figure 6 Te options menu A bus is made of a number of components suc as A strong cassis tat contains te eaviest items suc as engine, cooling system, controls, suspension, eels etc.

Use te options menu to select te double decker, single decker and cassis only configurations and discuss te position of te centre of gravity of te bus for eac.

6 Figure 6 Te options menu A bus is made of a number of components suc as A strong cassis tat contains te eaviest items suc as engine, cooling system, controls, suspension, eels etc. Te internal fittings suc as te seats. Te bodyork, ic is usually made of relatively tin metal seet. Use te options menu to select te double decker, single decker and cassis only configurations and discuss te position of te centre of gravity of te bus for eac. 2 Estimate te tipping angle for te tree different configurations. Remember tis ill appen en te eigt force acts outside te base of te object, ic, in tis case, is te eel. 3 Wat effect does alloing for upstairs passengers ave on te stability of te double decker configuration? (Ceck tis by turning on te Upstairs passengers option on te options menu.) 6 Centre of gravity of composite bodies

7 Notes and solutions Activity Figure 7 Force diagram It is stated tat te object is stationary, terefore tere sould be no net forces acting. On te left-and diagram, te eigt force mg acts vertically donards troug te centre of gravity of te object. Tis is balanced by an equal and opposite normal reaction force, R. Tis force acts beteen te base of te object and te plane and troug te centre of gravity. On te rigt-and diagram te eigt force mg acts vertically donards troug te centre of gravity of te object. Again tere is a normal reaction force, R, acting perpendicular to te plane. It is dran acting at te base of te object ere te line of action of te eigt force intersects te base. As te object is stationary, tere must also be a friction force acting along te plane to prevent sliding. Tis is also dran on te base of te object acting from te same point as te normal reaction. Activity 2 Stability Te folloing observations are made Tilting te object moves te line of action of te normal reaction and te eigt toards te corner. For angles up to 45 bot te eigt and te reaction fall itin te base of te object tere is no net moment about te loer corner of te object, and so it sos no tendency to tip, see Figure 8. Figure 8 Screen sot of resource for a stable configuration Centre of gravity of composite bodies 7

For angles above 45, te eigt force acts outside te base of te object. Hoever, te normal reaction cannot do tis it can only exert a force ere it is in contact it te surface.

8 For angles above 45, te eigt force acts outside te base of te object. Hoever, te normal reaction cannot do tis it can only exert a force ere it is in contact it te surface. Wen te normal reaction is concentrated at te corner and te eigt force acts outside te base of te object tere is a net turning moment and te object tips, see Figure 9. Figure 9 Screen sot of resource for an unstable configuration 2 Te flatter object (sape 2) as te same mass as te square object, but as it is flatter, as a loer centre of gravity. Te result of tis is tat te object can be tilted muc more before it tips over. Te narroer object (sape 3), also of te same mass, as its centre of gravity at te same eigt as te square block. Hoever, because it is narroer, te eigt acts outside te base at a loer angle, terefore te angle of tilt required before tipping over is less te square block. 8 Forces, Centre of centre gravity of gravity, of composite reactions bodies and stability

9 Activity 3 Finding te tipping angle Values of te tipping angles for te sapes in Table are son in Table 2. Object Widt, (cm) Heigt, (cm) = Calculated tipping angle Tipping angle from interactive resource θ θ (2 d.p.) (2 d.p.) (2 d.p.) (2 d.p.) 3 Table 2 A comparison sos tat te interactive resource gives larger values te calculation. For sapes 2 and 3 tis can be explained by te fact tat te resource only allos you to select integer values for te angle θ. Te nearest integer belo te calculated tipping angle ill not tip ile te nearest one above it ill definitely tip. For sape tere appears to be a disagreement. Tis is because te calculated value of θ represents te angle at ic te object is on te point of tipping. Tecnically, at tis angle te object ill not tip (as asserted by te interactive resource), oever, any infinitesimal increase in angle beyond tis ill lead to tipping. Hence θ is an upper limit before tipping starts. Stretc and callenge activity Te bus is constructed using a strong cassis tat contains te eaviest items suc as engine, cooling system, controls, suspension, eels etc. Tis as a centre of gravity reasonable close to te ground. On top of tis sits te coacork, including te seats. Tese add mass above te cassis and so raise te centre of gravity. Hoever, te coacork is ligt compared it te cassis so tat te centre of gravity remains relatively lo compared it te eigt of te bus, even en it is configured as a double decker. Te individual contributions from te cassis, loer body ork and upper body ork can be vieed by turning on te Composite body centre of gravity option. 2 Te bus ill tip at te point ere te line of action of te combined eigt passes troug te outer corner of te loer tyre. For te cassis only configuration, te tipping angle is estimated to be about For te single decker configuration te tipping angle is estimated to be about For te double decker configuration te tipping angle is estimated to be about Wen passengers are on te upper deck of a bus tey add significant eigt ic moves te centre of gravity upards. Activity 3 soed tat an object it a ig centre of gravity is less stable an object it a lo centre of gravity. Adding upstairs passengers (use te pull-out menu option to do tis) decreases te tipping angle to about 3.3. In te UK it is legislation tat a double decker ic is fully loaded on te top deck bus must not tip en tilted at an angle of 28 (cited ere.publications. parliament.uk/pa/cm98990/cmansrd/ /debate-.tml). Te bus in tis activity terefore passes! Forces, centre Centre of gravity, of gravity reactions of composite and stability bodies 9

Appendix Te main text states tat it can also be son tat te tipping angle, θ is related to te angle ϕ troug (refer to Figures 3 and 4). = =. Tis is an opportunity to use some angle formulae.

10 Appendix Te main text states tat it can also be son tat te tipping angle, θ is related to te angle ϕ troug (refer to Figures 3 and 4). = =. Tis is an opportunity to use some angle formulae. Approac Taking te gent of bot sides of θ + ϕ = 90 gives (θ + ϕ) = 90. Using te angle sum formula for ( ) + = 90 + = 90 Te rigt and side may be considered to be a problem a calculator ill give an error if you try to find te of 90. Hoever, if you look at a plot of θ, you ill see tat θ as θ 90. Rearranging te above + 90 = As 90 is infinitely large, any finite number divided by tis is zero, so tat + 90 = 0 = = = Substituting = = gives 0 Forces, Centre of centre gravity of gravity, of composite reactions bodies and stability

Approac 2 Consider te sape at a general angle as son in Figure 0.

Te disce from te corner to te centre of gravity is denoted d.

Its value is given by = 2 = (see solution for Activity 3, above). Using trigonometry x = d cos( + ).

11 Approac 2 Consider te sape at a general angle as son in Figure 0. Figure 0 Te orizontal disce from te corner to te line of action of te eigt force is denoted x in te diagram. Te disce from te corner to te centre of gravity is denoted d. Te angle θ is te tilt angle and ϕ is te angle beteen te bottom edge and te line joining te corner and 2 te centre of gravity. Its value is given by = 2 = (see solution for Activity 3, above). Using trigonometry x = d cos( + ). At te tipping point x = 0 as te line of action of te eigt passes troug te corner. Using tis and te angle sum formula for cos 0 = d cos( + ) 0 = cos( + ) ( d 0) 0 = cos cos sin sin sin sin = cos cos sin cos = cos sin = = i.e. = Forces, centre Centre of gravity, of gravity reactions of composite and stability bodies

Royal Academy of Engineering As te UK s national academy for engineering, e bring togeter te most successful and talented engineers for a sared purpose: to advance and promote excellence in

12 Royal Academy of Engineering As te UK s national academy for engineering, e bring togeter te most successful and talented engineers for a sared purpose: to advance and promote excellence in engineering. We ave four strategic callenges: Make te UK te leading nation for engineering innovation Supporting te development of successful engineering innovation and businesses in te UK in order to create ealt, employment and benefit for te nation. Address te engineering skills crisis Meeting te UK s needs by inspiring a generation of young people from all backgrounds and equipping tem it te ig quality skills tey need for a rearding career in engineering. Position engineering at te eart of society Improving public aareness and recognition of te crucial role of engineers everyere. Lead te profession Harnessing te expertise, energy and capacity of te profession to provide strategic direction for engineering and collaborate on solutions to engineering grand callenges. Royal Academy of Engineering Prince Pilip House, 3 Carlton House Terrace, London SWY 5DG Tel: +44 (0) raeng.org.uk Registered carity number Forces, centre of gravity, reactions and stability

dam wall Topic areas Prerequisites Problem statement Mechanical engineering: Mathematics: Forces Reaction Pressure Moments Algebra Areas and volumes

dam wall Topic areas Prerequisites Problem statement Mechanical engineering: Mathematics: Forces Reaction Pressure Moments Algebra Areas and volumes Force on a dam wall Topic areas Mechanical engineering: Forces Reaction Pressure Moments Mathematics: Algebra Areas and volumes Arithmetic Integration Prerequisites None for the main activity, however