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HNC CAD/CAM
SCIENTIFIC & ANALITYCAL METHODS
ASSIGNMENT 1
DAVID ANTUNA




                                  2010
David Antuna                                                                       HNC CAD/CAM

Table of Contents
                                                                                                    2

Summary: ...................................................................................... 3
Shear Force, Bending Moment & Stress for Beams: ......................... 4
Beam Calculation Software: ........................................................... 6
Shear Stress for Circular Shafts: ...................................................... 8
Friction & Acceleration: .................................................................. 9
Simple Harmonic Motion: ............................................................. 10
Appendix: .................................................................................... 11




Scientific & Analytical Methods Ass.1                                        ANT07078917
David Antuna                                                                      HNC CAD/CAM

Summary:
                                                                                           3
The aim of this report is to demonstrate the ability to apply and calculate the correct
formulas for different Scientific and Analytical scenarios like:

      Shear Force & Bending Moment for a Beam.
      The use of software for the calculation of the above.
      Shear Stress for solid shafts.
      Friction & Acceleration.
      Simple Harmonic Motion.




Scientific & Analytical Methods Ass.1                                       ANT07078917
David Antuna                                                                                                          HNC CAD/CAM

Shear Force, Bending Moment & Stress for Beams:
                                                                                                                                 4
        Determine distribution of shear force, bending moment and stress due to bending
         in simply supported beams.

       1. A cantilever of rectangular cross-section 74 mm wide and 110 mm deep carries a
          concentrated load of 3 kN at its free end. Neglecting the mass of the cantilever: Find
          the maximum length if the greatest permissible stress due to bending is 85 MPa.
          Draw the shear force and bending moment for the beam.

         74mm


                                                             BENDING EQUATION
                           110mm

                                                                    ������   ������
                                                                        = =
                                                                     ������  ������
When:

σ max=85 Mpa.
                                                                                 ������������³       ������.������������������������������.������������³
                                                                          ������ =           =
                                                                                 ������������               ������������
W= 3Kn
                                                                          I= 8.21x10¯⁶
       ������������³
������ =
        ������������

Y= 0.055m


        ������· ������������������������       ������.������������������������������¯������ ������ ������������������������������⁶
M=                     =
         ������������������������                  ������.������������������������¯²

M= 1.27x10⁴

                                        ������         ������.������������������������������⁴
M=WL there for L=                            =
                                        ������           ������������������������³

L= 4.23 m




Scientific & Analytical Methods Ass.1                                                                              ANT07078917
David Antuna                                                                      HNC CAD/CAM

   2. Find the maximum total uniformly distributed load which can be carried by a timber
      beam 300 mm deep, 200 mm wide and 3.5 m long which is simply supported at its          5
      ends. Neglect the mass of the beam itself and take the maximum stress due to
      bending to be 8 MPa. Draw the shear force and bending Moment for the beam.

                       200mm


                                        300mm
                                                                        3.5m


                                                        BENDING EQUATION

                                                              ������   ������
                                                                  = =
                                                               ������  ������
When:

σ= 8 Mpa = 8x10⁶ N/mm²

Y= 150x10¯³

       ������·������
M=
        ������

       ������������³          ������.������������������.������³
������ =            =                     = 4.50x10¯⁴
        ������������              ������������

L= 3.5 m

        ������             ������������        ������            ������������������������������
M= I            .·.           =I        .·. W=
        ������               ������        ������             ������������������

               ������.������������������������������¯⁴������������������������������⁶������������
.·. W=
                    ������.������������������������������������������������¯³

W= 5.49x10⁴ N




Scientific & Analytical Methods Ass.1                                          ANT07078917
David Antuna                                                                HNC CAD/CAM

Beam Calculation Software:
                                                                                          6
    Determine the sectional properties of structural sections.

   3. The below screen grabs are an example of how is possible to use different type of
      software to calculate Load, Shear Force & Bending Moments for any type of beam.
      In this case the software is SuperBeam 4.

Cantilever beam with a 3 KN concentrate load.




                      Figure 1 Use of Software to Calculate Beams




Scientific & Analytical Methods Ass.1                                  ANT07078917
David Antuna                                                           HNC CAD/CAM

UDL Timber Beam 3.5m long with a 8 MPa of bending stress.
                                                                                  7




                      Figure 2 Use of Software to Calculate Beams




Scientific & Analytical Methods Ass.1                               ANT07078917
David Antuna                                                                                                HNC CAD/CAM

 Shear Stress for Circular Shafts:
                                                                                                                       8
 Determine the distribution of shear stress and the angle and deflection due to torsion
  in circular shafts.

4. In a torsion experiment, a specimen is used whose shaft is 30 mm diameter. A torsion
   meter, which is clamped to the shaft shows that a 200 mm length of the shaft twists
   through an angle of 25° when a twisting moment of 400 Nm is applied.
   a. Find the value of the modulus of rigidity of the shaft material.
   b. Find the value of the maximum shear stress and hence draw a diagram showing the
   shear stress distribution over the cross section of the shaft.

                                                                R=0.015




                                                                                                     τ=




When:                                             TORSION EQUATION

                                                  ������       ������       ������������
R= 0.015m = 15x10¯³                                    =        =
                                                  ������       ������        ������
L= 200mm = 0.2m = 200x10¯³
             ������������������������
θ= 25° .·. θ= ������������������ = 0.436 rad
T= 400Nm
         ������������⁴            ������������������.������������⁴
������ =             = ������ =              =7.95x10¯⁸
         ������������                 ������������


       a. Modulus of Rigidity                                              b. Maximum Shear Stress

������        ������������                                                             ������     ������
     =                                                                          = ������
������         ������                                                              ������


                  ������������                                                                    ������������
.·. ������ =                                                                   .·. ������ =
                  ������������                                                                   ������
                                                                                   ������������������������������.������������������
                                                                           ������ =
                                                                                       ������.������������������������������¯⁸
                    ������������������������������.������
������ =                                                                       τ= 7.54x10⁷
           ������.������������������������������¯������ ������������.������������������

G= 438742138.4 N/m²
G= 4.38x10⁸ N/m²



Scientific & Analytical Methods Ass.1                                                                    ANT07078917
David Antuna                                                                         HNC CAD/CAM

Friction & Acceleration:
                                                                                                 9
 Determine the behaviour of dynamic mechanical Systems in which uniform
  acceleration is present.

5. A cast iron block of mass 30kg is pulled along a surface and accelerates at 0.3m/s2. If the
     coefficient of friction is 0.15N, and g=9.81m/s2.
     a) Find the necessary tractive force.
If the force in the last example is removed after 20 seconds, and the crate started moving
from rest.
     b) Calculate the final velocity, and how far it moved until the force was removed.

Sketch the crate as it accelerates and indicate all the forces acting upon it.

                                         W=30 Kg               a=0.3 m/s²




               FF=0.15 N                                 F=53.145 N



                                   FRICTION EQUATION
                                         F= FF+FI
                                       F= µMg+Ma

   a) Find the necessary tractive force.

F= FF+FI              FF=µFN
F= µMg+Ma                =µMg
                      FI =Ma

F= 0.15x30x9.81+30x0.3
F= 53.145 N

   a) Calculate the final velocity, and how far it moved until the force was removed.

                                        ������
V=u+at                           ������ = 2 (������ + ������)                     S= ut+½at²
                                        20
V=at           since U=0         ������ =        (6 + 0.3)                S= ½at²
                                        2
V=0.3x20                         S = 63m                              S= ½x0.3x20²
V=6 m/s        = 21.6 Km/Hr                                           S= 60m




Scientific & Analytical Methods Ass.1                                           ANT07078917
David Antuna                                                                   HNC CAD/CAM

Simple Harmonic Motion:
                                                                                               10

 Determine the behaviour of oscillating mechanical systems in which simple harmonic
  motion is present.

6. The crank of a scotch-yoke mechanism is 52 mm long and rotates at a uniform 250
   rev/mm. Find the slider's: (a) acceleration and velocity when 15 mm from the mid-point
   of its travel; (b) maximum acceleration, and state where it occurs; (c) maximum velocity,
   and state where it occurs.

                                                  ω




                            SIMPLE HARMONIC MOTION EQUATIONS

                                              X= aCosθ
                                           U= ������ ������² − ������²
                                               F= ω²X

When:
a= 52mm = 52x10¯³ m
                     250������2������
ω= 250 rev/min      = 60 rads/sec
X= 15mm = 15x10¯³ m

Find the Acceleration and Velocity.

            250������2������ 2
F= ω² X =                X 15x10¯³ .·. F= 10.28 m/s²
              60
                        250������2������ 2
U= ������ ������² − ������² =           52������10¯³ 2 − (15������10¯3 )² .·. U= 1.36 m/s²
                  60
Find the maximum acceleration, and state where it occurs./F=ω²X and occurs when x=a
               250������2������ 2
Fmax=ω²a =                  X 52x10¯³ .·. Fmax= 35.64 m/s²
                   60
Find maximum velocity, and state where it occurs. Occurs when X=0 From U= ������ ������² − ������²
            250������2������
Umax= ωa = ( 60 ) X 52x10¯³ .·. Umax= 1.36 m/s²




Scientific & Analytical Methods Ass.1                                     ANT07078917
David Antuna                                                                       HNC CAD/CAM

Appendix:
                                                                                              11
In this section you will find the diagrams for question 1 & 2 of this report.




Scientific & Analytical Methods Ass.1                                           ANT07078917

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Scientific & Analytical Methods Assignment 1

  • 1. HNC CAD/CAM SCIENTIFIC & ANALITYCAL METHODS ASSIGNMENT 1 DAVID ANTUNA 2010
  • 2. David Antuna HNC CAD/CAM Table of Contents 2 Summary: ...................................................................................... 3 Shear Force, Bending Moment & Stress for Beams: ......................... 4 Beam Calculation Software: ........................................................... 6 Shear Stress for Circular Shafts: ...................................................... 8 Friction & Acceleration: .................................................................. 9 Simple Harmonic Motion: ............................................................. 10 Appendix: .................................................................................... 11 Scientific & Analytical Methods Ass.1 ANT07078917
  • 3. David Antuna HNC CAD/CAM Summary: 3 The aim of this report is to demonstrate the ability to apply and calculate the correct formulas for different Scientific and Analytical scenarios like:  Shear Force & Bending Moment for a Beam.  The use of software for the calculation of the above.  Shear Stress for solid shafts.  Friction & Acceleration.  Simple Harmonic Motion. Scientific & Analytical Methods Ass.1 ANT07078917
  • 4. David Antuna HNC CAD/CAM Shear Force, Bending Moment & Stress for Beams: 4  Determine distribution of shear force, bending moment and stress due to bending in simply supported beams. 1. A cantilever of rectangular cross-section 74 mm wide and 110 mm deep carries a concentrated load of 3 kN at its free end. Neglecting the mass of the cantilever: Find the maximum length if the greatest permissible stress due to bending is 85 MPa. Draw the shear force and bending moment for the beam. 74mm BENDING EQUATION 110mm ������ ������ = = ������ ������ When: σ max=85 Mpa. ������������³ ������.������������������������������.������������³ ������ = = ������������ ������������ W= 3Kn I= 8.21x10¯⁶ ������������³ ������ = ������������ Y= 0.055m ������· ������������������������ ������.������������������������������¯������ ������ ������������������������������⁶ M= = ������������������������ ������.������������������������¯² M= 1.27x10⁴ ������ ������.������������������������������⁴ M=WL there for L= = ������ ������������������������³ L= 4.23 m Scientific & Analytical Methods Ass.1 ANT07078917
  • 5. David Antuna HNC CAD/CAM 2. Find the maximum total uniformly distributed load which can be carried by a timber beam 300 mm deep, 200 mm wide and 3.5 m long which is simply supported at its 5 ends. Neglect the mass of the beam itself and take the maximum stress due to bending to be 8 MPa. Draw the shear force and bending Moment for the beam. 200mm 300mm 3.5m BENDING EQUATION ������ ������ = = ������ ������ When: σ= 8 Mpa = 8x10⁶ N/mm² Y= 150x10¯³ ������·������ M= ������ ������������³ ������.������������������.������³ ������ = = = 4.50x10¯⁴ ������������ ������������ L= 3.5 m ������ ������������ ������ ������������������������������ M= I .·. =I .·. W= ������ ������ ������ ������������������ ������.������������������������������¯⁴������������������������������⁶������������ .·. W= ������.������������������������������������������������¯³ W= 5.49x10⁴ N Scientific & Analytical Methods Ass.1 ANT07078917
  • 6. David Antuna HNC CAD/CAM Beam Calculation Software: 6  Determine the sectional properties of structural sections. 3. The below screen grabs are an example of how is possible to use different type of software to calculate Load, Shear Force & Bending Moments for any type of beam. In this case the software is SuperBeam 4. Cantilever beam with a 3 KN concentrate load. Figure 1 Use of Software to Calculate Beams Scientific & Analytical Methods Ass.1 ANT07078917
  • 7. David Antuna HNC CAD/CAM UDL Timber Beam 3.5m long with a 8 MPa of bending stress. 7 Figure 2 Use of Software to Calculate Beams Scientific & Analytical Methods Ass.1 ANT07078917
  • 8. David Antuna HNC CAD/CAM Shear Stress for Circular Shafts: 8  Determine the distribution of shear stress and the angle and deflection due to torsion in circular shafts. 4. In a torsion experiment, a specimen is used whose shaft is 30 mm diameter. A torsion meter, which is clamped to the shaft shows that a 200 mm length of the shaft twists through an angle of 25° when a twisting moment of 400 Nm is applied. a. Find the value of the modulus of rigidity of the shaft material. b. Find the value of the maximum shear stress and hence draw a diagram showing the shear stress distribution over the cross section of the shaft. R=0.015 τ= When: TORSION EQUATION ������ ������ ������������ R= 0.015m = 15x10¯³ = = ������ ������ ������ L= 200mm = 0.2m = 200x10¯³ ������������������������ θ= 25° .·. θ= ������������������ = 0.436 rad T= 400Nm ������������⁴ ������������������.������������⁴ ������ = = ������ = =7.95x10¯⁸ ������������ ������������ a. Modulus of Rigidity b. Maximum Shear Stress ������ ������������ ������ ������ = = ������ ������ ������ ������ ������������ ������������ .·. ������ = .·. ������ = ������������ ������ ������������������������������.������������������ ������ = ������.������������������������������¯⁸ ������������������������������.������ ������ = τ= 7.54x10⁷ ������.������������������������������¯������ ������������.������������������ G= 438742138.4 N/m² G= 4.38x10⁸ N/m² Scientific & Analytical Methods Ass.1 ANT07078917
  • 9. David Antuna HNC CAD/CAM Friction & Acceleration: 9  Determine the behaviour of dynamic mechanical Systems in which uniform acceleration is present. 5. A cast iron block of mass 30kg is pulled along a surface and accelerates at 0.3m/s2. If the coefficient of friction is 0.15N, and g=9.81m/s2. a) Find the necessary tractive force. If the force in the last example is removed after 20 seconds, and the crate started moving from rest. b) Calculate the final velocity, and how far it moved until the force was removed. Sketch the crate as it accelerates and indicate all the forces acting upon it. W=30 Kg a=0.3 m/s² FF=0.15 N F=53.145 N FRICTION EQUATION F= FF+FI F= µMg+Ma a) Find the necessary tractive force. F= FF+FI FF=µFN F= µMg+Ma =µMg FI =Ma F= 0.15x30x9.81+30x0.3 F= 53.145 N a) Calculate the final velocity, and how far it moved until the force was removed. ������ V=u+at ������ = 2 (������ + ������) S= ut+½at² 20 V=at since U=0 ������ = (6 + 0.3) S= ½at² 2 V=0.3x20 S = 63m S= ½x0.3x20² V=6 m/s = 21.6 Km/Hr S= 60m Scientific & Analytical Methods Ass.1 ANT07078917
  • 10. David Antuna HNC CAD/CAM Simple Harmonic Motion: 10  Determine the behaviour of oscillating mechanical systems in which simple harmonic motion is present. 6. The crank of a scotch-yoke mechanism is 52 mm long and rotates at a uniform 250 rev/mm. Find the slider's: (a) acceleration and velocity when 15 mm from the mid-point of its travel; (b) maximum acceleration, and state where it occurs; (c) maximum velocity, and state where it occurs. ω SIMPLE HARMONIC MOTION EQUATIONS X= aCosθ U= ������ ������² − ������² F= ω²X When: a= 52mm = 52x10¯³ m 250������2������ ω= 250 rev/min = 60 rads/sec X= 15mm = 15x10¯³ m Find the Acceleration and Velocity. 250������2������ 2 F= ω² X = X 15x10¯³ .·. F= 10.28 m/s² 60 250������2������ 2 U= ������ ������² − ������² = 52������10¯³ 2 − (15������10¯3 )² .·. U= 1.36 m/s² 60 Find the maximum acceleration, and state where it occurs./F=ω²X and occurs when x=a 250������2������ 2 Fmax=ω²a = X 52x10¯³ .·. Fmax= 35.64 m/s² 60 Find maximum velocity, and state where it occurs. Occurs when X=0 From U= ������ ������² − ������² 250������2������ Umax= ωa = ( 60 ) X 52x10¯³ .·. Umax= 1.36 m/s² Scientific & Analytical Methods Ass.1 ANT07078917
  • 11. David Antuna HNC CAD/CAM Appendix: 11 In this section you will find the diagrams for question 1 & 2 of this report. Scientific & Analytical Methods Ass.1 ANT07078917