Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

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Mechanical Engineering : Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

The document Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Engineering Mechanics - Notes, Videos, MCQs & PPTs.
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Moment of a force about an axis

The moment of a force about an axis p is defined as the projection OB on p of the moment  (see Fig. 3.5).

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Figure 3.5: Moment of the force F about the axis p.

Denoting λ the unit vector of p, the moment Mp of force F about an axis p can be expressed as scalar product (dot product)

Mp = λ . Mo      (3.22)

or as the mixed triple product of the unit vector λ , the position vector r, and the force F:

Mp = λ . ( r *F)     (3.23)

Using the determinant form for the mixed triple product, we have the magnitude of the moment

Moment of a force about an axis

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev  (3.24)

λx,

where λyare direction cosines of axis p

λz

x, y, z are components of r

Fx,

Fy, Fz are components of F

Exercise 3.2.1

Sample problem A force F = 40 N is applied at a point M(xM;yM;zM) M(3;2;4) [m] (see Fig. 3.6). The line oF of action of the force F is described by direction angles (αFF;Cr ) (80o;60o; acute angle). An axis a which passes

through the origin O has direction angles (αa;βa;γa) ≡ (60o;100o; acute angle).

Determine:

  • The moment Mo of the force F about O.
  • The moments Mx, My, and Mz of the force F about axes x, y, and z, respectively.
  • The moment Mof the force F about a axis.

 Moment of a force about an axis

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Figure 3.6: Exercise 3.2.1. Moment of the force F about the axis a.

Notice: An acute angle is such an angle for which the condition 0 < α < 90o is valid. 

Solution

We determine the angle Cr first. Since the expression

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

is valid for any set of direction cosines, we have

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRevMoment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

According to the definition of the moment Mo of the force F, we may write

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRevMoment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRevMoment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Notice: Matlab works with radians and not with degrees. 

The magnitude of Mo is

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRevMoment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

88.06 Nm

The direction cosines of Mo are

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

As the components MOx, MOy, and MOz of Mo are equal to the moments Mx, My, and Mz of F about x, y, and z axes respectively, the following is valid:

Mx = MOx,    My = MOy,    Mz = MOz

According to the definition of the moment Ma of the force F about an axis a, we have

Ma = λT Mo

where the unit vector λ of a is

λ = [Cos αa, cosβa, cosγa]T

As

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

we conclude that

Ma = [0.5, 0.1736, 0.8484]  Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev  = 45.92 Nm

Using MATLAB the solution of the problem is much more convenient. The program for the purpose is called smoment.m. It can be found in program package.

Exercise 3.2.2 Moment of a force A force F = 50 N is applied at a point M(8;4;4) [m] and its line of action is described by direction angles (60o; 60o; acute angle). Determine the moments Mx, My, and Mz of the force F about the axes x, y, and z respectively and the moment Ma about the axis a passing through the point A(0;2;2) and having direction angles (30o;0o; acute angle).

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Figure 3.7: Exercise 3.2.2. Moment of the force F about the axis a.

Solution

Mx = 41.4 Nm, My = -182.8 Nm, Mz = 100 Nm, Ma = -98.47 Nm

Exercise 3.2.3 Moment of a force A force F = 50 N is applied at a point M(2;3;1) [m] and its line of action is described by direction angles (60o; 60o; acute angle). The axis a passes through the origin O and lies in xy-plane having the angle  δ= 30o with y axis. Determine

  • The moment Mo  of the force F about the origin O.
  • The moments Mx, My, and Mz of the force F about the axes x, y, and z respectively.
  • The moment Mof the force F about the axsis a.

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Figure 3.8: Exercise 3.2.3. Moment of the force F about the axis a. 

Solution 

Mx = 81.05 Nm, My = -45.7 Nm, Mz = -25 N m, MO = 96.34 Nm, Ma = -52.07 Nm 

Exercise 3.2.4 Moment of a force A force F = 40 N is applied at a point M(4;3;-2) [m] and its line of action is described by direction angles (30o; 90o; acute angle).
Determine the moment MA of the force F about the point A(-1;3;6) [m] and the moment Ma of the force F about the axis a passing through the point A and having direction angles (60o; 80o; acute angle).

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Figure 3.9: Exercise 3.2.4. Moment of the force F about the axis a.

Solution

MAx = 0 Nm, MAy = -377.13 Nm, MAz = 0 Nm, Ma = -65.49 Nm

Exercise 3.2.5 Moment of a force A tube is welded to the vertical plate yz at the point A. The tube is loaded by a force F at the point D. The line of action of the force  has an angle β with y-axis and an angle γ with z-axis. Determine the moment of the force

F about the point A. Further determine the moment Mx about x-axis and the moment MBC about BC axis of the same force F . How many solutions does the task have? It is known that  F = 200 N; β = 60o; γ= 80o;a = 0.2 m;b = 0.4 m;c = 0.8 m.

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Figure 3.10: Exercise 3.2.5. Moment of the force F.

Solution

MA = 152.2 Nm, Mx = - 66.1 Nm, MBC = 135.75 Nm 

Exercise 3.2.6 Moment of a force Determine the moment of the force F = 500 N about the axis AC. The angle  α = 30o, A(5;0;0), C(0;12;0), E(0;0;20), AB = 6.5, BD = 10 (all dimensions in m).

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Figure 3.11: Exercise 3.2.6. Moment of a force F.

Solution

MAC = 4510 Nm

Exercise 3.2.7 Tangent force A force T = 60 N acts at a tangent of a helix (see Fig.3.12). Find a generic expression for moments Mx, My, and Mz of the force T about x, y, and z axes respectively as a function of angle . Numerically compute the φ Numerically compute the magnitudes of these moments for φ =750o knowing that the radius of the helix is r =10 m and the pitch angle is  α =30o. Write a MATLAB program which can be used to calculate the above moments and use it for creating plots of moments values versus angle 0 ≤ φ ≤ 6π.

Moment of Force about a Specific Axis Mechanical Engineering Notes | EduRev

Figure 3.12: Exercise 3.2.7. Tangent force. 

Solution

Mx = 2222 Nm, My = -519.6 Nm, Mz = 3248 Nm

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