Understanding Bending Moment Calculations in Mechanical Systems

[Setareh7796]

Homework Statement

Finding the bending moment caused by force P at point A

Relevant Equations

My incorrect solution: P multiplied by perpendicular distance from point A so the equation becomes=M=P*b2

But the correct solution is M= P* (b2+b1). I don't understand how b2+b1 is the perpendicular distance.

 

[haruspex]

But the correct solution is M= P* (b2+b1).

The question is ambiguous. Strictly speaking, it is given by the vector equation, $\vec M=(\vec b_1+\vec b_2+\vec b_3)\times \vec P$. To see this, blank out everything in the diagram except point A and the force P.
But if they mean the net torque at A after including the reaction from the sleeve enclosing the crankshaft then your answer is correct.

 

[etotheipi]

But if they mean the net torque at A after including the reaction from the sleeve enclosing the crankshaft then your answer is correct.

@haruspex I wondered if you could explain this last part? I did look at this question but couldn't see how the torque of P about A could be anything other than $(\vec{b}_1 + \vec{b}_2 + \vec{b}_3) \times \vec{P}$. Thanks!

 

[haruspex]

@haruspex I wondered if you could explain this last part? I did look at this question but couldn't see how the torque of P about A could be anything other than $(\vec{b}_1 + \vec{b}_2 + \vec{b}_3) \times \vec{P}$. Thanks!

It is not shown, but in any sensible arrangement the shaft at A would be enclosed in a supporting sleeve so that it can only rotate about the X axis.
For the purposes of analysing the effectiveness of the mechanism, the only torque of interest is the torque about that axis. Any other torques produced by P would be balanced by reaction from the sleeve.

The book answer, if quoted correctly, is bizarre, so has it or the problem statement been misquoted? Maybe the original said "about the X axis". Or maybe it did use vector notation. But neither makes the book answer correct.

 

[etotheipi]

It is not shown, but in any sensible arrangement the shaft at A would be enclosed in a supporting sleeve so that it can only rotate about the X axis.
For the purposes of analysing the effectiveness of the mechanism, the only torque of interest is the torque about that axis. Any other torques produced by P would be balanced by reaction from the sleeve.

Ah right, thanks for clarifying .

The book answer, if quoted correctly, is bizarre, so has it or the problem statement been misquoted? Maybe the original said "about the X axis". Or maybe it did use vector notation. But neither makes the book answer correct.

I stared at it for quite a while and couldn't make any sense of it. The quantity $b_1 + b_2$ seems slightly irrelevant. Perhaps there are some details missing, or maybe it's just a mistake!

 

[Setareh7796]

It is not shown, but in any sensible arrangement the shaft at A would be enclosed in a supporting sleeve so that it can only rotate about the X axis.
For the purposes of analysing the effectiveness of the mechanism, the only torque of interest is the torque about that axis. Any other torques produced by P would be balanced by reaction from the sleeve.

The book answer, if quoted correctly, is bizarre, so has it or the problem statement been misquoted? Maybe the original said "about the X axis". Or maybe it did use vector notation. But neither makes the book answer correct.

The full question is:
For the purpose of analysis, a segment of a crankshaft in a vehicle is presented as shown in Figure Q5. The load P = 1 kN, and the dimensions are b1 = 80 mm, b2 = 120 mm and b3 = 40mm. The diameter of the shaft is d = 20 mm. Determine the maximum tensile, compressive and shear stresses at point A, located on the surface of the shaft at the z-axis

I forgot to mention point A is located on the z axis.
The first step to the full solution is to find the bending moment at point A and according to the correct solution bending moment is M= P *( b1+b2).

 

[haruspex]

The full question is:
For the purpose of analysis, a segment of a crankshaft in a vehicle is presented as shown in Figure Q5. The load P = 1 kN, and the dimensions are b1 = 80 mm, b2 = 120 mm and b3 = 40mm. The diameter of the shaft is d = 20 mm. Determine the maximum tensile, compressive and shear stresses at point A, located on the surface of the shaft at the z-axis

I forgot to mention point A is located on the z axis.
The first step to the full solution is to find the bending moment at point A and according to the correct solution bending moment is M= P *( b1+b2).

That makes a big difference.
If we are concerned with the different kinds of stress on the mechanism then the moments about different axes can be considered separately. The useful working torque about the crankshaft will be associated with shear stress, while torque about the vertical axis will create a bending moment.
That reduces the book error to a simple typo: bending moment should be P(b₁+b₃).