**6.2. ****STATIC FORCE ANALYSIS OF MACHINERY**

**6.2.1 ****Systems without Resisting Force**

Initially, we shall neglect the forces associated with the inertia of the links and the resisting forces. Later, we shall include the resistive forces and consider the inertia effects.

Static force analysis makes direct use of static force equilibrium equations. For an analytical solution formulation one must draw the free-body diagram of each rigid body and must identify the unknown forces. Next, one must write the static force equilibrium equations for each link. If a correct formulation is performed, the resulting equations will be equal to the number of unknown force components and these equations will be linear in terms of the unknowns. Let us illustrate this by a simple four-bar example shown below:

The system is in equilibrium under the action of the external force F_{14} and T_{12}. The magnitude and direction (h) of the force F_{14 }are known. We are to determine T_{12} and the forces acting at the joints. In free-body diagrams of the moving links are
shown.

On the free-body diagrams the unknown joint force components (F_{ijx} and F_{ijy}) are all shown acting in the positive x and y directions. a negative value is obtained in the result, it means that force component is in the opposite direction. In the example shown the equilibrium equations and the equalities due to Newton's third law (F_{ijx} = -F_{jix} ) result with 13 scalar equations in 13 unknowns (T_{12},F_{32x}, F_{32y}, F_{23x}, F_{23y}, F_{43x}, F_{43y}, F_{34x}, F_{34y}, G_{12x}, G_{12y} , G_{14x}, G_{14y}).

For link 2:

F_{32y} + G_{12y} = 0 |
(SF_{y}= 0) |
(1) |

F_{32x} + G_{12y} = 0 |
(SF_{x}= 0) |
(2) |

F_{32y}a_{2}cos(q_{12}) - F_{32x}a_{2}sin(q_{12}) + T_{12}= 0 |
(SM_{A0}= 0) |
(3) |

For link 3:

F_{23x} + F_{43x} = 0 |
(SF_{x}= 0) |
(4) |

F_{23y} + F_{43y} = 0 |
(SF_{y}= 0) |
(5) |

F_{43x}a_{3}sin(q_{13}) + F_{34y}a_{2}cos(q_{13})= 0 |
(SM_{A}= 0) |
(6) |

For link 4:

F_{34x} + G_{14x}-F_{14}cos(h)= 0 |
(SF_{x}= 0) |
7) |

F_{34y} + G_{14y}-F_{14}sin(h) = 0 |
(SF_{y}= 0) |
(8) |

F_{34x}a_{4}sin(q_{14}) + F_{34y}a_{4}cos(q_{14})+ F_{14}r_{4}(cos(h)sin(q_{14})-sin(h)cos(q_{14})) 0 |
(SM_{B0}= 0) |
(9) |

and due to action-reaction (Newton’s third law):

F_{32y } = - F_{23y} |
(10) |

F_{32x } =- F_{23x } |
(11) |

F_{43y } = - F_{34y } |
(12) |

F_{43x } = - F_{34x } |
(13) |

If the displacement analysis is performed beforehand, e.g. if q_{13}, q_{14} values are determined for different q_{12} values, the above equations can be used for every position of the mechanism. Another important point is that this formulation is very general. It can be used for mechanisms where there are several external forces acting on the same or different links and in cases where we have resisting forces at the joints. However, the resulting number of equations is usually too large (if there are* l* links in the mechanism there will be 3(*l*-1) equilibrium equations).

One can reduce the number of equations to be solved if the free-body diagrams are analysed to some detail. One need not write the forces in terms of its x and y components if the direction is known and one can identify the forces that are of equal magnitude before attempting for a solution. The free-body diagrams of the links in the four-bar mechanism are redrawn below.

In this case to simplify the calculations we note that **F _{13} = - G_{12} **

Hence:

**F _{43}= -F_{32} =-F_{34}= -G_{12}= -F_{23}**

Now, one can solve for the unknown forces if we write the 3 equilibrium equations for link 4 and one moment equilibrium equation for link 2, which are

- F_{14} cosh + F_{34}cosq_{13}+ G_{14} cosf = 0 or F_{14} cosh + F_{34}cosq_{13}+ G_{14x}= 0

- F_{14} sinh + F_{34}sinq_{13}+ G_{14} sinf = 0 or F_{14} cosh + F_{34}cosq_{13}+ G_{14y}= 0

r_{4}F_{14}(cosh sinq_{14} - sinh cosq_{14})+a_{4}F_{34}(sinq_{13} cosq_{14}- cosq_{13}sinq_{14}) = 0

And

a_{2}F_{32}(cosq_{13}sinq_{13}-sinq_{13} cosq_{12}) + T_{12} = 0
(**F _{32} = - F_{34}**)

There are four equations with four unknowns (F_{34}, T_{12}, G_{14},f or F_{34}, T_{12}, G_{14x}, G_{14y}). If the magnitudes come out negative, the direction of the force or torque is opposite to that indicated on the free-body diagram.
If we generalise the above result, we can see that:

a) For two-force members you don't have to write the equilibrium equations. You can simply state that the forces are equal and opposite and their line of action coincides with the line joining the points of applications.

b) For two-force plus a moment members you must write the moment equilibrium equation only. The two forces are equal and opposite and they form a couple equal and opposite to the moment applied.

c) In case of three or four force members, the three equilibrium equations (SF_{x} = 0, SF_{y} = 0, SM= 0) must be written.

Another simplification in the solution of moment equilibrium equations will result if we use unit vector in the direction of the forces and write a force vector in the form

**F** = F**v**

Where F is the magnitude of **F** and **v** is a unit vector in the direction of the force **F**.

We can similarly show the position vector **r** as :

**r** = r**u**

Where r is the distance from point C to a point on the line of action of **F** and u is a unit vector in the direction of **r**.
The moment of the force **F** about an axis passing through 0 and perpendicular to the plane will be:

** M** = **rxF** = r**u**xF**v** = rF(**uxv**)

Since

** u** = 1 <a = cos a ** i **+ sin a **j** ;

**v **= 1 < b = cos b i + sinb j

**u x v **= (sin b cos a - cos b sin a ) k

Or:

**u x v **= sin (b- a ) **k**

Hence
**M** = **rxF** = r F sin (b - a ) k

Now, the free-body diagrams of the previous example are redrawn

The moment equilibrium equations are:

a_{4}**u**_{4} x F_{34}**u**_{3} + r_{4}**u**_{4} x (-F_{14})**v**_{1 } (S**M**_{B0} = 0 )

Which result in:

a_{4 } F_{34 } sin(q_{13 }-q_{14 }) - r_{4 }F_{14 } sin( h-q_{14 }) = 0

-a_{2 } F_{32 } sin(q_{13 }-q_{12 }) + T_{12 } =0

One can see that the above equations are exactly the same equations that we have obtained previously.

If we were to solve the equilibrium equations graphically, note that link 4 is a three-force member and the forces must be concurrent. Since the line of action of the force F_{34 } and both the magnitude and direction of F_{14 } are known, the point of concurrency is at the intersection of the lines of action of these two forces Hence the line of action of G_{14 } is the line joining B_{0 } with the point of concurrency. Next, one can draw the force polygon:

** F _{14 }+G_{14 }+F_{34 }=0**

We want to determine the magnitudes of the forces **G _{14}** and

As another example, consider the slider-crank mechanism shown

. The system is in static equilibrium for all crank angles under the action of a known force F_{14 } acting on link 4 and unknown torque T_{12 } acting on link 2. We are to determine the joint forces and the input torque, T_{12 }.

The force **G _{14}** can be placed anywhere on link 4. Since link 4 is in static equilibrium, this will change the magnitude of the moment

(a) Point of Concurrency is within the Joint Boundary (b) Point of Concurrency Outside Joint Boundary

Physically, the joint forces must occur due to action and reaction between two contacting surfaces on two different links. In cases as shown in figure (b), since the forces acting on link 4 tends to rotate link 4, if we assume complete rigidity of the links, links 1 and 4 will be in point contact at the two edges of the boundaries as shown At the contact points There will be joint forces G’14 and G”14 whose directions are perpendicular to the slider axis (actually these forces will be perpendicular to the common tangent between the two contacting surfaces. In normal machine systems since the clearance at the joint is small compared to the dimensions of the joint, the angular rotation is negligible and this assumption is valid).

Also note that:

**G _{14} = G’_{14} + G”_{14} ** (In terms of magnitudes, G

G "_{14}d=G'_{14}b

G_{14} can be of a very small magnitude while the actual contact forces G’_{14} and G”_{14} can be of considerable magnitudes.

If one is only concerned with the force equilibrium, any one of the equilibrium diagrams for link 4 would be correct and valid. However, if one is interested with the actual joint forces, the type of contact and the actual magnitudes of the forces at the point(s) of contact must be determined. This will also have an important role when friction in these joints must be taken into account. The magnitudes of G’_{14} and G”_{14} will depend on the on the geometry of the joint as well as the magnitude and the direction of the other forces acting on the link. Also note that within one cycle of the mechanism the type contact between links 1 and 4 will change. There will be four different modes of contact as shown (G_{14}= - G_{41}). Note that the magnitudes of all the forces and the dimensions must all be positive for all the four modes. Otherwise the mechanism is not in the assumed mode of contact.

Modes of Contact for a Prismatic Joınt

**Example 6.1.**

B_{0}A = 158.75 mm; q_{14}= 49.11^{0} B_{0}C = 285.13 mm; q_{15} = 169.55^{0}

The free-body diagram of each link is shown above.. We note that links 3 and 5 are two-force and link 2 is a two-force plus a moment member. For link 6 if we assume that there is a single joint force between links 1 and 6, we note that the point of concurrency and the resultant joint force, G_{16} is outside the boundaries of the points of contact and that it is of mode is III .

We have:

**F _{56} = - F_{65} = F_{45} = -F_{54}**

**F _{34} = - F_{43} = F_{23} = -F_{32} = G_{12}**

** F _{16} **= -100 i (N)

**G” _{16}** = G”

Force equilibrium of link 6 results:

(G”_{16}- G’_{16})**j** - F_{56}** u _{5}** - 100 j = 0

or

(G”_{16}- G’_{16})- F_{56} sin(169.95^{0}) = 0

- F_{56} cos(169.95^{0}) -100 = 0

The moment equilibrium equation (SM_{C}= 0) yields (the distance from point B to the two boundaries are 70 and 30 mm respectively. These distances will be a function of the mechanism position):

G’_{16}*70 +G”_{16}*30 -100*20=0

We obtain:

**F _{56}**

Since link 5 is a two force member **F _{54}=-F_{45}=F_{65}=-F_{56}**=107.56 N < 169.95

The force equilibrium equation for link 4 is:

F_{54}**u _{5}** + G

Where

** v _{4} **= 1 < f= cosf i + sinf j

The force equilibrium equations can also be written as:

F_{54} cos 169.95^{0} + G_{14} cosf + F_{34} cos 40.89^{0} = 0

F_{54} sin l69.95^{0} + G_{14} sin f-F_{34} sin 40.89^{0} = 0

The moment equilibrium equation (SM_{B} = 0) is:

a_{4}**u _{4}**x F

Where:

u

and

B

When the moment equilibrium equation is simplified:

a_{4} F_{54} sin(q_{15}-q_{14} + r_{4} F_{34} sin(q_{14}-90^{0} -q_{14}) =0

or:

a_{4} F_{54} sin(120.44^{0}) - r_{4} F_{34} = 0

For link 2 :

**F _{32}**

And the moment equilibrium equation yields:

T_{12} + 120 * 33.14 sin(139.11^{0 }-30^{0 }) = 0

or

T_{12} = -3757.65 N-mm

or

T_{12} = 3.76 N-m (CW)

**Example 6.2.**

Consider the pneumatic press shown. It is a six link mechanism. The force F_{16} applied by the work piece onto the ram (link 6) is brought to equilibrium by the pneumatic pressure applied to the piston and cylinder.

Consider the free-body diagrams of the moving links. When link 6 is considered as a three-force member, the point of concurrency is point C, which lies outside the sliding joint boundaries. G_{16} is a fictitious force, which is the resultant of G'_{16} and G"_{16}. We can write :

**F _{56}+F_{16}+G_{16}**

G'_{16}-G"_{16} =G_{16}

aG'_{16}=cG"_{16}

When **F _{16}** and the position of the mechanism is known, the above equations can be used to determine

When we consider link 5, since there is a ternary joint at B between links 4,5 and 3, we must know how this joint is constructed.

As shown above, either the pin can be a rigid part of one of the links (say link 3) and the other links (4 and 5) are connected to this link (in this case there is no joint between links 4 and 5), or the pin is a separate free body to which all the three links are connected. It is a common practice to consider the free-body diagram of the pin when there is a joint of degree greater than one. The pin B will be under the action of the three concurrent forces **F _{3B}, F_{4B}, F_{5B}** (

(a) (b)

When we have the piston cylinder arrangement, instead of considering the free body diagrams of the piston and the cylinder separately, it is more convenient to draw the free-body diagram of the two links (piston and cylinder) together, as shown in (a). The hydraulic pressure is an internal force which is in equilibrium and the only external force acting on the system are the two joint forces **G _{12}** and