Transmission angle ofmechanism

Mechanism Design

Transmission angle ofmechanism

It would be useful to have a measure, criterion, orindex of how well a mechanisms might „run“*. Thetransmission angle is an important criterion for the

design of mechanisms by means of which thequality of motion transmission in a mechanism, at

its design stage can be judged.

*„Run“ is a term that more formally means the effectiveness with which motion is imparted to the output link

Definition:

• Transmission angle is the angle between the coupling member and the output member in a mechanism.

• The angle between the direction of absolute motion and direction of the relative motion of the point in which the driven member gets the motion impulse

Transmission angle of mechanism

ABSOLUTE MOTION: Motion with respect to a fixed frameRELATIVE MOTION: Motion with respect to a moving frame

The optimum value of the transmission angle is 90degree. Since the angle will be constantly changing duringthe motion cycle of the mechanism, there will be aposition at which the transmission angle will deviate mostfrom 90 degree. In practice it has been found out that ifthe maximum deviation of the transmission angle from 90degree exceeds 40 or 50 degree (depending on the typeof application), the mechanism will lock. In certain casesthis maximum deviation must be kept within 20 degreeand in certain other applications maximum deviations ofup to 70 degree may be permissible. One must considerthe practical application of a mechanism in order to give alimit to this deviation (whenever in doubt, try to keep thisdeviation to less that 40 or 50 degree).

Transmission angle of mechanism

Transmission angle of 4bar mechanism

We can express the transmission angle of 4barmechanism in terms of the crank angle and the linklengths as (by writing the cosine theorem for AB0

(lenght f) using the triangles A0AB0 and ABB0 andequating the length f=AB0).

Transmission angle of 4bar mechanism

The minimum and the maximum of the

transmission angle can be determined by taking

the derivative of the equation with respect to ψ

and equating to zero.

cos 𝜇 =𝑏2 + 𝑐2 − 𝑎2 − 𝑑2

2𝑏𝑐−

𝑎𝑑

𝑏𝑐cosψ

−sin 𝜇 𝑑𝜇 =𝑎𝑑

𝑏𝑐sinψ 𝑑ψ

using derivatives we can find the

slope of that function

Transmission angle of 4bar mechanism

The minimum and the maximum values of the transmission angle will be when (sinψ=0) or when ψ =0 or π (when the crank and the fixed link are collinear in extended or folded positions). The minimum and the maximum value of the transmission angle for the four-bar mechanism will be given by:

The minimum transmission angle occurswhen link 2 (crank) becomes collinear withlink 1 (ground link)

Transmission angle of 4bar mechanism

The critical transmission angle is either µmin or µmax, whichever deviates most from 90 degree.

Sometimes, for the transmission angles greater than 90 degree , instead of m (180 degree-m) is used

for the value of the transmission angle. In such a case, there are two minimum values of the

transmission angle ( mmin1=mmin, mmin2=180 degree-mmax) The most critical transmission angle is the

minimum of mmin1 and mmin2.

Transmission angle of 4bar mechanism

Transmission angle of 4bar mechanism

MSC.Adams – Creating a new Angle Measure1. Create new Angle Measure

2. Select Advanced function(Angle measure window opens)

3. Angle measure window

Transmission angle of crank mechanism

The transmission angle can be determined from the equation:

We can substitute:

We received:

Maximum deviation of the transmission angle occurs

when the derivative of µ with respect to ψ is zero.

Hence differentiating equation with respect to ψ :

cos 𝜇 =λ(sinψ + 𝑘)

−sin 𝜇 𝑑𝜇 =λ cosψ dψ

using derivatives we can find the

slope of that function

Transmission angle of crank mechanism

Maximum or minimum deviation occurs when

ψ is 90 degree or 270 degree and the value

of the maximum or minimum transmission

angle is given by:

Transmission angle

Instant centre of mechanism (I. C.) / Pole of mechanism

Instant centre of 4-bar mechanism.• Pin joint A – Instant centre 12 or 21 • Pin joint B – Instant centre 23 or 32• Pin joint C – Instant centre 34 or 43• Pin joint D – Instant centre 14 or 41

Link 2 can revolve with relativ to link 1 with point A as centre. Hence pin joint A becomes I.C. for links 1 and 2. Arguing in a similar way, it can be shownthat: Point B is I.C. for links 2 and 3,Point C is I.C. for links 3 and 4,Point D is I.C. for links 1 and 4.

In this 4-bar mechanism 12, 23, 34 and 14 are fixed I.C.

Instant centre / Pole of a kinematic link is the point about which the link of the mechanism is rotating at a particular instant.

Note: instant center is also called the pole, centro, instantenuous center of rotation, zero velocity point

Aronhold-Kennedy´s Theorem of three centres (Kennedy´s Rule)

Three bodies having relative motion with respect to one another, have three instant centres, all of which lie on the same straight line.

13=12+23, 14=13+34 (13=14+43)

24=23+34, 14=12+24 (24=21+14)

Instant centre of a kinematic link is the point about which the link of the mechanism is rotating at a particular instant. It may be fixed, permanent or variable.• Fixed IC is present when the link are in direct contact & one of the

links may be stationary/fixed.• Permanent IC is present when the link are connected through a pin

joint & are in relative motion. There relative position may be changing but the IC is always present at the centre of pin.

• Variable IC is present when the links are although in relative motion but they aren't in direct contact.

In this slider crank mechanism, 12 & 14 are fixed ICs while 23 & 34 are permanent ICs & can be easily found by visual inspection. But 24 & 14 are found by Kennedy's theorem.

Transmission angle – compound mechanism

Instant centre (pole) of absolute motion – member 5 (P51)

Transmission angle– compound mechanism

Transmission angle – compound mechanism

Transmission angle of crank mechanism

Literature: https://en.wikipedia.org/wiki/Instant_centre_of_rotation

Ashok G. Ambekar: Mechanism and machine theory. PHI Learning private Limited. Delhi. 2013