Mechanism Design. Transmission angle of mechanism

Transcription

1 Mechanism Design Transmission angle of mechanism

2 Transmission angle of mechanism It would be useful to have a measure, criterion, or index of how well a mechanisms might run *. The transmission angle is an important criterion for the design of mechanisms by means of which the quality 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

3 Transmission angle of mechanism 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 ABSOLUTE MOTION: Motion with respect to a fixed frame RELATIVE MOTION: Motion with respect to a moving frame

4 Transmission angle of mechanism The optimum value of the transmission angle is 90 degree. Since the angle will be constantly changing during the motion cycle of the mechanism, there will be a position at which the transmission angle will deviate most from 90 degree. In practice it has been found out that if the maximum deviation of the transmission angle from 90 degree exceeds 40 or 50 degree (depending on the type of application), the mechanism will lock. In certain cases this maximum deviation must be kept within 20 degree and in certain other applications maximum deviations of up to 70 degree may be permissible. One must consider the practical application of a mechanism in order to give a limit to this deviation (whenever in doubt, try to keep this deviation to less that 40 or 50 degree).

5 We can express the transmission angle of 4bar mechanism in terms of the crank angle and the link lengths as (by writing the cosine theorem for AB 0 (lenght f) using the triangles A 0 AB 0 and ABB 0 and equating the length f=ab 0 ). Transmission angle of 4bar mechanism

6 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 + c 2 a 2 d 2 2bc ad bc cos ψ using derivatives we can find the slope of that function sin μ dμ = ad bc sin ψ d ψ

7 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 occurs when link 2 (crank) becomes collinear with link 1 (ground link)

8 Transmission angle of 4bar mechanism The critical transmission angle is either µ min or µ max, 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 ( m min1 =m min, m min2 =180 degree-m max ) The most critical transmission angle is the minimum of m min1 and m min2.

9 Transmission angle of 4bar mechanism

10 1. Create new Angle Measure Transmission angle of 4bar mechanism MSC.Adams Creating a new Angle Measure 3. Angle measure window 2. Select Advanced function (Angle measure window opens)

11 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(sin ψ + k) using derivatives we can find the slope of that function sin μ dμ = λ cos ψ dψ

12 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 of crank mechanism

13 Transmission angle

14 Instant centre of mechanism (I. C.) / Pole of mechanism Instant centre / Pole of a kinematic link is the point about which the link of the mechanism is rotating at a particular instant. 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 shown that: 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. Note: instant center is also called the pole, centro, instantenuous center of rotation, zero velocity point

15 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. 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. 13=12+23, 14=13+34 (13=14+43) 24=23+34, 14=12+24 (24=21+14) 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.

16 Transmission angle compound mechanism Instant centre (pole) of absolute motion member 5 (P 51 )

17 Transmission angle compound mechanism

18 Transmission angle compound mechanism

19 Transmission angle of crank mechanism Literature: Ashok G. Ambekar: Mechanism and machine theory. PHI Learning private Limited. Delhi. 2013