Acceleration in four-bar mechanism

ACCELERATION IN FOUR-BAR MECHANISM

Fig.2.13(a) snows a four-bar chain ABCD in which AD is fixed and BC is the coupler. AB is the driver rotating at an angular speed of o rad/s in the clockwise direction. It is required to draw the acceleration diagram of the configuration.

Procedure:

Step 1: Configuration diagram: First of all, draw the configuration diagram, to some suitable scale, as shown in Fig.2.13(a).

Step 2: Velocity of input link: When length of input link AB and its angular velocity ω_AB are known, then the velocity of input link (ie., crank AB) is given by

Step 3: Velocity diagram: Now draw the velocity diagram, as shown in Fig.2.13(b), using the procedure given in Section 2.6.

Step 4: Velocity of various links:

By measurement from the velocity diagram, we get v_CB = vector bc; and v_CD = vector dc.

The angular velocities of links BC and CD can be determined by using the relations

Step 5: Construction of acceleration diagram:

Using the velocity of various links that are obtained with the help of velocity diagram, the values of radial and tangential components of acceleration of various links can be calculated as shown in Table 2.2 below.

Table 2.2. Radial and tangential components of acceleration of various links

Now using the known values of magnitude and direction of acceleration components, the acceleration diagram can be constructed, as shown in Fig.2.13(c), to some suitable scale, using the procedure given below.

1. Since the link AD is fixed, therefore take a' and d' as one point.

2. From point a', draw vector a'b' such that a'b' = a^r_BA = v²_BA/AB in the direction parallel to BA to represent the radial component of acceleration of link AB (ie., a^r_BA). Since a^t_BA = 0, therefore a_BA = a^r_BA

3. From point b', draw vector b'x such that b'x = a^r_CB = v²_CB / BC in the direction parallel to CB to represent the radial component of acceleration of link BC (i.e., a^r_CB). Now from point x, draw vector xc' perpendicular to BC to represent the tangential component of acceleration of link BC (ie., a^t_CB) whose magnitude is unknown.

4. From point d', draw vector d'y such that d'y = d^r_CD/DC in the direction parallel to CD to represent the radial component of acceleration of link CD (i.e., d^r_CD). Now from point y, draw vector yc' perpendicular to CD to represent the tangential component of acceleration of link (i.e., d^t_CD) whose magnitude is unknown.

5. The vectors xc' and yc' intersect at c'. Join a'c' and b'c'.

Step 6: Acceleration of various links:

Now by measurement from the acceleration diagram, the various components of acceleration of links can be found.

Acceleration of crank AB = a_BA = a^r_BA = vector a'b'

Radial component of acceleration of link BC = a^r_BC = vector b'x

Tangential component of acceleration of link BC = a^t_BC = vector xc'

Total acceleration of link BC = a_BC = vector b'c'

Radial component of acceleration of link CD = a^r_CD = vector a'y

Tangential component of acceleration of link CD = a^t_CD = vector yc'

Total acceleration of link CD = a_CD = vector a'c'

Also the angular accelerations of links BC and CD can be determined as