Velocity and Acceleration Diagrams

Flashcards covering the graphical construction and calculation of velocity and acceleration diagrams for four-bar link mechanisms and slider-crank mechanisms.

What key information can be obtained from a velocity and acceleration diagram of a mechanism?

The velocity and acceleration of any point relative to any other point for one particular position of the mechanism.

What types of mechanical transformations can a mechanism perform in a machine?

It can convert one speed to another speed, one force to another force, one torque to another torque, force to torque, one angular motion to another angular motion, angular motion to linear motion, and linear motion to angular motion.

What are the three sequential steps required to find the velocity and acceleration of a mechanism using the graphical method?

1. Sketch space diagram
2. Sketch velocity diagram
3. Sketch acceleration diagram

If an actual link length is $1500\,mm$ and the space diagram scale is $1\,cm : 50\,mm$, what length must be drawn in the space diagram?

$30\,cm$, calculated as $1500\,mm \times \frac{1\,cm}{50\,mm} = 30\,cm$.

What is the difference between absolute velocity and relative velocity?

Absolute velocity is measured from a fixed point (normally the ground or rigidly attached to the ground and not moving), whereas relative velocity is measured relative to another point that may itself be moving.

What is the direction of the relative velocity of one point on a rigid link relative to another point on the same link?

It must always be perpendicular ($90^\circ$) to the axis of the rigid link (tangential velocity).

What formula calculates the tangential velocity of point $B$ relative to pinned point $A$ on link $AB$ rotating at angular velocity $\omega$?

$v_b = \omega(AB)$ (also denoted as $(v_B)_A$).

In a velocity diagram, how are fixed points such as $A$ and $D$ represented?

Because there is no relative motion between fixed points, their relative velocity is zero and they are represented by a single coincident point ($a, d$).

In a four-bar mechanism velocity diagram, what proportion formula is used to locate an intermediate point $P$ on link $BC$?

$$\frac{bp}{bc} = \frac{BP}{BC}$$

How is the angular velocity of link $CD$ ($\omega_{cd}$) calculated from the velocity diagram?

$$\omega_{cd} = \frac{v_{cd}}{CD}$$

In velocity diagrams, how is the velocity of a piston or slider drawn and measured?

It is measured as radial velocity and drawn along the direction of motion (line of stroke) of the piston starting from the fixed point.

What two velocity components occur when a slider $C$ slides along a link $AB$ while the link rotates about point $A$?

Tangential velocity ($(v_c)_A\text{ Tangential}$, represented by vector $ac_1$) and radial velocity ($(v_c)_A\text{ Radial}$, represented by vector $c_1c_2$).

Why is it important to determine the acceleration of links in a mechanism?

Because acceleration produces inertia forces in the link which stress the component parts of the mechanism.

What are the two acceleration components of one point on a link relative to another?

The radial component (centripetal acceleration due to the angular velocity of the link) and the tangential component (due to the angular acceleration of the link).

What formulas are used to calculate the radial (centripetal) acceleration $(a_R)_{ab}$ of link $AB$?

$(a_R)_{ab} = \omega_{ab}^2(AB)$ or $(a_R)_{ab} = \frac{v_{ab}^2}{AB}$

What is the direction of the radial (centripetal) acceleration vector for a link $AB$ rotating about point $A$?

It acts parallel to the link and is directed towards the centre of rotation (towards point $A$).

Under what condition does tangential acceleration occur in a link, and how is it calculated?

It only occurs if the link has an angular acceleration ($\alpha\,rad/s^2$), and it is calculated using $(a_T)_{ab} = \alpha_{ab}(AB)$.

What is the orientation of the tangential acceleration vector relative to the rigid link and the velocity diagram?

It acts perpendicular to the link and parallel to the velocity diagram vector.

In the vector diagram shown, what does the resultant vector from $a$ to $b$ represent?

The true (resultant) acceleration of $B$ relative to $A$ ($(a_B)_A$), formed by vectorially adding radial acceleration $a_R$ ($ab_1$) and tangential acceleration $a_T$ ($b_1b$).

In Example 1, given crank $AB = 80\,mm$ rotating at a constant speed of $480\,rad/s$, what is the velocity $v_{ab}$?

$$v_{ab} = \omega_{ab} \times AB = 480 \times 0.08 = 38.4\,m/s$$

In Example 5, why is the tangential acceleration of the crank $(a_T)_{ab}$ equal to zero?

Because the crank rotates at a constant angular velocity ($\omega = 300\,rad/s$), meaning its angular acceleration is zero ($\alpha = 0$).

In Example 5, given $(a_T)_{bc} = 4000\,m/s^2$ and $BC = 0.17\,m$, what is the angular acceleration of link $BC$ ($\alpha_{bc}$)?

$$\alpha_{bc} = \frac{(a_T)_{bc}}{BC} = \frac{4000}{0.17} = 23529.41\,rad/s^2$$

Can an output link in a four-bar mechanism have an angular acceleration if the input link rotates at a constant speed?

Yes. Even if the input link rotates at constant speed ($\alpha = 0$), the varying geometry causes changing angular velocity in the follower link, resulting in an angular acceleration (such as $\alpha(CD) = 150000\,rad/s^2$ in Example 8).

What is the first step when constructing a velocity diagram for a slider-crank mechanism?

Determine and draw the piston's axis (line of stroke) and place the fixed point.