Chapter 15.7, Problem 15.245P

Two disks, each of 130-mm radius, are welded to the 500-mm rod CD. The rod-and-disks unit rotates at the constant rate ${\omega }_2=3$ rad/s with respect to arm AB. Knowing that at the instant shown ${\omega }_1=4$ rad/s, determine the velocity and acceleration of (a) point E, (b) point F.

Fig.P15.254

To determine

(a)

Velocity and acceleration of point E

Answer to Problem 15.245P

The velocity $v_E$ of point E is $-\left(0.61m/s\right)k$.

The acceleration $a_E$ of point E is $-\left(0.88m/s^2\right)i-\left(1.17m/s^2\right)j$.

Explanation of Solution

Given information:

The relative angular velocity of disk and rod with respect to arm AB is ${\omega }_2$.

The angular velocity of arm AB is ${\omega }_1$.

The velocity in a three dimensional motion of a particle relative to rotating frame is defined as:

$v_P=v_{P^1}+v_{P/F}$

In the above equation,

$v_P$ - Absolute velocity of particle P.

$v_{P^1}$ - Velocity of point $P^1$ of moving frame F Coinciding with P.

$v_{P/F}$ - Velocity of P relative to moving frame F.

The acceleration in a three dimensional motion of a particle relative to rotating frame is defined as:

$a_P=a_{P^1}+a_{P/F}+a_C$

In the above equation,

$a_P$ - Absolute acceleration of particle P.

$a_{P^1}$ - Acceleration of point P¹ of moving frame F coinciding with P

$a_{P/F}$ - Acceleration of P relative to moving frame F

$a_C$ -The Coriolis acceleration

The Coriolis acceleration is defined as

$a_C=2\Omega \times v_{P/F}$

Calculation:

The relative position vector $r_{E/B}$ of point E with respect to point B

$r_{E/B}=0.25i+0.13j$

The relative position vector $r_{E/D}$ of point E with respect to point D

$r_{E/D}=0.13j$

Assume Bxyz as the rotating frame of reference.

The constant angular velocity $\Omega$ of reference frame

$\Omega ={\omega }_{1}j=4j$

The relative angular velocity ${\omega }_2$ of disk and rod with respect to the frame

${\omega }_2={\omega }_{2}i=3i$

Assume $E^1$ as the point on frame that coincides with point E.

The velocity $v_{E^1}$ of point $E^1$

$\begin{array}{l}{v}_{E^1}=\Omega \times r_{E/B}\\ =4j\times \left[0.25i+0.13j\right]\\ =-1k\end{array}$

The relative velocity $v_{E/F}$ of point E with respect to frame

$\begin{array}{l}{v}_{E/F}={\omega }_2\times r_{E/D}\\ =3i\times \left[0.13j\right]\\ =0.39k\end{array}$

The velocity $v_E$ of point E

$\begin{array}{l}{v}_E=v_{E^1}+v_{E/F}\\ =-1k+0.39k\\ =-\left(0.61m/s\right)k\end{array}$

The acceleration $a_{E^1}$ of coinciding point $E^1$

$\begin{array}{l}{a}_{E^1}=\Omega \times v_{E^1}\\ =4j\times \left(-1k\right)\\ =-4i\end{array}$

The relative acceleration $a_{E/F}$ of point E with respect to frame

$\begin{array}{l}{a}_{E/F}={\omega }_2\times v_{E/F}\\ =3i\times 0.39k\\ =-1.17j\end{array}$

The Coriolis acceleration

$\begin{array}{l}2\Omega \times v_{E/F}=2\left(4j\right)\times \left(0.39k\right)\\ =3.12i\end{array}$

The acceleration $a_E$ of point E

$\begin{array}{l}{a}_E=a_{E^1}+a_{E/F}+2\Omega \times v_{E/F}\\ =-4i-1.17j+3.12i\\ =-\left(0.88m/s^2\right)i-\left(1.17m/s^2\right)j\end{array}$

Conclusion:

The velocity $v_E$ of point E is $-\left(0.61m/s\right)k$.

The acceleration $a_E$ of point E is $-\left(0.88m/s^2\right)i-\left(1.17m/s^2\right)j$.

To determine

(b)

Velocity and acceleration of point F

Answer to Problem 15.245P

The velocity $v_F$ of point F is $\left(0.52m/s\right)i-\left(0.39m/s\right)j-\left(1m/s\right)k$.

The acceleration $a_F$ of point F is $-\left(4m/s^2\right)i-\left(3.25m/s^2\right)k$.

Explanation of Solution

Given information:

The relative angular velocity of disk and rod with respect to arm AB is ${\omega }_2$.

The angular velocity of arm AB is ${\omega }_1$.

The velocity in a three dimensional motion of a particle relative to rotating frame is defined as:

$v_P=v_{P^1}+v_{P/F}$

In the above equation,

$v_P$ - Absolute velocity of particle P.

$v_{P^1}$ - Velocity of point $P^1$ of moving frame F Coinciding with P.

$v_{P/F}$ - Velocity of P relative to moving frame F.

The acceleration in a three dimensional motion of a particle relative to rotating frame is defined as:

$a_P=a_{P^1}+a_{P/F}+a_C$

In the above equation,

$a_P$ - Absolute acceleration of particle P.

$a_{P^1}$ - Acceleration of point P¹ of moving frame F coinciding with P

$a_{P/F}$ - Acceleration of P relative to moving frame F

$a_C$ -The Coriolis acceleration

The Coriolis acceleration is defined as

$a_C=2\Omega \times v_{P/F}$

Calculation:

The relative position vector $r_{F/B}$ of point F with respect to point B

$r_{F/B}=0.25i+0.13k$

The relative position vector $r_{F/D}$ of point F with respect to point D

$r_{F/D}=0.13k$

Assume Bxyz as the rotating frame of reference.

The constant angular velocity $\Omega$ of reference frame

$\Omega ={\omega }_{1}j=4j$

The relative angular velocity ${\omega }_2$ of disk and rod with respect to the frame

${\omega }_2={\omega }_{2}i=3i$

Assume $F^1$ as the point on frame that coincides with point F.

The velocity $v_{F^1}$ of point $F^1$

$\begin{array}{l}{v}_{F^1}=\Omega \times r_{F/B}\\ =4j\times \left[0.25i+0.13k\right]\\ =0.52i-1k\end{array}$

The relative velocity $v_{F/F}$ of point F with respect to frame

$\begin{array}{l}{v}_{F/F}={\omega }_2\times r_{F/D}\\ =3i\times \left[0.13k\right]\\ =-0.39j\end{array}$

The velocity $v_F$ of point F

$\begin{array}{l}{v}_F=v_{F^1}+v_{F/F}\\ =0.52i-1k-0.39j\\ =\left(0.52m/s\right)i-\left(0.39m/s\right)j-\left(1m/s\right)k\end{array}$

The acceleration $a_{F^1}$ of coinciding point $F^1$

$\begin{array}{l}{a}_{F^1}=\Omega \times v_{F^1}\\ =4j\times \left(0.52i-1k\right)\\ =-4i-2.08k\end{array}$

The relative acceleration $a_{F/F}$ of point F with respect to frame

$\begin{array}{l}{a}_{F/F}={\omega }_2\times v_{F/F}\\ =3i\times \left(-0.39j\right)\\ =-1.17k\end{array}$

The Coriolis acceleration

$\begin{array}{l}2\Omega \times v_{F/F}=2\left(4j\right)\times \left(-0.39j\right)\\ =0\end{array}$

The acceleration $a_E$ of point F

$\begin{array}{l}{a}_F=a_{F^1}+a_{F/F}+2\Omega \times v_{F/F}\\ =-4i-2.08k-1.17k+0\\ =-\left(4m/s^2\right)i-\left(3.25m/s^2\right)k\end{array}$

Conclusion:

The velocity $v_F$ of point F is $\left(0.52m/s\right)i-\left(0.39m/s\right)j-\left(1m/s\right)k$.

The acceleration $a_F$ of point F is $-\left(4m/s^2\right)i-\left(3.25m/s^2\right)k$.