Chapter 3, KINEMATICS Video Solutions, Computational Dynamics | Numerade (2024)

Figure P1 shows a rigid body $i$ that has a body fixed coordinate system $X^i Y^i$. The global position vector of the origin of the body coordinate system $O^i$ is defined by the vector $\mathbf{R}^i$, and the orientation of the body $i$ coordinate system in the global coordinate system is defined by the angle $\theta^i$. The local position vector of point $P^i$ on the body is defined by the vector $\overline{\mathbf{u}}_P^i$. If
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$\mathbf{R}^i=\left[\begin{array}{ll}5 & 3\end{array}\right]^{\mathrm{T}} \mathrm{m}, \theta^i=45^{\circ}$, and $\overline{\mathbf{u}}_P^i=\left[\begin{array}{ll}0.2 & 1.5\end{array}\right]^{\mathrm{T}} \mathrm{m}$, determine the global position vector of point $P^i$. If the body rotates with a constant angular velocity $\omega^i=150 \mathrm{rad} / \mathrm{s}$, determine the absolute velocity of point $P^i$ assuming that the absolute velocity of the reference point $\dot{\mathbf{R}}^i=\left[\begin{array}{ll}32 & -10\end{array}\right]^{\mathrm{T}} \mathrm{m} / \mathrm{s}$.

In problem 1 , let $\mathbf{R}^i=t\left[\begin{array}{ll}5 & 3\end{array}\right]^{\mathrm{T}} \mathrm{m}, \theta^i=\omega \mathrm{t}, \omega=50 \mathrm{rad} / \mathrm{s}$, and $\overline{\mathbf{u}}_P^f=\left[\begin{array}{ll}0.2\end{array}\right.$ $1.5]^{\mathrm{T}} \mathrm{m}$, determine the global position vector of point $P^i$ at $t=0,0.25$, and $1 \mathrm{~s}$. Also determine the absolute velocity of point $P^i$ at these points in time.
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In the slider crank mechanism shown in Fig. P2, the lengths of the crankshaft $O A$ and the connecting rod $A B$ are, respectively, 0.3 and 0.5 $\mathrm{m}$. The crankshaft is assumed to rotate with a constant angular velocity $\dot{\theta}^i=100 \mathrm{rad} / \mathrm{s}$ (counterclockwise). Assuming that the offset $h=0$, use analytical methods to determine the orientation and angular velocity and acceleration of the connecting rod and the position, velocity, and acceleration of the slider block when the angular orientation $\theta^2$ of the crankshaft is $45^{\circ}$. Also determine the absolute velocity and acceleration of the center of the connecting rod.
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Repeat problem 3 assuming that the offset $h=0.05 \mathrm{~m}$.
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The lengths of the crankshaft $O A$ and the connecting $\operatorname{rod} A B$ of the slider crank mechanism shown in Fig. P2 are 0.3 and $0.5 \mathrm{~m}$, respectively, and the offset $h=0.0$. The motion of the crankshaft is such that $\theta^2=150 t+$ $3.0 \mathrm{rad}$. Determine the orientation of the connecting rod and the position of the slider block at time $t=0,0.01$, and $0.03 \mathrm{~s}$.
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The lengths of the crankshaft $O A$, coupler $A B$, and rocker $B C$ of the fourbar linkage shown in Fig. P3 are, respectively, $0.3,0.35$, and $0.4 \mathrm{~m}$. The distance $O C$ is $0.32 \mathrm{~m}$. The crankshaft rotates with a constant angular velocity of $100 \mathrm{rad} / \mathrm{s}$. Determine the orientation and angular velocity and acceleration of the coupler and the rocker when the orientation of the crankshaft $\theta^2=60^{\circ}$. Also determine the orientation and angular velocity and angular acceleration of the crankshaft and the coupler when the orientation of the rocker $\theta^4=60^{\circ}$.
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Using Eq. 38, show that the velocities of two points $A$ and $B$ on a rigid body have equal components along the line $A B$.
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Repeat problem 1 assuming that the absolute velocity of the reference point is zero. Prove in this case, by using vector algebra, that the absolute velocity of point $P^i$ is perpendicular to the line $O^i P^i$, where $O^i$ is the reference point.
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Show that in the case of a general rigid body displacement, there exists a point on the rigid body whose velocity is instantaneously equal to zero. This point is called the instantaneous center of rotation.
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In problem 3 , let the velocity of the slider block be constant and equal to $5 \mathrm{~m} / \mathrm{s}$. Determine the angular velocities and angular accelerations of the crankshaft and the connecting rod. Also determine the absolute velocity and acceleration of the center of the connecting rod.
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Prove the identities of Eqs. 44 and 45.

Prove the identity of Eq. 66.

The motion of a rigid body $i$ is such that the location of the origin of its reference is defined by the vector $\mathbf{R}^i=\left[\begin{array}{ll}t & 8(t)^3\end{array}\right]^{\mathrm{T}} \mathrm{m}$, its angular velocity $\dot{\theta}^i$ $=-150 \mathrm{rad} / \mathrm{s}$ (clockwise), and its angular acceleration $\ddot{\theta}^i=0$. Determine the position, velocity, and acceleration of a point $P$ that moves with respect to the body such that its coordinates are defined in the body coordinate system by the vector $\overline{\mathbf{u}}_P^i=\left[\begin{array}{ll}1.5(t)^2 & -3(t)^3\end{array}\right]^{\mathrm{T}} \mathrm{m}$. Find the solution at time $t=0$, 1 , and $1.5 \mathrm{~s}$.

Solve problem 13 assuming that $\dot{\theta}^i=-150(t)^2 \mathrm{rad} / \mathrm{s}$.

Examine the singular configurations of the four-bar mechanism.

In the system shown in Fig. P4, $O A=0.2 \mathrm{~m}, O B=0.3 \mathrm{~m}$, and $B C=0.6$ $\mathrm{m}$. Link $O A$ is assumed to rotate with an angular velocity $50 \mathrm{rpm}$ counterclockwise. Find the velocity and acceleration of point $C$ and the angular velocity and acceleration of link $B C$.

Figure $\mathrm{P} 5$ shows a gear system that consists of gears $i, j$, and $k$, which are pinned at their centers to the rod $r$ at points $O, A$, and $B$, respectively. Gear $i$ is fixed with $r^i=0.3 \mathrm{~m}$, while $r^j=r^k=0.1 \mathrm{~m}$. If the $\operatorname{rod} r$ rotates
counterclockwise with a constant angular velocity of $15 \mathrm{rad} / \mathrm{s}$, determine the angular velocities and angular accelerations of the gears $j$ and $k$.

Solve problem 17 if the angular velocity and the angular acceleration of the rod are $15 \mathrm{rad} / \mathrm{s}$, and $120 \mathrm{rad} / \mathrm{s}^2$, respectively. The angular velocity and acceleration of the rod are assumed to be counterclockwise.

The motion of a rigid body $i$ is such that the global coordinates of point $P$ on the rigid body is given by $\mathbf{r}_P^j=\left[\begin{array}{ll}v t & 0\end{array}\right]^{\mathrm{T}}$, where $v$ is a constant. The angular velocity of the rigid body is assumed to be $\dot{\theta}^i=a_0+a_1 t$. Derive an expression for the kinematic constraint equations of this system in terms of the absolute coordinates $R_x^i, R_y^i$, and $\theta^i$. Assume that $\overline{\mathbf{u}}_P^i=\left[\begin{array}{ll}0.3 & 1.2\end{array}\right]^{\mathrm{T}}$ $\mathrm{m}$. Also determine the first and the second derivatives of the constraint equations. Use the resulting equations to determine the velocities $\dot{R}_x^i, \dot{R}_y^i$, and $\dot{\theta}^i$ and the accelerations $\tilde{R}_x^i, \ddot{R}_y^i$, and $\ddot{\theta}^i$ at $t=0$, and $2 \mathrm{~s}$. Use the data $v=5 \mathrm{~m} / \mathrm{s}, a_0=0, a_1=15 \mathrm{rad} / \mathrm{s}^2$.

The motion of two bodies $i$ and $j$ is such that $R_x^i=5 \mathrm{~m}=$ constant, $R_y^i=3$ $\sin 5 t \mathrm{~m}, \dot{\theta}^i-\dot{\theta}^j=5 \mathrm{rad} / \mathrm{s}=$ constant, and the position vector of point $P^j$ on body $j$ with respect to point $P^i$ on body $i$ is defined by
$$
\mathbf{r}_P^j-\mathbf{r}_P^j=\left[\begin{array}{l}
0.5 \sin 3 t \\
0.1 \cos 3 t
\end{array}\right]
$$
where
$$
\overline{\mathbf{u}}_P^i=\left[\begin{array}{ll}
0.7 & 1.2
\end{array}\right]^{\mathrm{T}}, \quad \overline{\mathbf{u}}_P^j=\left[\begin{array}{ll}
0.5 & -0.8
\end{array}\right]^{\mathrm{T}}
$$
Derive the vector of the constraint equations of this system and determine the number of degrees of freedom.

For the three-body system shown in Fig. P6, use three absolute coordinates
for each body to write the kinematic constraint equations of the revolute and prismatic joints. Also derive the constraint Jacobian matrix for the system.

Figure $\mathrm{P} 7$ shows two bodies $i$ and $j$ connected by a revolute-revolute joint that keeps the distance between points $P^i$ and $P^j$ constant. Derive the constraint equations for this type of joint using the absolute Cartesian coordinates. Derive also the constraint Jacobian matrix for this joint.

Derive the constraint Jacobian matrix of the reciprocating knife-edge follower cam system, and the roller follower cam system.

Derive the kinematic constraint Jacobian matrix of the offset flat-faced follower of Example 8 .

Determine the vectors $\mathbf{C}_t$ and $\mathbf{Q}_d$ of Eqs. 120 and 123, respectively, for the revolute-revolute joint of problem 22.

Determine the vectors $\mathbf{C}_t$ and $\mathbf{Q}_d$ of Eqs. 120 and 123, respectively, in the case of the reciprocating knife-edge follower cam system and the roller follower cam system of problem 23.

Determine the vectors $\mathbf{C}_t$ and $\mathbf{Q}_d$ of Eqs. 120 and 123, respectively, in the case of the offset flat-faced follower of Example 8.

Derive the constraint equations, the Jacobian matrix, the vector $\mathrm{C}_t$, and the
vector $\mathbf{Q}_d$ for the system shown in Fig. P8 using the absolute Cartesian coordinates.

Figure P9 shows a slider crank mechanism. The lengths of the crankshaft and the connecting rod are $0.2 \mathrm{~m}$ and $0.4 \mathrm{~m}$, respectively. The crankshaft is assumed to rotate with a constant angular velocity $\dot{\theta}^2=30 \mathrm{rad} / \mathrm{s}$. The initial angle $\theta_o^2$ of the crankshaft is assumed to be $30^{\circ}$. Using three absolutecoordinates $R_x^i, R_y^i$, and $\theta^i$ for each body in the system and the methods of constrained kinematics, determine the positions, velocities, and accelerations of the bodies at time $t=0,1$, and $2 \mathrm{~s}$.

Figure P10 shows a four-bar mechanism. The lengths of the crankshaft, coupler, and rocker are $0.2 \mathrm{~m}, 0.4 \mathrm{~m}$, and $0.3 \mathrm{~m}$, respectively. The crankshaft
of the mechanism is assumed to rotate with a constant angular velocity $\dot{\theta}^2$ $=15 \mathrm{rad} / \mathrm{s}$. The initial orientation of the crankshaft is assumed to be $\theta_o^2=$ $45^{\circ}$. By using three absolute coordinates $R_x^i, R_y^i$, and $\theta^i$ for each body in the system, determine the position, velocity, and acceleration of each body at time $t=0,1$, and $2 \mathrm{~s}$.
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