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Chapter 3: Problem 77

The robot arm is elevating and extending simultaneously. At a given instant, \(\theta=30^{\circ}, \dot{\theta}=40 \mathrm{deg} / \mathrm{s}\) \(\ddot{\theta}=120 \operatorname{deg} / \mathrm{s}^{2}, l=0.5 \mathrm{m}, \dot{l}=0.4 \mathrm{m} / \mathrm{s},\) and \(\ddot{l}=-0.3\) \(\mathrm{m} / \mathrm{s}^{2} .\) Compute the radial and transverse forces \(F_{r}\) and \(F_{\theta}\) that the arm must exert on the gripped part \(P,\) which has a mass of 1.2 kg. Compare with the case of static equilibrium in the same position.

Short Answer

Expert verified
The computed radial and transverse forces are non-zero, whereas in static equilibrium they would be zero.

Step by step solution

01

Understanding the Forces

The radial force \( F_r \) and transverse force \( F_\theta \) are given by the equations \( F_r = m(a_r) \) and \( F_\theta = m(a_\theta) \), where \( a_r \) and \( a_\theta \) are the radial and transverse accelerations of a point \( P \) on the arm, respectively.
02

Calculating Radial Acceleration

The radial acceleration \( a_r \) is given by the equation:\[ a_r = \ddot{l} - l \dot{\theta}^2 \]Substitute the given values:\[ \ddot{l} = -0.3 \space m/s^2, \quad l = 0.5 \space m, \quad \dot{\theta} = 40 \space deg/s = \frac{40\pi}{180} \space rad/s \]Calculate \(\dot{\theta}^2:\)\[ \dot{\theta}^2 = \left(\frac{40\pi}{180}\right)^2 \space rad^2/s^2 \]Calculate \( a_r \):\[ a_r = -0.3 - 0.5 \left(\frac{40\pi}{180}\right)^2 \]
03

Calculating Transverse Acceleration

The transverse acceleration \( a_\theta \) is given by the equation:\[ a_\theta = l \ddot{\theta} + 2 \dot{l} \dot{\theta} \]Substitute the given values:\[ \ddot{\theta} = 120 \space deg/s^2 = \frac{120\pi}{180} \space rad/s^2, \quad \dot{l} = 0.4 \space m/s \]Calculate \( a_\theta \):\[ a_\theta = 0.5 \left(\frac{120\pi}{180}\right) + 2 (0.4) \left(\frac{40\pi}{180}\right) \]
04

Finding Forces

Substitute the accelerations into the force equations, using the mass \( m = 1.2 \space kg \):For radial force:\[ F_r = m \times a_r \]For transverse force:\[ F_\theta = m \times a_\theta \]Calculate \( F_r \) and \( F_\theta \).
05

Comparing with Static Equilibrium

In static equilibrium, all accelerations are zero. Therefore, the forces are:\[ F_r = 0 \quad \text{and} \quad F_\theta = 0 \]Compare these results with the computed \( F_r \) and \( F_\theta \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radial Force Computation
The radial force in a robotic arm scenario helps us understand the impacts of radial acceleration on the object being moved. For students, it's vital to first comprehend that radial force is determined by how much the object is pushing or pulling along the length of the arm. Here's how we break this down:
  • Radial acceleration is calculated using the formula: \[ a_r = \ddot{l} - l \dot{\theta}^2 \]
  • The key terms include \( \ddot{l} \), which is the linear acceleration, \( l \), the length of the arm, and \( \dot{\theta} \), the angular velocity converted into radian per second.
  • After substituting the given values into this formula, you find the radial acceleration at that instant.
  • Finally, to compute the radial force \( F_r \), multiply the mass of the object \( m \) by \( a_r \): \[ F_r = m \times a_r \]
Keep in mind, in a rotating systems dynamics, radial forces change as both the speed (\( \dot{\theta} \)) and arm's linear motion (\( l \)) shift. This makes understanding radial force essentials in robotic arm manipulation.
Transverse Force Calculation
In the context of robot arm dynamics, transverse force plays a critical role in understanding rotational motion effects on the gripped object. The transverse force is about how the object moves perpendicular to the arm and is crucial for circular motion analysis.
  • Transverse acceleration \( a_\theta \) uses the formula:\[ a_\theta = l \ddot{\theta} + 2 \dot{l} \dot{\theta} \]
  • Here, \( \ddot{\theta} \) is the angular acceleration and the product \( 2\dot{l} \dot{\theta} \) accounts for the Coriolis effect that happens due to the combined radial and angular motions.
  • When you plug in the known values into the equation, you calculate the transverse acceleration, giving a sense of the sideways forces acting on the object.
  • The transverse force \( F_\theta \) is then found by multiplying the object's mass \( m \) by \( a_\theta \):\[ F_\theta = m \times a_\theta \]
Understanding transverse forces helps identify how the arm applies lateral pressure to move or stabilize an object facing dynamic rotations suitable for tasks like sketching arcs or swiping.
Robotic Arm Acceleration Analysis
Acceleration analysis in robotic arms involves understanding both linear and rotational dynamics. This gives insight into how the arm achieves its motions and applies necessary forces.
  • To begin with, acceleration analysis combines radial and transverse measures as outlined in their respective equations.
  • The total coordination in robotic systems involves plotting these accelerations over time to understand behavior patterns or optimize a task path.
  • Compared to a static equilibrium where no motion occurs (radial and transverse accelerations are zero), in operational mode, robotic systems need careful evaluation.
  • Such accelerations are influenced by factors such as programmed speed changes, task requirements, and structural constraints of the robotic arm design.
Incorporating acceleration analysis ensures accurate movements in automated systems, crucial for industrial operations, remote surgeries, or even simple tasks like object sorting performed by a robotic arm.

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Most popular questions from this chapter

The position vector of a particle is given by \(\mathbf{r}=\) \(8 t \mathbf{i}+1.2 t^{2} \mathbf{j}-0.5\left(t^{3}-1\right) \mathbf{k},\) where \(t\) is the time in seconds from the start of the motion and where \(\mathbf{r}\) is expressed in meters. For the condition when \(t=\) 4 s, determine the power \(P\) developed by the force \(\mathbf{F}=40 \mathbf{i}-20 \mathbf{j}-36 \mathbf{k} \mathrm{N}\) which acts on the particle.

The space shuttle launches an 800 -kg satellite by ejecting it from the cargo bay as shown. The ejection mechanism is activated and is in contact with the satellite for 4 s to give it a velocity of \(0.3 \mathrm{m} / \mathrm{s}\) in the \(z\) -direction relative to the shuttle. The mass of the shuttle is 90 Mg. Determine the component of velocity \(v_{f}\) of the shuttle in the minus \(z\) -direction resulting from the ejection. Also find the time average \(F_{\mathrm{av}}\) of the ejection force.

A 75 -g projectile traveling at \(600 \mathrm{m} / \mathrm{s}\) strikes and becomes embedded in the 40 -kg block, which is initially stationary. Compute the energy lost during the impact. Express your answer as an absolute value \(|\Delta E|\) and as a percentage \(n\) of the original system energy \(E\).

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The spring of stiffness \(k\) is compressed and suddenly released, sending the particle of mass \(m\) sliding along the track. Determine the minimum spring compression \(\delta\) for which the particle will not lose contact with the loop-the-loop track. The sliding surface is smooth except for the rough portion of length \(s\) equal to \(R,\) where the coefficient of kinetic friction is \(\mu_{k}\).
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