Lesson 4: Application Examples (Work, MA and Passive Resistances)

Lesson 4: Application Examples (Work, MA and Passive Resistances)#

This problem chapter focuses on applying energy concepts to the analysis of mechanisms, with particular focus on determining ideal forces (using the PVW) and the impact of passive resistances (friction and efficiency).

Example 4.1: Ideal Mechanical Advantage (IMA) Analysis via PVW#

Objective: Demonstrate the power of the Principle of Virtual Work (PVW) to compute forces in an ideal way (without friction), avoiding the computation of internal reactions.

Proposed Problem:

  • Option A (Revisit a problem): Return to Example 3.1 (Robotic Gripper). Ignoring friction, compute the actuator force \(F_a\) required for a clamping force \(F_g = 25N\) using the PVW.

    • \(\delta W = 0 \implies \delta W_{actuator} + \delta W_{gripper} = 0\)

    • \(\vec{F}_a \cdot \delta\vec{r}_a + \vec{F}_g \cdot \delta\vec{r}_g = 0\)

  • Option B (New problem): Use a slider-crank mechanism. Given an input force \(F_{in}\) on the slider, compute the torque \(M_{in}\) required on the crank to maintain equilibrium (IMA).

Skills to Develop:

  1. Identify velocities (or virtual displacements).

  2. Set up the virtual power (or work) equation.

  3. Compute the Ideal Mechanical Advantage (\(IMA = v_{in} / v_{out}\)).

Example 4.2: Friction in Prismatic Joints (Wedge)#

Objective: Apply Coulomb’s law (\(F_f = \mu N\)) to a simple system and compute the efficiency.

Proposed Problem:

  • (Based on Cardona 7.1 and USevilla Ch. 6)

  • A wedge with angle \(\alpha\) and friction coefficient \(\mu\) is used to raise a weight \(Q\).

  • Compute:

    1. The force \(F\) required to raise the weight (quasi-static equilibrium).

    2. The useful work (\(W_{useful} = Q \cdot \delta y\)) and the supplied work (\(W_{in} = F \cdot \delta x\)).

    3. The efficiency \(\eta\) of the system.

Skills to Develop:

  1. Free body diagram with friction forces opposing the motion.

  2. Force decomposition.

  3. Efficiency calculation.

Example 4.3: Friction in Revolute Joints (Friction Circle)#

Objective: Apply the friction circle model (Cardona 7.2) to analyze a linkage mechanism.

Proposed Problem:

  • A four-bar mechanism (e.g. slider-crank) must overcome a resistant force \(F_{res}\) on the slider at constant velocity.

  • The dimensions, the radius of all shafts (\(r_i\)), and the coefficient of friction (\(\mu\)) are known.

  • Compute:

    1. The ideal torque \(M_{ideal}\) (without friction, using the PVW).

    2. The friction torque \(M_{f,i} = R_i \cdot r_f\) at each joint (\(r_f = \mu \cdot r_i\)).

    3. The total dissipated passive power \(P_{passive} = \sum M_{f,i} \cdot \omega_{rel,i}\).

    4. The real required torque \(M_{real}\).

Skills to Develop:

  1. Prior kinematic analysis (for \(\omega_{rel}\)).

  2. Static (vector) analysis (as in Lesson 3) to obtain the reactions \(R_i\).

  3. Application of the friction circle model (Cardona 7.2).

  4. Computation of the dissipated power.

Example 4.4: Efficiency and Overall Power Analysis#

Objective: Integrate all the concepts for sizing an actuator.

Proposed Problem:

  • (Based on problems P9-1, P9-2 from T04_problemas.pdf, but simplified to steady-state operation).

  • A machine (e.g. a hoist) requires a useful power \(P_{useful} = 2 kW\) at a speed \(n_{mach} = 100 \text{ min}^{-1}\).

  • A motor running at \(n_{mot} = 750 \text{ min}^{-1}\) and a gearbox are used.

  • The gearbox efficiency is \(\eta_{gb} = 0.8\).

  • Compute:

    1. The useful torque on the machine shaft.

    2. The input power to the gearbox (power the motor must supply).

    3. The torque the motor must produce.

Skills to Develop:

  1. Working with the efficiency formula \(\eta = P_{useful} / P_{supplied}\).

  2. Relation between power, torque, and speed (\(P = M \cdot \omega\)).

  3. Understanding how efficiencies “chain together.”

Note on the T04_problemas.pdf Exercises (P9-1, P9-3…): As you can see, the idea from P9-1 was used for Example 4.4. However, many of those problems (such as P9-3) focus on acceleration (\(\Gamma_m = I \alpha + \Gamma_{rp}\)), which is dynamics (Cardona Ch. 9) and inertia analysis (Cardona 7.4+).

Following the course’s own guideline, those problems are a better fit for the “Machine Design” (“Cálculo de Máquinas”) course, while Examples 4.1 to 4.4 proposed here are ideal for “TMM”, since they focus on virtual work and passive resistances (Cardona 7.1-7.3).