Determine the moment of inertia I about axis O-M for the uniform slender rod bent into the shape shown. Plot I versus θ from θ = 0 to θ = 90° and determine the minimum value of I and the angle a which its axis makes with the x-direction. (Note: Because the analysis does not involve the z-coordinate, the expressions developed for area moments of inertia, Eqs. A /9, A /10, and A /11 in Appendix A of Vol. 1 Statics, may be utilized for this problem in place of the three-dimensional relations of Appendix B.) The rod has a mass ρ per unit length.

The uniform sector of Prob. 8/77 is repeated here with m = 4 kg, r = 325 mm, and β = 45°. If the sector is released from rest with θ₀ = 90°, plot the value of for the time period 0 ≤ t ≤ 6 s. Friction in the pivot at O results in a resistive torque of magnitude M = cθ˙, where the constant c = 0.35 N∙m∙s/rad. Compare your large-angle results with those for the small-angle approximation of sin θ ≅ θ and state the value of θ when t = 1 s for both large-angle and small-angle cases.

Shown in the figure are the elements of a displacement meter used to study the motion y_B = b sin wt of the base. The motion of the mass relative to the frame is recorded on the rotating drum. If l1 = 1.2 ft, l₂ = 1.6 ft, l₃ = 2 ft, W = 2 lb, c = 0.1 lb-sec/ft, and w = 10 rad/sec, determine the range of the spring constant k over which the magnitude of the recorded relative displacement is less than 1.5b. It is assumed that the ratio w/w_n must remain greater than unity.

The assembly shown consists of two sheaves of mass m₁ = 35 kg and m₂ = 15 kg, outer groove radii r₁ = 525 mm and r₂ = 250 mm, and centroidal radii of gyration (k_O)₁ = 350 mm and (k_O)₂ = 150 mm. The sheaves are fitted to a central shaft at O with bearings which allow them to rotate independently of each other. Attached to the central shaft is a carriage of mass m₃ = 25 kg. Each sheave has an inextensible cable wrapped securely within its outer groove. Each cable is attached to a spring at one end and to a fixed support at the other end. The springs have stiffnesses k₁ = 800 N/m and k₂ = 650 N/m. By the method of this article, determine the equation of motion for the system in terms of the variable x and state the period τ for small vertical oscillations about the equilibrium position. Neglect friction in the bearings at O.

Determine the amplitude of vertical vibration of the car as it travels at a velocity v = 40 km/h over the wavy road whose contour may be expressed as a sine or cosine function with a double amplitude 2b = 50 mm. The mass of the car is 1800 kg and the stiffness of each of the four car springs is 35 kN/m. Assume that all four wheels are in continuous contact with the road, and neglect damping. Note that the wheelbase of the car and the spatial period of the road are the same at L = 3 m, so that it may be assumed that the car translates but does not rotate. At what critical speed v_c is the vertical vibration of the car at its maximum?

The elements of the “swing-axle” type of independent rear suspension for automobiles are depicted in the figure. The differential D is rigidly attached to the car frame. The half-axles are pivoted at their inboard ends (point O for the half-axle shown) and are rigidly attached to the wheels. Suspension elements not shown constrain the wheel motion to the plane of the figure. The weight of the wheel– tire assembly is W = 100 lb, and its mass moment of inertia about a diametral axis passing through its mass center G is 1 lb-ft-sec². The weight of the half-axle is negligible. The spring rate and shockabsorber damping coefficient are k = 50 lb/in. and c = 200 lb-sec/ft, respectively. If a static tire imbalance is present, as represented by the additional concentrated weight w = 0.5 lb as shown, determine the angular velocity w which results in the suspension system being driven at its undamped natural frequency. What would be the corresponding vehicle speed v? Determine the damping ratio ζ. Assume small angular deflections and neglect gyroscopic effects and any car frame vibration. In order to avoid the complications associated with the varying normal force exerted by the road on the tire, treat the vehicle as being on a lift with the wheels hanging free.

The assembly of Prob. 8/40 is repeated here with the additional information that body ABC now has mass m4 and a radius of gyration k_O about its pivot at O, about which it is balanced. If a harmonic torque M = M₀ cos wt is applied to body ABC, determine the equation of motion for the system in terms of the variable x. State the critical frequency c of the harmonic torque which will result in an excessively large system response. Evaluate w_c for m₁ = 15 kg, m₂ = 12 kg, m₃ = 8 kg, m₄ = 6 kg, k₁ = 400 N /m, k₂ = 650 N /m, k₃ = 225 N /m, c₁ = 44 N ∙ s /m, c₂ = 36 N∙ s /m, c₃ = 52 N∙ s /m, a = 1.2 m, b = 1.8 m, c = 0.9 m, and k_O = 0.75 m. What is the damping ratio ζ for these conditions?

The owner of a 3400-lb pickup truck tests the action of his rear-wheel shock absorbers by applying a steady 100-lb force to the rear bumper and measuring a static deflection of 3 in. Upon sudden release of the force, the bumper rises and then falls to a maximum of 12 in. below the unloaded equilibrium position of the bumper on the first rebound. Treat the action as a one-dimensional problem with an equivalent mass of half the truck mass. Find the viscous damping factor ζ for the rear end and the viscous damping coefficient c for each shock absorber assuming its action to be vertical.

The thin circular disk of mass m and radius r is rotating about its z-axis with a constant angular velocity p, and the yoke in which it is mounted rotates about the x-axis through OB with a constant angular velocity w₁. Simultaneously, the entire assembly rotates about the fixed Y-axis through O with a constant angular velocity w₂. Determine the velocity v and acceleration a of point A on the rim of the disk as it passes the position shown where the x-y plane of the disk coincides with the X-Y plane. The x-y-z axes are attached to the yoke.

For a short interval of motion, collar A moves along its fixed shaft with a velocity v_A = 2 m/s in the Y-direction. Collar B, in turn, slides along its fixed vertical shaft. Link AB is 700 mm in length and can turn within the clevis at A to allow for the angular change between the clevises. For the instant when A passes the position where y = 200 mm, determine the velocity of collar B using nonrotating axes attached to B and fi nd the component w_n, normal to AB, of the angular velocity of the link. Also solve for v_B by differentiating the appropriate relation x² + y² + z² = l².