A uniform slender rod of mass M and length L is pinned at one end and attached to a point mass, m through a linear spring of stiffness k as shown in Figure 5. The pinned end of the rod is constrained by a torsional spring of stiffness Kr. Given the following parameters: M=5 kg, L = 1.2 m, K, = 300 Nm/radian, m = 5 kg, k = 1000 N/m, (i) determine the equations of motion of the system (ii) determine the natural frequencies and mode shapes of the system. (iii) If mass m is given a displacement of 3 mm to the right of its equilibrium position and the rod is simultaneously given an initial angular displacement of 10° (7/18 radians) in the anticlockwise direction, determine expressions for the free vibration response of both the rod and mass. - Pinned joint K, Rod of mass, M k w-

A uniform slender rod of mass M and length L is pinned at one end and attached to a point mass, m through a linear spring of stiffness k as shown in Figure 5. The pinned end of the rod is constrained by a torsional spring of stiffness Kr. Given the following parameters: M=5 kg, L = 1.2 m, K, = 300 Nm/radian, m = 5 kg, k = 1000 N/m, (i) determine the equations of motion of the system (ii) determine the natural frequencies and mode shapes of the system. (iii) If mass m is given a displacement of 3 mm to the right of its equilibrium position and the rod is simultaneously given an initial angular displacement of 10° (7/18 radians) in the anticlockwise direction, determine expressions for the free vibration response of both the rod and mass. - Pinned joint K, Rod of mass, M k w-

Mechanics of Materials (MindTap Course List)

9th Edition

ISBN:9781337093347

Author:Barry J. Goodno, James M. Gere

Publisher:Barry J. Goodno, James M. Gere

Chapter2: Axially Loaded Members

Section: Chapter Questions

Problem 2.2.12P: A small lab scale has a rigid L-shaped frame ABC consisting of a horizontal aim AB (length b = 30...

See similar textbooks

Similar questions

A small lab scale has a rigid L-shaped frame ABC consisting of a horizontal aim AB (length b = 30 cm) and a vertical arm BC (length c = 20 cm) pivoted at point B. The pivot is attached to the outer frame BCD that stands on a laboratory bench. The position of the pointer at C is controlled by two parallel springs, each having a spring constant k = 3650 N/m. that are attached to a threaded rod. The pitch of the threads is p = 1.5 mm. If the weight is 65 N. how many revolutions of the nut are required to bring the pointer back to the mark?
A spring with squared and ground ends is used for a load and deflection of 1.2 kN and 20 mm, respectively, with a spring index of 4. The wire diameter is 6 mm and the modulus of rigidity is 84 GPa. Which of the following is the active number of turns of the spring?
A HELICAL SPRING HAS A STIFFNESS OF 25 kN/m. IF A MASS OF 100kg IS ATTACHED TO ITS FREE END, PULLED DOWN, AND THENRELEASED, Req: (draw the figure first before solving the problem)A. FIND THE AMOUNT OF DEFLECTION, IN CM,B. DETERMINE THE PERIODIC TIME OF ITS MOTION,C. IF THE MAXIMUM DEFLECTION WAS 50 mm, FIND THEVELOCITY,D. FIND THE ACCELERATION OF THE MASS WHEN ITS IS 300 mmFROM THE EQUILIBRIUM POSITION.
A helical spring is made of 10 m diameter steel wire wound on a 100 mm diameter mandrel. If there are 10 active coils what is spring constant Take: C= 100 GPA What force must be applied to the spring to elongate it by 50 mm?
Problem 2. A uniform bar of mass m is pivoted at point O and supported at the ends by two springs, as shown in Figure 2. End P of spring PQ is subjected to a sinusoidal displacement, x(t) = x,sinwt. 1=1 m, k=1000 N/m, m=10 kg, xo=1 cm, and w=10 rad/s. a) Drive the equations of motion and find the natural frequency of the bar b) Find the steady-state angular displacement of the bar Vont do Simulate the system and plot displacements for the following cases (choose suitable simulation time for each case): 1) With the initial conditions theta0=thetad0=0, an excitation x=xesin(wt) is applied vertically at point P. Choose such w values so that the system is in resonancer experiencing beating. i. D ii. xt t) = xo sin wt Uniform bar, k mass m Figure 2. Spring-mass system
Problem 2. A uniform bar of mass m is pivoted at point O and supported at the ends by two springs, as shown in Figure 2. End P of spring PQ is subjected to a sinusoidal displacement, x(t) = x,sinwt. 1=1 m, k=1000 N/m, m=10 kg, xo=1 cm, and w=10 rad/s. a) Drive the equations of motion and find the natural frequency of the bar b) Find the steady-state angular displacement of the bar Simulate the system and plot displacements for the following cases (choose suitable simulation Vont time for each case): 1) With the initial conditions theta0=thetad0=0, an excitation xEsin(wi) Is applied vertically at point P. Choose such w values so that the system is i. in resonance, exPeriencing beating. do D ii. (1) = xo sin wt Uniform bar, mass m 31 Figure 2. Spring-mass system
Find the number of active coils of a No. 10 wire Helical Spring index of 7, steady load with spring rate of 46.5 lb/in. maximum allowable stress is 70 ksi. The stiffness of a spring is 98 N/mm of axial compression. Find the Joules of work to increase the compression from 60 mm to 90 mm.
K1 For the spring shown above, kl=100 N/mm, k2-200N/mm, k3-100N/mm, P-500N, ul-u3-0 Find: 1. The global stiffness matrix 2. Displacements of nodes 2 and 3
5. Design a helical compression spring for a maximum load of 1000 N and deflection of 25mm taking Wahl’s factor into consideration. Assume spring index as 5, maximum permissible shear stress for spring wire as 420 MPa and modulus of rigidity as 84 kN/mm2
Find the equivalent spring constant of a uniform mild steel rod length of 2.5 m, cross-sectional area of 100 cm2 and Young's modulus of 210 GPa subjected to an axial tensile (or compressive) force F= 500 N.
1. Design a compression helical spring to cary a load of 500 N with a deflection of 25 mm. The spring index may be taken as 8. Assume the following values for the spring material: Permissible shear stress 350 MPa Modulus of rigidity = 84 kN/mm2 4C – 1 + 4C – 4 0.615 Wahl's factor -, where C= spring index. [Ans. d= 5.893 mm; D=47.144 mm; n 6]
Consider the spring assemblage shown, using the direct stiffness method do the following: 1. Write the global stiffness matrix 2. Determine the nodal displacements 3. Determine the nodal forces 4. Determine the reaction of supports K1 = 15 kN/m f Node 1 F = 0.675 kN K2 = 45 kN/m No Node 2 X positive to the right vov Node 3 K3 = 15 N/m Node 4