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Consider the mass-spring system in Fig. P27.26. The frequencies for the mass vibrations can be determined by solving for the eigenvalues and by applying
Applying the guess
Use MATLAB's eig command to solve for the eigenvalues of the
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Chapter 27 Solutions
EBK NUMERICAL METHODS FOR ENGINEERS
- K Kt m₂ mi For this fig. > with damping Find the following: ~Equations of motion C ~Mass matrix ~Stiffness matrix ~Damping matrix ~Natural freqyancy ~All damping properties.. 72 ZIarrow_forwardT'sec In helical spring experiment a student plotted the graph of T2 versus the oscillating mass (M), and Used the slope to find the spring :constant, the value of k is 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 O. O a. 13.3 N/m O b. 6.8N/m O c. 5 N/m O d. 3.3 N/m O e. 24 N/m 5 10 15 20 25 30 35 40 45 50 55 60 65 M (9) 70 75 80arrow_forwardFor the model defined in Question 1, is the closed loop system with the negative feedback control law? 0 0 0 ;]* X 0 u=-Kx= O a. O b. O c. unstable 0 1 0 asymptotically stable not unstable O d. not asymptotically stablearrow_forward
- An object attached to a spring undergoes simple harmonic motion modeled by the differential equation d²x = 0 where x (t) is the displacement of the mass (relative to equilibrium) at time t, m is the mass of the object, and k is the spring constant. A mass of 3 kilograms stretches the spring 0.2 meters. dt² Use this information to find the spring constant. (Use g = 9.8 meters/second²) m k = + kx The previous mass is detached from the spring and a mass of 17 kilograms is attached. This mass is displaced 0.45 meters below equilibrium and then launched with an initial velocity of 2 meters/second. Write the equation of motion in the form x(t) = c₁ cos(wt) + c₂ sin(wt). Do not leave unknown constants in your equation. x(t) = Rewrite the equation of motion in the form ä(t) = A sin(wt + ), where 0 ≤ ¢ < 2π. Do not leave unknown constants in your equation. x(t) =arrow_forwardFind the subsequent motion of a string fixed at two ends and starting to move from rest with initial displacement h(x) given below. Use numerical method taking equal temporal and spatial division. Take C=1 m/s and carry out calculation for three time steps. 0.2 0.4 0.6 0.8 1.0 h(x) 0. 0.5 1arrow_forwardA mechanical system is presented as below. There are four simulation graphs for different values for m, b and c tested for a step response. Identify the graph for m=4 kg, b=0.3 N.s/m and k=1 N/m (Hint: you may need to use matlab simulation to find it). m 1.8 1.6 1.4 b ·f(t) Step Responsearrow_forward
- Equation of motion of a suspension system is given as: Mä(t) + Cx(t) + ax² (t) + bx(t) = F(t), where the spring force is given with a non-linear function as K(x) = ax²(t) + bx(t). %3D a. Find the linearized equation of motion of the system for the motion that it makes around steady state equilibrium point x, under the effect of constant F, force. b. Find the natural frequency and damping ratio of the linearized system. - c. Find the step response of the system ( Numerical values: a=2, b=5, M=1kg, C=3Ns/m, Fo=1N, xo=0.05marrow_forwardConsider the system presented in the following figure K, = 500N / m K, = 2000N / m M = 1Kg %3D X, = 20mm %3D Vo = Omm/s %3D The stiffness of the equivalent spring (in N/m) is equal to Choose... The natural frequency in rad/s is equal to The amplitude of vibration (in m) is Choose... Choose... The phase shift of the vibration is Choose...arrow_forwardThe velocity, v, of a falling parachutist is given by v= (1-em), 8. -(c/m)t C where g = 9.8067 m/s². Given the mass, m of the parachutist is 70kg, velocity, v = 40 m/s at time t = 10 s, find the drag coefficient, c by using the Bisection method.arrow_forward
- The stress profile shown below is applied to six different biological materials: Log Time (s] The mechanical behavior of each of the materials can be modeled as a Voigt body. In response to o,= 20 Pa applied to each of the six materials, the following responses are obtained: 2 of Maferial 6 Material 5 0.12 0.10 Material 4 0.08 Material 3 0.06 0.04 Material 2 0.02 Material 1 (a) Which of the materials has the highest Young's Modulus (E)? Why? Log Time (s) (b) Using strain value of 0.06, estimate the coefficient of viscosity (n) for Material 6. Stress (kPa) Strainarrow_forwardFind the solution of the following system using modal analysis method. Given: k₁=k2=k, J₁ = 3J2, Initial Conditions: x(0)=: [0 1] . v(0) [00] 0₁ (1) 1 Select one: O J₁ 0.2774cos0.4821 0.3613cos0.4821 O 0.4562sin1.4821 1.3613sin1.4821 O 0.4562cos1.4821 1.3613cos1.4821 h h k₂ J₂ t-0.2774cos 1.1976 t-0.6387cos 1.1976 t-0.1254cos 1.1976 t-0.6387cos 1.1976 t-0.1254cos 1.1976 t-0.6387cos 1.1976 0₂(1)arrow_forwardAn object is shot upward from the ground with an initial velocity of 640 ft/sec, and experiencés a constant deceleration of 32 ft/sec² due to gravity as well as a deceleration of (v(t) / 10) ft/sec due to air resistance, where v(t) is the object's velocity in ft/sec. (a) Set up and solve an initial-value problem to determine the object's velocity v(t) at time t. (b) At what time does the object reach its highest point?arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
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