2. Consider the mass-spring system shown in the figure below. It can be shown that the...
2. Consider the mass-spring system shown in the figure below. It can be shown that the motion of the mass is governed by the equation a=-sw^2, where s and a are the position and acceleration of the mass, respectively, and w is a constant (which is referred to as the natural frequency of the system).
-
Derive the equation describing the velocity of the mass in terms of the position. Assume that the
velocity of the mass is v(subzero) when s=0 .
-
Considering that v=ds/dt, , use the equation in part (a), and determine the position of the mass, s, as
-
a function of time. Assume that at s=0 at t=0.
.
Homework Answers
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2. Consider the mass-spring system shown in the figure below. It
can be shown that the...
Problem 5: The spring-mass system shown has spring constants ky = 24 kN/m and kz =...
Problem 5: The spring-mass system shown has spring constants ky = 24 kN/m and kz = 36 kN/m with a suspended mass of 35 kg at A. If the block is displaced 50 mm below its equilibrium position and released with no initial velocity, determine: a) The circular natural frequency, the natural frequency, and the period b) The position, velocity, and acceleration of the block after a time of 30 seconds k2 mm ki A
Q1. For the system shown in Figure 1 where the beam with mass m and length...
Q1. For the system shown in Figure 1 where the beam with mass m and length L is connected to the fixed surfaces through three springs with same stiffness k, (i) Calculate the total kinetic energy and total potential energy of the system; (ii) Derive the equation of motion in terms of rotation angle 0; (iii) Find the natural frequency of the system; (iv) Calculate the natural period if the stiffness k of all springs is doubled; (v) If the...
Exercises 1. (introduction) Sketch or plot the displacement of the mass in a mass-spring system for at least two per...
Exercises 1. (introduction) Sketch or plot the displacement of the mass in a mass-spring system for at least two periods for the case when Wn-2rad/s, 괴,-1mm, and eto =-v/5mm/s. 2. (introduction) The approximation sin θ ะ θ is reasonable for θ < 10°. If a pendulum of length 0.5m, has an initial position of 0()0, what is the maximum value of the initial angular velocity that can be given to the pendulum without violating this smll angle approximation? 3. (harmonic...
A vibratory system can be modeled as a mass spring dashpot system as shown in Figure. In a free vibration test, the mas...
A vibratory system can be modeled as a mass spring dashpot system as shown in Figure. In a free vibration test, the mass is disturbed from its equilibrium position. The corresponding time history plot is given as shown by the plot. Determine the following characteristics of the system: a) The natural frequency of the system b) The effective spring stifness c) The viscous damping coefficient c E 2 20kg 1.5 time (s)
A vibratory system can be modeled as a...
Consider a mass-spring system shown below with a hard spring. That is, it requires more force...
Consider a mass-spring system shown below with a hard spring. That is, it requires more force to deform the same amount as the spring stretches/compresses. elle m The equation of motion is given by mä+kx3 = mg, where x is the stretch of the spring from its undeformed length, m is the mass of the block, k is the spring constant, and g is the gravitational acceleration. After the equation of motion is linearized about its equilibrium position, it can...
2 with spring stiffness k 1000 N/m, Consider a mass-spring-damper system shown in Figure mass m...
2 with spring stiffness k 1000 N/m, Consider a mass-spring-damper system shown in Figure mass m = 10 kg, and damping constant c-150 N-s/m. If the initial displacement is xo-o and the initial velocity is 10 m/s (1) Find the damping ratio. (2) Is the system underdamped or overdamped? Why? (3) Calculate the damped natural frequency (4) Determine the free vibration response of the system.
Problem 4. Consider the spring-mass system shown in the figure. The displacement of the mass m...
Problem 4. Consider the spring-mass system shown in the figure. The displacement of the mass m as a function of time is as follows: x = Xocoswt) + cos(Wnt) ωη where xo is the initial displacement equals to 0.1 m, čo is the initial velocity equals to 1 m/s, and Wr is the natural frequency of the system equals to 4 rad/s. Calculate the acceleration (second time derivative of displacement) of the mass after 1 s with a time step...
Number 2 1. a) The displacement x of a forced spring-mass system is governed by dx...
Number 2
1. a) The displacement x of a forced spring-mass system is governed by dx d2x dt2 + (1 + t)x2 = sint t> 0 x(0) = 0 dx (0) = 0 dt dt Obtain the first four non-zero terms of the solution using the Taylor expansion approach. b) Calculate the position and velocity of the mass at t = 0.5 using the result of part (a). a) The displacement x of a forced spring-mass system is governed by...
3. The motion of a 1DOF mass-spring-damper system (see Figure 1) is modeled by the following seco...
3. The motion of a 1DOF mass-spring-damper system (see Figure 1) is modeled by the following second order linear ODE: dx,2 dt n dt2 (0) C dt where is the damping ratio an wn is the natural frequency, both related to k, b, and m (the spring constant, damping coefficient, and mass, respectively) (a) Use the forward difference approximations of (b) Using Δt andd to obtain a finite difference formula for x(t+ 2Δ) (like we did in class for the...
( 12 marks LO3) Consider an undan ed two-degree-of-freedom spring-mass system, shown in the f g re below. The motion of the system Es con pletely described by the coordinate 치(t) and x2(t). le H...
( 12 marks LO3) Consider an undan ed two-degree-of-freedom spring-mass system, shown in the f g re below. The motion of the system Es con pletely described by the coordinate 치(t) and x2(t). le Ho Assume: kI- k2 k3 2 Nm, m-m2-1 kg and F-F2- Use the provided white paper to work out your answers, then pick the proper choice from the drop down list The equation of motion of mass 1 is EQ 1-x+6x1-4x2 0 EO 2 x1+4x1-2x2 The...
Problem 5: The spring-mass system shown has spring constants ky = 24 kN/m and kz = 36 kN/m with a suspended mass of 35 kg at A. If the block is displaced 50 mm below its equilibrium position and released with no initial velocity, determine: a) The circular natural frequency, the natural frequency, and the period b) The position, velocity, and acceleration of the block after a time of 30 seconds k2 mm ki A
Q1. For the system shown in Figure 1 where the beam with mass m and length L is connected to the fixed surfaces through three springs with same stiffness k, (i) Calculate the total kinetic energy and total potential energy of the system; (ii) Derive the equation of motion in terms of rotation angle 0; (iii) Find the natural frequency of the system; (iv) Calculate the natural period if the stiffness k of all springs is doubled; (v) If the...
Exercises 1. (introduction) Sketch or plot the displacement of the mass in a mass-spring system for at least two periods for the case when Wn-2rad/s, 괴,-1mm, and eto =-v/5mm/s. 2. (introduction) The approximation sin θ ะ θ is reasonable for θ < 10°. If a pendulum of length 0.5m, has an initial position of 0()0, what is the maximum value of the initial angular velocity that can be given to the pendulum without violating this smll angle approximation? 3. (harmonic...
A vibratory system can be modeled as a mass spring dashpot system as shown in Figure. In a free vibration test, the mass is disturbed from its equilibrium position. The corresponding time history plot is given as shown by the plot. Determine the following characteristics of the system: a) The natural frequency of the system b) The effective spring stifness c) The viscous damping coefficient c E 2 20kg 1.5 time (s)
A vibratory system can be modeled as a...
Consider a mass-spring system shown below with a hard spring. That is, it requires more force to deform the same amount as the spring stretches/compresses. elle m The equation of motion is given by mä+kx3 = mg, where x is the stretch of the spring from its undeformed length, m is the mass of the block, k is the spring constant, and g is the gravitational acceleration. After the equation of motion is linearized about its equilibrium position, it can...
2 with spring stiffness k 1000 N/m, Consider a mass-spring-damper system shown in Figure mass m = 10 kg, and damping constant c-150 N-s/m. If the initial displacement is xo-o and the initial velocity is 10 m/s (1) Find the damping ratio. (2) Is the system underdamped or overdamped? Why? (3) Calculate the damped natural frequency (4) Determine the free vibration response of the system.
Problem 4. Consider the spring-mass system shown in the figure. The displacement of the mass m as a function of time is as follows: x = Xocoswt) + cos(Wnt) ωη where xo is the initial displacement equals to 0.1 m, čo is the initial velocity equals to 1 m/s, and Wr is the natural frequency of the system equals to 4 rad/s. Calculate the acceleration (second time derivative of displacement) of the mass after 1 s with a time step...
Number 2
1. a) The displacement x of a forced spring-mass system is governed by dx d2x dt2 + (1 + t)x2 = sint t> 0 x(0) = 0 dx (0) = 0 dt dt Obtain the first four non-zero terms of the solution using the Taylor expansion approach. b) Calculate the position and velocity of the mass at t = 0.5 using the result of part (a). a) The displacement x of a forced spring-mass system is governed by...
3. The motion of a 1DOF mass-spring-damper system (see Figure 1) is modeled by the following second order linear ODE: dx,2 dt n dt2 (0) C dt where is the damping ratio an wn is the natural frequency, both related to k, b, and m (the spring constant, damping coefficient, and mass, respectively) (a) Use the forward difference approximations of (b) Using Δt andd to obtain a finite difference formula for x(t+ 2Δ) (like we did in class for the...
( 12 marks LO3) Consider an undan ed two-degree-of-freedom spring-mass system, shown in the f g re below. The motion of the system Es con pletely described by the coordinate 치(t) and x2(t). le Ho Assume: kI- k2 k3 2 Nm, m-m2-1 kg and F-F2- Use the provided white paper to work out your answers, then pick the proper choice from the drop down list The equation of motion of mass 1 is EQ 1-x+6x1-4x2 0 EO 2 x1+4x1-2x2 The...
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