Concept explainers
The suspension of an automobile can be approximated by the simplified spring-and-dashpot system shown. (a) Write the differential equation defining the vertical displacement of the mass m when the system moves at a speed v over a road with a sinusoidal cross-section of amplitude δm and wavelength L. (b) Derive an expression for the amplitude of the vertical displacement of the mass m.
Fig. P19.151
(a)
Write the differential equation defining the vertical displacement of the mass m when the system moves at a speed v over a road with a sinusoidal cross section of amplitude
Answer to Problem 19.151P
The differential equation defining the vertical displacement of the mass m when the system moves at a speed v over a road with a sinusoidal cross section of amplitude
Explanation of Solution
Calculation:
Show the free body diagram of the system of automobile, spring and dashpot as in Figure (1).
The expression for the weight of the automobile (W) as follows:
Here,
The expression for the acceleration of the automobile (a) as follows:
Refer Figure (1),
The expression for the force by considering the vertical equilibrium condition as follows;
Substitute
Substitute
The expression for the time interval needed to travel
The expression for the forced circular frequency
Substitute
The expression for the motion of the wheel which is sine curve
Differentiate the above equation with respect to time ‘t’.
Substitute
Substitute
Therefore, the differential equation defining the vertical displacement of the mass m when the system moves at a speed v over a road with a sinusoidal cross section of amplitude
(b)
Derive an expression for the amplitude of the vertical displacement of the mass m.
Answer to Problem 19.151P
The expression for the amplitude of the vertical displacement of the mass m is
Explanation of Solution
Calculation:
The expression for the general solution from the identity as follows:
Here,
The expression for the force transmitted (F) to the automobile as follows:
Substitute
The expression for the differential equation of the motion for the damped forced vibration as follows:
Compare the equation (3) and (4).
The expression for the steady state of motion of the system as follows:
The expression for the steady state of motion of the system as follows:
Substitute
The expression for the phase angle
The expression for the Eulerian angle
Therefore, the expression for the amplitude of the vertical displacement of the mass m is
Want to see more full solutions like this?
Chapter 19 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
- A 400-kg motor supported by four springs, each of constant 150 kN/m, and a dashpot of constant c = 6500 N·s/m is constrained to move vertically. Knowing that the unbalance of the rotor is equivalent to a 23-g mass located at a distance of 100 mm from the axis of rotation, determine for a speed of 800 rpm (a ) the amplitude of the fluctuating force transmitted to the foundation, (b ) the amplitude of the vertical motion of the motor.arrow_forwardA 25-kg block is supported by the spring arrangement shown. If the block is moved vertically downward from its equilibrium position and released, determine (a) the period and frequency of the resulting motion, (b) the maximum velocity and acceleration of the block if the amplitude of the motion is 30 mm.arrow_forwardA motor of mass M is supported by springs with an equivalent spring constant k The unbalance of its rotor is equivalent to a mass m located at a distance r from the axis of rotation. Show that when the angular velocity of the motor is wf, the amplitude of the motion of the motor is wherearrow_forward
- A machine of mass 75 kg is mounted on springs and is fitted with a dashpot to damp out vibrations. There are three springs each of stiffness 10 N/mm and it is found that the amplitude of vibration diminishes from 38.4 mm to 6.4 mm in two complete oscillations. Assuming that the damping force varies as the velocity, determine : 1. the resistance of the dash-pot at unit velocity ; 2. the ratio of the frequency of the damped vibration to the frequency of the undamped vibration ; and 3. the periodic time of the damped vibration.arrow_forwardAn instrument package A is bolted to a shaker table as shown. The table moves vertically in simple harmonic motion at the same frequency as the variable-speed motor that drives it. The package is to be tested at a peak acceleration of 150 ft/s2. Knowing that the amplitude of the shaker table is 2.3 in., determine (a) the required speed of the motor in rpm, (b) the maximum velocity of the table.arrow_forwardA vibrometer used to measure the amplitude of vibrations consists essentially of a box containing a mass-spring system with a known natural frequency of 120 Hz. The box is rigidly attached to a surface that is moving according to the equation y= δm sin wf t. If the amplitude zm of the motion of the mass relative to the box is used as a measure of the amplitude δm of the vibration of the surface, determine (a) the percent error when the frequency of the vibration is 600 Hz,(b) the frequency at which the error is zero.arrow_forward
- A sensitive electronic system, of mass 25 kg, is supported by a spring-damper isolator that rests on the floor of a manufacturing plant. The operation of nearby rotating equipment causes the floor to vibrate with an amplitude of 8 mm at a frequency of 35 Hz. The electronic system can only operate effectively if the amplitude of its acceleration is less than 40 m/s2. It is known that the damping ratio of the isolator is 0.1. i. Determine the maximum stiffness of the isolator needed for the transmitted acceleration level to be acceptable and hence facilitate effective operation of the electronic system. Using the calculated stiffness value, also determine the largest deformation of the spring in millimetres when the system is in motion. ii. If the damping ratio is allowed to increase by only increasing the equivalent damping coefficient (c) of the isolator, discuss the effect this change would have on the response of the electronic system when operating in the same environment. In your…arrow_forwardDetermine the natural frequency of a pendulum whose length is 5 m.arrow_forwardA slender 10-kg bar AB with a length of l = 0.6 m is connected to two collars of negligible weight. Collar A is attached to a spring with a constant of k = 1.5 kN/m and can slide on a horizontal rod, while collar B can slide freely on a vertical rod. Knowing that the system is in equilibrium when bar AB is vertical and that collar A is given a small displacement and released, determine the period of the resulting vibrations.arrow_forward
- A mass is attached to the end of a spring and set into simple harmonic motion with an amplitude A on a horizontal frictionless surface. Determine the following in terms of only the variable A. (a) Magnitude of the position (in terms of A) of the oscillating mass when it's speed is 60% of it's maximum value. (B) magnitude of the position ( in terms of A) of the oscillating mass when the elastic potential energy of the spring is 60% of the total energy of the oscillating system.arrow_forwardVibration When suspended from a helical spring, a load of 91 kg is found to vibrate vertically with a periodic time of 0.75 s. Determine, for an amplitude of 51 mm : (a) the angular velocity of the generating vector; (b) the maximum acceleration; (c) the acceleration when the displacement is 127 mm; (d) the stiffness of the spring; (e) the static deflection in the spring; (f) the maximum force in the spring (122 N).arrow_forwardShow that for a small value of the damping factor c/cc , the maximum amplitude of a forced vibration occurs when Wf= Wn and that the corresponding value of the magnification factor is ½ (c/c2).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
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY