Example 3.5
A single-degree-of-freedom spring-mass-damper system has a mass og 60 kg and a spring stiffness of 6000 N/m. Determine:
(a)the critical damping coefficient
(b)the damped natural frequency when
(c)the logarithmic decrement.
Solution:
(a)
(b)
Damped natural frequency is
Example (damped vib).
FiG / -The amplitude of vibration is observed to decrease to 25% of the initial value after five consecutive cycles of motion as shown in Fig.- Find
Maximum amplitude in a cycle occurs when
Since and
logarithmic decrement.
We have
Energy dissipated by damping:
-Damping is present in all oscillatory systems.
-Its effect is to remove energy from the systems.
-Energy in a vibrating system is either dissipated into heat or radiated way.
- Dissipation of energy into heat can be experienced simply by bending a piece of metal back and forth a number of times.
-We are all aware of the sound which is radiated from an object for given a sharp blow.
-When a busy (float) is made to bob up and down in the water, waves radiate out and away from it, thereby resulting in its loss of energy.
-In vibration analysis, we are generally concerned with damping in terms of system response.
-The loss of energy from the oscillatory system results in the decay of amplitude of free vibration.
-In steady-state forced vibration, the loss of energy is balanced by the energy which is supplied by the excitation.
-A vibrating system may encounter many different types of damping forces, from internal molecular friction to sliding friction and fluid resistance.
-Generally their mathematical description is quite complicated and not suitable for vibration analysis.
-Thus simplified damping models have been developed that in many cases are found to be adequate in evaluating the system response.
-For example, the viscous damping model, designated by the dashpot, which leads to manageable mathematical solutions.
-Energy dissipation is usually determined under condition of cyclic oscillations.
-Depending on the type of damping present, the force-displacement relationship when plotted may differ greatly.
-In all cases, however, the force-displacement curve will enclose an area, referred to as the hystersis loop, that is proportional to the energy lost per cycle.
-The energy lost per cycle due to damping force is computed from the general equation.
(1)
-In general, depends on many factors, such as temperature, frequency, or amplitude.
-We consider the simplest case of energy dissipation, that of a spring-mass system with viscous damping.
-The damping force in this case is
-With the steady-state displacement and velocity
(2)
the energy dissipated per cycle, from eqn,(1), becomes (for viscous damping)
(3)
-Of particular interest is the energy dissipated in forced vibration at resonance.
-Substituting and , in eqn.(3)
We get,
(4)
-The energy dissipated per cycle by the damping force can be represented graphically as follows.
-The velocity can be written as
(5)
-The damping force becomes
(6)
or
or (7)
Eqn. (7) is the equation of an ellipse with and plotted along the vertical and horizontal axes, as shown in Fig 1(a).
-The energy dissipated per cycle is then given by the area enclosed by the ellipse.
-If we add to the spring force (i.e. of the losses spring), the hysteresis loop is rotated as shown in Fig 1.(b)
Location at which curve intersect x- axis
or
. It is the location where
-The Fig. 1(b) conforms to the Voigt model, which consists of a dashpot in parallel with a spring.
-Specific damping capacity: is defined as the loss per cycle divided by the peak potential energy U.
(8)
Loss coefficient : is defined as the ratio of damping energy loss per radian divided by the peak potential or strain energy U.
(9)
-For the case of linear damping where the energy loss is proportional to the square of he strain or amplitude (eq.4) the hysteresis curve is an ellipse.
-When the damping loss is not a quadratic function of the strain or amplitude, the hysteresis is no longer as ellipse.
Equivalent Viscous Damping:
-The primary influence of damping on oscillatory system is that of limiting the amplitude of response at resonance.
-The damping has little influence on the response in the frequency regions away from resonance.
-The equivalent damping is found by equating the energy dissipated by the viscous damping to that of the non-viscous damping force with assumed harmonic motion.
(11)
where must be evaluated from the particular type of damping force.
-In the case of viscous damping, the amplitude at resonance, equation was found to be
(10)
-For other types of damping, no such simple expression exists. It is possible, however, to approximate the resonant amplitude by substituting an equivalent damping in the above equation.
Example 3.3
A single –degree-of-freedom viscosity damped system has a spring stiffness of 6000 N/m, critical damping constant 0.3 Ns/mm, and a damping ratio of 0.3. If the system I given an initial velocity of 1 m/sec, determine the maximum displacement of the system.
Solution:
The natural frequency of the system is given by
Given
or
hence
Damping ratio
or
Now assuming and m/s, the general expression for displacement is:
For maximum displacement and
Hence m