## Structural Dynamics Examples

Undamped Free Vibration

**Example 1 - Calculating the natural circular frequency, period, and frequency of a simply supported beam with a concentrated load in the middle**

A simply supported beam having a concentrated weight at its midspan is shown below.

Assume: Mass of beam negligible in comparison to mass of concentrated weight

Neglect effects of damping.

Determine equation of motion, natural circular frequency, period, and natural frequency

Let W = 10 kips, I = 200 in^4, E = 29,000 ksi, L = 20'

**Example 2 - Calculating the natural circular frequency, period, and frequency of a cantilevered beam with a concentrated load at the end**

A cantilevered beam having a concentrated weight at its end is shown below.

Assume: Mass of beam negligible in comparison to mass of concentrated weight

Neglect effects of damping.

Determine equation of motion, natural circular frequency, period, and natural frequency

Let W = 10 kips, I = 200 in^4, E = 29,000 ksi, L = 20'

**Example 3 - Calculating the natural circular frequency, period, and frequency of a fixed fixed beam with concentrated load at midspan**

A simply supported beam having a concentrated weight at the given location is shown below.

Assume: Mass of beam negligible in comparison to mass of concentrated weight

Neglect effects of damping.

Determine equation of motion, natural circular frequency, period, and natural frequency

Let W = 10 kips, I = 200 in^4, E = 29,000 ksi,

L = 20', a = 15', b = 5'

**Example 4 - Calculating the natural circular frequency, period, and frequency of a cantilevered beam with a concentrated load at any point along its length**

A fixed-fixed beam having a concentrated weight at its midspan is shown below.

Assume: Mass of beam negligible in comparison to mass of concentrated weight

Neglect effects of damping.

Determine equation of motion, natural circular frequency, period, and natural frequency

Let W = 10 kips, I = 200 in^4, E = 29,000 ksi, L = 20'

**Example 5 - Calculating the natural circular frequency, period, and frequency of a simply supported beam with a concentrated load at any point along its length**

A cantilevered beam having a concentrated weight at the given location is shown below.

Assume: Mass of beam negligible in comparison to mass of concentrated weight

Neglect effects of damping.

Determine equation of motion, natural circular frequency, period, and natural frequency

Let W = 10 kips, I = 200 in^4, E = 29,000 ksi,

L = 20', a = 15', b = 5'

**Example 6 - Calculating the natural circular frequency, period, and frequency of a steel shear frame structure with pinned supports**

For the steel shear frame structure, determine the natural circular frequency, natural frequency, and period of vibrations for oscillations in the horizontal direction.

The horizontal girder supports a dead weight of 30 kips uniformly distributed along its length. Assume the horizontal girder to be infinitely rigid with respect to the columns. Neglect the mass of the columns that bend abut their strong axis.

**Example 7 - Calculating the natural circular frequency, period, and frequency of a steel shear frame structure with fixed supports**

For the steel shear frame structure, determine the natural circular frequency, natural frequency, and period of vibrations for oscillations in the horizontal direction.

The horizontal girder supports a dead weight of 30 kips uniformly distributed along its length. Assume the horizontal girder to be infinitely rigid with respect to the columns. Neglect the mass of the columns that bend abut their strong axis.

**Example 8 - Calculating the natural circular frequency, period, and frequency of a steel shear frame structure with fixed and pinned supports**

For the steel shear frame structure, determine the natural circular frequency, natural frequency, and period of vibrations for oscillations in the horizontal direction.

The horizontal girder supports a dead weight of 30 kips uniformly distributed along its length. Assume the horizontal girder to be infinitely rigid with respect to the columns. Neglect the mass of the columns that bend abut their strong axis.

Damped Free Vibration

**Example 1 - Calculating the amplitude of free vibration of a damped spring-mass system after "n" oscillation**

The damped spring-mass system shown in the figure below has the following properties:

Mass, m = 0.0259 lb-sec^2/in

Stiffness, k = 20 lb/in

Damping Coefficient, c = 0.1 lb-sec/in.

If the mass is suddenly released with zero initial velocity from an initial displacement of 2.0 in, find the maximum amplitude of free vibration of the mass after 5 and 20 oscillations.

**Example 2 - Calculating the damping coefficient and mass displacement of a critically damped spring-mass system**

The critically damped spring-mass system shown in the figure below has the following properties:

Weight, W = 10 lbs

Stiffness, k = 20 lb/in

Initial Displacement, xo = 2 in

Initial Velocity, vo = 0 in/sec

Determine the damping coefficient for the system and the mass displacement after 0.1 and 0.3 seconds

**Example 3 - Calculating logarithmic decrement and damping factor for a viscously damped system**

For a viscously damped system, a certain vibration displacement is measured to be 80% of the immediately preceding amplitude within a free vibration trace. Find the damping factor for this system.

**Example 4 - Calculating the damping factor and damping coefficient for a viscously damped system**

For a viscously damped system, free vibration trace measurements show a 60% reduction in vibration amplitude after 15 cycles . The critical damping constant, Cc, for the system is 50 N-sec/m. Find the damping factor and damping coefficient for this system.

**Example 5 - Calculating the structural damping coefficient and equivalent viscous damping coefficient for a SDOF bridge span**

The main span of a bridge structure has the following properties based on vibration tests:

Effective Mass = 400 x 10^3 kg

Effective Stiffness = 40,000 kN/M

Ratio of successive displacement amplitudes from free vibration trace = 1.05. The span is considered a SDOF system to in order to calculate the fundamental frequency,

Determine the structural damping coefficient and the equivalent viscous damping coefficient.

**Example 6 - Calculating the structural damping coefficient and equivalent viscous damping coefficient for a simply supported steel beam**

A simply supported W24x62 steel beam spans a length of 50 ft and supports a concentrated 20 kip load at its center. Free oscillations of the beam experience an amplitude decay of 0.75% per cycle.

Assuming that the mass of the beam is negligible compared to the concentrated load, find the structural damping coefficient and equivalent viscous damping coefficient.

Young's Modulus, E = 29000 ksi

**Example 7 - Calculating the vibration displacement amplitude for a SDOF system subjected to Coulomb frictional damping**

The single degree of freedom system (SDOF) shown below has the following characteristics:

Weight, W = 2 kips

Stiffness, k = 2 kips/in

Coefficient of Friction = 0.15

Initial Displacement, x0 = 4 inches

Initial Velocity, v0 = 0 inches/sec

Find the vibration displacement amplitude after 4 cycles and the number of cycles of motion needed to for the mass to come to rest

Response to Harmonic Excitation

**Example 1 - Response of undamped spring-mass system to harmonic force excitation**

The undamped spring-mass system shown in the figure below has the following properties:

Mass, m = 5 kg

Spring Stiffness, k = 5 N/mm

Initial Displacement, x(0) = 10 mm

Initial Velocity, v(0) = 150 mm/sec

It is excited by a harmonic force with the following characteristics:

Amplitude, Fo = 100 N

Excitation Frequency = 20 rad/sec

Find the (I) frequency ratio, (II) the amplitude of the forced response, (III) the displacement of the mass at time t = 2 sec., and (IV) the velocity of the mass at time t = 4 sec.

**Example 2 - Determining beat period of an undamped system subject to harmonic steady state force**

An undamped system is harmonically forced which leads to a beating condition. Based on the following parameters:

Natural Frequency of the System - 1500 cycles / minute

Excitation Frequency - 1475 cycles / minute

Weight of System - 10,000 lbs

Amplitude of Steady-State Force - 5000 lbs

Determine:

A) Beat period of the resulting motion

B) Number of oscillations within each beat

C) Maximum amplitude of the oscillations

**Example 3 - Calculating the vibration displacement amplitude for a SDOF system subjected to harmonic force at resonance**

The single degree of freedom (SDOF) system shown below has the following properties:

Total Weight = 1000 lbs

Spring Stiffness = 2500 lbs/in

If the system is excited at resonance by a harmonic force with an amplitude of Fo = 500 lbs. Find the displacement amplitude of the force response after

A) 1 1/4 cycles

B) 10 1/4 cycles

C) 20 1/4 cycles

**Example 4 - Calculating the steady-state displacement parameters for a frame structure with rigid deck subjected to harmonic excitation**

The structure shown below has the following properties:

Mass = 100 kg

Translational Stiffness = 50,000 N/m

Damping Factor = 0.1

It is subjected to a harmonic force having an amplitude of 500 N and an exciting frequency of 10 rad/sec. For the steady-state vibration:

A) Find the amplitude of the steady-state displacement

B) The phase of the steady-state displacement with respect to the harmonic exciting force

C) Maximum Velocity of the response

Also, find the equation of total displacement of the structure as a function of time if the initial displacement is 5 cm and the initial velocity is 0 cm / sec.

**Example 5 - Calculating the resonant and maximum amplitude of a structure excited by a harmonic force**

Calculate the resonant amplitude and maximum amplitude of steady-state vibration for a structure with the following features:

Structure Weight = 1000 lbs

Stiffness = 800 lbs/in

Damping Coefficient = 5 lb-sec/in

It is excited by a harmonic force of amplitude Fo = 500 lbs

**Example 6 - Calculating the vibration displacement amplitude for a rotating machine experiencing a mass unbalance**

The rotating machine shown in the figure below has a total weight of 500 lbs. When the operating speed reaches 2000 rpm, the machine exhibits an unbalancing force of 5 lbs at an eccentricity of 5 inches. Resonance was seen to occur at an operating speed of 1000 rpm, and the damping factor is estimated to be .001. Find the vibration amplitude for the system at 2000 rpm. Also, find the stationary value of the steady-state vibration amplitude if the operating speed is very high.

**Example 7 - Calculating the vibration displacement amplitude for a system undergoing base excitation**

A vehicle is equipped with an instrument panel weighing 40 lbs through the use of a flexible mount that has an equivalent stiffness k = 500 lbs/in . The vehicle vibrates with a steady state amplitude of 0.25 in at a frequency of 40 Hz because of the engine vibration. Calculate the steady-state amplitude of vibration of the instrument panel. Consider damping to be negligible.

**Example 8 - Steady state response and transmissibility of a rotating machine experiencing a mass unbalance**

The rotating machine shown in the figure below weighs 150 lbs and is mounted on a structure that has a stiffness k = 1000 lbs/in and damping factor equal to 0.1. It's operating speed is 500 rpm, and the machine experiences a rotating imbalance of 10 lb-in.

Find the:

A) Amplitude of steady-state vibration

B) Transmissibility

C) Maximum dynamic force transmitted to the base

**Example 9 - Calculating required spring constant to achieve a specified level of isolation**

An instrument package weighing 300 lbs is mounted on a train console. When the train is running and in operation, the console experience a vibration amplitude of 0.5 in at a frequency of 40 Hz. Damping effects are negligible

Calculate the:

A) Required spring constant of the mount to achieve 80% isolation

B) The vibration amplitude of the instrument package.