Book summary “Mechanical vibrations”: summary

Introduction

You face constant mechanical vibration problems when designing rotating machinery, engines, and supporting structures. Unwanted oscillations cause structural fatigue, excessive noise, bearing failures, and catastrophic resonance. J. P. Den Hartog provides a comprehensive engineering framework in Mechanical Vibrations to analyze and solve these problems across industrial applications.

The central idea is that you can master linear, torsional, and nonlinear oscillations through rigorous mathematical modeling and practical mitigation techniques. You will learn to calculate natural frequencies, isolate vibrating equipment, balance rotors, and install dampers.

The book moves from single-degree systems and harmonic motion to complex multi-degree systems, continuous beams, reciprocating engine balancing, and shaft critical speeds.

Consider how you can apply these classical vibration methods by exploring questions such as:

• How do you calculate natural frequencies for complex elastic structures?
• What mechanisms cause destructive resonance in rotating machinery?
• How do you design effective vibration absorbers and dampers?

1. Kinematics and Single-Degree Systems

Harmonic motions, rotating vector representations, and differential equations form the foundational framework for analyzing single-degree-of-freedom mechanical systems.

A harmonic motion is the simplest form of periodic motion. You define it through amplitude, period, and circular frequency. A body suspended from a spring vibrates vertically between positions 1 and 1.5 inches above the ground twenty times per second, yielding an amplitude of 0.25 inches, a period of 1/20 seconds, a frequency of 20 cycles per second, and an omega of 126 radians per second.

You can visualize reciprocating motions by treating them as projections of rotating vectors. Adding two motions $x_1 = 5 \sin 25t$ and $x_2 = 10 \sin (25t + 1)$ using a vector diagram where one vector has length 5 and the other has length 10 turned 1 radian yields a sum vector length of 13.4 inches. Complex numbers provide an efficient numerical method for handling these rotating vectors.

Beats occur when you superpose two vibrations of nearly equal frequencies. A body describing $x_1 = 3 \sin 40t$ and $x_2 = 4 \sin 41t$ has a maximum amplitude of 7 inches, a minimum of 1 inch, a beat circular frequency of 1 radian per second, and a full beat period of 6.28 seconds.

Hydraulic-turbine penstock vibrations caused by runner and guide vane interactions can be mitigated by modifying the number of runner buckets. In a Francis turbine operating at 400 r.p.m. with 18 guide vanes and a 17-bucket runner, violent vibrations occurred at 113.3 cycles per second. Replacing the 17-bucket runner with a 16-bucket runner caused impulses from adjacent vanes to arrive in phase opposition, reducing penstock vibration amplitude to one-third.

Over a full cycle, a harmonic force does net work only when it has a component in phase with the velocity. A force of $10 \sin 2\pi 60t$ acting on a displacement of $\sin [2\pi 60t - 45^\circ]$ at 60 cycles per second performs 133 inch-pounds of work during the first second. You can decompose any non-harmonic periodic motion via Fourier series into a sum of fundamental and higher harmonic sine and cosine waves.

The number of degrees of freedom equals the independent coordinates required to specify position. A 12-cylinder gas engine with a rigid crank shaft and rigidly mounted cylinder block has only one degree of freedom because the crank angle determines every moving part, but mounting the block on flexible springs increases it to seven degrees of freedom. You derive the differential equation of motion directly from Newton's second law by summing inertia, damping, spring, and external forces.

2. Free and Forced Vibrations

Free undamped vibrations depend solely on mass and spring stiffness, while viscous damping causes exponential decay and external periodic forces create resonance.

Free undamped vibrations follow harmonic sine and cosine functions whose natural frequency depends solely on the system mass and spring stiffness. You can calculate the natural circular frequency as the square root of spring stiffness divided by mass. The acceleration of gravity $g$ equals 386 in./sec.² in these calculations.

Viscous damping causes free vibrations to decay exponentially, with consecutive amplitude maxima decreasing in a geometric series defined by the logarithmic decrement. In a specific beam example, the rate of decay is 0.038 and the ratio of consecutive amplitudes is 0.962.

When the frequency of an external periodic force matches the natural frequency of an undamped system, the vibration amplitude theoretically becomes infinite, a phenomenon known as resonance. For instance, a 200 hp motor generator set running at 725 r.p.m. connected by a shaft near resonance exhibits a magnification factor of 6.0 and a maximum shear stress of 14,500 lb./in.² in the example shaft.

Mechanical systems involving mass, spring stiffness, and viscous damping share the exact same mathematical differential equation as series alternating-current electrical circuits containing inductance, capacitance, and resistance. Moving the upper support of a spring-mass system harmonically is completely equivalent to applying an external harmonic force directly to the suspended mass.

A complex geared torsional system can be reduced to an equivalent non-geared single-degree-of-freedom system by scaling the inertias and spring constants. You apply specific reduction steps:

• Divide the system into separate parts each having the same speed within itself.
• Choose one part as the base and assign speed ratios n to each of the other parts.
• Remove all gears and multiply all spring constants k and inertias I by n squared.

Frequency-measuring instruments for low frequencies spanning 6 cycles per second to 60 cycles per second can be constructed using cantilever spring-steel reeds that exhibit large resonance amplitudes at specific matching frequencies. Frahm's tachometer contains a box of small cantilever spring-steel strips with white tips. You operate a single-reed frequency meter using clear instructions:

• Press the clamped end of a single-reed frequency meter firmly against the vibrating object.
• Turn the screw slowly to vary the free length of the reed until it shows a large amplitude at resonance.

Seismic instruments utilize an elastically suspended mass inside a housing to measure either absolute displacements or accelerations depending on the instrument's natural frequency relative to the vibration. Geiger vibro- and torsiographs use a pulley and a heavy flywheel connected by a torsional spring. Accelerometers maintain a critical damping range of 0.5 to 0.7 to avoid phase distortion.

3. Vibration Isolation and Absorbers

Electrical measuring instruments, soft spring isolation, and tuned dynamic vibration absorbers protect structures and machines from transmitting high-frequency disturbances.

Seismo-electric pickups convert mechanical vibrations into electrical voltage using a seismic mass and a magnetic field. Units like the Sperry-M.I.T. pickup operate on principles similar to radio loudspeakers, generating an alternating voltage proportional to relative velocity when a seismic magnet moves relative to a voice coil. Draper and Bentley developed this linear vibration pickup, marketed by the Sperry Gyroscope Company and Consolidated Engineering Corporation with dimensions of about 1 in. and a weight not exceeding 2 oz. An integrating circuit can transform a velocity record directly into an amplitude record by feeding the voltage into a C-R-series circuit where the resistance voltage is many times, say ten times, greater than the condenser voltage.

The electrical-resistance-strain-sensitive wire gage enables the measurement of vibrations in high-centrifugal-field environments like airplane propellers. Simmons and Datwyler first used the wire strain gage, further developed by Ruge and De Forest and marketed as the SR-4 gage by the Baldwin Southwark Co. This gage features a wire thickness of 0.001 in., a total electrical resistance of about 500 ohms, and a strain-sensitivity factor of about 3. A stroboscope uses intermittent flashes of light to make rapid vibratory motions appear stationary or move slowly. The General Radio Company markets an instrument called Strobotac, developed by Edgerton, requiring at least 15 flashes per second for a non-flickering illusion and a flash duration of 1/135,000 sec. or less for a 10,000 r.p.m. machine with 4 in. radius and less than 1/32 in. blurring.

Mounting an unbalanced machine on very soft springs prevents vibrations from being transmitted to the supporting structure. A natural frequency one-fifth of disturbing frequency yields a transmissibility of 1 part in 24. Single-phase electrical machinery produces a periodically fluctuating torque at twice the line voltage frequency. Large single-phase generators in a New York City basement caused humming complaints from residents blocks away whose floor happened to be tuned to 120 cycles per second from a line voltage frequency of 60 cycles per second. Automobile engines can be isolated from the chassis frame by mounting them on journals and a cantilever spring along the principal axis of rotation. This floating power arrangement accommodates an engine weight of 400 lb., a radius of gyration of 6 in., and a cantilever length of 4 in.

A system with two degrees of freedom possesses two distinct natural frequencies and natural modes of motion. Wilberforce's spring demonstrates energy transfer between an up-and-down motion and a twisting motion through a coupling spring. Attaching a small auxiliary spring-mass system tuned to the disturbing frequency eliminates vibrations in a main machine. Invented by Frahm, the dynamic vibration absorber utilizes an absorber mass ratio of one-fifth or one-tenth of the main mass. Electric hair clippers use a vibration absorber tuned to 120 cycles per second to prevent housing vibration in the barber's hands.

4. Damped Absorbers and Ship Stabilization

Damped vibration absorbers, Lanchester friction elements, and anti-rolling systems eliminate resonant vibrations and control large-scale marine and automotive oscillations.

Adding a secondary spring-mass-dashpot system to a machine reduces resonant vibrations of the main mass to a minimum if properly tuned and damped. You can plot the vibration absorber's amplitude ratio by treating mechanical systems using complex numbers and equivalent electrical circuits. All resonance curves for various damping values intersect at two fixed points, P and Q.

Optimizing the system requires adjusting the absorber's tuning so the fixed points are at equal heights and choosing an optimum damping coefficient so the curve passes horizontally through one of them. Applying a damper where the main mass is 20 times the damper mass and tuned to the same frequency illustrates how resonance peaks are brought down to a minimum.

A Lanchester damper replaces the absorber spring with a linear dashpot or friction element, offering simpler construction at the cost of lower efficiency. When the spring constant k equals zero, the device acts as a viscous or Coulomb friction Lanchester damper. Shrunk-on steel or cast-iron rings on the inside rim of gears act as Lanchester dampers to eliminate ringing noise.

Ship rolling in rough seas can be mitigated using passive or active anti-rolling tanks, gyroscopes, or underwater hydrofoils. Frahm anti-rolling tanks use communicating water tanks to act as a secondary absorber system. Frahm anti-rolling tanks were installed on the large German liners Bremen and Europa, and hydrofoil stabilizers were used on British destroyers.

Automobile ride comfort is evaluated through the jerk metric, and hydraulic shock absorbers introduce necessary damping across soft body springs. A car possesses 18 degrees of freedom, dominated by body bobbing, pitching, and axle bouncing. Cars running over concrete highways with joints spaced at 40 feet experience violent pitching at medium speeds like 30 m.p.h. when shock absorbers are absent.

Isolating powerful engines on light, non-rigid foundations like ship hulls or aircraft wings requires significantly more flexible springs than rigid foundations. Mounting an engine on a light ship foundation with a mass ratio of 0.1 requires a static spring sag of 2.5 inches instead of 0.25 inches.

5. Multi-Degree Systems and Continuous Beams

Multi-degree systems, continuous strings, beam bending frequencies, and Stodola's iterative method enable precise calculation of complex modal shapes.

When an alternating force acts on a specific mass in a multi-degree-of-freedom system, you find resonance frequencies where that driven mass remains completely stationary while the rest of the system vibrates. Influence numbers and dimensionless amplitudes help you solve these undamped equations. In a stretched string with three equidistant masses excited at the first mass, the first mass remains stationary at two distinct frequencies corresponding to $x = 1$ and $x = 3$ while the other masses vibrate.

In systems with damping, motion requires a complex frequency formulation that combines the natural frequency and the rate of decay. Substituting an exponential-harmonic assumption yields a sixth degree frequency equation where the root $s$ is a complex number representing both the decay exponent $p$ and the natural frequency $q$. For small damping values, you can use an energy method based on undistorted normal modes. Equating the work dissipated per cycle to the kinetic energy or work input at a $90\text{ deg.}$ phase angle bypasses cumbersome differential solutions.

Transverse vibrations of strings, longitudinal vibrations of bars and organ pipes, and torsional vibrations of uniform shafts share an identical partial differential equation. An air column in an organ pipe behaves identically to a steel column undergoing longitudinal vibrations, substituting gas pressure modulus for elasticity modulus. Rayleigh's method provides an approximate procedure for finding the lowest fundamental frequency by equating maximum potential and kinetic energies based on an assumed deflection shape. Applying Rayleigh's method to a vibrating string using a parabolic arc shape yields a frequency that is only $0.7\text{ per cent}$ greater than the exact value, while an improbable triangular-like shape results in a $10\text{ per cent}$ error.

To apply Rayleigh's method to complex continuous systems, you follow these exact steps:

• Assume a shape for the first normal elastic curve.
• Calculate the maximum potential energy based on the assumed shape and elastic properties.
• Calculate the maximum kinetic energy based on the assumed shape and mass distribution.
• Equate the maximum potential and kinetic energies to solve for the frequency squared.

The natural frequencies of a uniform beam in bending increase proportionally to the square of integers rather than linearly. Consecutive natural frequencies for a simply supported beam scale as $1, 4, 9, 16$, unlike strings where they scale as $1, 2, 3, 4$. Stodola's method offers an iterative procedure to calculate the fundamental frequency and deflection shape of non-uniform beams and turbine rotors using a modulus of elasticity of $30 \times 10^6\text{ lb. per square inch}$ for steel shaft examples.

When you evaluate variable shafts, you execute a structured iteration process:

• Draw an initial reasonable assumed deflection curve for the shaft.
• Multiply the curve by the mass per unit length and a trial frequency squared to obtain an assumed inertia loading.
• Integrate the inertia loading twice to find the bending moment curve using a $22.5\text{ in.-lb.}$ horizontal distance scale.
• Divide the bending moment by the variable bending stiffness $EI$ to form the loading diagram for the second integration.
• Integrate twice more to construct the resulting new deflection curve.
• Compare the initial and final deflection ordinates to determine the first approximation for the natural frequency.

Normal elastic curves form an orthogonal set of functions that allow any arbitrary deflection curve to be expanded into a convergent series using Maxwell's reciprocity theorem and influence functions. Because Stodola's iteration automatically magnifies the lowest mode, you must purify trial curves to isolate higher modes. You eliminate lower-mode contamination by following a specific sequence:

• Assume a trial shape for the higher mode containing potential lower-mode impurities.
• Evaluate the contamination coefficient by integrating the product of the known lower mode, the trial shape, and the mass distribution.
• Subtract the lower-mode proportion from the assumed shape to yield a purified curve.
• Apply the Stodola iteration cycle to the purified curve.

6. Reciprocating Engines and Torsional Vibrations

Crank mechanism dynamics, multicylinder inertia balance, and Holzer's tabular method reveal the natural frequencies of complex torsional systems.

Vibration phenomena in reciprocating machines originate from transmission to the foundation by the engine as a whole and torsional oscillations in the shafting, driven by periodic accelerations and gas pressure variations. These effects stem from a combination of periodic accelerations of moving parts like pistons, rods, and cranks, alongside periodic variations in cylinder steam or gas pressure, governed by Newton's laws.

The displacement, velocity, and acceleration of a piston in a crank mechanism can be expressed through primary terms varying at crankshaft frequency and secondary terms varying at double frequency. Using an exact displacement formula expanded into a power series, the motion of the piston deviates from a simple sinusoid due to the finite length of the connecting rod, introducing primary and secondary components. You encounter a 1 part in 2,000 error margin when using the approximation formula, while usual engines maintain a $r/l$ ratio of $1/4$.

Multicylinder engines can achieve inertia balance of primary and secondary forces and moments by properly arranging crank angles and counterbalancing masses. Using vector diagrams where crank arrangements represent rotating force and moment vectors, you can analyze and eliminate unbalance across different cylinder configurations.

A triple expansion steam engine with piston weights in the ratio $1:1/2:2$ requires specific crank angles calculated via equilibrium equations to achieve primary force balance. The solution results in a $28\text{ deg}$ angle for $\beta$ and a $47\text{ deg}$ angle for $\alpha$ in the triple expansion steam engine example.

To execute this balance:

• Draw the vector of the main unit length vertically in the equilibrium diagram.
• Arrange remaining vectors so their horizontal components balance and vertical components sum to the required units.
• Calculate cosines and angles using trigonometric relations.

The Holzer method provides a systematic tabular procedure of successive approximations to determine the natural frequencies and complete mode shapes of torsional vibration in complex machine systems. By assuming an initial frequency and a unit angular amplitude for the first disk, the method calculates inertia torques, shaft flexibilities, and windup angles sequentially across all disks, checking the remainder torque at the end.

A V-8 4-cycle Diesel engine rated at $50\text{ hp}$ per cylinder at $2,000\text{ r.p.m.}$ and driving a $250\text{ kw}$ generator is analyzed using a Holzer table to find its first and second natural frequencies and mode shapes, yielding successive $\omega^2$ tries of $300,000$ and $310,000$ for the first mode.

Apply this tabular process:

• Guess an initial frequency and set the angular amplitude of the first disk to $1$ radian.
• Calculate the inertia torque of the first disk and the resultant shaft torque.
• Compute the shaft windup angle and subtract it from the previous disk amplitude to find the next disk amplitude.
• Proceed sequentially across all disks to find the final remainder torque.
• Adjust the frequency guess and repeat until the remainder torque approaches zero.

7. Torsional Critical Speeds and Dampers

Torque harmonic components and critical speeds require careful analysis, propeller damping, silicone fluid dampers, and optimized firing orders to avoid structural failure.

Engine torque is irregular and can be broken down into harmonic components. Four-cycle internal-combustion engines uniquely produce half-order harmonics such as 0.5, 1.0, and 1.5 due to their two-revolution firing cycle of 720 deg. Two-cycle and steam engines only exhibit integer orders. F. P. Porter published harmonic coefficients for eight widely different engine types in the Transactions of the A.S.M.E. in 1943.

Torsional vibration amplitudes become significant when a torque harmonic frequency coincides with a natural frequency of the system. You define these operating points as critical speeds. The author observed an anchor chain 500 feet away on a Diesel oil tanker visibly and audibly vibrating at the engine torsional frequency.

Hydrodynamic action of a ship propeller provides substantial damping for torsional vibrations because retarding torque increases with propeller speed variations. F. M. Lewis studied a large number of engines and found that resonant magnification typically causes maximum shaft torque to be about 50 times the calculated exciting torque, ranging from 25 to 100 times.

Lanchester and Houde viscous dampers dissipate torsional vibration energy through relative slip and friction. The Houde damper, manufactured by the Houdaille-Hersey Corporation, uses a silicone fluid with a viscosity that varies less drastically with temperature than oil.

Foettinger's hydraulic coupling transmits torque through fluid circulation and Coriolis forces while supplying natural positive damping to torsional vibrations. A hydraulic seal at D with little friction permits the cover C to hold the fluid in place while the follower turns slightly slower than the driver.

Tuned centrifugal pendulums act like an infinite mass for the specific harmonic order to which they are tuned. Sarazin in France and Chilton in the United States independently invented the bifilar type where a U-shaped weight rolls without slipping on pins through circular holes.

Changing the firing order in a multi-cylinder engine shifts minor critical speed severities in the spectrum. A 10-cylinder engine allows 181,440 possible firing orders.

Apply these steps during design:

• Specify the damping fluid and clearance between flywheel and housing to hit the optimum damping value.
• Replace the damper with a moment of inertia equal to that of the hub and housing plus half that of the loose flywheel when starting Holzer calculations.
• Install three pendulums located 120 degrees apart to prevent all motion in the plane of the crank throw when facing mixed torque, lateral, and radial excitations.
• Examine possible firing sequences during a fresh design stage on paper.
• Construct vector diagrams for remaining acceptable crankshaft configurations after eliminating poor engine inertia balance designs.
• Select the configuration with the smallest vector resultant for the target running speed range.

8. Critical Speeds and Rotor Balancing

Shaft critical speeds, static and dynamic unbalance correction, and flexible rotor field balancing eliminate dangerous centrifugal force forces in rotating machinery.

Critical speeds occur when the angular speed of a shaft coincides with the natural frequency of the non-rotating shaft on its elastic bearings, leading to violent vibrations. A disk with mass and eccentricity generates a centrifugal force acting as a vibrating force. When this force is in resonance with the natural frequency, the deflection tends to become infinite. Below the critical speed the heavy side flies out, whereas above the critical speed the light side flies out.

Holzer's method, traditionally used for torsional calculations, can be extended to flexural vibrations to find critical speeds of beams or shafts. The beam is divided into sections with concentrated masses at the ends. Using equilibrium and deformation equations across cuts, a Holzer table is constructed. Modern digital computers are essential to perform these extensive iterations efficiently.

To find flexural critical speeds:

• Divide the beam into a convenient number of sections.
• Calculate the mass of each section, divide it into halves, and concentrate these halves at the two ends of each section.
• Assume a frequency and proceed from section to section along the beam using equilibrium and deformation equations.

A rotor can be in static balance while remaining dynamically unbalanced if equal and opposite masses create a rotating moment on the bearings. Static balance alone is insufficient for elongated bodies because offsetting masses can preserve the center of gravity on the shaft center line while still generating centrifugal force couples during rotation. Any unbalance in a rigid rotor can be corrected by placing weights in two accessible end planes.

Unbalance in a rotor plane can be reliably determined by observing vibration amplitudes through three or four specific test runs.

Automatic balancing devices can employ free-to-rotate arms or balls that find the position of complete balance above the critical speed. Thearle invented a machine with two arms that are clamped during startup and released above the critical speed. Centrifugal forces automatically drive the arms to the position where all vibration ceases.

Flexible rotors undergo deformation at higher speeds, requiring field balancing or multi-plane balancing under service conditions. At speeds above half the first critical, a flexible rotor bows and sets up additional centrifugal forces. Because correction weights vary depending on the operating speed and critical mode, shop balancing is often insufficient for large turbogenerators.

Local heating at the high spot of a rotor can cause a slow retrograde or advancing spiral wandering of unbalance over time. Unsymmetrical heating from rubbing or steam condensation expands rotor fibers, shifting the center of gravity and creating a very slow rotating unbalance. This results in periodic, slowly varying vibration intensities that require careful monitoring.

Secondary critical speeds occur at half the primary critical speed due to gravity combined with unbalance or non-uniform shaft stiffness. Non-uniform bending stiffness in a shaft or gravity acting on an eccentric rotating disk generates alternating disturbing forces with a frequency of 2ω. When the shaft runs at half its critical speed, these impulses trigger secondary vibrations.

10 Key Ideas

1. Harmonic motions, rotating vector representations, and differential equations form the foundational framework for analyzing single-degree-of-freedom mechanical systems.
2. Free undamped vibrations depend solely on mass and spring stiffness, while viscous damping causes exponential decay and external periodic forces create resonance.
3. Electrical measuring instruments, soft spring isolation, and tuned dynamic vibration absorbers protect structures and machines from transmitting high-frequency disturbances.
4. Damped vibration absorbers, Lanchester friction elements, and anti-rolling systems eliminate resonant vibrations and control large-scale marine and automotive oscillations.
5. Multi-degree systems, continuous strings, beam bending frequencies, and Stodola's iterative method enable precise calculation of complex modal shapes.
6. Crank mechanism dynamics, multicylinder inertia balance, and Holzer's tabular method reveal the natural frequencies of complex torsional systems.
7. Torque harmonic components and critical speeds require careful analysis, propeller damping, silicone fluid dampers, and optimized firing orders to avoid structural failure.
8. Shaft critical speeds, static and dynamic unbalance correction, and flexible rotor field balancing eliminate dangerous centrifugal forces in rotating machinery.
9. A rotor can be in static balance while remaining dynamically unbalanced if equal and opposite masses create a rotating moment on the bearings.
10. Non-uniform bending stiffness in a shaft or gravity acting on an eccentric rotating disk generates alternating disturbing forces with a frequency of two omega.