Passive measures for vibration minimisation using the example of a load-bearing structure with supported machine :: Tutorial (Fraunhofer LBF)

Problem description

Mechanical substitute system

Real structures often have a high degree of complexity. For this reason, it makes sense to perform basic analyses on simplified structures. For this purpose, the real system is transformed into a mechanical substitute system. This must reproduce the properties of the real system to be analysed with sufficient accuracy. The engine is assumed to be a 12-cylinder engine with a total weight of 600 kg. Each cylinder has a mass of 16 kg with a stroke of 190 mm. The typical frequency range is 13 – 34 Hz, which  corresponds to a speed of 800 – 2000 rpm.

The idealised motor can be represented as an unbalance exciter with the total mass m_m, the rotational frequency Ω and the unbalance mass m_u, which moves around the radius r. The resulting force of the unbalance exciter is given by . In the simplest case, the ship’s floor is modelled as an elastic beam with the modulus of elasticity E, the area moment of inertia I and the length-related mass μ. This beam can be converted into an equivalent spring-mass-damper system with one degree of freedom. For this, the vibration properties of the beam are first calculated and then the parameters of the spring-mass damper system (m_s, k_s and d_s) are determined so that they reflect the vibration behaviour of the system at a specific position. In the following, it is assumed for the sake of simplicity that the motor is located in the centre of the beam (position 1, z = L/2) and that this position corresponds to the position at which the movement of the structure is measured (position 2) (see figure 2). [1]

Vibration characteristics of the beam: An I-section with the dimensions H = 0.2 m, B = 0.2 m, t = 0.02 m and a length L = 4.8 m is selected for the geometry of the beam (figure 3). The material is assumed to be steel with a modulus of elasticity of E = 2.1e+11 N/m² and a density of ρ = 7850 kg/m³. With these values, the moment of inertia can be calculated with = 7.1893e-05 m⁴ and the cross-sectional area with = 0.0112 m². [2] With the length-related mass μ = ρA = 87.92 kg/m and assuming negligibly low damping, the first three natural frequencies result from

with n = 1, 2, 3 to f₁ = 28.252 Hz, f₂ = 113.01 Hz and f₃ = 254.27 Hz. Figure 3 shows the corresponding eigenmodes a₁, a₂ and a₃. The first resonant frequency of the basic structure is of particular interest in the following, as it lies within the frequency range of the motor.

In order to obtain an equivalent single-mass oscillator, an equivalent mass and an equivalent stiffness are calculated, taking into account the bending shape and the potential and kinetic energy. The equivalent mass

results for the sinusoidal eigenmode shown in Figure 3 at the location to = 211.01 kg. Taking into account the concentrated total mass of the motor m_m, the equivalent total mass is = 811.01 kg. The equivalent stiffness is = 6.649e+06 N/m. A hysteretic damping of approximately η = 4% is assumed as the damping of the structure. The equivalent modal damping thus results in 1000 kg/s. [3]

Implementation in Simulink

The time domain simulations are carried out with Simulink. For this purpose, the model shown in Figure 4 is created, whereby the simulations are performed with the fixed-step solver and a fixed sampling rate of ts = 0.001 s for a time from 0 to t_end = 30 s. To implement the model, the blocks Stiffness1DOF, RigidBody1DOF and UnbalancedMassExcitation from the “Structure and Vibration” toolbox are used and parameterised with the previously determined values. A speed range of 0 to 3000 rpm is selected for the unbalance exciter with an acceleration rate of 100 1/min/s. The parameters of the blocks are summarised in Table 1:

For further processing of the data outside of Simulink, the signals are exported as “Structure with Time” (simout_0) using the Simulink To Workspace block. Figure 5 shows the results of the simulation in the time domain. The top two axes show the speed of the motor and the force resulting from the unbalance. The lower axis shows the speed of the beam at position 1. It is clearly recognisable that this increases sharply after about 9 seconds. This is due to the direct coupling of the machine mass and the beam.

% Display simulation data in time domain
time_0 = simout_0.time;
rpm_0  = simout_0.signals.values(:,1);
F_0    = simout_0.signals.values(:,2);
x_0    = simout_0.signals.values(:,3);

h = figure();
subplot(3,1,1);
title('Simulation result - Time domain')
ma_graphics.plot(time_0,rpm_0);
ylabel('rpm');

subplot(3,1,2);
ma_graphics.plot(time_0,F_0);
ylabel('F');

subplot(3,1,3);
ma_graphics.plot(time_0,x_0);
ylabel('$\dot{x}$','Interpreter','latex','FontSize',15);
xlabel('time');
ylim([-13,13]);
ah = findobj(h,'type','axes');
set(ah,'XGrid','on','YGrid','on');

In order to analyse the results more precisely, the simulation results are transformed into the frequency range using the function ma_frequency_response.

% Display simulation results in frequency domain
ma_frequency_response(time_0, F_0, x_0,{0,50});

The resonance at a frequency of approx. 14 Hz is clearly recognisable. The observed resonance corresponds to the result of the analytically calculated natural frequency of the system = 14.411 Hz.

Passive isolation using elastic bearings

By using elastic bearings, the dynamic forces that are introduced into a structure by a machine can be reduced. In the following, the procedure is shown using the marine engine considered in this tutorial. In addition to the dynamic behaviour of the machine, the properties of the basic structure, in this case the ship, must also be taken into account. In the case of a rigid mounting, as considered in the previous section, the displacements of the structure x_s and the motor x_m are identical. Vibration isolation can be achieved with an additional spring-damper element (k_i, d_i). A suitable choice of stiffness and damping parameters is crucial for this. To describe the isolation effect, the system consisting of motor mass and isolator-spring-damper is initially idealised (see figure 7).

The transmission behaviour of the isolator shown in Figure 7 is represented by the function

where and . The isolation effect occurs above the effective frequency . Figure 8 shows the transmission behaviour of the isolator as a function of the damping factor ξ. In the undamped case (ξ = 0), the amplitude at resonance approaches infinity. Above the isolation frequency, the transmission decreases with a value of 40 dB/decade. By increasing the damping, the amplitude at the point of resonance is reduced, but at the same time the isolation effect is reduced. [4]

In order to model vibration isolation by means of an elastic bearing of the motor, the model shown in Figure 2 is extended by a spring-damper element (see Figure 9). The stiffness of the isolation k_i must be selected taking into account the machine mass m_m = 600 kg. It is important to select k_i in such a way that the isolation effect covers the operating speed range . In this specific case, this means that the insulation frequency ω_i must be selected so that the effective isolation effect () starts below the lower speed limit of 800 rpm or a circular frequency of ω = 83.776 Hz. Assuming that k_i < k_s, the required insulation frequency can be determined with . This means that the stiffness must be less than 2.1055e+06 N/m. In this example, a value of 2000000 N/m is selected. This configuration results in an isolation frequency of = 9.1888 Hz.

The damping of the isolation d_i is important in order to limit the amplitude of the resonance frequency, as the excitation passes through it during run-up. On the other hand, it is important not to set the damping too high with regard to the isolation effect. In the following, a damping factor ζ of approx. 2% is assumed. This results in the damping constant of the isolator d_i with . In this example, an damping constant d_i = 1400 kg*s is assumed.

% Display simulation data in time domain

time_i = simout_i.time;
rpm_i  = simout_i.signals.values(:,1);
F_i    = simout_i.signals.values(:,2);
x_i    = simout_i.signals.values(:,3);

h = figure();
subplot(3,1,1);
title('Simulation result - Time domain')
ma_graphics.plot(time_i,rpm_i);
ylabel('rpm');

subplot(3,1,2);
ma_graphics.plot(time_i,F_i);
ylabel('F');

subplot(3,1,3);
ma_graphics.plot(time_i,x_i);
ylabel('$\dot{x}$','Interpreter','latex','FontSize',15);
xlabel('time');
ylim([-6,6]);
ah = findobj(h,'type','axes');
set(ah,'XGrid','on','YGrid','on');

% Display simulation results in frequency domain

[FREQ_0,AMPL_0,PHAS_0] = ma_frequency_response(time_0, F_0, x_0,{0,50},'window',true);
[FREQ_i,AMPL_i,PHAS_i] = ma_frequency_response(time_i, F_i, x_i,{0,50},'window',true);

figure();
ma_graphics.semilogy(FREQ_0,AMPL_0);
hold on;
ma_graphics.semilogy(FREQ_i,AMPL_i);
grid on
xlabel('Frequency in Hz')
ylabel('Amplitude in m/N')
legend('Stiff coupling','Isolation','Location','SouthWest')
xlim([5 50])

% display operation range
fu_min = 13;
fu_max = 33;

rectangle('Position',[fu_min,1e-6,fu_max-fu_min,1.9e-4],'FaceColor',[0 .5 .5 0.2],'EdgeColor','b', 'LineWidth',1);
% push rectangle (last plot) to the back
h = get(gca,'Children');
set(gca,'Children',[h(2:end)',h(1)]);
text(14,1.5e-6,'Operation range')

The next section examines the extent to which the resonances that occur can be reduced by using a vibration damper. First, however, the class ma_mosys of the ma_Toolbox is used to analytically analyse the simulation results.

% Define the mass matrix of the system, where the first entry defines the mass annotation
jData = [1 m_s;
         2 m_m];

% Define arrangement of the masses and the spring and damper values of the respective connections
kData = [1 inf k_s d_s; 
         1 2   k_i d_i];

% Force application points of the system
originFvalue = 2;

% Output velocity masses of the system
outputPvalue = 1;

system = ma_mosys(jData,kData,'originF',originFvalue,'outputP',outputPvalue);

% Create the state-space model
system.getSS;

% Use the matlab built-in function eig to calculate the eigen values
eigenValues = eig(system.SS);
eigenFrq_i = unique(abs(eigenValues))/(2*pi);

freq_analytic=0:0.1:50;
[mag_analytic, phase_analytic]=bode(system.SS,freq_analytic*2*pi);
mag_analytic = reshape(mag_analytic,1,length(freq_analytic));
 
h = figure();
ma_graphics.semilogy(FREQ_i,AMPL_i); 
hold on;
ma_graphics.semilogy(freq_analytic,mag_analytic);
hold off;
grid on
xlabel('Frequency in Hz')
ylabel('Amplitude in m/N')
legend('Simulink','Analytic','Location','SouthWest')
xlim([5 50])

Vibration reduction with a vibration absorber

It is possible to reduce vibrations in a structure through additional structural dynamic measures, such as a vibration absorber. This reduces the amplitude of a structural resonance. The procedure is first illustrated using the rigidly mounted motor. For this purpose, the equivalent system shown in Figure 2 is extended by a corresponding absorber (see Figure 14). In the equivalent spring-damper system, the motor mass and the absorber are assumed to be superimposed in the position z = L/2.

For an effective reduction effect, the absorber must be tuned to the natural frequency of the structure. The appropriate selection of the parameters m_t, k_t is therefore crucial. However, the tuning of the damping also influences the properties of the absorber. As analytical optimisation of the parameters is complicated, the parameters are estimated and the quality of the solution is evaluated and adjusted using the graphical representation of the transmission. The stiffness is to be calculated taking into account the dominant resonant frequency of the rigidly coupled motor f₁ = 14.411 Hz. The stiffness can be estimated from . The degree of damping D should be between 5% and 20%. By this the damping d_t can be determined by . When parameterising vibration dampers, it should be noted that the parameters m_t, k_t and d_t are mutually dependent. An iterative approach is therefore recommended to optimise the parameters. The resulting curve provides information about the quality of the parameters. Figure 15 a – c shows the influence of the parameters on the curve. One parameter was varied in each case to show the influence on the curve, while the other two parameters were adjusted to achieve the best possible curve. [3]

% Display simulation data in time domain

simout_label = {'d_t = 5%','d_t = 20%','Iteratively optimised'};

[FREQ_0,AMPL_0,PHAS_0]=ma_frequency_response(time_0, F_0, x_0,{0,50},'window',true);
h = figure();
ma_graphics.semilogy(FREQ_0,AMPL_0);
hold on;
for ind = 1:3
    [FREQ_t(:,ind),AMPL_t(:,ind),PHAS_t(:,ind)]=ma_frequency_response(time_t(:,ind), F_t(:,ind), x_t(:,ind),{0,50},'window',true);
     ma_graphics.semilogy(FREQ_t(:,ind),AMPL_t(:,ind));
     grid on
     xlabel('Frequency in Hz')
     ylabel('Amplitude in m/N')    
end
legend('Stiff coupling',simout_label{:},'Location','best')
xlim([5 50])
a=rectangle('Position',[f_u_min,1e-6,f_u_max-f_u_min,1.9e-4],'FaceColor',[0 .5 .5 0.2],'EdgeColor','b', 'LineWidth',1);
% push rectangle (last plot) to the back
ah = get(gca,'Children');
set(gca,'Children',[ah(2:end)',ah(1)]);
text(14,1.5e-6,'Operation range')

% Define the mass matrix of the system, where the first entry defines the mass annotation
jData = [1 m_s+m_m;
         2 m_t];

% Define arrangement of the masses and the spring and damper values of the respective connections
kData = [1 inf k_s d_s; 
         1 2   k_t_opti d_t_opti];

% Force application points of the system
originFvalue = 1;

% Output velocity masses of the system
outputPvalue = 1;

system = ma_mosys(jData,kData,'originF',originFvalue,'outputP',outputPvalue);

% Create the state-space model
system.getSS;

% Use the matlab built-in function eig to calculate the eigen values
eigenValues = eig(system.SS);
eigenFrq_t = unique(abs(eigenValues))/(2*pi);

freq_analytic=0:0.1:50;
[mag_analytic, phase_analytic]=bode(system.SS,freq_analytic*2*pi);
mag_analytic = reshape(mag_analytic,1,length(freq_analytic));
 
h = figure();
ma_graphics.semilogy(FREQ_t(:,3),AMPL_t(:,3)); 
hold on;
ma_graphics.semilogy(freq_analytic,mag_analytic);
hold off;
grid on
xlabel('Frequency in Hz')
ylabel('Amplitude in m/N')
legend('Simulink','Analytic')
xlim([5 50])

Combination of vibration isolation with an absorber

In the previous sections, vibration isolation and the use of a vibration absorber were considered separately. In order to achieve optimum vibration reduction, both structural-dynamic measures are combined in the following (Figure 19).

Firstly, the spring and damping values of the absorber must be adapted to the resonant frequency of the elastically mounted motor (see section: Passive isolation using elastic mounts). The mass of the absorber is again assumed to be m_t = 100 kg. As only the mass of the motor is taken into account in this case, this corresponds to a mass ratio of 17%. Based on the resonant frequency of 32.541 Hz prevailing in the operating range, the value for k_t is initially estimated to be 4.1803e+06 N/m with . This results in a value of 2044.6 kg*s for the damping constant d_t at a damping factor of 5% and a value of 8178.3 kg*s at 20% damping. Here too, the values for k_t and d_t were then iteratively optimised and determined as 2090200 N/m and 7228.8 kg*s.

simout_label = {'d_t_i = 5%','d_t_i = 20%','Iteratively optimised'};
h = figure();
ma_graphics.semilogy(FREQ_0,AMPL_0);
hold on;
for ind = 1:3
[FREQ_t_i(:,ind),AMPL_t_i(:,ind),PHAS_t_i(:,ind)]=ma_frequency_response(time_t_i(:,ind), F_t_i(:,ind), x_t_i(:,ind),{0,50},'window',true);
    ma_graphics.semilogy(FREQ_t_i(:,ind),AMPL_t_i(:,ind)); 
end
legend('Stiff coupling',simout_label{:},'Location','best')
xlim([5 50])
rectangle('Position',[f_u_min,1e-6,f_u_max-f_u_min,1.9e-4],'FaceColor',[0 .5 .5 0.2],'EdgeColor','b', 'LineWidth',1);
% push rectangle (last plot) to the back
hc = get(gca,'Children');
set(gca,'Children',[hc(2:end)',hc(1)]);
text(14,1.5e-6,'Operation range')
grid on;
xlabel('Frequency in Hz')
ylabel('Amplitude in m/N')

Concluding considerations

This tutorial showed how unwanted vibrations can be reduced by structural dynamic measures. Using the model of an internal combustion engine in a ship, the influence of an elastic mounting and a vibration absorber was discussed and it was shown how the dynamic behaviour can be simulated and evaluated using elements of the ma_Toolbox. Figure 22 compares the simulation results of the various measures. Elastic mounting of the motor isolates the excitation. This significantly reduces the vibrations introduced into the basic structure. However, a new resonance occurs in the higher frequency range, the amplitude of which is greater in this range than with the rigid mounting. By using a vibration absorber, it is possible to significantly reduce the amplitude in the range of the natural frequency of the basic structure. The influence of the stiffness and damping factors of the absorber was discussed in this context. A low damping factor results in new resonant frequencies, while a higher damping factor significantly reduces the effectiveness of the absorber. The combination of both measures leads to the best results over the entire operating range. In this case, the vibration absorber is designed for the resonance frequency resulting from the elastic mounting.

Literature

[1] Grote, K.-H.; Feldhusen, J. (Hg.) (2014): DUBBEL – Taschenbuch für den Maschinenbau, DOI: https://doi.org/10.1007/978-3-642-38891-0

[2] Böge, A.; Böge, W. (Hg.) (2021): Handbuch Maschinenbau – Grundlagen und Anwendungen der Maschinenbau-Technik, DOI: https://doi.org/10.1007/978-3-658-30273-3

[3] Weber, B. (2012): Tragwerksdynamik, DOI: https://doi.org/10.3929/ethz-a-007362062

[4] Preumont, A. (2011): Vibration Control of Active Structures – An Introduction, DOI: https://doi.org/10.1007/978-94-007-2033-6For